Essays on Identification in Macroeconomics · 2020. 7. 13. · Abstract This dissertation consists of three independent chapters on questions of identi cation and causal e ect estimation

Essays on

Identification in Macroeconomics

Christian Klaus Wolf

A Dissertation

Presented to the Faculty

of Princeton University

in Candidacy for the Degree

of Doctor of Philosophy

Recommended for Acceptance by

the Department of

Economics

Advisers: Giovanni Violante & Mark Watson

June 2020

c© Copyright by Christian Klaus Wolf, 2020.

All rights reserved.

Abstract

This dissertation consists of three independent chapters on questions of identification and

causal effect estimation in macroeconomics.

In the first chapter I propose a new method to estimate the aggregate effects of a large

family of consumption and investment demand shocks. My approach has two steps: first, I

recover direct partial equilibrium spending responses through cross-sectional variation, and

second, I estimate the “missing intercept” of general equilibrium effects as the response of

private spending to exogenous changes in aggregate public spending. I justify the second

step through a formal demand equivalence result.

The second chapter revisits the classical question of monetary policy transmission. I

show that the seemingly disparate findings of the recent empirical literature on the aggregate

effects of monetary policy shocks are in fact all consistent with the same standard macro

models. Taken together, empirical estimates paint a consistent picture of significant short-

term stimulative effects of monetary easing.

In the third chapter, which is coauthored with Mikkel Plagborg-Møller, we prove that

local projections (LPs) and Vector Autoregressions (VARs) estimate the same impulse re-

sponses. Our result implies that LP and VAR estimators are not conceptually separate

procedures; instead, they belong to a spectrum of dimension reduction techniques with com-

mon estimand but different bias-variance properties. In particular, it follows that VAR-based

structural estimation can equivalently be performed using LPs, and vice versa.

iii

Acknowledgements

I thoroughly enjoyed my six years in Princeton, and there are many people that I have to

thank for that.

I am indebted to my advisors for their continued support. To Gianluca Violante, for

believing in me, and for giving me a push when I needed it. To Mark Watson, for helping

me to never lose track of the big picture in my research. To Ben Moll, for giving me a

chance to work with him very early on, and for his support whenever I hit a rough patch.

And to Mikkel Plagborg-Møller, for being the best co-author that I could possibly hope for.

I’m also thankful to many others who were advisors in all but name: Mark Aguiar, Markus

Brunnermeier, Oleg Itskhoki, Greg Kaplan, Richard Rogerson, Chris Sims, Chris Tonetti

and Tom Winberry time and time again helped me with their fantastic feedback. Finally,

I owe a lot to Julia Shvets, who many years ago fostered my excitement for economics and

encouraged my application to graduate school.

The past couple of years would not have been the same without my fellow students. I am

grateful to have been in the same cohort as Joshua Bernstein, Yann Koby, Franz Ostrizek,

Elia Satori, and Fabian Trottner, and I will miss the traditional Friday lunches with Riccardo

Cioffi, Simon Schmickler, George Sorg Langhans and Maxi Vogler. Finally, I feel lucky to

have shared an office with Joseph Abadi, Tyler Abbot, Nick Huang and Rob Sperna Weiland.

I am grateful for the academic and financial support provided by the Department of Eco-

nomics. Laura Hedden kindly guided me through administrative matters, Steve Redding was

the most efficient placement director that I could have asked for, and seminar participants

at the Princeton macroeconomics, econometrics and finance workshops provided fantastic

comments through the years. I also thank the Macro Financial Modeling Project and Eq-

uitable Growth for financial support, and the Bundesbank and European Central Bank for

hosting me in the summers of 2017 and 2018, respectively.

iv

My time in Princeton would not have been the same without my girlfriend, Rowan. I am

grateful that our paths crossed here, and I look forward to our future together. Finally, I

owe a lot to my family, especially my mother, Andrea, and my father, Hans Klaus. Without

them, none of this would have been possible.

The individual chapters in this dissertation also greatly benefited from the comments of

many other people. Chapter 1: I received helpful comments from Andy Atkeson, Adrien

Auclert, Thorsten Drautzburg, Gregor Jarosch, Nobuhiro Kiyotaki, Moritz Lenel, Alisdair

McKay, Emi Nakamura, Ezra Oberfield, Jonathan Payne, Monika Piazzesi, Diego Perez,

Matt Rognlie, Martin Schneider, Jon Steinsson, Ludwig Straub, Ivan Werning, conference

participants at the 2019 NBER Summer Institute and the 2019 Chicago Fed Rookie Con-

ference, and seminar participants at several venues. Chapter 2: I thank the editor, Giorgio

Primiceri, as well as three anonymous referees. I also received useful feedback from Jonas

Arias, Thorsten Drautzburg, Jim Hamilton, Marek Jarocinski, Peter Karadi, Matthias Meier,

Ulrich Muller, Emi Nakamura, Harald Uhlig, and seminar participants at several venues.

Chapter 3: Mikkel and I thank Domenico Giannone, Marek Jarocinski, Oscar Jorda, Pe-

ter Karadi, Lutz Kilian, Dake Li, Pepe Montiel Olea, Valerie Ramey, Giovanni Ricco, Neil

Shephard, Jim Stock, conference participants at the 2019 NBER Summer Institute and the

2020 ASSA Annual meeting, and seminar participants at various venues.

v

To my parents.

vi

Contents

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

1 The Missing Intercept: A Demand Equivalence Approach 1

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Consumption demand equivalence . . . . . . . . . . . . . . . . . . . . . . . . 9

1.2.1 The benchmark model . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.2.2 Direct responses and general equilibrium feedback . . . . . . . . . . . 14

1.2.3 A simple example of demand equivalence . . . . . . . . . . . . . . . . 15

1.2.4 A general equivalence result . . . . . . . . . . . . . . . . . . . . . . . 20

1.3 A two-step approach to estimating consumption demand counterfactuals . . 27

1.3.1 The two-step methodology . . . . . . . . . . . . . . . . . . . . . . . . 27

1.3.2 Application: income tax rebates . . . . . . . . . . . . . . . . . . . . . 31

1.4 Approximation accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

1.4.1 The estimated HANK model . . . . . . . . . . . . . . . . . . . . . . . 39

1.4.2 Labor supply . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

1.4.3 Interest rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

1.4.4 Beyond one-good economies . . . . . . . . . . . . . . . . . . . . . . . 45

1.5 Investment demand counterfactuals . . . . . . . . . . . . . . . . . . . . . . . 47

1.5.1 Investment demand equivalence . . . . . . . . . . . . . . . . . . . . . 48

vii

1.5.2 Application: bonus depreciation . . . . . . . . . . . . . . . . . . . . . 51

1.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

2 SVAR (Mis-)Identification and the Real Effects of Monetary Policy Shocks 57

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

2.2 VAR analysis in structural models . . . . . . . . . . . . . . . . . . . . . . . . 64

2.2.1 Model laboratories . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

2.2.2 Structural models and VAR analysis . . . . . . . . . . . . . . . . . . 66

2.2.3 Interpreting SVAR estimands . . . . . . . . . . . . . . . . . . . . . . 68

2.3 Sign restrictions and masquerading shocks . . . . . . . . . . . . . . . . . . . 72

2.3.1 The identified set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

2.3.2 The Haar prior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

2.4 Zero restrictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

2.5 Recent advances in identification . . . . . . . . . . . . . . . . . . . . . . . . 85

2.5.1 Taylor rule restrictions . . . . . . . . . . . . . . . . . . . . . . . . . . 85

2.5.2 External instruments . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

2.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

3 Local Projections and VARs Estimate the Same Impulse Responses 91

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

3.2 Equivalence between local projections and vector autoregressions . . . . . . . 95

3.2.1 Main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

3.2.2 Extension: Non-recursive specifications . . . . . . . . . . . . . . . . . 100

3.2.3 Extension: Finite lag length . . . . . . . . . . . . . . . . . . . . . . . 101

3.2.4 Graphical illustration . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

3.3 Efficient estimation of impulse responses . . . . . . . . . . . . . . . . . . . . 106

3.3.1 Sample equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

viii

3.3.2 Bias-variance trade-off . . . . . . . . . . . . . . . . . . . . . . . . . . 106

3.4 Structural identification of impulse responses . . . . . . . . . . . . . . . . . . 108

3.4.1 Structural model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

3.4.2 Implementing “SVAR” identification using LPs . . . . . . . . . . . . 110

3.4.3 Identification and estimation with instruments . . . . . . . . . . . . . 116

3.4.4 Estimands in non-linear models . . . . . . . . . . . . . . . . . . . . . 120

3.5 Empirical application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

3.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

A Appendix for Chapter 1 127

A.1 Model details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

A.1.1 The benchmark model . . . . . . . . . . . . . . . . . . . . . . . . . . 127

A.1.2 Parametric special cases . . . . . . . . . . . . . . . . . . . . . . . . . 134

A.2 Empirical appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

A.2.1 Direct response: micro consumption elasticities . . . . . . . . . . . . 145

A.2.2 Direct response: micro investment elasticities . . . . . . . . . . . . . 147

A.2.3 The missing intercept: VAR estimation . . . . . . . . . . . . . . . . . 148

A.2.4 Joint Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

A.3 Proofs and auxiliary lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

A.3.1 Proof of Lemma 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

A.3.2 Proof of Proposition 1 . . . . . . . . . . . . . . . . . . . . . . . . . . 154

A.3.3 Auxiliary Lemma for Proposition 2 . . . . . . . . . . . . . . . . . . . 155




A.3.7 Auxiliary Lemma for Proposition 5 . . . . . . . . . . . . . . . . . . . 163

ix


A.3.9 Proof of Corollary 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

A.4 Additional results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

A.4.1 Approximation accuracy . . . . . . . . . . . . . . . . . . . . . . . . . 167

A.4.2 Correcting for wealth effects in labor supply . . . . . . . . . . . . . . 190

A.4.3 General equilibrium amplification . . . . . . . . . . . . . . . . . . . . 191

A.4.4 Impulse response matching . . . . . . . . . . . . . . . . . . . . . . . . 193

A.4.5 Demand equivalence along transition paths . . . . . . . . . . . . . . . 194

A.5 Application: income redistribution . . . . . . . . . . . . . . . . . . . . . . . 195

B Appendix for Chapter 2 198

B.1 Identified sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198

B.2 Supplementary results on invertibility . . . . . . . . . . . . . . . . . . . . . . 201

B.3 Shock volatility and Bayesian posteriors . . . . . . . . . . . . . . . . . . . . 205

C Appendix for Chapter 3 209

C.1 Equivalence result with finite lag length . . . . . . . . . . . . . . . . . . . . . 209

C.2 Long-run identification using local projections . . . . . . . . . . . . . . . . . 211

C.3 Best linear approximation under non-linearity . . . . . . . . . . . . . . . . . 212

Bibliography 214

x

Chapter 1

The Missing Intercept: A Demand

Equivalence Approach

1.1 Introduction

A large literature in macroeconomics tries to estimate the aggregate effects of shocks to con-

sumption and investment expenditure.1 For most of these demand shifters, the experimental

ideal – exogeneity at the macro level – is not attainable. In response, researchers increasingly

leverage the cross-sectional variation available in micro data. Appealingly, because these es-

timates rely exclusively on cross-sectional information, they do not require macroeconomic

identification restrictions. The well-known shortcoming is that such estimates are not inter-

pretable as macro counterfactuals, simply because any potential general equilibrium effects

– price changes, aggregate employment responses, tax financing, and so on – are differenced

out. Previous work has tried to identify this “missing intercept” through fully specified

structural models, with little systematic guidance on what model to choose, how to estimate

it, and how to communicate uncertainty across the range of plausible models.

1Well-known examples include tax rebates (Parker et al., 2013), redistribution (Jappelli & Pistaferri,2014), credit tightening (Mian et al., 2013; Guerrieri & Lorenzoni, 2017) and bonus depreciation (Zwick &Mahon, 2017).

1

1. The Missing Intercept

I develop an alternative semi -structural method, applicable to a general family of con-

sumption and investment demand shifters. My method is structural in that it builds on a

property of models: I prove that, in a broad class of business-cycle models, identical changes

in partial equilibrium private and public net excess demand also elicit identical general equi-

librium price responses – intuitively, identical pressure on the economy’s resource constraint

is accommodated in the same way. This “demand equivalence” justifies a two-step empiri-

cal strategy: First, a researcher leverages cross-sectional heterogeneity in shock exposure to

recover the partial equilibrium response of private spending demand to the shifter. Second,

she estimates the aggregate effects of an equally large shock to public spending using the

existing toolkit for fiscal shocks (e.g. Ramey, 2018). By demand equivalence, summing (i)

her micro estimates and (ii) the aggregate response of private spending to the fiscal shock

will recover the full effect of the demand shifter on private spending.

I apply my method to study tax rebates. Micro data suggest a large but short-lived direct

consumption response, while macro experiments for a similarly short-lived, deficit-financed

change in public spending imply only little crowding-out of private consumption. By demand

equivalence, it follows that full macro counterfactuals are close to the direct micro estimates.

In order for a structural model to be inconsistent with such a near-zero “missing intercept,”

it must either break demand equivalence or feature fiscal multipliers far from one. I find a

similarly small missing intercept in a second application to investment tax stimulus.

I first discuss the restrictions on household behavior and fiscal policy required for identical

general equilibrium propagation of shocks to public spending and to private consumption de-

mand. For a general class of quantitative business-cycle models, linearized impulse responses

to macro shocks can be characterized implicitly as solutions to a linear infinite-horizon system

of market-clearing conditions. If two shocks perturb the same market-clearing conditions by

the same amount, then, by the chain rule, the general equilibrium adjustment to these com-

mon perturbations must also be the same. This invariance result, together with three further

2


assumptions, allows me to prove the (first-order) equivalence of shocks to private consump-

tion and to public spending. First, households and government need to consume the same

final good. If so, identical changes in private or public spending will lead to identical partial

equilibrium excess demand for that common good. Second, households and government must

borrow and lend at the same interest rate. The identical expansions in private and public

demand can then be discounted at that common rate, and so can be financed using identical

paths of taxes and transfers. Third, household labor supply must not respond differentially

to the two shocks. Sufficient conditions are either the absence of wealth effects in labor

supply or fully demand-determined employment. The first two assumptions required for my

exact equivalence result are satisfied in many popular structural general equilibrium models,

including standard medium-scale New Keynesian models (Smets & Wouters, 2007), but also

models with rich micro household and firm heterogeneity (e.g. Guerrieri & Lorenzoni, 2017;

Khan & Thomas, 2013). While the third assumption is sometimes violated, I argue both

empirically and theoretically that the associated error is robustly small.

I leverage the consumption demand equivalence result to formally justify my two-step

empirical procedure. First I show that, if cross-sectional heterogeneity in exposure to a con-

sumption demand shifter is independent of household characteristics, then the econometric

estimands of micro difference-in-differences regressions are interpretable as the direct (partial

equilibrium) response of consumption demand to the shifter. Second, by demand equiva-

lence, the “missing intercept” can be recovered as the response of consumption to a public

spending shock – or a combination of such shocks – that induces the same path of (partial

equilibrium) net excess demand. For measurement, I link this response to the econometric

estimands of the popular Structural Vector Autoregressive (VAR) or Local Projection (LP)

approaches to fiscal shock transmission (Hall, 2009; Ramey, 2018). Under my identifying

assumptions, the sum of the micro and macro estimates is then indeed interpretable as the de-

sired semi-structural aggregate consumption counterfactual for the private demand shifter.

3


Equivalently, the researcher could have written down any particular parametric model in

my equivalence class, parameterized the model to be consistent with the estimated micro

and macro moments, and solved it – the equivalence result guarantees that she would have

recovered the exact same macro counterfactual.

I demonstrate the feasibility and applicability of my methodology through the study

of a popular consumption stimulus policy: a lump-sum, one-off income tax rebate. I first

review previous empirical work (Parker et al., 2013; Jappelli & Pistaferri, 2014) and show

that the direct partial equilibrium response of consumption to the stimulus is indeed either

equal or at least tightly linked to the econometric estimands of those studies. Their dif-

ferent quasi-experiments consistently paint the picture of a large but short-lived expansion

in consumption demand. Next, I construct a government spending news variable based on

professional forecast errors, and treat this forecast error as a macro instrumental variable for

government spending shocks (Stock & Watson, 2018). Following Plagborg-Møller & Wolf

(2019b), I project on this news variable using a recursive VAR. I find that the forecast error

impulse leads to an uptick in government spending as short-lived as the private consumption

demand increase, a persistent rise in government debt, and a fiscal multiplier of around one

– output rises briefly, and consumption is flat. Summing the micro and macro estimates, I

conclude that a one-off, deficit-financed transfer briefly but significantly stimulates aggregate

consumption, with the overall response close to the direct effect estimated using micro data.

While the output of the two-step procedure is only exactly interpretable as a valid coun-

terfactual under my three key assumptions, I show that approximate demand equivalence is

supported in the data and obtains in several quantitative model extensions. As discussed

above, equivalence fails in an important benchmark class of models only because of short-

term wealth effects in labor supply. Micro evidence is inconsistent with this labor supply

channel (Cesarini et al., 2017), and general equilibrium model closures with even moderate

degrees of nominal rigidity are well-known to largely neuter the aggregate effects of tran-

4


sitory labor supply shifts (Christiano, 2011a). I use a heterogeneous-agent New Keynesian

(“HANK”) model, estimated to be consistent with salient features of cross-sectional earnings

risk, the aggregate wealth distribution, and the time series distribution of macro aggregates,

to illustrate the quantitative irrelevance of the labor supply channel. In the same model,

saving and borrowing rates for households and the government are – in line with the evi-

dence in Fagereng et al. (2018) – sufficiently similar to ensure that my second assumption is

also nearly satisfied. Finally, to gauge the importance of the single common good assump-

tion, I extend the model to a multi-sector economy, allowing for (i) a durable consumption

good, (ii) productive benefits and consumption complementarities for public spending, and

(iii) imperfect factor mobility across sectors with heterogeneous production functions. In

empirically disciplined variants of these models, the approximation remains accurate.

My methodology extends with little change to shifters of investment demand. In response

to the shock, investment increases today (excess demand) while capital and so production

build up gradually (excess supply). I give sufficient conditions under which the investment

demand shifter is accommodated in general equilibrium exactly like an expansion in gov-

ernment expenditure today (excess demand), followed by a contraction in the future (excess

supply). Importantly, these sufficient conditions impose no material restrictions on the pro-

duction block of the economy; in particular, exact investment demand equivalence holds in

most recent quantitative studies on the aggregate effects of firm-level investment frictions,

including models with very rich firm heterogeneity.2 Finally, I apply my results to study the

aggregate effects of bonus depreciation stimulus: I find a large partial equilibrium increase

in investment demand (Zwick & Mahon, 2017; Koby & Wolf, 2020), accommodated through

a sharp rise in output, with little investment crowding-out and consumption relatively flat.

2For example, exact equivalence applies in the popular structural models of Khan & Thomas (2008),Khan & Thomas (2013), or Winberry (2018).

5


Before proceeding further, I briefly comment on the scope and limitations of my analy-

sis. First, my methodology requires first-stage micro regressions whose estimands are inter-

pretable as direct partial equilibrium effects. This is arguably the case for across-household

or across-firm regressions, but not for cross-regional regressions (Mian et al., 2013; Mian &

Sufi, 2014). I generalize my results to such cross-regional regressions in the companion note

Wolf (2019b). Second, the general principle underlying my approach – to leverage macro evi-

dence on the general equilibrium propagation of plausibly equivalent shocks – is applicable to

a rich family of consumption and investment shifters. It does not, however, solve the missing

intercept problem for all possible shocks and policies. Third, while my demand equivalence

results are only valid to first order, I impose no restriction on where the underlying Taylor

series approximation is taken. Evidence on state dependence in the transmission of fiscal

shocks thus applies without change to generic consumption and investment demand shifters.

Finally, my two-step procedure relies sensitively on the assumption that all agents only inter-

act through (a small set of) aggregate prices and quantities. Appealingly, I take little stand

on the precise nature of that interaction, so my theory covers both conventional neoclassi-

cal as well as quite different Keynesian adjustment mechanisms. Less appealingly, strategic

interaction between agents breaks the neat separation into partial equilibrium impacts and

general equilibrium accommodation that lies at the heart of my approach.

Literature. This paper contributes to several strands of the literature.

First, my methodology connects two empirical literatures. A fast-growing line of work

uses variation at the individual or regional level to estimate spending responses to policy

changes and other macro shocks. For example, Johnson et al. (2006), Agarwal et al. (2007),

and Parker et al. (2013) leverage cross-sectional heterogeneity in policy exposure to study

the response of household consumption expenditure to lump-sum payments. Estimates of

household-level marginal propensities to consume have also been used to gauge the likely

6


effects of income redistribution, either outright through policy changes or through gradual

increases in inequality (Jappelli & Pistaferri, 2014). In Mian & Sufi (2009) and Mian & Sufi

(2014), cross-regional heterogeneity in shock exposure is used to recover the direct consump-

tion effects of changes in household balance sheets. Analogous micro causal effects can also

be estimated for investment (e.g. Zwick & Mahon, 2017). As all of these studies control for

macro fluctuations through time fixed effects, they are silent on any possible general equilib-

rium feedback. My key insight is that a second empirical literature – that on the aggregate

effects of variations in government spending – can be informative about this “missing in-

tercept.” Comprehensive literature summaries are Hall (2009) and Ramey (2018); overall,

earlier empirical work quite consistently estimates output multipliers around 1, and zero (or

slightly negative) responses of private spending. In connecting these two literatures, my two-

step procedure is semi-structural in exactly the same way as conventional Structural Vector

Autoregressive (SVAR) analysis (Sims, 1980): It relies on general identifying restrictions,

rather than being tied to any particular parametric model of the macro-economy.

Second, the theoretical demand equivalence result itself builds on the burgeoning suffi-

cient statistics literature in macroeconomics. Earlier contributions show that the estimands

of micro difference-in-differences studies are linked to partial equilibrium spending elastici-

ties (e.g. Kaplan & Violante, 2014; Berger et al., 2017). Very recently, several studies have

tried to clarify the relationship between these partial equilibrium elasticities and the asso-

ciated aggregate counterfactuals. Among those, my analysis relates most closely to Auclert

& Rognlie (2018), Auclert et al. (2018), and Guren et al. (2019). Auclert et al. show that,

in models with demand-determined labor, passive monetary policy, and without investment,

consumption demand and government spending shocks have identical effects on aggregate

output – a special case of what I call “demand equivalence.” I extend the equivalence re-

sult to a larger family of models (and to investment demand), find support for approximate

equivalence in micro and macro data, and measure the common general equilibrium effects

7


through macro quasi-experiments. In ongoing related work, Guren et al. (2019) use estimates

of local government spending multipliers to cleanse regional consumption responses to house

price changes from regional spending multipliers. Relative to their analysis, I use the reverse

logic, aggregating partial equilibrium consumption effects up to macro outcomes, show that

the equivalence logic applies to generic consumption and investment demand shifters, and

study its accuracy in a larger model space, and with an emphasis on full impulse response

dynamics.3 Overall, my results have a “sufficient statistics” interpretation similar to Chetty

(2009) or Arkolakis et al. (2012): To deviate much from the conclusion of a near-0 miss-

ing intercept for private demand shifters, structural modelers must either leave the demand

equivalence class or implicitly impose that fiscal multipliers are far from 1.

Third, my results connect to the large literature on estimation of quantitative business-

cycle models. Dominant approaches are limited-information moment-matching, notably of

impulse response functions, as well as full likelihood-based estimation (Christiano et al., 2005;

Smets & Wouters, 2007; Nakamura & Steinsson, 2018b). The equivalence result provides a

novel justification for impulse response matching: By commonality of general equilibrium

feedback, impulse responses to particular aggregate structural shocks can be informative for

many different counterfactuals. This idea has a clear conceptual antecedent in the microe-

conomic program evaluation literature: Marschak’s Maxim suggests that economists should

try to identify the combinations of structural parameters needed for policy analysis, rather

than the hard-to-estimate parameters themselves (Marschak, 1974; Heckman, 2010). De-

mand equivalence suggests that fiscal multipliers are precisely such a useful combination. As

such, using the language of Nakamura & Steinsson (2018b), they can be a useful “identified

moment” for the quantitative discipline of structural macro models.

3A common early antecedent to this line of work is Hall (2009), who argued that “the effects of higherconsumer purchases [are likely to be] similar to the effects of higher government purchases” – an identicaloutput multiplier. My additive decomposition relies on the same intuition and is equivalent to scaling microestimates by one plus the consumption or investment multipliers associated with fiscal purchases.

8


Outline. Section 1.2 establishes the consumption demand equivalence result. In Sec-

tion 1.3, I leverage commonality in general equilibrium propagation to propose a two-step

procedure for estimation of consumption demand counterfactuals, with an application to in-

come tax rebates. Section 1.4 then shows that the proposed approximation remains accurate

under more general assumptions. The generalization to investment demand, including an

application to investment tax stimulus, is discussed in Section 1.5. Section 3.6 concludes, and

supplementary details, proofs and a third application are all relegated to several appendices.

1.2 Consumption demand equivalence

This section develops an exact equivalence result for the general equilibrium propagation of

private consumption demand and public spending shocks. Section 1.2.1 outlines a benchmark

quantitative business-cycle model. In Section 1.2.2 I proceed to formally define my notions

of direct “partial equilibrium” responses and indirect “general equilibrium” feedback. To

build intuition, Section 1.2.3 develops the equivalence result in a stylized special case with

closed-form solution. In Section 1.2.4 I return to the rich benchmark class of models and

give a simple set of sufficient conditions for exact demand equivalence.

1.2.1 The benchmark model

Time is discrete and runs forever, t = 0, 1, . . .. The model economy is populated by house-

holds, firms, and a government. There is no aggregate uncertainty, but households and

firms are allowed to face idiosyncratic risk. I study perfect foresight transition paths back

to steady state after one-time unexpected aggregate innovations at time 0; for vanishingly

small innovations, these transition paths are mathematically equivalent to standard impulse

response functions computed from the first-order perturbation solution to an otherwise iden-

9


tical model with aggregate risk.4 Anticipating my main empirical application, I will mostly

focus on two such innovations: first, a one-off transfer to households, and second, a tran-

sitory expansion in government spending. To nevertheless emphasize the generality of the

demand equivalence result, I also consider a third shock: fluctuations in household patience

as a simple reduced-form stand-in for various more plausibly structural shocks to household

spending (e.g. changes in borrowing constraints, redistribution, . . . ).

Notation. The realization of a variable x at time t along the equilibrium perfect foresight

transition path will be denoted xt, while the entire time path will be denoted x = {xt}∞t=0.

Hats denote deviations from the deterministic steady state, bars denote steady-state values,

and tildes denote logs. I study three structural shocks indexed by s ∈ {τ, g, v} – tax rebates,

government spending, and household impatience. I write individual shock paths as εεεs, and

use subscripts εεε for transitions after a path εεε ≡ (εεε′τ , εεε′g, εεε′v)′. I reserve the simpler s subscripts

for one-time single shocks – that is, shock paths with εs,0 = 1 and εu,τ = 0 for (u, τ) 6= (s, 0).

Households. A unit continuum of households i ∈ [0, 1] has preferences over consumption

cit and labor ìt. They are subject to idiosyncratic productivity risk eit and potentially differ

in their baseline discount factor βi. The discount factor of every household is further subject

to an additional common shifter ζt, with ζζζ = ζζζ(εεεv). Households can self-insure by investing

in liquid nominal bonds bhit, with nominal returns ibt and subject to a borrowing constraint b.5

Household income consists of labor earnings as well as (potentially type-specific) lump-sum

rebates τit and dividend income dit. Total hours worked ìt are determined by demands of a

unit continuum k ∈ [0, 1] of price-setting labor unions, as in Erceg et al. (2000); the problem

of labor unions will be considered later. Given a path of prices, rebates, dividends, hours

4This result is an implication of certainty equivalence coupled with Taylor’s theorem (Boppart et al., 2018).For ordinary business-cycle fluctuations, such first-order perturbations offer an accurate characterization ofthe model’s global dynamics (e.g. Fernandez-Villaverde et al., 2016; Ahn et al., 2017; Auclert et al., 2019).

5I consider an extension with liquid and illiquid assets in Section 1.4.3.

10


worked and inflation (πt), the consumption-savings problem of household i is thus

max{cit,bhit}

E0

[∞∑t=0

βtiζt(εεεv)u(cit, cit−1, ìt)

](1.1)

such that

cit + bhit = (1− τ`)wteitìt +1 + ibt−1

1 + πtbhit−1 + τit + dit

and

bhit ≥ b

Productivity eit follows a (stochastic) law of motion with∫ieitdi = 1 at all times.

Labor unions behave as in conventional New Keynesian models (Erceg et al., 2000; Auclert

et al., 2018). Worker i provides ìkt units of labor to union k, giving total hours worked for

household i of ìt ≡∫kìktdk. The total effective amount of labor intermediated by union k

is `kt ≡∫ieitìktdi; each union then sells its labor services to a competitive labor packer at

price wkt. The labor packer aggregates union-specific labor to aggregate labor services,

`ht ≡(∫

k

`εw−1εw

kt dk

) εwεw−1

sold at the aggregate wage index wt, and where εw denotes the elasticity of substitution

between different types of labor. Union k chooses its wage rate wkt subject to wage-setting

adjustment costs, and satisfies the corresponding demand for its labor services. I assume

that it does so by demanding a common amount of hours worked from its members.6 Since

the wage-setting problem is standard, I relegate details to Appendix A.1.1. For the purposes

6A uniform hiring rule is the natural assumption in sticky-wage heterogeneous-household models, but isof course awkward in the flexible-wage limit, as it then does not nest the alternative natural case of flexiblelabor supply for each individual household. I consider a model without unions in Appendix A.4.1.1.

11


of the analysis here, it suffices to note that union behavior can be summarized through a

simple wage New Keynesian Phillips curve – effectively, an aggregate labor supply relation.

Fiscal policy. The fiscal authority consumes the same final good as households. Fiscal

consumption gt and total lump-sum transfers τt ≡∫ 1

0τitdi are financed through debt issuance

and taxes on labor income. The government budget flow constraint is

1 + ibt−1

1 + πtbt−1 + gt + τt = τ`wt`t + bt

I assume that total government spending g = g(εεεg) follows some exogenous process, and

that the government freely sets a discretionary part of tax rebates τττx = τττx(εεετ ). Given paths

for spending targets (εεεg, εεετ ), initial nominal debt b−1 and a path of prices and quantities

(w, `, ib,πππ), a government debt financing rule is a path τττ e such that τττ = τττ e + τττx, the flow

government budget constraint holds at all periods t, and limt→∞

(∏ts=0

1+πs1+ibs−1

)bt = 0.

Rest of the economy. Since my focus is on the equivalence of private and public expan-

sions in demand, I only sketch the rest of the model, with a detailed outline provided in

Appendix A.1.1. The corporate sector is populated by three sets of firms: a unit continuum

of heterogeneous, perfectly competitive intermediate goods producers j, a unit continuum of

monopolistically competitive retailers with nominal price rigidities, and a final goods aggre-

gator. Intermediate goods producers accumulate capital, hire labor, issue risk-free debt, and

sell their composite intermediate good, possibly subject to (both convex and non-convex)

capital adjustment costs as well as generic constraints on equity and debt issuance. Retailers

purchase the intermediate good, costlessly differentiate, monopolistically set prices, and sell

their differentiated good on to the competitive aggregator.

The last remaining entity is the monetary authority. This monetary authority sets nom-

inal rates on liquid bonds ib in accordance with a conventional Taylor rule.

12


Equilibrium. I assume that there exists a unique deterministic steady state.7 To allow

interpretation of perfect foresight transition paths as conventional first-order perturbation

solutions, I impose that the economy is indeed initially in steady state, and then study perfect

foresight transition equilibria back to the initial deterministic steady state. The definition

of equilibrium perfect foresight transition paths is then standard (see Appendix A.1.1); I

discuss an extension to transition paths with other starting points in Appendix A.4.5.

Nested Models. My benchmark model is designed to nest several important earlier con-

tributions to quantitative business-cycle analysis. In the absence of uninsurable household

earnings risk and household borrowing limits, and without firm-level productivity differences

and financial frictions, it becomes a standard New Keynesian model (e.g. Smets & Wouters,

2007). However, the environment is also rich enough to allow for non-trivial micro hetero-

geneity at the household and firm level. On the household side, income risk and limited

self-insurance can endogenously generate hand-to-mouth behavior. With flexible prices, the

model is identical to Aiyagari (1994) or Krusell & Smith (1998); with nominal rigidities, it

is a HANK model in the mold of McKay et al. (2016) and Guerrieri & Lorenzoni (2017). On

the firm side, I allow for a rich set of real and financial frictions to the capital accumulation

process, as for example in Khan & Thomas (2008), Khan & Thomas (2013) and Winberry

(2018). In other words, the benchmark model is as rich as most models that – in the absence

of the identification results developed here – would be used to structurally pin down the

missing general equilibrium intercept of, say, income tax rebate shocks.

The results in Section 1.2.4 will show what extra restrictions on this canonical model

class are needed to attain an exact demand equivalence result and so justify my claims

about model identification and empirical counterfactuals in Section 1.3.

7More precisely, I make implicit assumptions on functional forms and parameter values that guaranteethat there is a unique deterministic steady state. In all numerical exercises, I have verified the uniquenessof the steady state and the (local) existence and uniqueness of transition paths.

13


1.2.2 Direct responses and general equilibrium feedback

The demand equivalence result will assert a commonality in the general equilibrium propaga-

tion of different shocks. A precise statement of such equivalence requires a formal definition

of direct partial equilibrium responses and indirect general equilibrium adjustment.

I assume that the consumption-savings problem (1.1) has a unique solution for any

path of prices, quantities and shocks faced by households. Aggregating the solutions

across households, we obtain an aggregate consumption function c = c(sh;εεε), where

sh = (ib,πππ,w, `, τττ e,d) collects household income and saving returns – objects that adjust in

general equilibrium. The total impulse response of consumption to the shock path εεε is

cε ≡ c(shε ;εεε) − c(sh; 0)

I decompose this aggregate impulse response into two parts: a direct “partial equilibrium”

impulse and an indirect “general equilibrium” feedback part.8

Definition 1. Let the direct (partial equilibrium) response of consumption to a shock path εεε

be defined as

cPEε ≡ c(sh;εεε) − c(sh; 0) (1.2)

Similarly, let the indirect (general equilibrium) feedback be

cGEε ≡ c(shε ; 0) − c(sh; 0) (1.3)

8My definition of the partial equilibrium consumption response abstracts from endogenous adjustmentsin earnings. I do so for three reasons. First, many empirical estimates of household spending responsesto sudden income changes are actually interpretable as such netted spending elasticities (e.g. see Auclert,2019). Second, in models with union-intermediated labor supply – like the one considered here –, replicat-ing cross-sectional micro regressions invariably differences out labor responses (see Proposition 3). Third,microeconomic evidence suggests that short-run wealth effects are very weak anyway (Cesarini et al., 2017;Fagereng et al., 2018). Nevertheless, in Appendix A.4.1.1, I repeat all of my analysis in an alternative modelwithout unions, but with a non-standard preference parameterization allowing for (data-consistent) weakshort-run wealth effects (Jaimovich & Rebelo, 2009; Galı et al., 2012).

14


It is immediate that, to first order, the aggregate impulse response admits an additive

decomposition into partial equilibrium response and general equilibrium feedback:

cε = cPEε + cGEε (1.4)

The decomposition (1.4) is only interesting to the extent that its components can be tied to

empirically measurable objects. The remainder of this section establishes conditions under

which the consumption response to particular government spending shocks cg is informative

about the general equilibrium feedback term cGEd of private demand shocks d ∈ {τ, v} – the

demand equivalence result. In Section 1.3 I then argue that (i) cross-sectional regressions

estimate the direct spending response cPEd and (ii) it is in practice often possible to recover

the aggregate effects of public spending shocks that can proxy for cGEd .

1.2.3 A simple example of demand equivalence

The intuition for the demand equivalence result is easily illustrated using a particular special

case of my benchmark model – a simple spender-saver real business-cycle (RBC) model. In

this model, a mass λ of households are spenders (so βi = 0), while the remaining households

are savers (βi > 0). Both types have log consumption utility and inelastically supply their

labor endowment, and savers hold all risk-free real bonds and receive firm dividends. The

firm sector admits aggregation to a representative firm which hires labor and accumulates

capital; for simplicity I assume that capital depreciates fully within the period and that the

production function is Cobb-Douglas, y = kα`1−α. The fiscal authority issues risk-free bonds,

consumes the final good, and imposes (different) lump-sum taxes on savers and spenders.

There are no nominal rigidities, so central bank behavior is irrelevant for all real quantities.

In this environment I compare the transmission of two structural shocks: (i) a one-off

income tax rebate ετt (to spenders) and (ii) a one-period expansion in aggregate government

15


spending εgt. I assume that the tax increases (transfer cuts) τττ e used to finance the two

policies fall on savers. All equations are stated in Appendix A.1.2.1.

Demand Equivalence. I begin with a concrete numerical example. I set the saver dis-

count factor to β = 0.99, the capital share to α = 1/3, and assume that a mass λ = 0.3

of households is hand-to-mouth. Figure 1.1 shows consumption impulse responses for one-

period tax rebate and government spending shocks.

Figure 1.1: Demand Equivalence, Spender-Saver RBC

Note: Impulse response decompositions after equally large, one-off tax rebate and governmentspending shocks in the simple spender-saver RBC model. The direct response and the indirectgeneral equilibrium feedback are computed following Definition 1.

The left panel shows the consumption response to a one-off transfer, normalized to in-

crease partial equilibrium consumption demand by one per cent. In line with Definition 1,

this aggregate impulse response is decomposed into direct partial equilibrium (green) and

indirect general equilibrium (orange) responses. By assumption, spenders consume all of the

rebate today. The grey line then shows that, after general equilibrium price adjustments,

aggregate consumption only moderately rises on impact, then falls, and gradually returns to

16


steady state. General equilibrium adjustment thus substantially crowds-out consumption.

Intuitively, this is so because a rise in interest rates leads savers to postpone consumption;

at the same time, investment is crowded out, future output drops and income declines.

The right panel then shows the consumption response to a one-period expansion in gov-

ernment spending, normalized to increase total fiscal consumption of the final good by one

per cent of steady-state private consumption. By definition, household consumption does not

respond directly to this second shock (the green line). In general equilibrium, consumption

drops substantially; exactly as for the tax rebate, this is largely due to higher rates crowding

out both saver consumption and aggregate investment, and thus further pushing down fu-

ture income. Crucially, the response of aggregate consumption to the public spending shock

appears to be identical to the general feedback associated with the tax rebate shock – a

property of the model that I will refer to as “demand equivalence”. As it turns out, demand

equivalence is not an artifact of the particular parameterization chosen for Figure 1.1, but a

general feature of my simple spender-saver model.

Proposition 1. Suppose that cPEτ = gg. Then

cGEτ = cGEg (1.5)

and so

cτ = cPEτ︸︷︷︸PE response

+ cg︸︷︷︸= GE feedback

(1.6)

Irrespective of the model parameterization, the total response of consumption to a gov-

ernment spending shock can proxy for the missing general equilibrium intercept of the private

spending change.

17


Proof. It is straightforward to establish the decomposition in Proposition 1 through the

familiar closed-form solution of log-linearized RBC models. As I show in Appendix A.1.2.1,

the response paths of (log-linearized) capital and consumption follow

ˆkt = α× ˆkt−1 −1− αβ

1− λ(1− αβ)× (ετt + εgt) (1.7)

ˆct = α× ˆkt−1 +αβ

1− λ(1− αβ)× (ετt + εgt)− εgt (1.8)

The key observation is that both shocks enter the law of motion for the capital stock (1.7)

identically. In other words, consumption demand and government spending shocks have

equal effects on capital accumulation, and so also output, interest rates, and wages. The sole

difference between the two shocks lies in how the common amount of net output (output less

investment) is split between household and government consumption, as evident from (1.8).

Unfortunately, this proof strategy is not particularly constructive – it relies on the explicit

closed-form solution of the model, which of course will not be available for quantitatively

relevant model variants. Instead, I find it convenient to write the equilibrium as a dynamic

system of market-clearing equations (and prices adjusting to clear those markets).

Lemma 1. Consider a shock path εεε. Sequences of real rates r and taxes on savers −τττ e are

part of a perfect foresight equilibrium if and only if

c(r,w(r),d(r), τττ e;εεε) + g(εεε) = y(r)− i(r) (1.9)

τττ e = τττ e(εεε) (1.10)

where y(•) and i(•) are firm policy functions, and optimal firm behavior implicitly pins down

wages w(•) and dividends d(•) as functions of r.

Given a path of real interest rates r, optimal firm behavior gives production y, investment

i, and payments to households as dividends d and wages w. Similarly, given total household

18


income and returns to saving, optimal household behavior implies a path for consumption

demand c; finally, the path of government spending g is exogenous. It is of course immediate

that any possible equilibrium path of interest rates r and saver transfers (taxes) τττ e must be

such that the output market clears (equation (1.9)) and the government budget constraint

holds (equation (1.10)). Lemma 1 then merely asserts that these conditions are also suffi-

cient : Equilibria in the economy are fully characterized by adjustments of one intertemporal

price – real interest rates – to clear one market – the output market.

Once this result is established, it is a small leap to go to Proposition 1: Totally differen-

tiating both sides of (1.9) - (1.10) we find that, to first order,

∂c∂εεε

+ ∂g∂εεε

∂τττe∂εεε

× εεε

︸︷︷︸excess demand

=

∂y∂r− ∂i

∂r− ∂c

∂r− ∂c∂τττe

0 I

×

r

τττ e

︸︷︷︸

GE adjustment

(1.11)

The initial disturbance εεε leads to some time path of initial excess demand or supply, and

some shortfall in the intertemporal government budget. Now suppose that a path r and

τττ e solves (1.11) for a tax rebate shock εεετ . Then the same path (r, τττ e) also solves (1.11)

for a government spending shock with the same intertemporal demand profile – that is, if

cPEτ = gg. Intuitively, for general equilibrium feedback, it does not matter why there is a

given amount of excess demand, or why there is a shortfall in the intertemporal government

budget constraint – it just matters how much.9

Interpretation. The decomposition (1.6) shows that, at least in the spender-saver model,

government spending impulse responses are a useful sufficient statistic for the general equi-

9Formally, the heuristic argument given here ensures only that a solution for the rebate transition pathis also a solution for a particular public spending transition path; it is, however, silent on the existenceand uniqueness of such transition paths. For the simple spender-saver application, existence and uniquenessare verified in the usual way for the recursive representation of the analogous linearized stochastic differenceequation (Blanchard & Kahn, 1980), which implies that the infinite-dimensional general equilibrium feedbackmap in (1.11) has a unique left-inverse. I provide further details in Appendix A.1.2.1.

19


librium feedback effects associated with an income tax rebate. By the statement of Proposi-

tion 1, this result is not tied to any particular model parameterization; for example, it holds

for arbitrary values of the saver discount rate or the capital share in production.

The proof strategy suggests that the demand equivalence result should in fact be quite a

bit more general: Ultimately, the proof only relies on the sequential equilibrium characteri-

zation (1.9) - (1.10), and so should – for example – be invariant to largely arbitrary changes

in production functions and preferences. Of course, the ability to characterize a model’s

equilibrium through such a single market-clearing condition in a single intertemporal price

is highly restrictive. However, as I show next, a variant of this proof strategy can be applied

to justify the decomposition (1.6) in a very rich family of structural macro models.

1.2.4 A general equivalence result

This section establishes my most general consumption demand equivalence result. I will first

state the result and its underlying assumptions, and then provide further intuition by linking

the proof strategy back to the simple model.

Consumption demand equivalence relies on three key assumptions. The first assumption

is implicitly embedded in the model of Section 1.2.1, but I explicitly state it here for emphasis.

Assumption 1. Households and the government consume a single, homogeneous final good.

The second assumption relates to the interest rates faced by households and government.

The model already imposes that all agents borrow and lend at a common interest rate;

Assumption 2 re-states this property for emphasis, and then provides an additional restriction

on the actual financing of government expenditure shocks.

Assumption 2. Households and government borrow and lend at the same interest rate. The

path of taxes and transfers used to finance a given public expenditure shock εεετ or εεεg depends

20


only on the present value of the expenditure, not its time path. A spending path with zero

net present value is purely deficit-financed, and so elicits no direct tax response.

The third assumption restricts the economy’s labor market. In response to the partial

equilibrium increase in consumption demand cPEd , the average marginal utility of consump-

tion declines, and so sticky-wage unions may try to bargain for higher wages. I denote

the desired adjustment in aggregate hours worked at unchanged wages by ˆPEd , defined for-

mally in Appendix A.1.1. My third assumption provides two possible sufficient conditions

to guarantee that ˆPEd = 0.

Assumption 3. There are either no wealth effects in labor supply, or wages are perfectly

sticky (i.e., wage adjustment costs are infinitely large).

These assumptions are sufficient for the following generalized equivalence result.

Proposition 2. Consider the structural model of Section 1.2.1. Suppose that, for each one-

time shock {τ, g, v}, the equilibrium transition path exists and is unique. Under Assump-

tions 1 and 2, the responses of consumption to a private demand shock d (either impatience

v or tax rebate τ) and to a government spending shock g with gg = cPEd satisfy, to first order,

cd = cPEd + cg + error(

ˆPEd

)(1.12)

where the error function is equal to 0 if ˆPEd = 0. Under the additional Assumption 3,

cd = cPEd︸︷︷︸PE response

+ cg︸︷︷︸= GE feedback

(1.13)

The proof strategy for Proposition 2 is almost identical to that of the spender-saver RBC

model. Equilibria in the richer model can generally not be characterized as solutions to a

single market-clearing condition in a single price; instead, as I show formally in Lemma 2,

21


they are solutions to a rich set of market-clearing conditions and other restrictions. Assump-

tions 1 to 3 are simply sufficient to ensure that private and public spending shocks perturb

the same market-clearing conditions by the same amount, and thus elicit the same general

equilibrium adjustment, exactly as in the proof of Proposition 1.

Assumption 1 – in conjunction with the requirement that gg = cPEd – ensures that the

private and public demand shocks lead to the same excess demand pressure for the common

final good. Since households and governments borrow and lend at identical rates, these

identical net excess demand paths can in principle be financed using identical paths of taxes

and transfers. Without Ricardian equivalence, however, the precise timing of the financing

matters. Assumption 2 then simply ensures that, indeed, the two shocks are financed in

exactly the same way.10 Under these restrictions alone, the general equilibrium propagation

of private and public spending shocks may still differ, as households may also decide to

directly adjust their desired labor supply following the shock εεεd. Assumption 3 – a restriction

on household behavior – is enough to rule this out: Following the shock εεεd, households either

do not wish to or are not able to directly adjust their hours worked, i.e. ˆPEd = 0. Together,

Assumptions 1 to 3 ensure exact demand equivalence.

In the proof of Proposition 2, I establish the existence of a “demand multiplier” D –

a map transforming partial equilibrium net excess demand paths (such as cPEd or gg) into

general equilibrium impulse responses. As such, it builds on results in Auclert & Rognlie

(2018) and Auclert et al. (2018). In particular, in Auclert et al. (2018), the intertemporal

Keynesian cross matrixM – a special case of the multiplier D – governs the transmission of

private and public demand shocks, establishing demand equivalence. Their result applies in

a model with passive monetary policy, demand-determined labor, and without investment;

10Note that impatience shocks – shocks that just shift the intertemporal profile of private consumptionspending – have zero net present value. As a result, the analogous government spending change also has zeronet present value, and need not be (and I assume is not) financed through any change in taxes or transfers.

22


Proposition 2 provides the generalization to the model of Section 1.2.1.11 The intuition

for such a common “demand multiplier” is particularly transparent in the standard static

Keynesian cross, and was for example previously discussed in Hall (2009).

Approximation Error. The decomposition in (1.12) shows that, in the family of business-

cycle models nested by the outline in Section 1.2.1, the only channel through which demand

equivalence may fail is differential labor adjustment: Households that receive the rebate may

decide to optimally work less. I assess the plausibility of this mechanism in two ways. First,

in the remainder of this section, I analyze its strength in two particular numerical examples.

Consistent with conventional wisdom in the recent business-cycle literature (e.g. Christiano,

2011a,b) I find that even moderate degrees of price and wage stickiness are sufficient to largely

neuter the aggregate effects of transitory changes in labor supply. Second, in Section 1.4.2, I

provide direct empirical discipline on the error term, and conclude that it is robustly small.

My first example is a heterogeneous-agent New Keynesian (HANK) model. The model

falls into the benchmark class of Section 1.2.1 and features uninsurable income risk, moder-

ate degrees of nominal price and wage stickiness, and several further frictions familiar from

standard business-cycle models (e.g. investment adjustment costs, variable capital utiliza-

tion, and a rich Taylor rule). Its parameterization is close to that of the estimated HANK

model of Section 1.4; I relegate further details on model structure and parameterization to

that section as well as Appendix A.1.2.2. Importantly, Assumption 1 holds and fiscal policy

in the model is consistent with Assumption 2, but household preferences – of the typical

separable kind – feature strong short-run wealth effects, and wages re-set every 2.5 quarters

on average.

11Auclert & Rognlie (2018) is, to the best of my knowledge, the first paper to discuss general equilibriummultipliers for perfect foresight transition paths. In particular, they prove that different kinds of consumptiondemand shocks are propagated identically in general equilibrium; Proposition 2 shows under what conditionsthose same multipliers also apply to public demand shocks – that is, demand equivalence.

23


I compare impulse responses to a one-off income tax rebate and to a transitory increase in

government spending. The two shocks give identical partial equilibrium net spending paths

(normalized to 1% of steady-state consumption on impact) and are financed using identical

delayed increases in taxes. Results are displayed in Figure 1.2.

Figure 1.2: Approximate Demand Equivalence, HANK Model

Note: Impulse response decompositions and demand equivalence approximation in a simple HANKmodel, with details on the parameterization in Appendix A.1.2.2. The direct response and theindirect general equilibrium feedback are computed following Definition 1.

The left panel decomposes the total consumption response to the rebate into direct partial

equilibrium effect and indirect general equilibrium feedback, in line with Definition 1. Since

the model features a high average MPC, Keynesian multiplier effects dominate, and so gen-

eral equilibrium effects further amplify the initial stimulus. The right panel approximates the

consumption response by summing (i) the direct consumption response cPEτ and (ii) the ag-

gregate general equilibrium response of consumption to a similarly transitory and identically

financed expansion in government spending, cg. Under Assumption 3, the decomposition

would be exact, so the grey and black dotted lines would be indistinguishable. Instead,

after receiving the rebate, households would like to work less, simultaneously pushing down

24


consumption. The approximation cPEτ + cg thus over-states the true consumption response;

the associated error, however, is small, at just below 3 per cent of the true peak consumption

response. Even with (moderately) sticky wages, labor is largely demand-determined, and so

small and transitory shifts in labor supply are quantitatively irrelevant.

The second example is the popular quantitative business-cycle model of Justiniano et al.

(2010). I solve the model at their estimated posterior mode, but – to allow for non-trivial

effects of aggregate income tax rebates – add a small fringe λ of hand-to-mouth households.

Figure 1.3: Approximate Demand Equivalence, Justiniano et al. (2010)

Note: Impulse response decompositions and demand equivalence approximation in the model ofJustiniano et al. (2010), solved at the posterior mode and with a fraction λ→ 0 of spenders. Thedirect response and the indirect general equilibrium feedback are computed following Definition 1.

For the numerical experiment in Figure 1.3, I let λ → 0, but keep the effective size of

the rebate εεετ × λ fixed. Specifically, I consider a sequence of rebates given to spenders,

inducing a spending response similar to the intertemporal demand profile in my estimated

HANK model.12 Figure 1.3 plots the resulting aggregate impulse response decompositions.

12It is straightforward to show that, in the limit λ→ 0 but with εεετ×λ = constant, the aggregate dynamicsof the model are identical to that of Justiniano et al. (2010), and the income tax rebate shock enters thehousehold consumption-savings problem exactly like a standard impatience shock.

25


As before, general equilibrium feedback is the result of a complicated interaction of several

model features. The flat orange line reveals that, at the model’s estimated mode parame-

terization, the crowding-in effects associated with higher household income just happen to

be almost exactly offset by interest rate crowding-out. Crucially, however, in response to an

equally large increase in government spending, consumption also barely moves, and so the

additive approximation of Proposition 2 is again highly accurate, with a maximal error of

only 0.4 per cent of the true peak consumption response.13 Relative to the HANK model,

the approximation is even more accurate since wages are (much) stickier.

For additional insights on this near-equivalence, it is instructive to further dissect the

approximation error in (1.12). As I show in the proof of Proposition 2, the error term can

be re-expressed as the full general equilibrium response of consumption to a particular labor

supply (wage cost-push) shock. The near-equivalence in Figures 1.2 and 1.3 is consistent

with the discussion in Christiano (2011a): Even with moderate wage and price stickiness,

the effects of transitory shifts in labor supply are largely neutralized in general equilibrium.

However, my analytical results on the composition of the error term also justify a more direct

measurement strategy: As I show in Section 1.4.2, micro and macro data can jointly provide

direct empirical discipline on the size of the error, and robustly imply that it is small.

Conclusions. The analysis in this section has demonstrated that, in a large and empirically

relevant class of structural models, private and public spending shocks share (either exactly

or approximately) identical general equilibrium propagation.

This equivalence result is, however, completely silent on the strength of those common

general equilibrium effects. In the simple spender-saver model of Section 1.2.3, partial equi-

librium spending responses are crowded-out; in my two quantitative examples, feedback

13Unsurprisingly, in both models, the approximation deteriorates for highly persistent shocks, as wagesare not permanently sticky. Instead, the implied persistent shifts of labor supply materially affect aggregatequantities, and so the approximation error becomes larger. I provide an illustration in Appendix A.4.1.3.

26


effects are instead relatively weak. In Appendix A.4.3, I show two extreme examples, one

with full crowding-out, the other with strong amplification, yet both featuring exact demand

equivalence. Ultimately, the strength of general equilibrium effects – and so the size of the

missing intercept – is an empirical question. The next section presents my empirical strategy.

1.3 A two-step approach to estimating consumption

demand counterfactuals

This section develops a two-step methodology to estimate semi-structural macro counterfac-

tuals for generic consumption demand shifters. I describe the approach in Section 1.3.1, and

then in Section 1.3.2 apply it to study the effects of income tax rebates.

1.3.1 The two-step methodology

Consider a researcher interested in the response of aggregate consumption to a generic “con-

sumption demand” shifter – a shock that directly affects incentives for household spending.

Examples of such shifters are plentiful in recent work; among the most notable are income

tax rebate stimulus (Parker et al., 2013), household deleveraging due to tightened borrowing

conditions (Mian et al., 2013; Berger et al., 2017), changes in household bankruptcy exemp-

tions (Auclert et al., 2019) and redistribution across households through taxation (Jappelli

& Pistaferri, 2014). As is well-known, estimation of the aggregate effects of such shocks is

severely complicated by their likely endogeneity to wider macroeconomic conditions.14

In response to these challenges, most recent work has tried to estimate shock propagation

using household-level data, exploiting plausibly exogenous heterogeneity in shock exposure.

14More specifically, direct projection on measures of aggregate shocks (proxy variables) is ruled out by theirendogeneity (Ramey, 2016). Other, more involved macro structural identification approaches are hinderedby (i) likely non-invertibility due to the relative infrequency of shocks (Plagborg-Møller, 2019) and (ii) thedearth of plausible macro exclusion restrictions (e.g. zero restrictions in structural VAR representations).

27


In the remainder of this section I argue that (i) the econometric estimands of such cross-

sectional regressions are often interpretable as direct (partial equilibrium) shock responses

cPEd and (ii) we can use estimates of the aggregate effects of changes in government spending

to proxy for their missing general equilibrium intercept cGEd .

Model. As before, I develop all arguments in the context of the structural model of Sec-

tion 1.2.1. In my theoretical analysis of demand equivalence, the proof strategy dictated

a focus on perfect foresight transition paths. For standard cross-sectional and macro re-

gression estimands to be well-defined, however, I need a proper notion of aggregate risk. I

thus now consider the linear vector moving-average representation induced by the first-order

perturbation solution of the model, assuming that the shocks εst, s ∈ {τ, g, v} are mutually

i.i.d. and N(0, 1).15 I use s subscripts to indicate impulse response functions to such one-

time structural shocks; by certainty equivalence, these impulse responses are to first order

identical to the transition paths for one-off structural shocks studied in Section 1.2, thus

justifying the re-use of notation.

Finally, to introduce cross-sectional heterogeneity in shock exposure, I further assume

that the rebate and impatience shocks faced by household i satisfy εsit = ξsit×εst, where ξsit

is i.i.d. across households and time (and uncorrelated with any household characteristics),

with E(ξsit) = 1 and Var(ξsit) > 0. In the proof of Proposition 3 I show that, under my

assumptions on the exposure term ξsit, all aggregates are – to first order – unaffected by this

cross-sectional heterogeneity in shock exposure.

15Previous studies that exploit the first-order equivalence of perturbation and perfect foresight transitionsfor estimation include Mankiw & Reis (2007) and Auclert et al. (2019). It is immediate from the propertiesof linear VMA representations that arbitrary further shocks could be added without affecting my results.

28


Micro regressions. A typical regression exploiting microeconomic heterogeneity in house-

hold exposure to the demand shocks d ∈ {τ, v} takes the form

cit+h = αi + δt + βdh × εdit + uit+h, h = 0, 1, 2, . . . (1.14)

where αi and δt are individual and time fixed effects.16

It is straightforward to show that, under my assumptions, regressions such as (1.14) esti-

mate average household-level causal effects that are interpretable as direct partial equilibrium

shock responses, consistent with Definition 1.

Proposition 3. Suppose an econometrician observes a panel of household consumption {cit}

and measures of shock exposure {εdit} generated from the linear vector moving average repre-

sentation of the structural model of Section 1.2.1. Then the ordinary least-squares estimand

of βββd ≡ (βd0, βd1, . . .)′ satisfies

βββd =

∫ 1

0

∂ci∂εd0

di = cPEd (1.15)

In words, regressions such as (1.14) do not estimate the true macro counterfactual cd,

but instead give a household-level average treatment effect that is interpretable as a partial

equilibrium response, cPEd – precisely the object defined in my decomposition in Definition 1.

Obtaining such estimates from a sequence of cross-sectional micro regressions like (1.14) is

the first step of my methodology.

General equilibrium effects. To map the micro estimates βββd into full general equilib-

rium counterfactuals, researchers would typically use full structural models, calibrated to be

consistent with the micro estimates themselves as well as various other formally or infor-

16The regression in (1.14) is at the individual level. My analysis here thus does not apply to cross-regionalregressions, as for example in Mian et al. (2013). I generalize my method to such regressions in Wolf (2019b).

29


mally targeted macro moments (e.g. Kaplan & Violante, 2018). The equivalence result in

Proposition 2 suggests that, for a large class of models, evidence on the aggregate effects of

public spending shocks should be a highly informative macro moment – in fact informative

enough to give some counterfactuals without ever having to solve any particular model.

The second step of my proposed methodology leverages this insight. Suppose that the

econometrician can jointly estimate the response of the macro-economy to a list of nk different

kinds of government spending shocks {εgk}nkk=1, where these shocks induce potentially different

paths of aggregate government spending and tax financing. Furthermore suppose that, for

some linear combination of shocks with weights {γk}, it is the case that

βββd =

nk∑k=1

γk × ggk (1.16)

In words, a linear combination of government spending shocks available from macro experi-

ments gives similar partial equilibrium excess demand pressure as the private demand shock

εd. This is a restrictive requirement, but I will later demonstrate through several appli-

cations that such “demand matching” is possible in practice for many interesting partial

equilibrium demand paths βββd and so shocks εd.17,18 It then remains to gauge the accuracy of

the financing Assumption 2. For example, if the researcher is interested in counterfactuals

for a deficit-financed rebate, then the composite public spending shock∑nk

k=1 γk× ggk should

also be deficit-financed. If so, then we can invoke Proposition 2 to conclude that

cGEd =

nk∑k=1

γk × cgk (1.17)

17If researchers are willing to ignore anticipation effects of news shocks, then any path βββd can be replicatedexactly with evidence on just a single government spending shock ggk . In my applications I do not need thisadditional assumption, but it is interesting to note that, at least for the government spending shocks studiedin previous work, anticipation effects do appear rather limited (e.g. Ramey, 2011).

18In Appendix A.4.1.8 I document approximate equivalence when (1.16) is roughly satisfied.

30


Putting all the pieces together, we get the full general equilibrium counterfactual

cd = βββd︸︷︷︸PE response

+

nk∑k=1

γk × cgk︸︷︷︸GE feedback

(1.18)

Since by assumption the econometrician is able to jointly estimate the response of the macro-

economy to the list {εgk}nkk=1 of fiscal shocks, she can straightforwardly construct frequentist

standard errors or Bayesian confidence sets for the full general equilibrium term.19

In the remainder of this paper I illustrate my method with three examples. First, in Sec-

tion 1.3.2, I use it to estimate the aggregate effects of a deficit-financed income tax rebate

(Parker et al., 2013). Second, in Appendix A.5, I study the effects of a one-off (budget-

neutral) income re-distribution from rich (low-MPC) to poor (high-MPC) households.20 Fi-

nally, in Section 1.5, I establish a theoretical investment demand equivalence result, and use

my two-step approach to estimate the aggregate effects of bonus depreciation stimulus.

1.3.2 Application: income tax rebates

I combine micro and macro evidence to estimate the response of aggregate consumption and

output to a one-off income tax rebate (i.e., lump-sum transfer). My main finding is that full

general equilibrium counterfactuals are close to direct micro estimates: The partial equilib-

rium increase in consumption demand is accommodated one-for-one through an increase in

output, with relatively limited general equilibrium crowding-in or -out.

19Except for a brief discussion in Section 1.3.2, I will largely ignore estimation uncertainty for the directresponse. Under my assumptions, sampling uncertainty for the micro and macro parts is independent, soconstruction of joint confidence sets is in principle straightforward. Intuitively, this is so because macro shocksare differenced out in micro regressions, and micro shocks have no aggregate effects (see Appendix A.2.4).

20Such budget-neutral re-distribution is also the topic of Auclert & Rognlie (2018). Due to the scarcityof evidence on heterogeneity in dynamic intertemporal MPCs across households, I in this application relyon a standard partial equilibrium consumption-savings problem to construct cPEd . The mapping into macrocounterfactuals is then again completed using the semi-structural second step from (1.17).

31


Direct Response. I first require an estimate of the direct spending response cPEτ . For a

one-off, one-quarter stimulus payment, this direct spending response is given as

cPEτt ≡MPCt,0 × τ0

where

MPCt,0 ≡∫ 1

0

∂cit∂τ0

di

is the average marginal propensity to consume at time t out of an income gain at time 0.

Several recent studies have used rich household spending data to estimate objects that are

either exactly or approximately interpretable as the desired average MPC (e.g. Johnson et al.,

2006; Parker et al., 2013; Jappelli & Pistaferri, 2014; Fagereng et al., 2018).21 A common

finding in this literature is that households spend most of a (small) one-time income receipt

on impact, and that the spending response decays back to zero relatively quickly. Johnson

et al. (2006) and Parker et al. (2013), who specifically focus on the consumption response to

income tax rebates, estimate a differenced version of the micro regression (1.14); building on

Proposition 3, Appendix A.2.1 shows that – at least under some assumptions on household

expectation formation – their regression estimates MPC0,0 (and MPC1,0).

The point estimates of Parker et al. (2013) suggest that, following the rebate stimulus of

2008, total consumption expenditures increased by about 50 to 90 per cent of payments in

the quarter of the receipt. Given the overall size and (staggered) timing of the stimulus, this

spending response corresponds to around 1.5 per cent of personal consumption expenditure

on impact, and 0.7 per cent in the following quarter.22 In the left panel of Figure 1.4, the

21By my definition of the consumption function c(•) in Section 1.2, the MPC should be interpreted asan MPC after adjusting for any endogenous response of earnings. In the notation of Auclert (2019), it is

the adjusted MPC. As discussed there, popular empirical studies arguably estimate this adjusted object.Furthermore, estimated earnings responses are usually small anyway, as discussed further in Section 1.4.2.

22These estimates include the durables spending response. As I show in Appendix A.4.1.4, demand equiva-lence extends without change to a model with durable and non-durable consumption. To ensure consistency,my VAR analysis also throughout contains measures of total consumption. For completeness, however, I

32


green x’s show the corresponding direct consumption responses cPEτ0 and cPEτ1 ; the solid green

line shows what I take as the estimate of the full partial equilibrium spending response cPEτ .

Figure 1.4: Measuring cPEτ & cg

Note: The left panel shows direct consumption responses to income tax rebate (green) vs. directgovernment spending response to identified spending shock (black). Estimated consumption re-sponses from Parker et al. (2013) (Table 3). The dashed lines for the government spending shockcorrespond to 16th and 84th percentile confidence bands. The right panel shows the response ofconsumption to the same identified spending shock.

The Missing Intercept. It remains to estimate the aggregate effects of a similarly transi-

tory and deficit-financed expansion in government spending. Previous studies often find that

government spending expansions – both transitory and more persistent – are accommodated

roughly one-for-one through increases in output, with relatively little feedback to private

spending (Hall, 2009; Gechert, 2015; Caldara & Kamps, 2017; Ramey, 2018).

My identification of government spending shock propagation relies on professional forecast

errors for federal spending. Formally, I treat the forecast errors as a (noisy) measure of

have repeated my analysis using evidence on non-durables consumption only. The direct partial equilibriumconsumption response is smaller and more persistent, and the aggregate non-durables consumption responseto a similarly persistent government spending is an even more tightly estimated 0.

33


exogenous innovations to public expenditure; intuitively, this assumption can be justified by

likely lags in the response of fiscal policy to any changes in macroeconomic fundamentals.

In the language of macro identification, I assume that residualized forecast errors are valid

external instruments. In the context of my structural model, the IV relevance and exclusion

restrictions can be phrased follows:

Assumption 4. Suppose that an econometrician observes time series of macroeconomic

aggregates yt and professional forecast errors of real federal spending zt, where the residualized

forecast error zt ≡ zt − E (zt | {zt−`, yt−`}∞`=1) satisfies

E(zt · εgt) 6= 0, E(zt · εju) = 0 for all (j, u) 6= (g, t) (1.19)

As shown in Plagborg-Møller & Wolf (2019b), estimating a recursive vector autoregres-

sion (VAR) in instrument and macro aggregates (zt, y′t)′, with the instrument ordered first,

correctly identifies the impulse responses of all macro aggregates yt to a structural innovation

εgt in aggregate government spending, up to a scale parameter that is independent of horizon

and response variable:

Proposition 4. (Plagborg-Møller & Wolf, 2019b) Suppose that the researcher estimates a

VAR in (zt, yt)′, where yt is a vector of observed macroeconomic aggregates and zt satisfies

Assumption 4. Let θθθy denote the vector of impulse responses of y to the first shock in a

recursively identified SVAR. Then the ordinary least-squares estimand of θθθy satisfies

θθθy = constant × yg (1.20)

where the constant term is a scalar, independent of the individual response variable in y or

the impulse response horizon.

34


Relative impulse responses are thus identified – and since impulse responses will be re-

scaled to match cPEτ , relative responses are sufficient to recover cg.23 Also note that the

consistency proof requires no assumptions on invertibility of the shock εgt, mitigating con-

cerns about timing and anticipation (Ramey, 2011; Leeper et al., 2013). I thus estimate a

recursive VAR in forecast errors zt and aggregates yt, where the vector yt includes measures

of overall government spending, output, consumption, investment, hours worked, taxes, and

government debt. To plausibly estimate cg in a stable macroeconomic regime, I restrict my

sample to range from the third quarter of 1981 to the fourth quarter of 2007. Further details

on exact variable definitions, data construction and the estimation procedure are relegated to

Appendix A.2.3. The appendix also discusses several robustness checks, notably with respect

to the vector of macro aggregates yt, lag length selection, prior selection, and controls.

The results are also included in Figure 1.4. The left panel shows that, in response to the

shock, government spending increases sharply, but returns to baseline quickly. Importantly,

the time profile of the demand expansion quite closely mirrors the micro-estimated expansion

in private consumption spending.24 The right panel shows the corresponding response of

aggregate consumption – cg. Consumption appears to be somewhat crowded-in on impact,

and mildly crowded-out in the following quarters. Overall, cg is close to 0 throughout, and in

fact reasonably tightly estimated. Finally, in Appendix A.2.3, I show that the expansion in

government spending leads to a delayed increase in taxes, as well as a persistent rise in total

government debt. By Assumption 2, my counterfactuals for a transitory income tax rebate

23Strictly speaking, the estimated impulse response gg takes into account general equilibrium feedbackto government spending; for the demand matching (1.16), any such feedback needs to be filtered out. Inprinciple this can be done in two ways. First, researchers may simply assume that there is no such feedback.Second, if the structural government spending equation can be identified (e.g. as in Blanchard & Perotti,2002), then feedback effects can be removed manually. For simplicity I choose the first path. Encouragingly,however, results using the second approach are almost identical, simply because I find limited feedback frommacro aggregates to government expenditure, exactly as in previous work (e.g. Caldara & Kamps, 2017).

24While the two demand paths are quite similar, they of course do not align perfectly. In Appendix A.4.1.8,I discuss the accuracy of my approximation under imperfect demand matching.

35


should thus be interpreted as pertaining to a particular, quite persistently deficit-financed

one-off transfer to households.25

Macro Counterfactuals. To construct a valid general equilibrium counterfactual, it sim-

ply remains to sum the estimated cPEτ and cg. The results are displayed in the left panel of

Figure 1.5. Note that, for construction of the plot, I take the point estimate of cPEτ as given,

and only account for macroeconomic estimation uncertainty.26

Figure 1.5: Income Tax Rebate, Aggregate Impulse Responses

Note: Consumption and output responses to an income tax rebate shock. The full consumptionresponse is computed following the exact additive decomposition of Proposition 2, while the outputresponse is simply equal to the response after a government spending shock. The dashed lines againcorrespond to 16th and 84th percentile confidence bands.

The left panel shows the full general equilibrium counterfactual for consumption. The

aggregate effect of the policy – according to my decomposition given as the simple sum

25Other fiscal spending episodes could be used to construct counterfactuals for other financing schemes.For a detailed review of different spending episodes and their financing, see Ferriere & Navarro (2018).

26This is in keeping with my emphasis on the “missing intercept.” However, since the direct spendingresponse is only a function of the impact response coefficient of Parker et al. (2013), and since this coefficientis statistically significant, it is immediate that the full impact response is – by independence – also significant.

36


cPEτ + cg – appears to be quite close to the (large) micro-estimated direct spending response

cPEτ documented in Parker et al. (2013). Thus, perhaps surprisingly, the various price and

multiplier effects cited in previous empirical and theoretical work seem to roughly cancel.

The right panel then shows the corresponding impulse response of output which, by the

demand equivalence result, is identical for tax rebate and government spending expansion.

Here I find a significant (if short-lived) response, with output on impact rising by somewhat

less than 1 per cent, and then returning to baseline. Overall, deficit-financed income tax

rebates appear to provide meaningful stimulus to aggregate consumption and output.

My analysis suggests that, at least for income tax rebate stimulus, the “missing intercept”

of general equilibrium feedback is a relatively tightly estimated zero. This conclusion is

an immediate implication of the theoretical demand equivalence result in conjunction with

a relatively standard piece of empirical evidence – deficit-financed government spending

multipliers around 1, with limited feedback to private spending. While direct micro estimates

are thus actually a reliable guide to full general equilibrium counterfactuals, arriving at this

conclusion nevertheless required important macroeconomic identifying assumptions, notably

on demand equivalence and the identification of aggregate public spending shocks.27

Implications for Structural Modeling. My results have implications for structural

modeling similar to those of the sufficient statistics characterizations in Chetty (2009) and

Arkolakis et al. (2012): Any structural analysis that estimates a consumption response to

tax rebates different from Figure 1.5 either (i) breaks demand equivalence, (ii) is inconsistent

with micro evidence on large direct spending responses, or (iii) is inconsistent with macro

evidence that suggests around unit fiscal multipliers and thus relatively limited feedback to

private spending.

27My pre-crisis VAR implicitly measures “normal-time” general equilibrium effects. However, as I discussin Appendix A.4.5, the equivalence result applies to (small) shocks around any given current state of theeconomy. With an extended sample containing the recent period of low rates, I find suggestive evidence ofslightly larger multipliers, consistent with Ramey & Zubairy (2018) and Debortoli et al. (2019).

37


The estimated HANK model of Section 1.4.1 is an example of a structural model that

satisfies (i) and (ii): it features approximate demand equivalence (recall Figure 1.2) and large

direct spending responses due to large household-level marginal propensities to consume (see

Appendix A.1.2.2).28 However, the overall model generates fiscal multipliers somewhat in

excess of 1, and so – consistent with the equivalence result – it suggests that direct micro

effects are a slight under-estimate of full counterfactuals. With somewhat less sticky prices

and more aggressive monetary policy, the model almost perfectly matches my empirically

estimated fiscal policy impulse responses (see Appendix A.4.4). This re-estimated HANK

model is a promising laboratory for further structural analysis.29

1.4 Approximation accuracy

The theoretical equivalence result of Section 1.2 relies on three main assumptions: (i) the

existence of a common final good, (ii) identical borrowing and lending rates for households

and government, and (iii) zero (short-run) wealth effects of labor supply, or fully rigid wages.

All three assumptions are presumably incorrect. This section systematically studies the role

of each in ensuring that the general equilibrium counterfactuals computed in Section 1.3

remain at least approximately valid.

I do so in two steps. First, in Section 1.4.1, I present a rich structural model, estimated to

be consistent with evidence on both individual household consumption and savings behavior

as well as the time series properties of macroeconomic aggregates – in other words, a model

suitable for structural fiscal policy counterfactuals. In this model, the approximation is

28Auclert et al. (2018) show that HANK models of the kind considered in my structural analysis can closelymatch empirically documented paths of intertemporal MPCs. An analogue of Figure 1.4 with a model-baseddirect spending response (instead of the estimated one) thus unsurprisingly looks very similar.

29More conventional medium-scale DSGE models usually imply multipliers below 1 (Gechert, 2015; Ramey,2016). Hand-to-mouth behavior is thus central to the documented consistency between model and data.

38


accurate. Then, in Sections 1.4.2 to 1.4.4, I consider several departures from this benchmark,

allowing me to study in isolation the importance of each individual assumption.

1.4.1 The estimated HANK model

My main test for accuracy of the proposed approximation is an estimated HANK model,

featuring a conventional consumption-savings problem under imperfect insurance embedded

into an otherwise standard medium-scale DSGE model. Arguably, the model is rich enough

to serve as a quantitative laboratory for structural analysis of the aggregate effects of generic

private and public spending shocks; as such, it is an example of a structural model that could

plausibly be used to identify the missing general equilibrium intercept of, say, transitory

transfer payments to households. I provide a brief outline of the model and my estimation

strategy here, and relegate further details to Appendix A.1.2.2.

The household block is slightly more general than that of the benchmark model in Sec-

tion 1.2.1. In particular, household borrowing in the liquid asset is now only possible at a

penalty rate ibt + κ. As a result, households and government discount at different interest

rates, and so equally large expansions in private and public spending cannot be financed us-

ing identical paths of taxes and transfers. The rest of the economy is designed to be as close

as possible to the medium-scale structural model of Justiniano et al. (2010). First, I allow

for investment adjustment costs, variable capacity utilization, and a rich monetary policy

rule. Second, in addition to the impatience and government spending shocks discussed in

Section 1.2.1, I also include shocks to total factor productivity and the marginal efficiency of

investment, to price and wage mark-ups, and to monetary policy. As I restrict attention to

first-order transition paths, these additional shocks of course do not affect the propagation

of private and public spending shocks; I only include them for estimation purposes.

39


I calibrate the model’s steady state using targets familiar from the HANK literature (e.g.

Kaplan et al., 2018). Importantly, because household self-insurance is severely limited, the

average MPC is high, at around 30% quarterly out of a lump-sum 500$ income gain. Model

parameters governing dynamics are then estimated using conventional likelihood methods

(An & Schorfheide, 2007; Mongey & Williams, 2017) on a standard set of macroeconomic

aggregates.30 The key exception is the degree of wage stickiness which – in light of its

centrality to my results – is directly calibrated to be consistent with recent micro evidence

(Grigsby et al., 2019; Beraja et al., 2019), with wage re-sets every 2.5 quarters on average.

Results. I solve the model at the estimated posterior mode, and implement the demand

equivalence approximation for a one-off income tax rebate following the method of Sec-

tion 1.3. Results are displayed in Figure 1.6.

Figure 1.6: Approximate Demand Equivalence, Estimated HANK Model

Note: Impulse response decompositions and demand equivalence approximation in the estimatedHANK model, with details on the parameterization in Appendix A.1.2.2. The direct response andthe indirect general equilibrium feedback are computed following Definition 1.

30Specifically, I closely follow Justiniano et al. (2010) and include measures of output, inflation, a short-term interest rate, consumption, investment, hours worked, and a measure of aggregate wages.

40


The plot looks extremely similar to Figure 1.2. Of course this is unsurprising – the models

are almost identical, differing only in the presence of a borrowing wedge. In terms of error

metrics, the approximation here is in fact better than in the previous model: the maximal

error (relative to the true peak response of aggregate consumption) is around 2.2 per cent,

and around 3 per cent in the simpler model. The intuition is as follows: In the model with

borrowing wedge, indebted households use the rebate to pay down (high-return) debt. As a

result, the average return faced by households is higher than that faced by the government.

Taxes thus have to rise by more to finance the government spending expansion gg compared

to the consumption stimulus cPEτ , and so the demand equivalence approximation tends to

under -state the aggregate effects of a tax rebate, partially offsetting the labor supply error.

I further elaborate on this intuition and on the size of the associated error in Section 1.4.3.

Figure 1.6 only reveals that my approximation is accurate at the estimated posterior mode

of a particular structural model. In Appendix A.4.1.7 I go one step further and show that

most of the estimated parameters governing model dynamics – including the monetary rule,

the nature of investment adjustment costs, and the degree of variable capacity utilization in

production – are in fact largely orthogonal to the accuracy of the approximation. Formally, I

randomly draw model parameters from large supports, solve the implied model, and compute

the approximation accuracy. I find that, of all estimated parameters, only the degree of

price rigidity has a material impact on the accuracy of the approximation.31 However, even

with near-flexible prices, and fixing the relatively moderate calibrated wage rigidity, the

approximation error remains at only 9 per cent of the peak consumption response.

31With rigid prices, labor is demand-determined. Shifts in labor supply thus only affect relative wage anddividend pay-outs, and so the approximation is accurate (up to a redistributive effect).

41


1.4.2 Labor supply

The demand equivalence approximation is accurate in all models studied so far because of

sufficiently strong nominal rigidities. With fully flexible prices and wages, the strong wealth

effects of my conventional separable preferences will invariably break the demand equivalence

logic. Figure 1.7 provides an illustration, using the estimated structural HANK model of the

previous section, but with flexible prices and wages.

Figure 1.7: Failure of Demand Equivalence, Flex-Price HA Model

Note: Impulse response decompositions and demand equivalence approximation in the estimatedHANK model, but with flexible prices and wages. The direct response and the indirect generalequilibrium feedback are computed following Definition 1.

The right panel shows that the quality of the demand equivalence approximation dete-

riorates sharply. Intuitively, following the tax rebate, households consume more and so –

because the marginal utility of consumption is lower – optimally choose to work less. Labor

supply is not similarly reduced after expansions in government spending, so the demand

equivalence approximation over-states the aggregate consumption response.

The mechanism underlying this inaccuracy is, however, sharply at odds with all kinds of

micro and macro evidence. First, my estimation exercise based on standard aggregate time

42


series data as usual calls for nominal rigidity. Without such rigidity, the model would feature

large countercyclical (and counterfactual) swings in wages and inflation. Second, direct

estimates of wage rigidity using microeconomic or cross-regional data suggest moderate, but

non-trivial, amounts of stickiness (Grigsby et al., 2019; Beraja et al., 2019). As emphasized

above, even very moderate degrees of nominal wage stickiness – with re-sets occurring every

2.5 quarters – are enough to make the demand equivalence approximation highly accurate.

Third, the flexible-price model implies that, in response to the rebate, households would like

to reduce their earnings by almost as much as they increase their spending. As emphasized by

Auclert & Rognlie (2017), such large negative earnings responses are an inescapable feature

of macro models with large wealth effects of labor supply, large average MPCs, and flexible

wages. Micro data instead suggest an earnings response an order of magnitude smaller than

the average MPC (e.g. Cesarini et al., 2017; Fagereng et al., 2018).32 In models with flexible

wages but data-consistent small short-run wealth effects, the approximation is instead again

highly accurate (see Appendix A.4.1.1).

Finally, as I show in Appendix A.4.2, it is actually possible to adapt my two-step method-

ology to directly account for the labor supply error term in the decomposition of Proposi-

tion 2. As discussed in Section 1.2.4, this error term is identical to the response of aggregate

consumption to a particular kind of “labor wedge” shock – the sudden desire of households

to work less. Combining (i) micro estimates of the size of this labor wedge shock and (ii) ev-

idence on the aggregate effects of distortionary labor income taxes (Mertens & Ravn, 2013),

we can thus recover a direct empirical correction for the error. Unsurprisingly, since empiri-

cal estimates of the desired labor supply contraction are small, the results of this augmented

procedure are almost identical to the benchmark estimates of Section 1.3.2.

32The strength of wealth effects in labor supply has also been estimated in macro data. Such studiesroutinely favor near-zero wealth effects (Schmitt-Grohe & Uribe, 2012; Born & Pfeifer, 2014).

43


1.4.3 Interest rates

If households and government borrow and lend at different rates, then identical changes in

private and public partial equilibrium net excess demand cannot be financed using identical

paths of taxes and transfers, violating Assumption 2. In particular, if household returns

are high (low) relative to government returns, then taxes need to increase by less (more) to

finance private relative to public spending. These lower (higher) taxes will sooner or later

feed back into consumption; if this happens immediately, then the simple demand equivalence

approximation will tend to under-state (over-state) the true impulse response of aggregate

consumption to the private demand shifter.33

In the estimated structural model of Section 1.4.1, some households pay down high-return

liquid debt, so the implicit household discount rate is high. The associated bias, however, is

negligible; in a model variant with fully rigid wages, the maximal error is equal to 0.7 per cent

of the true peak consumption response. The intuition for this quantitative near-irrelevance is

simple: Suppose that, in response to a shock, direct (partial equilibrium) household spending

increases by 1$ for one year. My approximation compares the aggregate effects of this shock

to those of an identical expansion in aggregate public spending. Crucially, even if the wedge

between average household and government discount rates is an (arguably implausible) five

per cent, the difference in present discounted values of the two spending expansions is just five

cents – relatively small compared to the initial size of the stimulus. The implied difference

in tax financing is thus also small, and so the approximation remains accurate.

Empirical evidence suggests that, in response to lump-sum transfer receipts, households

mostly adjust their – arguably low-return – liquid deposits (Fagereng et al., 2018, Table 4).

In Appendix A.4.1.2, I analyze the inaccuracy associated with such low household returns

33A very similar logic also applies to open economies: If home bias in private consumption demand is low(high) relative to home bias in government spending, then taxes need to increase by more (less) to financeprivate relative to public spending, and so the demand equivalence approximation will tend to over-state(under-state). The full formal argument is omitted in the interest of space but available upon request.

44


in a rich two-asset HANK model, similar to Kaplan et al. (2018). To threaten the quality

of my approximation as much as possible, I assume that households earn the government

interest rate on illiquid assets, and face a substantial return penalty of 1 per cent per quarter

for transacting in liquid assets. As a result, household returns are weakly – and for most

transactions strictly – below the interest rate paid on government debt. The separate biases

associated with wealth effects in labor supply and return heterogeneity thus both push the

demand equivalence approximation to over-state the aggregate consumption response. Even

under these extreme assumptions, however, the approximation remains quite accurate, with

a maximal error relative to the peak consumption response of around 7 per cent.34

1.4.4 Beyond one-good economies

Exact equivalence requires households and government to consume a single, homogeneous

final good. This section considers various deviations from this benchmark: (i) durable and

non-durable consumption goods, (ii) valued and productive government spending and (iii)

multiple goods with imperfect factor mobility and heterogeneous production functions.

Durables. All models considered so far abstract from the empirically relevant distinction

between durable and non-durable consumption goods. As it turns out, even in a generalized

model with separate durable and non-durable consumption, demand equivalence obtains

under exactly the same assumptions as those discussed in Section 1.2.4. I relegate the

formal argument to Appendix A.4.1.4, and only briefly discuss the intuition here.

The key assumption – routinely made in previous work featuring durable and non-durable

consumption (e.g. Barsky et al., 2007; Berger & Vavra, 2015) – is that both the durable and

the non-durable good can be produced costlessly from a common final good. If that is the

34If a researcher has a strong prior about return heterogeneity, then the implied difference in net presentvalues can simply be returned to households as an additional rebate stimulus, perfectly analogous to myanalysis of the effects of consumption complementarities in Appendix A.4.1.5.

45


case, then a generalized demand equivalence result applies to total household expenditure

on non-durable and durable consumption.

Useful Government Spending. In the benchmark model, government spending is so-

cially useless – it is neither valued by households, nor does it have productive benefits. In Ap-

pendix A.4.1.5 I study the extent to which my approximation is affected by non-separabilities

in the private valuation of government spending (following Leeper et al., 2017) and produc-

tive benefits of government investment. I only briefly summarize my main conclusions here,

with details and further intuition presented in the appendix.

I first show that, if private and public consumption are complements (substitutes), then

the demand equivalence approximation is likely to over-state (under-state) the consumption

response to the demand shifter. However, given a standard parametric form for the non-

separability (as in Leeper et al., 2017), it is easy to correct for this bias. Next, to gauge the

importance of productive benefits of government spending, I review the empirical evidence

on fiscal multipliers for public investment. The key take-away is that such multipliers are

usually estimated to be larger than standard spending multipliers, with a cross-study average

of around 1.5 (e.g. Gechert, 2015; Ramey, 2016). These findings caution against the use of

public investment multipliers for my approximations. I illustrate the associated inaccuracy

in an extension of my benchmark model where government expenditure directly shows up in

aggregate production functions (following Leeper et al., 2010).

Multi-Goods Models. Private and public consumption baskets are different. Previous

work has identified at least three channels through which such heterogeneity may break de-

mand equivalence. First, if factors of production do not move freely between different sectors,

then differences in consumption baskets will lead to heterogeneous relative price responses

(Ramey & Shapiro, 1998). Second, if different goods have different production technologies,

46


then the income generated by private and public demand shocks may flow to different fac-

tors of production, leading to heterogeneous general equilibrium propagation (Baqaee, 2015;

Alonso, 2017). And third, if firm investment demand features a higher intertemporal elas-

ticity than private consumption demand, then government purchases of consumption and

investment goods have different aggregate demand effects (Boehm, 2016).

The strength of these mechanisms is best tested directly with evidence on the aggregate

effects of different kinds of government purchases. Reassuringly, with the notable excep-

tion of productive long-term investment, previous empirical work finds largely homogeneous

multipliers by the type of spending (Gechert, 2015; Ramey, 2016). Since the resulting es-

timates are noisy, however, Appendix A.4.1.6 also presents indirect model-based evidence;

specifically, I study the accuracy of the demand equivalence approximation in several multi-

sector generalizations of the benchmark model, disciplined by empirical evidence on (i) the

strength of relative price effects, (ii) heterogeneity in network-adjusted labor shares, (iii)

the intertemporal elasticity of consumption and investment demand. I give intuition for the

signs of the associated biases, but largely find that – in the empirically disciplined variants

of these extended models – the asymmetry in multipliers is sufficiently small so as to not

materially threaten the accuracy of the demand equivalence approximation, with maximal

prediction errors always below 10 per cent.

1.5 Investment demand counterfactuals

This section extends my methodology to study the general equilibrium propagation of shocks

to investment demand. Section 1.5.1 establishes the theoretical equivalence result, and Sec-

tion 1.5.2 leverages it to derive semi-structural aggregate counterfactuals for investment tax

stimulus through accelerated bonus depreciation.

47


1.5.1 Investment demand equivalence

I again consider the benchmark model of Section 1.2.1. Relative to consumption demand

shocks, the additional challenge of fluctuations in investment demand is that, through firms’

production technologies, investment will sooner or later translate into additional hiring and

production. This adds two complications: First, firm behavior will induce a net excess

demand path i−y, rather than a pure spending response. Second, tax financing of investment

stimulus, changes in firm dividend pay-outs, as well as any potential expansion in labor hiring

will have redistributional implications and directly feed back into consumption demand.

In this section, I will give sufficient conditions under which the net excess demand path

i − y fully determines general equilibrium feedback, thus allowing this aggregate feedback

to be exactly replicated by a mix of expansionary and contractionary government spending

shocks. Intuitively, the expansionary shock mirrors the impact excess (investment) demand,

while the contractionary news shock synthesizes the implied future expansion in supply. As

I will show, these sufficient conditions do not at all constrain the richness of the model’s

investment block, but do impose some meaningful restrictions on household behavior.

The Equivalence Result. The production block of the economy was sketched in Sec-

tion 1.2.1, with a detailed outline in Appendix A.1.1. For purposes of the analysis here, it

suffices to note that I allow for a rich set of real and financial frictions, including (convex and

non-convex) capital adjustment costs as well as a generic set of constraints on firm equity

issuance and borrowing. The production block of the economy is thus general enough to nest

essentially all recent contributions to the quantitative heterogeneous-firm investment litera-

ture (e.g. Khan & Thomas, 2008, 2013; Winberry, 2018; Koby & Wolf, 2020); importantly,

my equivalence results do not require any restrictions on this rich firm side.

48


Anticipating the empirical application, I establish an exact equivalence result for invest-

ment tax credit shocks εεεq – shocks that reduce the cost of capital purchases at time t by an

amount τqt = τqt(εεεq).35 I obtain this investment demand equivalence result under four key

restrictions on the non-production block of the economy. As before, I state the assumptions

and the result first, and provide intuition as well as a proof sketch afterwards.

The first assumption – a single common final good – is again implicit in the model set-up.

Assumption 5. A single, homogeneous final good is used for both (government) consumption

and investment.

Implicitly, I assume that all meaningful capital adjustment costs are internal to the firm,

and that the aggregate supply of capital (out of the common final good) is perfectly elastic.

This assumption is consistent with the empirical findings in House & Shapiro (2008), Edger-

ton (2010) and House et al. (2017). The second assumption then rules out any redistributional

effects associated with the firm subsidy.

Assumption 6. All households i ∈ [0, 1] have identical preferences, receive equal lump-sum

government rebates τt and firm dividend income dt, and face no idiosyncratic earnings risk.

This assumption effectively imposes a standard representative-household structure. The

third assumption again concerns household labor supply decisions.

Assumption 7. The Frisch elasticity of labor supply is either infinite (linear labor disutility),

or wages are perfectly sticky.

Linear labor disutility – clearly at odds with micro data on household labor supply –

is sometimes justified at the aggregate level as a by-product of labor indivisibility (Hansen,

1985; Rogerson, 1988). Finally, the fourth assumption restricts monetary policy feedback.

35More generally, my results can be interpreted as applying to any kind of shock that appears as a reduced-form wedge in firm investment optimality conditions.

49


Assumption 8. The monetary authority’s interest rate rule does not include an endogenous

response to fluctuations in the level of aggregate output.

I define direct (partial equilibrium) responses and indirect (general equilibrium) feedback

for firm investment and production exactly analogously to Definition 1, using the implied

aggregate investment and production functions i(•) and y(•), respectively. Assumptions 5

to 8 are enough for an investment demand equivalence result.

Proposition 5. Consider the structural model of Section 1.2.1. Suppose that, for each

one-time shock {q, g}, the equilibrium transition path exists and is unique. Then, under

Assumptions 5 to 8, the responses of investment and output to an investment tax credit

shock q and to a government spending shock g with gg = iPEq − yPEq satisfy, to first order,

iq = iPEq + ig (1.21)

yq = yPEq + yg (1.22)

The proof strategy is identical to that of the consumption demand equivalence result.

First, Assumption 5 implies that I can consider a single aggregate output market-clearing

condition. Second, Assumption 6 ensures the absence of any distributional effects that may

lead to differential partial equilibrium consumption demand responses to the investment

demand and public spending shocks, allowing me to restrict attention to the firm net demand

path i − y. Third, Assumption 7 is sufficient to ignore labor market adjustments, either

because households are willing to supply the additional demanded labor, or because they

have no choice. And fourth, Assumption 8 is needed to ensure that only the level of net excess

demand matters, not its composition. Without this assumption, the central bank would lean

against any excess demand associated with higher output supply, breaking equivalence.36

36Alternatively, equivalence would obtain if the monetary authority were to respond to the output gap.

50


Accuracy. Assumptions 5 to 8 are routinely imposed in quantitative general equilibrium

models of investment; in particular, they hold – and thus exact equivalence applies – in the

well-known models of Khan & Thomas (2008), Khan & Thomas (2013), Winberry (2018),

Ottonello & Winberry (2018) and Bloom et al. (2018). As such, the decomposition in (1.21)

- (1.22) provides a useful exact identification result for a popular class of models.

To further gauge the accuracy of my approximation, Appendix A.4.1.9 studies shocks to

investment demand in the estimated HANK model of Section 1.4.1. This model is a useful

laboratory because it violates Assumptions 6 to 8: households are subject to non-trivial

earnings risk and receive heterogeneous firm profit payments, the Frisch elasticity of labor

supply is relatively small (it is 1), wages are quite flexible, and the monetary authority

responds to fluctuations in aggregate output. Even though each of these model ingredients

individually biases my approximation upwards, I find that it remains accurate, in particular

at short horizons.

1.5.2 Application: bonus depreciation

The investment equivalence result justifies a two-step procedure to study generic investment

demand shocks, exactly analogous to my analysis of consumption shifters in Section 1.3. In

this section I leverage the additive decomposition in Proposition 5 to construct a general

equilibrium counterfactual for investment bonus depreciation stimulus – that is, the ability

to tax-deduct investment expenditure at a faster rate, as implemented in the U.S. in the

two most recent recessions (Zwick & Mahon, 2017). It is well-known that, in the absence of

firm-level financial frictions, such accelerated bonus depreciation schedules are isomorphic to

the investment tax credits covered by the investment equivalence result (Winberry, 2018).37

37As in Section 1.3, the analysis in this section implicitly relies on the stochastic VMA representation ofthe model, and considers estimation of impulse responses to one-off structural shocks.

51


Direct Response. My estimates of the direct response of investment to the shock rely

heavily on Zwick & Mahon (2017) and Koby & Wolf (2020), who exploit cross-sectional

firm-level heterogeneity in the exposure to bonus depreciation investment stimulus. In Koby

& Wolf (2020), we estimate dynamic regressions akin to (1.14) and give sufficient conditions

under which the regression estimands are identical to or at least informative about the desired

partial equilibrium investment spending responses iPEq . The discussion is largely analogous

to that in Proposition 3, so I relegate further details to Appendix A.2.2.

Given a path for the direct investment spending response iPEq , I can recover the implied

partial equilibrium production path using standard estimates of the capital elasticity of pro-

duction. In particular, assuming a simple Cobb-Douglas production function y = (kα`1−α)ν

as well as competitive spot labor markets, it is straightforward to show that

ˆyPEqt =αν

1− (1− α)ν× ˆkPEqt−1

Thus, given estimates of the capital depreciation rate δ, the capital share α, and the returns

to scale parameter ν, it is possible to recover the implied partial equilibrium production

path. Consistent with my estimated HANK model, I set δ = 0.016, α = 0.2 and ν = 1.

I take the regression estimates of iPEqt for t = 0, 1, 2, 3 straight from Koby & Wolf (2020,

Table 1). The green x’s in the investment panel of Figure 1.8 show the estimated path

of direct investment spending responses to a one-quarter bonus depreciation shock worth

around 8 cents, a shock similar in magnitude to (but less persistent than) the stimulus

of 2008-2010. The solid green line extrapolates the empirical estimates to a full response

path using a Gaussian basis function, similar to Barnichon & Matthes (2018). I take this

extrapolated path to be the empirical estimate of the full spending response path iPEq .

Investment demand increases substantially and persistently in response to the stimulus.

Since capital is pre-determined, and since all prices faced by firms (except for taxes and

52


so effective capital goods prices) are fixed by the nature of the partial equilibrium exercise,

output does not increase on impact, but instead only gradually increases over time. Together,

the investment and output responses translate into a more complicated intertemporal net

excess demand profile, displayed in the top left panel: Net excess demand is large and positive

on impact (due to higher investment demand), but turns negative over time, as additional

capital becomes productive and so expands the productive capacity of the economy.

Figure 1.8: Investment Tax Credit, Impulse Responses

Note: Investment, output and consumption responses to an investment tax incentive shock, withthe partial equilibrium net output response path matched to a linear combination of governmentspending shocks. The investment and output responses are computed in line with Proposition 5,while the consumption response is simply equal to the response after the identified combinationof government spending shocks. The dashed lines again correspond to 16th and 84th percentileconfidence bands.

53


The Missing Intercept. Following Proposition 5, it remains to replicate the estimated

net excess demand path through a suitable list of government spending shocks:

iPEq − yPEq =

nk∑k=1

γk × ggk (1.23)

It is unlikely that any single estimated spending shock can replicate the reversal documented

in Figure 1.8. Encouragingly, much previous work on fiscal multipliers actually estimates the

effects of delayed increases in government spending (Ramey, 2011; Caldara & Kamps, 2017)

– that is, government spending news shocks. In principle, combining these delayed spending

responses with the immediate spending effect estimated in Section 1.3.2 should allow me to

replicate the net demand effects of the investment tax credit.

To operationalize this insight, I consider the same VAR as before, but now study the

responses to residualized innovations in both the instrument equation as well as the equa-

tion for government expenditure itself. The first innovation is simply the shock studied in

Section 1.3.2, while the second innovation is similar to the popular recursive identification

scheme of Blanchard & Perotti (2002), augmented to include forecast errors as a control

for anticipation effects. Consistent with previous work, I find the effects of the Blanchard-

Perotti shock to be delayed, and so a linear combination of the two shocks allows me to

match the implied net excess demand path of the investment demand shock, as shown in the

top left panel of Figure 1.8. Further details on the empirical implementation (in particular

the construction of standard errors) are provided in Appendix A.2.3.

Macro Counterfactuals. All results for general equilibrium counterfactuals are displayed

in Figure 1.8. With the requirement that gg = iPEq −yPEq satisfied, the investment and output

panels implement the additive decompositions in (1.21) and (1.22), respectively. My main

finding is that the substantial partial equilibrium investment demand responses estimated

54


in Zwick & Mahon (2017) and Koby & Wolf (2020) also survive in general equilibrium.

The increase in investment demand is accommodated through a sharp immediate increase

in output as well as a smaller and somewhat delayed drop in consumption. Taken together,

the large direct investment spending responses estimated in micro data as well as extant

evidence on the transmission of aggregate government spending shocks suggest that bonus

depreciation investment incentives provide a sizable macroeconomic stimulus.

Implications for Structural Modeling. My results contrast sharply with the predic-

tions of the standard neoclassical model closure routinely entertained in quantitative models

featuring rich investment micro-heterogeneity (e.g. Khan & Thomas, 2013; Bloom et al.,

2018). As is well-known (e.g. Barro & King, 1984), investment demand shocks in such

models are accommodated through large drops in consumption and only moderate impact

increases in output. Real interest rates thus increase, leading to substantial general equilib-

rium crowding-out of rate-sensitive firm investment (Khan & Thomas, 2008). The results in

Figure 1.8 are instead consistent with models featuring (i) relatively price-inelastic invest-

ment,38 (ii) strong aggregate demand effects and (iii) little consumption crowding-out, for

example due to hand-to-mouth spending or strong habit formation.

1.6 Conclusion

I develop a new approach to the estimation of aggregate counterfactuals for a general family

of consumption and investment demand shifters. Micro data can help us learn about the

extent to which these demand shifters directly stimulate household and firm spending, and

extant evidence on the transmission of public spending shocks to private expenditure contains

38It may seem strange to claim that the large partial equilibrium investment responses documented inthe cross-sectional regressions are consistent with price-inelastic investment. As we show in Koby & Wolf(2020), while these responses are indeed large in economic terms, they are orders of magnitude smaller thanpredicted by standard neoclassical models of investment.

55


valuable information about the “missing intercept” of general equilibrium accommodation.

Applied to income tax rebates and investment bonus depreciation incentives, my methodol-

ogy suggests that both policies substantively stimulate aggregate private spending, and that

this expansion in spending is accommodated in general equilibrium through a one-to-one

increase in production, rather than being crowded out through price responses. Any cali-

brated structural model that implies large general equilibrium amplification or dampening

either does not feature demand equivalence, or is inconsistent with conventional estimates

on the size of the fiscal multiplier.

The methodology promises to be useful beyond the applications considered in this paper.

In the companion paper Wolf (2019b), I generalize my results to map cross-regional regression

estimates into macro counterfactuals, with an application to household deleveraging due to

tighter borrowing conditions (Mian et al., 2013; Guerrieri & Lorenzoni, 2017). Examples

of other interesting macro shocks covered by my two-step method include firm uncertainty

(Bloom, 2009; Bloom et al., 2018), shocks to firm credit conditions (Khan & Thomas, 2013)

and household debt relief (Auclert et al., 2019). I leave those extensions to future work.

56

Chapter 2

SVAR (Mis-)Identification and the Real

Effects of Monetary Policy Shocks

This paper is forthcoming at the American Economic Journal: Macroeconomics. Below I

reproduce the latest public working paper version of the article.

2.1 Introduction

A central question in empirical macroeconomics is the response of the economy to changes in

monetary policy. Going back to Sims (1980), a long literature has tackled this question using

structural vector autoregressions (SVARs), with policy shocks identified through zero restric-

tions on the contemporaneous response of macro aggregates to policy changes. This early

literature suggests that a policy tightening indeed reduces real activity, if only moderately

so and with a delay. Recent work challenges this consensus. Uhlig (2005) casts doubt on the

conventional timing restrictions, proposes a weaker identification procedure based on uncon-

troversial sign restrictions, and finds that, if anything, contractionary monetary shocks boost

output. Yet more recently, refinements of Uhlig’s identification scheme (Arias et al., 2019) or

identification based on external instruments (Gertler & Karadi, 2015) tend to qualitatively

57

2. Monetary Policy SVARs

re-store conventional wisdom, and in fact suggest somewhat larger and faster real effects

than previously believed. At the same time, more and more studies have started to outright

question the ability of SVARs to reliably identify shock transmission (e.g. Plagborg-Møller,

2019; Nakamura & Steinsson, 2018b), raising concerns about the informativeness of macroe-

conomic aggregates for hidden structural shocks – the so-called non-invertibility problem.

Evidently, a consensus on the real effects of monetary policy remains elusive.

In this paper I show that, when viewed through the lens of standard structural models,

these apparent inconsistencies across different empirical methods are not at all surprising,

but exactly what we should expect. The argument is simple: I fix a single common structural

model as my data-generating process, characterize the probability limits of various popular

empirical strategies, and show that the estimators disagree in exactly the same fashion as

they do in real data. Sign restrictions, as in Uhlig (2005), are vulnerable to expansionary

demand and supply shocks “masquerading” as contractionary monetary policy shocks, which

then seemingly boost – rather than depress – output. Standard impact zero restrictions on

output impulse responses give classical results because they implicitly safeguard against this

particular form of mis-identification, but at the cost of understating the (short-horizon) real

effects of monetary policy. Direct restrictions on the implied Taylor rule of the monetary

authority (as in Arias et al. (2019)) or external instruments (IVs) instead robustly estimate

the true model-implied aggregate effects of monetary policy shocks. In principle, both ap-

proaches are vulnerable to non-invertibility concerns, but in practice either solution can work

well, as monetary policy shocks are often near -invertible.

My analysis builds on a fully specified structural model of monetary policy transmission

in the mold of Woodford (2003), Galı (2008) or Smets & Wouters (2007). In line with

empirical practice, I assume that the econometrician observes data on output, inflation and

the policy rate generated from the model, estimates their VAR representation, and identifies

structural shocks using different identification schemes. To avoid conflating estimation and

58


identification uncertainty, I allow the econometrician to observe an infinitely long sample,

so she is able to perfectly recover the true population reduced-form VAR representation.

Against this reduced-form VAR, I then characterize the probability limits of various popular

estimators of monetary policy transmission. In particular, for each estimator, I am able

to write the identified shocks – what the researcher will call a “monetary policy shock” –

as a linear combination of the true shocks of the underlying model. Perfect identification

corresponds to a coefficient of 1 on the shock of interest, and 0 on all other shocks.

I first address the non-invertibility problem. I show that, for any possible SVAR identifi-

cation scheme, the coefficient on the actual monetary policy shock is bounded above by the

R2 in a regression of that shock on past and current values of the observed macro variables.

Under invertibility (R2 = 1), identification can thus in principle succeed; if instead the R2 is

small, then conventional SVAR identification schemes will invariably fail. My first result is

that, because monetary policy shocks are – at least in my models – the only shock to drive

interest rates and inflation in opposite directions, any VAR that includes these macro aggre-

gates is likely to give a high R2. For example, even for a small VAR in (y, π, i), estimated on

data generated by the model of Smets & Wouters, the R2 is 0.8702. With the invertibility

assumption nearly satisfied, we know that some SVAR identification scheme will at least

approximately recover true impulse responses. In the remainder of the paper, I ask whether

any of the popular standard SVAR estimators in fact attain this near-perfect identification.

I begin with the identification scheme of Uhlig (2005). He defines as a candidate mone-

tary policy shock any shock that moves interest rates and inflation in opposite directions.1

Since different linear combinations of reduced-form forecasting errors are consistent with

these restrictions, his procedure will only provide set identification – it will not identify a

single SVAR, but a set, and thus a set of candidate “monetary policy shocks.” As discussed

1In fact Uhlig imposes some further restrictions, designed chiefly to disentangle monetary policy shocksfrom money demand shocks. My models feature no such shocks, so I abstract from his additional restrictions.

59


above, the true monetary policy shock uniquely moves policy rates and inflation in opposite

directions; appealingly, this not only ensures near-invertibility, but also implies that the ac-

tual true shock will lie in the set of acceptable candidate shocks, while any of the other pure

shocks will not. Troublingly, however, accurate identification is still not guaranteed, as the

identified set may contain linear combinations of other structural shocks. I find that this

“masquerading” problem is prominent in my structural models, where many acceptable can-

didate “monetary policy shocks” counterfactually increase aggregate output. Intuitively, the

right linear combination of expansionary demand and supply shocks can also push inflation

and interest rates in opposite directions, but of course boosts output.

In large-sample Bayesian analysis of sign-restricted VARs, posterior uncertainty over the

identified set is exclusively governed by the prior (Baumeister & Hamilton, 2015; Watson,

2019). I show that the Haar prior – the most popular prior in applied work (Uhlig, 2005;

Rubio-Ramırez et al., 2010) – automatically puts more mass on more volatile structural

shocks. But since, in my structural models, demand and supply shocks are more volatile than

monetary policy shocks, most posterior mass is automatically put on the “masquerading”

shock combinations that counterfactually increase real output. This conclusion agrees exactly

with posterior uncertainty over identified sets reported in Uhlig’s analysis.2

I next consider the performance of zero or near-zero identifying restrictions. Uhlig (2005),

in his review of the classical zero restriction literature, finds the zero output restriction to be

central to the old conventional wisdom. My model-based analysis reveals that this key role for

the impact output restriction is not an accident, but in fact an economically sensible feature

of identified sets. The logic is simple: In purely sign-identified SVARs, counterfactual positive

output responses are generated by masquerading expansionary supply and demand shocks.

These shocks move interest rates in opposite directions, but output in the same direction,

2My results should not be taken to imply that the Haar prior is incorrectly imposed in popular work, northat the derived Bayesian posterior sets are invalid. I merely clarify the role of this particular choice of priorin shaping posterior uncertainty over the identified sets implied by sign restrictions alone.

60


and thus imply very large impact output multipliers of monetary policy interventions. Even

moderate bounds on these multipliers are enough to eliminate most combinations of positive

demand and supply shocks from the identified set, thus substantially tightening inference

around the truth. Literal zero restrictions of course afford most tightening, but – at least in

models with small, but non-zero impact effects – lead the researcher to robustly understate

the short-horizon real effects of monetary policy shocks.

Two alternative recently proposed identification schemes do not suffer from this defect.

First, Arias et al. (2019) combine the benchmark identification scheme of Uhlig with addi-

tional sign restrictions on implied Taylor rule coefficients, and show that their identification

scheme restores conventional wisdom. Yet again, this finding can be rationalized through

standard structural models: I show that, in regions of the identified set where demand and

supply shocks masquerade as contractionary monetary policy shocks, the coefficient on out-

put in the implied mis-identified “Taylor rule” is invariably (and counterfactually) negative.

Restricting the coefficient to be positive thus markedly improves identification. Second, sev-

eral researchers have proposed to identify monetary policy SVARs using external instruments

(e.g. Gertler & Karadi, 2015). Plagborg-Møller & Wolf (2019a) show that, even with a valid

external instrument, the standard SVAR-IV estimator is biased under non-invertibility. The

bias, however, is proportional to the reciprocal of the R2 in a regression of the monetary

policy shock on lags of the macro aggregates, and so, by near-invertibility, likely to be small.

My results have important implications for macro-econometric practice in general, and

the study of monetary policy transmission in particular. The review of the agnostic identifi-

cation scheme in Uhlig (2005) reveals that, for tight and reliable inference, it is not enough to

ensure that the imposed sign restrictions are uniquely satisfied by the shock of interest. Lin-

ear combinations of other structural shocks can masquerade as the shock of interest and thus

lead inference astray. In particular, if these rival shocks are more volatile than the shock of

interest, then the popular Haar prior is likely to focus attention on the mis-identified region

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of the identified set. For monetary policy transmission, my results encouragingly suggest

that, first, recent advances in identification effectively address the masquerading problem,

and second, even small sets of macro observables may carry a lot of information about policy

shocks. Viewed in this light, I conclude that existing empirical work quite consistently paints

the picture of significant, medium-sized effects of monetary policy on the real economy.

Literature. My work relates to several strands of literature. First, I provide a unifying

model-based perspective on recent advances in the empirical study of monetary policy trans-

mission. In particular, my results lend support to recent empirical work identifying medium-

sized real effects of monetary policy via a variety of quite different identification schemes: the

narrative evidence reviewed in Coibion (2012), the external SVAR-IV approach of Gertler

& Karadi (2015) and Jarocinski & Karadi (2019), the Taylor rule restrictions of Arias et al.

(2019), and heteroskedasticity-based identification of Brunnermeier et al. (2017). In its at-

tempt to reconcile different empirical findings, my work shares similarities with Mertens &

Ravn (2014) and Caldara & Kamps (2017). I show that restrictions on either impact output

responses or on the VAR-implied Taylor rule parameter are robustly sufficient to generate

negative output responses in monetary policy SVARs.

Second, I offer several novel results on the relation between structural macro models and

SVAR representations, in particular for the non-invertible case. The mapping from model

parameters to VAR coefficients, and from primitive structural shocks to SVAR-identified

shocks, is characterized in detail in Fernandez-Villaverde et al. (2007) and Giacomini (2013).

Relative to those papers, I offer additional insights by tying the connection between SVAR

and model shocks to quantitative measures of the degree of invertibility. In particular, and

perhaps somewhat surprisingly, I show that standard macro aggregates can be informative

for monetary policy shocks even if those shocks are largely irrelevant for aggregate business-

cycle fluctuations (Ramey, 2016; Plagborg-Møller & Wolf, 2019a).

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Third, I add several cautionary results to the fast-growing literature on sign-based set

identification in empirical macro-econometrics. The sign restrictions methodology for the

identification of SVARs was pioneered by Faust (1998), Canova & De Nicolo (2002) and Uhlig

(2005). A comprehensive algorithm for inference, relying on the Haar prior, is developed in

Rubio-Ramırez et al. (2010). Similar to Paustian (2007) and Castelnuovo (2012), my results

reveal the common minimal requirement for sign-based analysis – that only the shock of

interest satisfy all imposed sign restrictions – to be necessary, but not sufficient for reliable

inference (also see Uhlig, 2017). Relative to these earlier contributions, my analysis adds

further insights by explicitly characterizing the model-implied (mis-)identified set in terms

of the underlying true structural disturbances, and then using the “masquerading shocks”

interpretation to rationalize the importance of the Haar prior in shaping posterior uncertainty

over this identified set. Relatedly, Paustian (2007) and Canova & Paustian (2011) emphasize

that sign restrictions are likely to perform well for sufficiently volatile shocks. My analysis

of Bayesian posteriors over identified sets shows that this conclusion is exclusively driven by

the prior: If, in a given model and with a given SVAR identification scheme, the researcher

is unable to sign the response of a variable of interest to a certain shock, then she would be

unable to sign the response even if the shock of interest were arbitrarily volatile. Equivalently,

for a judiciously chosen prior, the Bayesian posterior probability assigned to a positive (say)

impulse response for the variable of interest can always be made arbitrarily large or small,

whatever the underlying relative shock volatilities (also see Giacomini & Kitagawa, 2016).

Outline. Section 2.2 presents my model laboratories, characterizes the mapping from

structural model to SVAR estimand, and argues for robust (near-)invertibility of conven-

tional monetary policy shocks. Sections 2.3 to 2.5 then interpret recently popular empirical

estimators through the lens of the model laboratories. Finally Section 3.6 concludes. Ap-

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pendix B provides further details and selected proofs, and a supplementary appendix is

available online.3

2.2 VAR analysis in structural models

A wide class of popular structural models admits a VAR representation for observable macro

aggregates, and so gives well-defined population estimands for different SVAR estimators of

structural shock transmission. In Section 2.2.1, I outline two particular model laboratories.

Section 2.2.2 characterizes the probability limits of generic SVAR estimators applied to arti-

ficial model-generated data. Finally, in Section 2.2.3, I leverage knowledge of the underlying

data-generating process to link SVAR-estimated “structural” shocks to the true disturbances

of the structural model, and in particular connect my results to SVAR non-invertibility.

This section mostly reviews relatively standard material; in particular, the only result

novel to this paper is my characterization of SVAR estimands under non-invertibility. As

such, the analysis here merely collects the tools necessary for my model-based interpretation

of SVAR-implied identified sets in Sections 2.3 to 2.5.

2.2.1 Model laboratories

For most of this paper, I will study the properties of popular SVAR identification strategies

through the lenses of two structural models. First, I consider a simple variant of the canonical

three-equation New Keynesian model (Galı, 2008; Woodford, 2003). This model is simple

enough to conveniently and transparently provide closed-form illustrations of my results.

Second, I use the quantitatively more realistic model of Smets & Wouters (2007) to show that

3See https://www.christiankwolf.com/research. My webpage also contains codes for replication ofall exercises reported here and in the Online Appendix.

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https://www.christiankwolf.com/research


all intuitions also survive in a richer environment. In particular, the Smets-Wouters model

allows me to judge the likely importance of VAR mis-specification due to non-invertibility.

Section B.1 of the Online Appendix extends my results to other environments; notably, I

consider a dynamic three-equation model as well as various alternative model variants with

passive monetary policy rules (Castelnuovo & Surico, 2010; Leeper & Leith, 2016).

The Three-Equation Model. Detailed derivations of the conventional three-equation

New Keynesian model are offered in Galı (2008) and Woodford (2003). To allow the cleanest

possible study of the various popular SVAR estimators, I consider a particularly simple static

variant of this model, without any exogenous or endogenous persistence:

yt = Et (yt+1)− (it − Et (πt+1)) + σdεdt (IS)

πt = βEt (πt+1) + κyt − σsεst (NKPC)

it = φππt + φyyt + σmεmt (TR)

where (εdt , εst , ε

mt )′ ∼ N(0, I). y is real output, i is the nominal interest rate (the federal funds

rate), and π is inflation. The model has three structural disturbances: a demand shock εd,

a supply shock εs and a monetary policy shock εm. The first equation is a standard IS-

relation (demand block), the second equation is the New Keynesian Phillips curve (supply

block), and the third equation is the monetary policy rule (policy block). For most of my

analysis, I do not rely on any specific assumptions on model parameterization; I only make

the conventional assumptions β ∈ (0, 1), κ > 0, φπ > 1, φy ≥ 0, and σd, σs, σm > 0.

65


It is straightforward to show that this benchmark model is static and admits the closed-

form solutionyt

πt

it

=1

1 + φy + φπκ

σd φπσ

s −σm

κσd −(1 + φy)σs −κσm

(φy + φπκ)σd −φπσs σm

εdt

εst

εmt

(2.1)

For the study of different SVAR estimators, I assume that the econometrician observes data

on output, inflation, and the policy rate. However, she is not aware that the data are actually

generated according to (2.1), and so does not exploit the structure of the model for inference.

The Smets-Wouters Model. The structural model of Smets & Wouters (2007) is perhaps

the most well-known example of an empirically successful business-cycle model. For further

details, I refer the reader to the original paper. In most of my analysis here, I consider their

posterior mode parameterization; as a further robustness check, Section B.4 in the Online

Appendix presents results taking into account posterior estimation uncertainty.4 Exactly as

before I assume that the econometrician observes data on aggregate output, inflation, and

the interest rate, but does not know the true underlying model.

2.2.2 Structural models and VAR analysis

When solved through standard first-order perturbation techniques, my laboratories – as well

as many other business-cycle models – give linear evolution equations for all model variables.

Splitting variables into observables and unobservable states generates a linear state-space

4My implementation of the Smets-Wouters model is based on Dynare replication code kindly providedby Johannes Pfeifer. The code is available at https://sites.google.com/site/pfeiferecon/dynare.

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https://sites.google.com/site/pfeiferecon/dynare


model. As is conventional, I restrict attention to Gaussian linear state-space models:5

st = Ast−1 +Bεt (2.2)

xt = Cst−1 +Dεt (2.3)

where st is a ns-dimensional vector of state variables, xt is an nx × 1 vector of observables

and εt is an nε×1 vector of structural shocks. The disturbances εt are Gaussian white noise,

with E[εt] = 0, E[εtε′t] = I and E[εtε

′t−j] = 0 for j 6= 0.

Under weak conditions, the state-space system (2.2) - (2.3) implies a VAR representation

for the observables xt. As most material in this section is relatively standard, I only state

the main results here, and refer the interested reader to Section B.2 of the Online Appendix

and the literature referenced therein. The implied reduced-form VAR representation is6

xt =∞∑j=1

Bjxt−j + ut (2.4)

where the coefficient matrices Bj, j = 1, 2, . . . are complicated functions of the fundamental

model matrices (A,B,C,D), and the ut are the (Gaussian) forecast errors on observables xt

given information up to time t− 1, with disturbance variance E(utu′t) ≡ Σu.

The Computational Experiment. I assume that the econometrician observes macro ag-

gregates xt, but does not exploit the structure of the model – that is, the matrices (A,B,C,D)

– for inference. Since I allow her to observe an infinitely large sample generated from (2.2)

- (2.3), I simply treat the reduced-form VAR representation (2.4) as known. Further details

on the computation of this VAR(∞) are presented in Section B.7 of the Online Appendix.

5The Gaussianity assumption is made for notational simplicity only. Equivalently, I could restrict struc-tural identification to only come from the second-moment properties of the data.

6Note that I use {Bj}∞j=1 for VAR coefficient matrices and B for the shock impact matrix in the stateequation (2.2). I do so to be as close as possible to textbook notation.

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SVAR Identification. Structural VAR analysis posits that the true structural shocks εt

can be obtained as a linear combination of contemporaneous reduced-form disturbances ut.

With et denoting the SVAR-identified “structural” shocks, a structural VAR representation

of the same system is thus

A0xt =∞∑j=1

Ajxt−j + et (2.5)

where A−10 A−1′

0 = Σu, et ≡ A0ut is Gaussian white noise with E[et] = 0, E[ete′t] = I and

E[etet−j] = 0 for j 6= 0, and Aj ≡ A0Bj.

As is well-known, a continuum of SVARs are consistent with a given reduced-form VAR

representation. It is straightforward to see that, under the Gaussianity assumption, the

SVAR (2.5) is identified up to orthogonal rotations – pre-multiplying both sides of (2.5)

with a matrix Q in the space of nx-dimensional orthogonal rotation matrices O(nx) does

not change the likelihood of the model. In other words, SVARs are identified up to nx(nx−1)2

restrictions (Rubio-Ramırez et al., 2010). Outside identifying information is then used to

restrict attention to a strict subset of the set O(nx), often a singleton. In what follows, I

will refer to this smaller set of SVARs as the identified set, and to outside identifying in-

formation as the identification scheme. A formal definition of identified sets is relegated to

Appendix B.1. Given a model-implied reduced-form VAR representation and a structural

identification scheme, it is straightforward to numerically characterize the (population) iden-

tified set of SVARs, as well as any corresponding impulse response functions, forecast error

variance decompositions, or other objects of interest. Again, further computational details

are provided in Section B.7 of the Online Appendix.

2.2.3 Interpreting SVAR estimands

The analysis in Section 2.2.2 did little to exploit the structure of the underlying model –

any reduced-form VAR can be mapped into identified sets, both those estimated on actual

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data and those derived from fully-specified structural models. A controlled data-generating

process does, however, have one important advantage: It allows us to interpret SVARs by

linking their identified “structural” shocks, et, to the true shocks of the model, εt.

The nature of the link depends on the invertibility (or lack thereof) of the model (2.2) -

(2.3). A linear state-space system is said to be invertible for the structural shocks εt if and

only if, given knowledge of the system matrices (A,B,C,D), the infinite past of observables

{xt−`}∞`=0 is sufficient to perfectly identify the hidden shocks εt.7 A natural quantitative

measure of invertibility is the R2 in an (infeasible) regression of a structural shock εj,t on

current and past macro aggregates {xτ}−∞<τ≤t (Plagborg-Møller & Wolf, 2019a).

The Invertible Case. Under invertibility – that is, if the R2 is 1 for all structural shocks

εt –, the link between SVAR shocks and true disturbances is very simple:

et = P × εt (2.6)

where P ∈ O(nx) is an orthogonal matrix. In short, with invertibility, the identified shocks

are linear combinations of the true (contemporaneous) underlying structural shocks, with the

weights given by the entries of the rotation matrix P . The right kind of SVAR identification

scheme then identifies P = I as the true rotation, with SVAR-identified structural shocks et

equal to the true structural shocks εt.

It is straightforward to show that, with (yt, πt, it) observable, the three-equation model

of Section 2.2.1 is invertible. In particular, writing out (2.6), we see that the identified set

associated with any SVAR identification scheme is simply a collection of unit-length weight

7Strictly speaking, for SVAR analysis of a given shock εj,t to work, the system needs to only be invertiblefor that shock. In that case, the static relation (2.6) will apply for shock j, while the richer dynamic relation(2.8) will apply to other shocks, with potentially non-zero weights for higher horizons ` > 0.

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vectors pm = (pmd, pms, pmm)′, with implied “monetary policy shocks” given as

emt = pmd × εdt + pms × εst + pmm × εmt (2.7)

Whether or not we are close to the ideal of pm = (0, 0, 1)′ depends, of course, on the details

of the chosen identification scheme.

The Non-Invertible Case. In the general non-invertible case, the link between identified

and true structural shocks is more complicated. Following Fernandez-Villaverde et al. (2007)

and Lippi & Reichlin (1994):

et = P (L) × εt =∞∑`=0

P` × εt−l (2.8)

where the entries of the polynomial P (L) are complicated functions of the chosen iden-

tification scheme as well as the fundamental model matrices (A,B,C,D). In a natural

generalization of (2.6), the (k, j)th entry of P` is now the weight of kth identified SVAR

shock on the `th lag of the jth true underlying structural disturbance.8

Standard small-scale VARs induced by the large Smets-Wouters model are not invertible,

so the more complicated expression (2.8) applies. Two new results, developed in more detail

in Appendix B.2, clarify when and how non-invertibility threatens SVAR-based identification

of (monetary policy) shock transmission.

First, I tie the weights in the matrix polynomial P (L) of (2.7) to the R2 in a regression

of shock j on current and lagged macro aggregates xt.

Proposition 6. Let the SVAR (2.5) be derived from a structural model (2.2) - (2.3). The

weight of the kth identified SVAR shock on the jth contemporaneous structural shock, P0(k, j)

8Since εt and et are both orthonormal white noise, we see immediately, following Lippi & Reichlin (1994),that P (L) is a Blaschke matrix – that is, the matrix-polynomial generalization of an orthogonal matrix.

70


is subject to the following upper bound:

P0(k, j) ≤√R2j ≡

√1− Var (εj,t | {xτ}−∞<τ≤t) (2.9)

Furthermore, there exists a SVAR, consistent with the model-implied reduced-form VAR

(2.4), such that this upper bound is attained.

If R2j is small, then – for any possible SVAR identification scheme – the identified shock

ek,t will bear little relation to the true structural shock εj,t. Conversely, if R2j is close to

1, then some SVAR identification scheme will (nearly) identify the true shock. Thus, in a

precise sense, SVAR identification can work if and only if R2j is sufficiently close to 1.

Second, I establish that, in the Smets-Wouters model, the R2m for monetary policy shocks

in a VAR in (y, π, i) is robustly close to 1. In this model, macro fluctuations are driven by

seven distinct shocks; out of these, monetary policy shocks are among the least important,

as measured by conventional forecast error variance decompositions. It thus seems a priori

unlikely that a small trivariate VAR should contain much information about monetary policy

shocks, casting doubt on the viability of SVAR inference. This simple intuition, however,

turns out to be incorrect. At my benchmark parameterization, the R2m is 0.8702, so the

maximal attainable weight on the monetary policy shock is√

0.8702 = 0.9328.9

The intuition underlying this result is subtle: Monetary policy shocks are not important

drivers of any individual macro aggregate, but they induce highly atypical co-movement

patterns. Notably, monetary policy shocks are unique in that they push interest rates and

inflation in opposite directions. Thus, while a divergence of interest rates and inflation is not

definitive proof, it is at least suggestive of monetary policy shocks. I provide further details

9As I show in Section B.3 of the Online Appendix, this result is not sensitive to the assumption of infiniteVAR lag lengths. For example, the R2

m is already equal to 0.8662 for a trivariate VAR with four lags.Furthermore, I also show that a high R2

m is not special to the model’s posterior mode, but is in fact a featureof most draws from the estimated model posterior.

71


for this argument in Appendix B.2. In particular, the appendix discusses an instructive

illustration using forward guidance shocks: In response to a credible promise of an interest

rate hike tomorrow, interest rates and inflation move in the same direction today. Upon

observing this co-movement, the econometrician initially concludes that the economy was

almost surely hit by a conventional demand shock, and so the monetary policy R2 is small.

As soon as the promised rate hike materializes, however, inflation and policy rate diverge,

the econometrician realizes that actually a forward guidance shock may have occurred, and

the R2 jumps back up.

Outlook. The results in this section establish that, for both model laboratories sketched

in Section 2.2.1, SVAR-based inference can in principle succeed. Whether any given identi-

fication scheme succeeds is, of course, a different question. In the remainder of this paper, I

will use the structural shock decompositions in (2.6) and (2.9) to evaluate and economically

interpret the performance of several popular approaches to SVAR identification.

2.3 Sign restrictions and masquerading shocks

This section uses the controlled model laboratories of Section 2.2.1 to judge and economically

interpret the popular agnostic sign identification scheme of Uhlig (2005). In Section 2.3.1,

I characterize the entire SVAR-implied identified set; in particular, I study the largest and

smallest output responses possibly consistent with the imposed sign-identifying information.

Section 2.3.2 then analyzes the distribution over this identified set induced by the popular

Bayesian implementation of sign restrictions – that is, the Haar prior.

72


2.3.1 The identified set

Uhlig (2005) proposes an agnostic identification scheme. He defines as a monetary policy

shock any shock that, for a pre-specified (often quite large) number of periods, moves in-

terest rates and inflation in opposite directions.10 Notably, the response of output is left

unrestricted, contrary to popular recursive schemes. In my candidate data-generating pro-

cesses, monetary policy shocks indeed are the only shocks to satisfy these restrictions, so the

proposed identification scheme is in principle promising.

Static Model. I begin with the simple three-equation model. Since the model is static, I

only restrict the inflation and interest rate responses on impact. It is straightforward to see

that the proposed sign restrictions are not strong enough to uniquely pin down the sign of

the output response. I provide an informal discussion here, and relegate the formal proof to

Section B.4.1 of the Online Appendix.

By construction, the monetary policy shock is the only pure shock to lie in the identified

set. However, linear combinations of (expansionary) demand and supply shocks can do so as

well and thus “masquerade” as contractionary policy shocks. By definition, any candidate

“structural” shock emt ≡ pmdεdt + pmsε

st + pmmε

mt , where the unit-length vector of weights

pm = (pmd, pms, pmm)′ is such that

pmd × κσd − pms × (1 + φy)σs − pmm × κσm ≤ 0 (2.10)

pmd × (φy + φπκ)σd − pms × φπσs + pmm × σm ≥ 0, (2.11)

10In his benchmark analysis, Uhlig considers a few additional constraints, designed chiefly to disentanglemonetary policy and money demand shocks. As my candidate models feature no such shocks, I ignore theserestrictions. Also, it is well-known that Uhlig’s results continue to hold with my smaller set of restrictionson estimated three-variable SVARs (e.g Castelnuovo, 2012; Wolf, 2017).

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will lie in the identified set. The corresponding (scaled) output response is

pmd × σd + pms × φπσs − pmm × σm (2.12)

Clearly, as long as the impact impulse response matrix displayed in (2.1) is full rank, the

two inequality restrictions cannot possibly be informative about the sign of the impact out-

put response. In particular, straightforward algebra shows that, as long as φy > 0, positive

weights on demand and supply shocks are consistent with the imposed identifying restric-

tions, but with the obvious incorrect implications for the output response to the identified

shock. It is important to note that this logic works completely independently of relative

shock volatilities. In particular, even if the monetary shock were the overwhelming driver of

macro fluctuations (σm � σd, σs), the sign-restricted identified set would continue to con-

tain incorrect positive output responses. Thus, at least in this simple model, sign-identifying

information alone are not enough to pin down the sign of the unrestricted output response.

Smets-Wouters. The previous conclusions may appear particular to the simple model

considered so far. Realistic data-generating processes are not static, and actual applications

of sign-identifying schemes restrict impulse responses for many periods, not just one. I thus

extend the inflation and interest rate restrictions to hold for six quarters, and apply them

to identify structural VARs generated from the more realistic medium-scale DSGE model of

Smets & Wouters (2007). Figure 2.1 displays identified sets of impulse response functions.

Consistent with the intuition from the static model, and similar to the earlier simulation-

based evidence of Castelnuovo (2012), I conclude that the impact output response is not well-

identified; in particular, the identified set again contains both positive and negative values.

To allow an economic interpretation of this identified set, I use the results of Section 2.2.3

to link the mis-identified SVAR shocks to the true underlying structural shocks. Figure 2.2

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Figure 2.1: Identified Set of Impulse Responses: Uhlig (2005) Sign Restrictions

Note: Identified sets of the responses of output, inflation and policy rate to a one standard deviationshock to the monetary policy rule, identified through sign restrictions on inflation and policy rate(imposed for six quarters).

visualizes my results by matching the impact output response of identified shocks to their

decomposition in terms of the underlying disturbances.11

Recall that the Smets-Wouters model features seven shocks; to ease visual interpreta-

tion, I have summed the weights on the three demand and supply shocks, respectively. The

plot reveals that the right tail of positive output responses largely reflects positive demand

and supply shocks masquerading as contractionary monetary policy shocks. The right lin-

ear combination of these shocks also pushes inflation down and interest rates up, but of

course boosts output. Section B.4.5 of the Online Appendix shows that the exact same

masquerading shocks logic features just as prominently in a dynamic three-equation model.

11To ease visual interpretability, I adjust raw shock weights in two ways. First, I only show impact weights,and ignore any weights on lagged true structural shocks. Appendix B.2 explains why this simplification isharmless. Second, there is in fact no strict one-to-one mapping between impact output responses andcorresponding shock weights. I thus draw many entries from the model’s identified set, and smooth theresulting series of shock weights as a function of the impact output response. I show a plot of unsmoothedsampled weights in Section B.4 of the Online Appendix. Finally, note that the weight vector (0, 0, 1) doesnot lie in the identified set, precisely because the model is non-invertible.

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Figure 2.2: Identified Set of Shock Weights: Uhlig (2005) Sign Restrictions

Note: Identified set of static shock weights as a function of the output response at horizon 0. Forthe demand and supply shocks I sum all relevant weights, adjusting for the fact that the cost-push and wage-push shocks are negative supply shocks. I smooth the resulting series to facilitateinterpretation. The true impact response of output is -0.219.

Implications. The analysis in this section has implications for sign-based SVAR inference

in general and for the identification of monetary policy shocks in particular. First, both

model laboratories suggest that the common minimal requirement of sign restrictions – that

they be exclusively satisfied by the shock of interest – is necessary, but not sufficient for

successful identification.12 Monetary policy shocks are arguably unique in having opposite

effects on inflation and interest rates, but, unfortunately, this is only enough to ensure that

SVAR analysis can in principle succeed (cf. Section 2.2.3), not that weak sign restrictions

alone give tight identified sets.

12Wolf (2017) studies the identification of technology shocks as a second illustration, and Section B.4.3shows that even the simultaneous identification of multiple structural shocks does not safeguard against themasquerading threat. Kilian & Murphy (2012) arrive at a similar conclusion in the context of oil shocks.

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Second, the wide identified sets in Uhlig (2005) can be given an economic “masquerading

shocks” interpretation. Through the lens of popular structural models, the large positive

output responses in Uhlig’s identified sets are readily explained as particular linear combina-

tions of positive demand and supply shocks masquerading as contractionary policy shocks. I

also showed that, at least in the static three-equation model, such contamination of identified

sets will persist even when monetary policy shocks are (counterfactually) extremely volatile.

I further expand on this point in the next subsection.

2.3.2 The Haar prior

In addition to its width, a second defining feature of the identified set for output responses

in Uhlig (2005) is that – at least under the Haar prior – most posterior mass is actually put

on positive output responses (see Figure 7 in his paper). The model-based perspective taken

here can also rationalize this finding and offer broader lessons for the role of the Haar prior

in applied macro-econometrics with sign restrictions.

The Haar prior is a uniform prior over orthogonal rotation matrices P ∈ O(nx). Under

invertibility, by (2.6), we can directly interpret the entries of these rotation matrices as

weights on the underlying true structural shocks.13 For example, in the static model of

Section 2.2.1, the uniform Haar prior randomly draws shock weights p, spaced uniformly

over the unit sphere. But if all shocks receive equal prior weight, yet some shocks have

much larger effects on macro aggregates than others, then the prior distribution for impulse

responses of these aggregates is automatically dominated by the most volatile shocks. In

the remainder of this section, I show that this observation has two important implications.

First, it can rationalize the substantial posterior mass on positive output responses observed

in Uhlig (2005). Second, it clarifies that earlier results on the promise of sign restrictions

13Formally, this result uses translation-invariance of the Haar prior, ensuring uniformity for any basismatrix b(Σu) (see the discussion in Appendix B.1).

77


for volatile shocks (Paustian, 2007; Canova & Paustian, 2011) are exclusively driven by the

imposition of a probabilistic prior over the identified set, and not by the sign-identifying

information itself.

Static Model. I again begin with an illustration in the simple static model, summarized in

Figure 2.3. Panel (a) shows the top right part of the unit circle, corresponding to candidate

“structural shocks” that assign positive weights to the true demand shock (x-axis) and

the true supply shock (y-axis); the weight on the true monetary policy shock is implicitly

assumed to be positive and then simply recovered residually (recall that the weight vector

p must have unit length). The light grey region – the interior of the unit circle – is the set

of all possible shock vectors with positive weights on true demand and supply shocks. The

orange region shows, for a benchmark parameterization chosen to replicate the relative shock

volatilities in Smets & Wouters (2007), combinations of those shock weights that (i) lie in the

identified set and (ii) increase output – that is, the undesirable masquerading shocks. The

dark grey region gives the analogous combinations of masquerading shocks for a different

model parameterization, now with more volatile monetary policy shocks. Finally, panel (b)

shows the posterior probability of a negative output response to identified monetary policy

shocks (under the Haar prior) as a function of relative shock volatilities.

Figure 2.3 illustrates the two main results of this section. First, in the baseline calibration,

the orange region of “masquerading” demand and supply shocks features prominently in the

top right part of the unit circle, and the posterior probability of correctly signing the output

response is small. Intuitively, because demand and supply shocks are much more volatile

than monetary policy shocks, very large weights on monetary policy shocks are needed

to dominate the output response. Such large weights are unlikely according to the prior,

so most posterior mass will instead be put on the large orange area of masquerading shock

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Figure 2.3: Identified Sets and the Haar Prior

(a) Mis-Identification Regions (b) Probability of Negative Output Response

Note: Identified sets in the static three-equation model, with φπ = 1.5, φy = 0.2, κ = 0.2, σd = 1,σs = 1. In the benchmark calibration σm = 0.2; in the high-volatility calibration, σm = 6. Panel(a) shows regions of masquerading shocks giving positive output responses; panel (b) gives theprobability of a negative identified output response as a function of relative shock volatilities.

combinations. Exactly in line with this intuition, the posterior distribution over the identified

set in Uhlig (2005) is dominated by positive output responses.

Second, as relative shock volatilities are re-scaled, the shape of the posterior over the

identified set changes dramatically. Consider first the two identified sets of masquerading

demand and supply shocks in panel (a), constructed for two different values of monetary

policy shock volatility. From the inequality constraints (2.10) - (2.11), we know that there

exists a simple one-to-one mapping between all points in these two identified sets.14 Their

posterior probabilities, however, are very different. In the benchmark parameterization,

positive demand and supply shocks in the identified set occupy a large region in the unit

circle, and so are regarded as likely by the Haar measure. As the monetary policy shock

becomes more volatile, the associated weights on demand and supply shocks necessarily

14Let p be a weight vector in the original identified set, giving a positive output response. Now let

p∗i = pi× σi

σi(i ∈ (d, s,m), and where σ and σ are the old and new shock volatilities, respectively). Then the

vector p ≡ p∗

||p∗|| lies in the identified set for the rescaled model, and also gives a positive output response.

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become larger – graphically, the orange area maps into a smaller and smaller sliver of the unit

circle, and thus the masquerading shock combinations are regarded as increasingly unlikely.

In the limit, as the monetary policy shock becomes infinitely more volatile than demand

and supply shocks, the orange and dark grey regions actually get mapped into a measure-0

subspace at the boundary of the unit circle, and so receive a posterior probability of 0. Panel

(b) provides an illustration across a large range of possible relative shock volatilities.

My results can also help to rationalize the conclusions in Paustian (2007) and Canova &

Paustian (2011). If the shock of interest is sufficiently volatile, then conventional Bayesian

posteriors over identified sets are likely to put most mass on correctly signed impulse re-

sponses. However, it is also immediate that this conclusion is exclusively driven by the

particular choice of prior. As I show in Appendix B.3, it is always possible to construct an

alternative prior such that, whatever the relative shock volatility, the posterior probability

of a correctly signed impulse response remains arbitrarily small.15

Smets-Wouters. The insights from the simple static model generalize without change to

the environment of Smets & Wouters (2007). Figure 2.4 provides a graphical illustration.

Since monetary policy shocks are on average relatively small, most posterior mass over the

identified set concentrates on positive output responses, fully consistent with the empirical

findings in Uhlig (2005). As the relative volatility of the monetary policy shock increases,

posterior mass mostly shifts to negative output responses. Nevertheless, even for an extreme

counterfactual increase of monetary policy shock volatility, the identified set itself continues

to include strictly positive output responses, so any conclusions about statistical significance

of a negative output response are necessarily exclusively driven by the prior.

15Finally, my results are also informative about the role played by the uniform Haar prior in allowing sign-restricted inference to be informative about quantities. For example, as the monetary policy shock becomesdominant relative to other shocks, the identified set for the output response converges to [− 1

1+φy+φπκσm, 0],

and the distribution over this identified set can be derived following the steps in Baumeister & Hamilton(2015). The quantity information contained in pure sign restrictions is thus simply that the impulse responseis somewhere between zero and the truth; all further information comes from the prior.

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Figure 2.4: Scaled Volatilities in Smets & Wouters (2007)

(a) Benchmark Model (b) Scaled Volatilities

Note: Identified set of the output response, with identifying restrictions as in Figure 2.1. Posterioruncertainty via imposition of the uniform Haar prior; the solid and dotted lines give 16th, 50th and84th percentile bands. In the model with scaled volatilities, the relative volatility of the monetarypolicy shock is scaled up by a factor of 30.

Implications. Large-sample Bayes inference over identified sets is dominated by the prior

(Moon & Schorfheide, 2012; Baumeister & Hamilton, 2015; Watson, 2019). Taking a popula-

tion perspective, my analysis precisely characterizes the additional probabilistic identifying

information embedded in the popular Haar prior. In particular, I show that this flat prior

over orthogonal rotation matrices can equivalently be interpreted as a flat prior over hidden

shock weights, thus automatically over-weighting particularly volatile macro shocks.

Whether or not the Haar prior is a sensible prior is invariably an application-dependent

question. In my analysis of monetary policy shock identification, I find that, due to the

relatively low volatility of policy shocks, researchers relying on the Haar prior are likely

to mis-characterize the sign of the aggregate output response. Since pure sign-identifying

information is also consistent with (correct) negative output responses, any conclusions about

statistical significance of positive responses are exclusively driven by the prior.16

16In actual empirical practice, such identification uncertainty is further conflated with reduced-form pa-rameter estimation uncertainty. My analysis is exclusively concerned with population limits, and so only

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2.4 Zero restrictions

The classical approach to monetary policy shock identification is the imposition of zero

impact response restrictions on output and inflation. Uhlig (2005) shows that the zero output

restriction is central to recovering the conventional negative – if small – output effects of a

contractionary monetary shock. In this section I provide a rationale for this centrality of the

zero output restriction, but also show that, at least at short horizons, the estimated output

response is likely to understate the policy’s true aggregate effects.

The impact output restriction. Uhlig (2005) shows that conventional wisdom (e.g.

Christiano et al., 1996) relies sensitively on the impact zero output restriction – in his words

a “rather spurious identification restriction.” Expanding on the analysis of Section 2.3, I

will now show that the strong bite of the impact output restriction is not an accident, but

an intuitively sensible feature of identified sets.

In my structural models, the large positive output responses identified by the pure sign-

restricting scheme of Uhlig (2005) correspond to large weights on positive demand and supply

shocks. These shocks both push output up, but move interest rates in opposite directions,

and so necessarily imply a large ratio (or multiplier) |dy0di0|. Restricting this impact multiplier

thus promises to chop off the right tail of mis-identified masquerading shocks displayed in

Figure 2.2; as Figure 2.5 shows, this is exactly what happens in the model of Smets &

Wouters. Panel (a) shows the identified set for the output response if the baseline sign

restrictions of Uhlig (2005) were to be complemented with a hard zero restriction on the

impact response of output. Consistent with the intuition given above, the identified set is now

tight around the familiar hump-shaped negative response of output to an identified monetary

policy shock. The plot of shock weights in panel (b) confirms that the large mis-identified

speaks to one part of the inference problem. Nevertheless, the larger Monte Carlo exercise in Section B.4.4suggests that identification uncertainty is, in relevant applications, large relative to estimation uncertainty.

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region of masquerading expansionary demand and supply shocks is eliminated. Section

B.4.6 of the Online Appendix further shows that, even with moderate bounds on the impact

multiplier |dy0di0|, identified sets tighten significantly around negative output responses.17

Figure 2.5: Identified Sets with Zero Restrictions

(a) Output Response (b) Shock Weights

Note: Identified sets of output and shock weights. Inflation and interest rates are restricted tomove in opposite directions for six quarters; additionally, the impact output response is restrictedto be 0. Panel (a) also shows the point-identified recursive impulse responses (with the policy shockordered last) as well as the true impulse response. Panel (b) shows shock weights as a function ofthe average output response over the first year.

My analysis suggests that the centrality of a zero impact output restriction – or of

weaker bounds – to the sign of the identified output response path should not come as a

surprise. However, to the extent that the true impact output restriction is not literally zero,

monetary policy shocks will still be mis-identified. In particular, for the first few quarters,

real effects will mechanically be understated. Panel (a) in Figure 2.5 shows exactly this.

At the same time, farther-out dynamics may be mis-identified in other less obvious ways;

in the model of Smets & Wouters, the identified sets indicate greater persistence of policy

shocks than is actually the case. As it turns out, the economics underlying this long-horizon

mis-identification are particularly transparent for standard recursive identification schemes.

17The same happens in actual data, as shown in Wolf (2017).

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Recursive identification and shock persistence. Recursive identification of monetary

policy shocks complements the impact zero output restriction with an impact zero inflation

restriction. Together, these two pieces of identifying information are enough to provide

point identification. The corresponding output impulse response is also displayed in panel

(a) of Figure 2.5.18 Recursively identified monetary policy shocks appear to depress output,

but, relative to the true model-implied impulse response, real effects are understated at

short horizons and overstated at long horizons. In fact, recursive identification distorts long-

horizon impulse responses more than any other SVAR in the identified set of Figure 2.5.

As before, understatement at short horizons is simply a mechanical implication of the

impact zero restriction. The subsequent pattern of dynamic mis-identification is more subtle

and intimately related to the relative persistence of the underlying structural shocks. In Sec-

tion B.5 of the Online Appendix I show that, if all true model shocks had equally persistent

effects on macro aggregates, then the recursively identified “monetary policy” impulse re-

sponses for output and inflation would be exactly 0 at all times. Intuitively, if a given linear

combination of shocks – all with equally persistent dynamic effects – implies a zero response

on impact, then it will necessarily imply a zero response forever. This simple logic can help

clarify the dynamics displayed in Figure 2.5: Relative to all other SVARs in that identified

set, a recursive SVAR gives the largest possible (i.e., zero) inflation impact response, and it

achieves this zero impact response through a large positive weight on contractionary supply

shocks. Crucially, in the structural model of Smets & Wouters, technology shocks – which

account for most low-frequency variation in macroeconomic aggregates – are extremely per-

sistent. These persistent supply shocks then dominate long-run dynamics, and in particular

result in the displayed substantial overstatement of the output drop at long horizons.

18Since this paper is chiefly concerned with the real effects of monetary policy shocks, I do not furtherdiscuss the “price puzzle” – another well-known anomaly of recursively identified monetary policy SVARs.In Section B.6 I study the identified inflation response and discuss the extent to which the model-basedperspective taken here can also rationalize the price puzzle.

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Implications. The theory presented here rationalizes the centrality of zero impact output

restrictions to conventional wisdom, but cautions against interpretation of the resulting

estimates as accurate representations of the economy’s true shock propagation. Recursively

identified shocks are likely to understate the aggregate effects of policy interventions at

short horizons, and their implied long-horizon dynamics are sensitive to the persistence of

the other underlying macro shocks.19 The next section considers alternative identification

schemes that are less vulnerable to such criticisms.

2.5 Recent advances in identification

Following the concerns expressed in Uhlig (2005), the past few years have seen a flurry of

research trying to identify the real effects of monetary policy without any direct restrictions

on the response of output. Two particularly prominent examples are sign restrictions on the

VAR-implied Taylor rule, as in Arias et al. (2019), and the use of external instruments, as

in Gertler & Karadi (2015) or Jarocinski & Karadi (2019). Most of these methods indicate

somewhat larger effects of policy shocks on real outcomes, in particular at short horizons. In

this section, I argue that, first, these results are again not at all surprising through a model

lens, and second, the resulting identified sets are likely to be quite informative about the

true real effects of monetary policy disturbances.

2.5.1 Taylor rule restrictions

Arias et al. (2019) show that restrictions on the output coefficient in an implied Taylor rule

substantially tighten Uhlig’s identified set around negative effects of monetary policy shocks;

equivalently, their analysis suggests that many of the candidate “monetary policy shocks” in

19Of course, these concerns would be less acute in models like Christiano et al. (2005), which have beenexplicitly constructed to ensure consistency of the usual recursive estimators.

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Uhlig (2005) must imply Taylor rules with a negative output response. This section makes

two observations. First, I show that the long tail of masquerading supply and demand shocks

in my structural models also induces implied Taylor rules with negative output coefficients.

It is thus not surprising that, just like in the data, an additional restriction on implied Taylor

rule coefficients materially tightens identified sets in my models. Second, I show that this

conclusion is a robust implication of basic properties of New Keynesian models.

Figure 2.6 displays the identified set under the identification scheme of Arias et al. (2019).

Building on the baseline sign restrictions of Uhlig (2005) – but then additionally imposing

that the output and inflation coefficients in the SVAR-implied Taylor rule are strictly positive

– leads to a substantial tightening of the identified set around significant negative output

responses. Exactly as in Arias et al. (2019), I find that this tightening is almost exclusively

driven by the restriction on the implied Taylor rule output coefficient.20

Figure 2.6: Identified Sets with Taylor Rule Restrictions

(a) Output Response (b) Shock Weights

Note: Identified sets of output and shock weights. Inflation and interest rates are restricted tomove in opposite directions for six quarters; additionally, the implied Taylor rule coefficients oninflation and output are restricted to be positive.

20Differently from their analysis, and consistent with the discussion in Section 2.3.2, I do not impose theuniform Haar prior, but instead display entire identified sets.

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The results of Figure 2.6 indicate that restrictions on implied Taylor rule coefficients

contain substantial additional identifying information. The intuition is as follows. As shown

in Figure 2.2, most of the mis-identified “masquerading” shock combinations that coun-

terfactually increase aggregate output are in fact mixtures of positive demand and supply

shocks. Equivalently, the mis-identified monetary policy shocks are linear combinations of

residuals in the model’s IS and NKPC curves. But if the shocks are a linear combination

of these residuals, then the implied Taylor rule itself is a linear combination of those same

IS and NKPC equations. For example, in the static three-equation model of Section 2.2.1,

straightforward manipulations show that the implied (mis-identified) Taylor rule is

it =pmmφπ + pmspmd + pmm︸︷︷︸

φπ

× πt +pmmφy − pmd − pmsκ

pmd + pmm︸︷︷︸φy

× yt + emt

Importantly, for mis-identified masquerading shocks with pmm ≈ 0 and pmd, pms > 0, the

SVAR-implied Taylor rule coefficient φy is necessarily negative. It is thus unsurprising that

the additional restriction φy > 0 substantially tightens identified sets and largely removes

masquerading supply and demand shocks.

2.5.2 External instruments

A popular alternative to the use of direct identifying restrictions – “internal instruments,”

in the language of Stock & Watson (2018) – is the use of instrumental variables, or “external

instruments.” An external instrument is a variable correlated with the shock of interest, and

uncorrelated with any other structural shocks. For the study of monetary policy shocks, the

most popular instrument is that of Gertler & Karadi (2015).

In this section I study the performance of the popular SVAR-IV estimator (Stock, 2008;

Stock & Watson, 2012; Mertens & Ravn, 2013) in the structural model of Smets & Wouters

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(2007). Specifically, I assume that, in addition to the usual macro aggregates xt, the econo-

metrician now also observes an artificially generated external instrument zt satisfying

zt =∞∑`=1

(Ψ`zt−` + Λ`xt−`) + αεm,t + σvvt, (2.13)

where (i) all roots of the lag polynomial 1−∑∞

`=1 Ψ`L` are outside the unit circle, (ii) {Λ`}`

is absolutely summable, and (iii) vt is uncorrelated at all leads and lags with the structural

shocks εt. The econometrician then implements the SVAR-IV estimator using the linear

projection zt ≡ zt − E [zt | {zτ , xτ}−∞<τ<t] = αεm,t + σvvt as an external instrument.21

Even with a valid instrument, however, non-invertibility can threaten the consistency of

the SVAR-IV estimator. In Plagborg-Møller & Wolf (2019a), we prove two related results.

First, we show that, under non-invertibility, the weight of identified on true contemporaneous

monetary policy shock is

P0(k,m) =√R2m =

√1− Var (εm,t | {xτ}−∞<τ≤t) (2.14)

Thus, the SVAR-IV estimator attains the theoretical bound in (2.9), and so – in a very

particular sense – provides the best possible approximation to the true unknown monetary

policy shock. Of course, with a low R2m, this approximation could still be quite poor. Second,

we partially characterize the resulting bias. In particular, we show that impact impulse

response estimates are biased up (in absolute value) by a factor of 1/√R2m.

Taken together, these theoretical results as well as my earlier conclusions about likely

near-invertibility of monetary policy shocks imply that SVAR-IV estimators of monetary

policy transmission are likely to perform reasonably well. Figure 2.7 shows that this is

exactly what happens in the model of Smets & Wouters (2007). Even with only three

21Note that the probability limit of the SVAR-IV estimator is independent of the particular numericalvalues of α and σv (as long as α 6= 0). I thus do not need to take a stand on what those values actually are.

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macro observables, the impact bias is small, and the dynamics of output and inflation are

captured adequately, at least at short horizons. If anything, the real effects of policy shocks

are somewhat overstated. With an augmented set of observables, identification obviously

improves further; for example, if the researcher were to additionally include investment and

a measure of total labor, then the R20,m would rise to 0.9302, and the weight on the true shock

would increase to 0.9645. If the researcher were to go even further and include measures of

consumption and real wages, then the system becomes invertible and identification is perfect.

Figure 2.7: Identified Set of Impulse Responses

Note: Impulse response functions identified via valid external instruments. The small-scale VARcontains output, inflation and the interest rate; the large-scale VAR adds investment and hoursworked.

2.6 Conclusion

In this paper I interpret various different empirical approaches to the study of monetary

policy transmission through the lens of fully specified structural models. This model-based

perspective suggests two important conclusions. First, theory and empirics are internally

consistent. I find that different estimators, all applied to the same standard structural

model, can give estimates of impulse responses that look as disparate as those estimated on

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actual data. I conclude that the data are consistent with monetary policy having significant

real effects, and in fact somewhat larger and somewhat less persistent than often estimated.

Second, pure sign restrictions are quite weak identifying information. Because of what I call

masquerading shocks, the common minimal requirement for sign-based inference – that the

shock of interest be the only one to simultaneously satisfy all imposed sign restrictions –

is not sufficient. The masquerading shock problem is particularly acute when the shock of

interest is not very volatile, as then the uniform Haar prior will concentrate most posterior

mass on rival large masquerading shocks.

The identification of monetary policy transmission can, of course, be improved further.

A valid external instrument is clearly the ideal solution, implemented either using LP-IV

or SVAR-IV methods (Stock & Watson, 2018; Plagborg-Møller & Wolf, 2019b). Existing

high-frequency instruments, however, may fail to adequately disentangle true policy shocks

and information effects (Jarocinski & Karadi, 2019; Nakamura & Steinsson, 2018a). Alter-

natively, model-consistent set-identifying information in the spirit of Uhlig (2005) and Arias

et al. (2019) promises to be robust and, in the latter case, informative across a wide range of

structural models, but may not yield tight enough inference. Further identifying restrictions

may thus be needed.

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Chapter 3

Local Projections and VARs Estimate the

Same Impulse Responses

I thank my co-author Mikkel Plagborg-Møller for his permission to reproduce our work here.

3.1 Introduction

Modern dynamic macroeconomics studies the propagation of structural shocks (Frisch, 1933;

Ramey, 2016). Central to this impulse-propagation paradigm are impulse response functions

– the dynamic response of a macro aggregate to a structural shock. Following Sims (1980),

Bernanke (1986), and Blanchard & Watson (1986), Structural Vector Autoregression (SVAR)

analysis remains the most popular empirical approach to impulse response estimation. Over

the past decade, however, starting with Jorda (2005), local projections (LPs) have become

an increasingly widespread alternative econometric approach.

How should we choose between SVAR and LP estimators of impulse responses? Unfor-

tunately, so far there exists little theoretical guidance as to which method is preferable in

practice. Conventional wisdom holds that SVARs are more efficient, while LPs are more

robust to model misspecification. Examples of such statements are found in Jorda (2005, p.

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3. Local Projections and VARs

162), in the literature reviews of Ramey (2016, p. 83) and Nakamura & Steinsson (2018b,

pp. 80–81), and in the textbook treatment of Kilian & Lutkepohl (2017, ch. 12.8).1 Stock &

Watson (2018, p. 944), however, caution that these remarks are not based on formal analysis

and call for further research. It is also widely believed that LPs invariably require a measure

of a “shock,” so that SVAR estimation is the only way to implement more exotic struc-

tural identification schemes such as long-run or sign restrictions.2 Finally, when applied to

the same empirical question, LP- and VAR-based approaches sometimes give substantively

different results (Ramey, 2016). Existing simulation studies on their relative merits reach

conflicting conclusions and disagree on implementation details (Meier, 2005; Kilian & Kim,

2011; Brugnolini, 2018; Nakamura & Steinsson, 2018b; Choi & Chudik, 2019).

The central result of this paper is that linear local projections and VARs in fact estimate

the exact same impulse responses in population. Specifically, any LP impulse response func-

tion can be obtained through an appropriately ordered recursive VAR, and any (possibly

non-recursive) VAR impulse response function can be obtained through a LP with appro-

priate control variables. This result is nonparametric, in that it essentially only requires the

data to be weakly stationary and the lag structures in the two specifications to be unre-

stricted.3 Intuitively, a VAR model with sufficiently large lag length captures all covariance

properties of the data. Hence, iterated VAR(∞) forecasts coincide with direct LP forecasts.

Since impulse responses are just forecasts, LP and VAR impulse response estimands coincide

in population. Furthermore, we prove that if only a fixed number p of lags are included in

the LP and VAR, then the two impulse response estimands still agree out to horizon p (but

not further), again without imposing any parametric assumptions on the data generating

1In the online postscript to her handbook chapter, Ramey corrects the claim and restates the relationshipbetween LP and VAR estimands following the findings of this paper.

2See the reviews by Ramey (2016) and Kilian & Lutkepohl (2017, ch. 12.8).3Although linear LP and VAR estimators may in principle be viewed as “parametric” procedures, we do

not assume that the data generating process can be summarized by any finite-dimensional parametric model.

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process. In summary, if VAR and LP results differ in population or in sample, it is due to

extraneous restrictions on the lag structure.

The nonparametric equivalence of VAR and LP estimands has several implications for

structural estimation in applied macroeconometrics.

First, LP and VAR estimators are not conceptually different methods; instead, they

belong to a spectrum of linear projection techniques that share the same estimand but differ

in their finite-sample bias-variance properties. Standard LPs effectively provide no dimension

reduction, while conventional low-order VARs extrapolate shock propagation from the first

few autocorrelations of the data. The relative mean-square error of the two methods – and

of intermediate dimension reduction techniques, such as shrinkage – necessarily depends

on assumptions about the data generating process (DGP). VAR estimators are optimal

if the true DGP is exactly a finite-order VAR, but this is rarely the case in theory or

practice. The formal equivalence of LP and VAR impulse response estimation to direct and

iterated forecasting, respectively, means that applied researchers can look to the existing

forecasting literature for guidance on how to choose between the menu of available estimators

(Schorfheide, 2005; Marcellino et al., 2006; Pesaran et al., 2011).

Second, structural estimation with VARs can equally well be carried out using LPs, and

vice versa. Structural identification – which is a population concept – is logically distinct from

the choice of finite-sample dimension reduction technique. In particular, we show concretely

how various popular “SVAR” identification schemes – including recursive, long-run, and

sign identification – can just as easily be implemented using local projection techniques.

Ultimately, our results show that LP-based structural estimation can succeed if and only if

SVAR estimation can succeed.

Third, valid structural estimation with an instrument (IV, also known as a proxy variable)

can be carried out by ordering the IV first in a recursive VAR a la Ramey (2011). This is

because the LP-IV estimand of Stock & Watson (2018) can equivalently be obtained from a

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recursive (i.e., Cholesky) VAR that contains the IV. Importantly, the “internal instrument”

strategy of ordering the IV first in a VAR yields valid impulse response estimates even if the

shock of interest is non-invertible, unlike the well-known “external instrument” SVAR-IV

approach (Stock, 2008; Stock & Watson, 2012; Mertens & Ravn, 2013).4 In particular, this

result goes through even if the IV is contaminated with measurement error unrelated to the

shock of interest.

Fourth, in population, linear local projections are exactly as “robust to non-linearities”

in the DGP as VARs. We show that their common estimand may be formally interpreted as

a best linear approximation to the underlying, perhaps non-linear, data generating process.

In summary, in addition to clarifying misconceptions in the literature about the LP and

VAR estimands, our results allow applied researchers to separate the choice of identification

scheme from the choice of estimation technique. Researchers who prefer the intuitive regres-

sion interpretation and generally low bias of the LP impulse response estimator can apply our

methods for imposing “SVAR” identifying restrictions such as short-run, long-run, and sign

restrictions. Researchers who instead prefer the explicit multivariate model and generally

low variance of the VAR estimator can apply our results on how to use instruments/proxies

without requiring invertible shocks, as in LP-IV.

Literature. While the existing literature has pointed out connections between LPs and

VARs, our contribution is to formally establish a nonparametric equivalence result and derive

implications for estimation efficiency and structural identification. Jorda (2005) and Kilian

& Lutkepohl (2017, Ch. 12.8) show that, under the assumption of a finite-order VAR model,

VAR impulse responses can be estimated consistently through LPs. In this context, Kilian

& Lutkepohl also discuss the relative efficiency of the two estimation methods and mention

4In contemporaneous work, Noh (2018) also recommends including the IV as an internal instrument in aVAR; our result offers additional insights by drawing connections to LP-IV and to the general equivalencebetween LPs and VARs.

94


the literature on direct versus iterated forecasts. In contrast, our equivalence result is non-

parametric, and we further demonstrate how structural VAR orderings map into particular

choices of LP control variables, and vice versa.5 Moreover, to our knowledge, our results on

long-run/sign identification, LP-IV, and best linear approximations have no obvious parallels

in the preceding literature.6

In this paper we focus exclusively on identification and point estimation of impulse re-

sponses. Plagborg-Møller & Wolf (2019a) provide identification results for variance/historical

decompositions when an instrument/proxy is available. We do not consider questions related

to inference, and instead refer to the discussions in Jorda (2005), Kilian & Lutkepohl (2017),

and Stock & Watson (2018).

Outline. Section 3.2 presents our core result on the population equivalence of local pro-

jections and VARs. Finite-sample estimation is discussed in Section 3.3, while Section 3.4

traces out implications for structural estimation. We illustrate our equivalence results with

a practical application to IV-based identification of monetary policy shocks in Section 3.5.

Section 3.6 concludes with several recommendations for empirical practice. Some proofs are

relegated to Appendix C.

3.2 Equivalence between local projections and vector

autoregressions

This section presents our core result: Local projections and VARs estimate the same impulse

response functions in population. First we establish that local projections are equivalent

5Jorda et al. (2019) informally discuss the connection between control variables in local projections andrecursive SVARs.

6Kilian & Lutkepohl (2017, Ch. 12.8) present alternative arguments for why it is a mistake to assert thatfinite-order LPs are generally more “robust to model misspecification” than finite-order VAR estimators.They do not appeal to the nonparametric equivalence of the LP and VAR estimands, however.

95


with recursively identified VARs when the lag structure is unrestricted. Then we extend the

argument to (i) non-recursive identification and (ii) finite lag orders. Finally, we illustrate

the results graphically. Our analysis in this section is “reduced form” in that it does not

assume any specific underlying structural model; we merely work with linear projections of

stationary time series. We will discuss implications for structural identification in Section 3.4.

3.2.1 Main result

Suppose the researcher observes data wt = (r′t, xt, yt, q′t)′, where rt and qt are, respectively,

nr × 1 and nq × 1 vectors of time series, while xt and yt are scalar time series. We are

interested in the dynamic response of yt after an impulse in xt. The vector time series rt and

qt (which may each be empty) will serve as control variables. The distinction between them

relates to whether they appear as contemporaneous controls or not, as will become clear in

equations (3.1) and (3.2) below.

For now, we only make the following nonparametric regularity assumption.7

Assumption 9. The data {wt} are covariance stationary and purely non-deterministic, with

an everywhere nonsingular spectral density matrix and absolutely summable Wold decompo-

sition coefficients. To simplify notation, we proceed as if {wt} were a (strictly stationary)

jointly Gaussian vector time series.

In particular, we assume nothing about the underlying causal structure of the economy,

as this section is concerned solely with properties of linear projections. The Gaussianity

assumption is made purely for notational simplicity, as this allows us to write conditional

7The restriction to non-singular spectral density matrices rules out over-differenced data. We conjecturethat this restriction could be relaxed using the techniques in Almuzara & Marcet (2017).

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expectations instead of linear projections. If we drop the Gaussianity assumption, all calcu-

lations below hold with projections in place of conditional expectations.8

We will show that, in population, the following two approaches estimate the same impulse

response function of yt with respect to an innovation in xt.

1. Local projection. Consider for each h = 0, 1, 2, . . . the linear projection

yt+h = µh + βhxt + γ′hrt +∞∑`=1

δ′h,`wt−` + ξh,t, (3.1)

where ξh,t is the projection residual, and µh, βh, γh, δh,1, δh,2, . . . the projection coefficients.

The LP impulse response function of yt with respect to xt is given by {βh}h≥0. Notice

that the projection (3.1) controls for the contemporaneous value of rt but not of qt.

2. VAR. Consider the multivariate linear “VAR(∞)” projection

wt = c+∞∑`=1

A`wt−` + ut, (3.2)

where ut ≡ wt−E(wt | {wτ}−∞<τ<t) is the projection residual, and c, A1, A2, . . . the pro-

jection coefficients. Let Σu ≡ E(utu′t), and define the Cholesky decomposition Σu = BB′,

where B is lower triangular with positive diagonal entries. Consider the corresponding

recursive SVAR representation

A(L)wt = c+Bηt,

8Throughout we write any linear projection on the span of infinitely many variables as an infinite sum.This is justified under Assumption 9, since we can invert the Wold representation to obtain a VAR(∞)representation.

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where A(L) ≡ I −∑∞

`=1 A`L` and ηt ≡ B−1ut. Notice that rt is ordered first in the VAR,

while qt is ordered last. Define the lag polynomial

∞∑`=0

C`L` = C(L) ≡ A(L)−1.

The VAR impulse response function of yt with respect to an innovation in xt is given by

{θh}h≥0, where

θh ≡ Cnr+2,•,hB•,nr+1,

since xt and yt are the (nr + 1)-th and (nr + 2)-th elements in wt. The notation Ci,•,h,

say, means the i-th row of matrix Ch, while similarly B•,j is the j-th column of matrix B.

Note that our definitions of the LP and VAR estimands include infinitely many lags of wt in

the relevant projections. We consider the case of finitely many lags in Section 3.2.3, while

all finite-sample considerations are relegated to Section 3.3. Note also that we take the use

of the control variables rt and qt as given in this section, as controls are common in applied

work. We will discuss structural justifications for the use of controls in Section 3.4.

Although LP and VAR approaches are often viewed as conceptually distinct in the liter-

ature, they in fact estimate the same population impulse response function.

Proposition 7. Under Assumption 9, the LP and VAR impulse response functions are

equal, up to a constant of proportionality: θh =√E(x2

t ) × βh for all h = 0, 1, 2, . . . , where

xt ≡ xt − E(xt | rt, {wτ}−∞<τ<t).

That is, any LP impulse response function can equivalently be obtained as an appropriately

ordered recursive VAR impulse response function. Conversely, any recursive VAR impulse

response function can be obtained through a LP with appropriate control variables. We

comment on non-recursive identification schemes below. The constant of proportionality

in the proposition depends on neither the response horizon h nor on the response variable

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yt. The reason for the presence of this constant of proportionality is that the implicit LP

innovation xt, after controlling for the other right-hand side variables, does not have variance

1. If we scale the innovation xt to have variance 1, or if we consider relative impulse responses

θh/θ0 (as further discussed below), the LP and VAR impulse response functions coincide.

The intuition behind the result is that a VAR(p) model with p→∞ is sufficiently flexible

that it perfectly captures all covariance properties of the data. Thus, iterated forecasts based

on the VAR coincide perfectly with direct forecasts E[wt+h | wt, wt−1, . . . ]. Although the

intuition for the equivalence is simple, its implications do not appear to have been generally

appreciated in the literature, as discussed in Section 3.1.

Proof. The proof of the proposition relies only on least-squares projection algebra. First

consider the LP estimand. By the Frisch-Waugh theorem, we have that

βh =Cov(yt+h, xt)

E(x2t )

. (3.3)

For the VAR estimand, note that C(L) = A(L)−1 collects the coefficient matrices in the

Wold decomposition

wt = χ+ C(L)ut = χ+∞∑`=0

C`Bηt, χ ≡ C(1)c.

As a result, the VAR impulse responses equal

θh = Cnr+2,•,hB•,nr+1 = Cov(yt+h, ηx,t), (3.4)

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where we partition ηt = (η′r,t, ηx,t, ηy,t, η′q,t)′ the same way as wt = (r′t, xt, yt, q

′t)′. By ut = Bηt

and the properties of the Cholesky decomposition, we have9

ηx,t =1√

E(u2x,t)× ux,t, (3.5)

where we partition ut = (u′r,t, ux,t, uy,t, u′q,t)′ and define10

ux,t ≡ ux,t − E(ux,t | ur,t) = xt. (3.6)

From (3.4), (3.5), and (3.6) we conclude that

θh =Cov(yt+h, xt)√

E(x2t )

,

and the proposition now follows by comparing with (3.3).

In conclusion, LPs and VARs offer two equivalent ways of arriving at the same population

parameter (3.3), up to a scale factor that does not depend on the horizon h. Our argument

was nonparametric and did not assume the validity of a specific structural model.

3.2.2 Extension: Non-recursive specifications

Our equivalence result extends straightforwardly to the case of non-recursively identified

VARs. Above we restricted attention to recursive identification schemes, as the VAR directly

contains a measure of the impulse xt. In a generic structural VAR identification scheme, the

impulse is some – not necessarily recursive – rotation of reduced-form forecasting residuals.

9B is lower triangular, so the (nr + 1)-th equation in the system Bηt = ut is Bnr+1,1:nrηr,t +Bnr+1,nr+1ηx,t = ux,t, with obvious notation. Since ηx,t and ηr,t are uncorrelated, we find Bnr+1,nr+1ηx,t =ux,t − E(ux,t | ηr,t) = ux,t − E(ux,t | ur,t) = ux,t. Expression (3.5) then follows from E(η2x,t) = 1.

10Observe that ux,t − xt = E(xt | rt, {wτ}−∞<τ<t)− E(xt | {wτ}−∞<τ<t) = E(ux,t | rt, {wτ}−∞<τ<t) =E(ux,t | ur,t, {wτ}−∞<τ<t) = E(ux,t | ur,t).

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Thus, let us continue to consider the VAR (3.2), but now we shall study the propagation of

some rotation of the reduced-form forecasting residuals,

ηt ≡ b′ut, (3.7)

where b is a vector of the same dimension as wt. Under Assumption 9, we can follow the

same steps as in Section 3.2.1 to establish that the VAR-implied impulse response at horizon

h of yt with respect to the innovation ηt equals – up to scale – the coefficient βh of the linear

projection

yt+h = µh + βh(b′wt) +

∞∑`=1

δ′h,`wt−` + ξh,t, (3.8)

where the coefficients are least-squares projection coefficients and the last term is the pro-

jection residual. Thus, any recursive or non-recursive SVAR(∞) identification procedure

is equivalent with a local projection (3.8) on a particular linear combination b′wt of the

variables in the VAR (and their lags). For recursive orderings, this reduces to Proposition 7.

3.2.3 Extension: Finite lag length

Whereas our main equivalence result in Section 3.2.1 relied on infinite lag polynomials, we

now prove an equivalence result that holds when only finitely many lags are used. Specifically,

when p lags of the data are included in the VAR and as controls in the LP, the impulse

response estimands for the two methods agree out to horizon p, but generally not at higher

horizons. This result is still entirely nonparametric, in the sense that we do not impose that

the true DGP is a finite-order VAR.

First, we define the finite-order LP and VAR estimands. We continue to impose the

nonparametric Assumption 9. Consider any lag length p and impulse response horizon h.

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1. Local projection. The local projection impulse response estimand βh(p) is defined

as the coefficient on xt in a projection as in (3.1), except that the infinite sum is

truncated at lag p. Again, we interpret all coefficients and residuals as resulting from

a least-squares linear projection.

2. VAR. Consider a linear projection of the data vector wt onto p of its lags (and a

constant), i.e., the projection (3.2) except with the infinite sum truncated at lag p. Let

A`(p), ` = 1, 2, . . . , p, and Σu(p) denote the corresponding projection coefficients and

residual variance. Define A(L; p) ≡ I −∑p

`=1A`(p) and the Cholesky decomposition

Σu(p) = B(p)B(p)′. Define also the inverse lag polynomial∑∞

`=0 C`(p)L` = C(L; p) ≡

A(L; p)−1 consisting of the reduced-form impulse responses implied by A(L; p). Then

the VAR impulse response estimand at horizon h is defined as

θh(p) ≡ Cnr+2,•,h(p)B•,nr+1(p),

cf. the definition in Section 3.2.1 with p =∞.

Note that the VAR(p) model used to define the VAR estimand above is “misspecified,” in

the sense that the reduced-form residuals from the projection of wt on its first p lags are not

white noise in general.

We now state the equivalence result for finite p. The statement of the result is a simple

generalization of Proposition 7, which can be thought of as the case p =∞.

Proposition 8. Impose Assumption 9. Define xt(`) ≡ xt − E(xt | rt, {wτ}t−`≤τ<t) for all

` = 0, 1, 2, . . . . Let the nonnegative integers h, p satisfy h ≤ p. If xt(p) = xt(p − h), then

θh(p) =√E(xt(p)2)× βh(p).

Proof. Please see Appendix C.1.

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Thus, under the conditions of the proposition, the population LP and VAR impulse response

estimands agree at all horizons h ≤ p, although generally not at horizons h > p. This finding

would not be surprising if the true DGP were assumed to be a finite-order VAR (as in Jorda,

2005, and Kilian & Lutkepohl, 2017, Ch. 12.8), but we allow for general covariance stationary

DGPs. The reason why the result still goes through is that a VAR(p) obtained through least-

squares projections perfectly captures the autocovariances of the data out to lag p (but not

further), and these are precisely what determine the LP estimand.11

Proposition 8 assumes xt(p) = xt(p − h) to obtain an exact result, but the conclusion

is likely to hold qualitatively under more general conditions. If xt is a direct measure of

a “shock” and thus uncorrelated with all past data, then xt(`) = xt for all ` ≥ 0, so the

conclusion of the proposition holds exactly. More generally, the LP estimand projects yt+h

onto xt(p) (and controls); thus, the projection depends on the first p+ h autocovariances of

the data. The estimated VAR(p) generally does not precisely capture the autocovariances

of the data at lags p+ 1, . . . , p+ h, and so the LP and VAR potentially project on different

objects. However, at short horizons h� p, it will usually be the case in empirically relevant

DGPs that xt(p) ≈ xt(p − h), since it is typically only the first few lags of the data that is

useful for forecasting xt. In this case, the conclusion of Proposition 8 will hold approximately.

We provide an illustration in Section 3.2.4.

In conclusion, even if we use “too short” a lag length p, the LP and VAR impulse response

estimands only disagree at horizons longer than p. This is a comforting fact in applications

where the main questions of interest revolve around short-horizon impulse responses.

11Baek & Lee (2019) prove a similar result for the related but distinct setting of single-equation Autore-gressive Distributed Lag models with a white noise exogenous regressor.

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3.2.4 Graphical illustration

We finish the section by illustrating graphically the previous theoretical results. We do so

in the context of a particular data generating process: the structural macro model of Smets

& Wouters (2007). We abstract from sampling uncertainty and throughout assume that the

econometrician actually observes an infinite amount of data.12 Since this section is merely

intended to illustrate the properties of different projections, we do not comment on the

relation of the projection estimands to true structural model-implied impulse responses. We

formally discuss structural identification in Section 3.4.

Figure 3.1: Illustration: Population equivalence of VAR and LP estimands

0 2 4 6 8 10 12 14 16

-0.5

0

0.5

1

1.5

0 2 4 6 8 10 12 14 16

-0.5

0

0.5

1

1.5

Note: LP and VAR impulse response estimands in the structural model of Smets & Wouters (2007).Left panel: response of output to a government spending innovation. Right panel: response ofoutput to an interest rate innovation. The horizontal line marks the horizon p after which thefinite-lag-length LP(p) and VAR(p) estimands diverge.

12We use the Dynare replication of Smets & Wouters (2007) kindly provided by Johannes Pfeifer. The codeis available at: https://sites.google.com/site/pfeiferecon/dynare. We truncate the model-impliedvector moving average representation at a large horizon (H = 350), and then invert to obtain a VAR(∞).

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https://sites.google.com/site/pfeiferecon/dynare


The left panel of Figure 3.1 shows LP and VAR impulse response estimands of the

response of output to a government spending innovation. We assume the model’s government

spending innovation is directly observed by the econometrician, who additionally controls

for lags of output and government spending. This experiment is therefore similar in spirit

to that of Ramey (2011). As ensured by Proposition 7, the LP(∞) and VAR(∞) estimands

– i.e., with infinitely many lags as controls – agree at all horizons. Since by assumption the

“impulse” variable xt is a direct measure of the government spending innovation, we have

xt(`) = xt for all ` ≥ 0. Thus, any LP(p) estimand for finite p also agrees with the LP(∞)

limit at all horizons. Finally, we observe that the impulse responses implied by a VAR(4)

exactly agree with the true population projections up until horizon h = 4, as predicted by

Proposition 8.

The right panel of Figure 3.1 shows LP and VAR impulse response estimands for the re-

sponse of output to an innovation in the nominal interest rate. Here the model’s innovation

is not directly observed by the econometrician, only the interest rate. The LP specifications

control for the contemporaneous value of output and inflation as well as lags of output, infla-

tion, and the nominal interest rate; as discussed, this set of control variables is equivalent to

ordering the interest rate last in the VAR. Thus, the experiment emulates the familiar mon-

etary policy shock identification analysis of Christiano et al. (2005), although we, at least for

the purposes of this section, interpret the projections purely in a reduced-form way. Again,

the LP(∞) and VAR(∞) estimands agree at all horizons. Now, however, the “impulse” xt(p)

upon which the different methods project is different. Hence, LP(p) and VAR(p) estimands

differ from each other, as well as from the population limit LP(∞)/VAR(∞) estimands.

Formally, Proposition 8 only assures that the estimated impact impulse responses of LP(p)

and VAR(p) agree exactly. Nevertheless, and consistent with the intuition in Section 3.2.3,

all impulse response estimands are nearly identical until the truncation horizon p = 4.

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3.3 Efficient estimation of impulse responses

This section discusses our equivalence result in the context of finite-sample estimation of

impulse responses. We first provide a sample analogue of our population equivalence result

when the lag length is large. Then we discuss the bias-variance trade-off associated with

estimation of impulse response functions. While we maintain a reduced-form perspective in

this section, in Section 3.4 we will apply the insights to structural estimators.

3.3.1 Sample equivalence

In addition to being identical conceptually and in population, we show in the Online Appendix

that local projection and VAR impulse response estimators are nearly identical in sample

when large lag lengths are used in the regression specifications. Formally, let βh(p) and

θh(p) denote the least-squares estimators of the LP and VAR specifications (3.1)–(3.2) if we

include p lags of the data in the VAR and on the right-hand side of the local projection.

Under standard nonparametric regularity conditions, the sample analogue of the population

equivalence result in Section 3.2.1 holds: There exists a constant of proportionality κ such

that, at any fixed horizon h, the distance |θh(p) − κβh(p)| tends to zero in probability

asymptotically, provided that the lag length p tends to infinity with the sample size at an

appropriate rate. We relegate the details of this result to the Online Appendix.13

3.3.2 Bias-variance trade-off

Empirically relevant short sample sizes force researchers to economize on the number of

lags, and the relative accuracy of LP and VAR estimators with a small/moderate number of

lags invariably depends on the underlying data generating process (DGP). This is perfectly

13The appendix is available at: http://scholar.princeton.edu/mikkelpm/lp_var

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http://scholar.princeton.edu/mikkelpm/lp_var


analogous to the choice between “direct” and “iterated” predictions in multi-step forecasting

(Marcellino et al., 2006; Pesaran et al., 2011). Schorfheide (2005) proves that the mean-

square error ranking of LP (i.e., direct) and VAR (i.e., iterated) forecasts depends on how

large in magnitude the partial autocorrelations of the DGP are at lags longer than the lag

length used for estimation.14 Hence, although Meier (2005), Kilian & Kim (2011), and

Choi & Chudik (2019) exhibit simulation evidence that VAR estimators (or other iterated

estimators) outperform the LP estimator, this conclusion must necessarily depend on the

choice of DGP. Indeed, Brugnolini (2018) and Nakamura & Steinsson (2018b) exhibit DGPs

where the LP estimator instead outperforms VARs.

The forecasting literature has generally found that LP (direct) methods tend to have

relatively low bias, whereas VAR (iterated) methods have relatively low variance. The

trade-off is most relevant at longer response horizons, as shown by our finite-p equivalence

result in Proposition 8. The VAR(p) model extrapolates long-horizon impulse responses from

the autocovariances at lags 0, 1, . . . , p, and thus may potentially be substantially biased if

p is not very large. For the same reason, though, VAR(p) estimators tend to deliver much

smaller estimation variance than LPs at long horizons. Hansen (2010, 2016), Pesaran et al.

(2011), and Kilian & Lutkepohl (2017, ch. 2.6) discuss methods for choosing the lag length

p for VAR and LP estimators in a way that is informed by the bias-variance trade-off.

More generally, effective finite-sample estimation of impulse responses involves an un-

avoidable bias-variance trade-off, and many dimension reduction or penalization approaches

may be sensible depending on the application. Bayesian VARs reduce effective dimensional-

ity by imposing priors on longer-lag coefficients, e.g., through a Minnesota prior (Giannone

et al., 2015); model averaging across restricted and unrestricted VARs has similar effects

(Hansen, 2016). Dimension reduction can also be achieved through penalized local projection

(Plagborg-Møller, 2016, Ch. 3; Barnichon & Brownlees, 2019) or by shrinking unrestricted

14See also Chevillon (2007), McElroy (2015), and references therein.

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local projections towards low-order VAR estimates (Miranda-Agrippino & Ricco, 2018b). Al-

ternatively, impulse response estimation could be based on plugging a shrinkage/regularized

autocovariance function estimate into the explicit formula (3.3) for the LP/VAR estimand.

We believe that the different estimation methods in the literature are best viewed as

sharing the same large-sample estimand but lying along a spectrum of small-sample bias-

variance choices. Low-order VAR(p) models only have a conceptually special status insofar

as we think the finite-p assumption is literally true, which is typically not the case. In

general, the relative accuracy of the methods depends on smoothness/sparsity properties

of the autocovariance function of the data. From the point of view of point estimation,

no single method dominates for all empirically relevant data DGPs. In principle, standard

VAR model diagnostic checks or pseudo-out-of-sample forecast performance can be used

as a means to select between impulse response estimators. However, we recommend that

researchers compare results from different methods, since any disparities may indicate that

further thought about the DGP and/or the shrinkage procedure is warranted.

To summarize: Guided by the previously cited forecasting literature, the choice of es-

timation method should depend on (i) the researcher’s preferences over bias and variance

and on (ii) features of the DGP. In contrast, in the next section we argue that the choice of

structural identification scheme should not determine the choice between LPs and VARs (or

other dimension reduction techniques).

3.4 Structural identification of impulse responses

We now show that our result on the equivalence of LP and VAR impulse response functions

has important implications for structural identification. We have seen that LP and VAR

methods only differ to the extent that they represent different approaches to finite-sample

dimensionality reduction. The problem of structural identification is a population concept

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and thus logically distinct from that of dimensionality reduction. In this section we apply

our equivalence result to popular SVAR and LP identification schemes – including short-

run restrictions, long-run restrictions, sign restrictions, and external instruments – and we

discuss how to think about non-linear models.

3.4.1 Structural model

To discuss structural identification, we now impose a linear but otherwise general semipara-

metric Structural Vector Moving Average (SVMA) model. This model does not restrict

the linear transmission mechanism of shocks to observed variables (we address non-linear

models in Section 3.4.4). SVMA models have been analyzed by Stock & Watson (2018),

Plagborg-Møller & Wolf (2019a), and many others. The class of SVMA models encompasses

all discrete-time, linearized DSGE models as well as all stationary SVAR models.

Assumption 10. The data {wt} are driven by an nε-dimensional vector εt = (ε1,t, . . . , εnε,t)′

of exogenous structural shocks,

wt = µ+ Θ(L)εt, Θ(L) ≡∞∑`=0

Θ`L`, (3.9)

where µ ∈ Rnw×1, Θ` ∈ Rnw×nε, and L is the lag operator. {Θ`}` is assumed to be absolutely

summable, and Θ(x) has full row rank for all complex scalars x on the unit circle. For

notational simplicity, we further assume normality of the shocks:

εti.i.d.∼ N(0, Inε). (3.10)

Under these assumptions wt is a nonsingular, strictly stationary jointly Gaussian time series,

consistent with Assumption 9 in Section 3.2. The (i, j) element Θi,j,` of the nw × nε moving

average coefficient matrix Θ` is the impulse response of variable i to shock j at horizon `.

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The researcher is interested in the propagation of the structural shock ε1,t to the observed

macro aggregate yt. Since yt is the (nr + 2)-th element in wt, the parameters of interest

are Θnr+2,1,h, h = 0, 1, 2, . . . . In line with applied work, we also consider relative impulse

responses Θnr+2,1,h/Θnr+1,1,0. This may be interpreted as the response in yt+h caused by a

shock ε1,t of a magnitude that raises xt by one unit on impact.

3.4.2 Implementing “SVAR” identification using LPs

In this subsection we show that LP methods are as applicable as VAR methods when im-

plementing common identification schemes. Our main result in Section 3.2.1 implies that

LP-based causal estimation can succeed if and only if SVAR-based estimation can succeed.

We will exhibit several concrete and easily implementable examples of this equivalence.

Identification under invertibility. Standard SVAR analysis assumes (partial) invert-

ibility – that is, the ability to recover the structural shock of interest, ε1,t, as a function of

only current and past macro aggregates:

ε1,t ∈ span ({wτ}−∞<τ≤t) . (3.11)

A given SVAR identification scheme then identifies as the candidate structural shock a

particular linear combination of the Wold forecast errors:

ε1,t ≡ b′ut, (3.12)

where the chosen identification scheme gives the vector b as a function of the reduced-form

VAR parameters (A(L),Σu), or equivalently the Wold decomposition parameters (C(L),Σu).

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Under invertibility, there must exist a vector b such that ε1,t = ε1,t, ensuring that SVAR

identification can in principle succeed (Fernandez-Villaverde et al., 2007; Wolf, 2019a).

We now illustrate through three examples that common SVAR identification schemes are

as simple to implement using LP methods. We first consider a standard recursive scheme

covered by our benchmark analysis in Section 3.2.1. The second and third examples involve

long-run and sign restrictions and require the general equivalence result of Section 3.2.2.

Example 1 (Recursive identification). Christiano et al. (2005) identify monetary policy

shocks through a recursive ordering. They assume that their observed data {wt} follow an

invertible SVMA model, i.e. the condition (3.11) holds for all shocks in the system (3.9).

They then additionally impose a temporal ordering on the set of variables wt: Output, con-

sumption, investment, wages, productivity, and the price deflator do not respond within the

period to changes in the policy rate (Federal Funds Rate), which itself in turn does not react

within the period to changes in profits and money growth. In the notation of Section 3.2.1,

the assumed ordering corresponds to the Federal Funds Rate as the impulse variable xt, all

aggregates ordered before the Federal Funds Rate as the controls rt, and all other variables

collected in the vector qt. Christiano et al. implement their structural analysis through the

recursive VAR (3.2). By our main result, they could have equivalently estimated the regres-

sion (3.1) and collected the regression coefficients {βh}h≥0. The population estimand would

have been the same, but in finite samples the mean-square error ranking of the two estimators

is ambiguous, as discussed in Section 3.3.

Example 2 (Long-run identification). Blanchard & Quah (1989) identify the effects of de-

mand and supply shocks using long-run restrictions in a bivariate system. Let gdpt and unr t

denote log real GDP (in levels) and the unemployment rate, respectively. Then ∆gdpt ≡

gdpt−gdpt−1 is log GDP growth. Blanchard & Quah impose that wt ≡ (∆gdpt, unr t)′ follows

the SVMA model in Assumption 10 with nε = 2 shocks, where the first shock is a supply

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shock, the second shock a demand shock, and both shocks are invertible, cf. (3.11). They

then additionally impose the identifying restriction that the long-run effect of the demand

shock on the level of output is zero, i.e.,∑∞

`=0 Θ1,2,` = 0.

While Blanchard & Quah impose their long-run restriction on a SVAR model to estimate

impulse responses, the extended equivalence result in Section 3.2.2 implies that the same

restriction can be equivalently implemented using an LP approach. To see how, consider, for

a large horizon H, the “long difference” projection

gdpt+H − gdpt−1 = µH + β′Hwt +∞∑`=1

δ′H,`wt−` + ξH,t. (3.13)

Intuitively, this projection uncovers the linear combination of the data that best explains long-

run movements in GDP. By assumption, such explanatory power can only come from the

supply shock. Thus, to estimate impulse responses with respect to the supply shock, we can

run the local projection (3.8) with b = βH and with yt given by the response variable of interest

(either ∆gdpt or unr t). Indeed, we show formally in Appendix C.2 that, as H → ∞, this

procedure correctly identifies the impulse responses Θi,1,h with respect to the supply shock, up

to a constant scale factor. In this way, relative impulse responses Θi,1,h/Θ1,1,0 are correctly

identified.15 To estimate relative impulse responses Θi,2,h/Θ1,2,0 to the demand shock, the

researcher can choose any vector b such that b′b = 0, and then implement the local projection

(3.8) with b in lieu of b.

In finite samples, the mean-square error performance of the proposed procedure relative

to the conventional SVAR(p) approach of Blanchard & Quah (1989) will depend on the

tuning parameters H and p, and on whether the low-frequency properties of the data are well

approximated by a low-order VAR model.16 For researchers who prioritize bias over variance,

15Absolute impulse responses can be identified by rescaling the identified shock so it has variance 1.16Christiano et al. (2006) and Mertens (2012) make the related point that SVAR-based long-run identifi-

cation need not rely on the VAR-implied long-run variance matrix. Alternative nonparametric estimators ofthe latter may have attractive bias-variance properties, depending on the true DGP.

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the LP approach to long-run restrictions has the advantage that it does not extrapolate long-

run impulse responses from short-run autocorrelations, as a VAR does.

Example 3 (Sign identification). Uhlig (2005) set-identifies the effects of monetary policy

shocks by sign-restricting impulse responses. For concreteness, suppose we are interested in

the impulse response of yt (say, real GDP growth) to a monetary shock at horizon h. As

before, assume that the full set of observed data {wt} follows an SVMA system (3.9) where

all shocks are invertible. As a very simple example of sign restrictions, we may impose the

identifying restriction that the scalar variable rt (say, the nominal interest rate) responds

positively to a monetary shock at all horizons s = 0, 1, . . . , H.

The traditional SVAR approach to sign identification proceeds as follows. By invertibility,

the monetary shock ε1,t is related to the Wold forecast errors ut through ε1,t = ν ′ut, where

ν ∈ Rnw is an unknown vector. If we knew ν, the structural impulse responses of any variable

wi,t to ε1,t could be obtained as the linear combination ν of the reduced-form impulse responses

of wi,t from a VAR in wt. To impose the sign restrictions, we search over all possible vectors

ν such that (i) the rt impulse responses are positive at all horizons s = 0, 1, . . . , H and (ii)

the impact rt impulse response is normalized to 1 (other normalizations are also possible).

Once we have determined the set of possible ν’s, we can then use the VAR to compute the

corresponding set of possible impulse responses of yt with respect to ν ′ut.

By the logic in Section 3.2.2, we can alternatively impose sign restrictions using an LP

approach. We simply estimate the reduced-form impulse responses using LPs instead of a

VAR. Consider the coefficient vector βh obtained from the projection

yt+h = µh + β′hwt +∞∑`=1

δ′h,`wt−` + ξh,t.

The above LP yields the reduced-form impulse responses βh of yt to the Wold forecast errors

ut. Exactly as in the VAR approach, we now seek the linear combination ν ′βh that equals

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the structural impulse response to the monetary shock ε1,t = ν ′ut. To find the set of ν’s

consistent with the sign restrictions, the natural analogue of the VAR approach is as follows.

For each horizon s = 0, 1, . . . , H, store the coefficient vector βs from the projection

rt+s = µs + β′swt +∞∑`=1

δ′s,`wt−` + ξs,t.

The coefficients βs measure the reduced-form impulse responses of rt to ut, so sign restrictions

on the structural impulse responses of rt amount to linear inequality restrictions on these

coefficients. Consequently, the largest possible response of yt+h to a monetary shock that

raises rt by one unit on impact can be obtained as the solution to the linear program17

supν∈Rnw

ν ′βh subject to β′0ν = 1,

β′sν ≥ 0, s = 1, . . . , H.

To compute the smallest possible impulse response, replace the supremum with an infimum.18

In population, this LP-based procedure recovers exactly the same identified set as analogous

sign restrictions in an SVAR. It is straight-forward to implement more complicated identifi-

cation schemes by adding additional equality or inequality constraints of the above type.

These three examples demonstrate that invertibility-based identification need not be

thought of as “SVAR identification,” contrary to standard practice in textbooks and parts

of the literature. As a matter of identification, the two methods succeed or fail together.

Ideally, researchers ought to decide on the identification scheme separately from how they

decide on the finite-sample dimension reduction technique. The former choice should be

17To consider impulse responses to a one-standard-deviation monetary shock, replace the equality con-straint in the linear program by the constraint ν′Var(ut)

−1ν = 1. The resulting linear-quadratic programwith inequality constraints is similar to those in Gafarov et al. (2018) and Giacomini & Kitagawa (2018).

18We focus on computing the bounds of the identified set. An alternative approach is to sample from theidentified set, as is standard in the Bayesian SVAR literature (Rubio-Ramırez et al., 2010).

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based on economic theory. The latter choice should be based on the researcher’s preferences

over bias and variance as well as on features of the DGP, as discussed in Section 3.3.

Beyond invertibility. If the invertibility assumption (3.11) is violated, then identifica-

tion strategies that erroneously assume invertibility – independent of whether they are imple-

mented using VARs, LPs, or any other dimensionality reduction technique – will not measure

the true impulse responses.19 Instead, these methods will measure the impulse responses to a

white noise disturbance that is a linear combination of current and lagged structural shocks:

ε1,t = ϑ(L)εt. (3.14)

The properties of the lag polynomial ϑ(L) are characterized in detail in Fernandez-Villaverde

et al. (2007) and Wolf (2019a). Combining (3.9) and (3.14), we see that, in general, both

LP and VAR impulse response estimands are linear combinations of contemporaneous and

lagged true impulse responses. Thus, projection on a given identified impulse ε1,t correctly

identifies impulse response functions (up to scale) if and only if ε1,t affects the response

variable yt only through the contemporaneous true structural shock ε1,t. Trivially, this is the

case if ε1,t is a function only of ε1,t (the invertible case); less obviously, the same is also true

if ε1,t is only contaminated by shocks that do not directly affect the response variable yt.20

Instrumental variable identification, discussed in the next section, is the leading example of

this second case.

19Several recent papers have demonstrated how to perform valid semi-structural identification withoutassuming invertibility, cf. the references in Plagborg-Møller & Wolf (2019a). Often such methods rely on LPor VAR techniques to compute relevant linear projections, without interpreting the VAR disturbances (i.e.,Wold innovations) as linear combinations of the contemporaneous true shocks.

20In particular, this means that neither invertibility nor recoverability (as defined in Plagborg-Møller &Wolf, 2019a) are necessary for successful semi-structural inference on impulse response functions.

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3.4.3 Identification and estimation with instruments

Instruments (also known as proxy variables) are popular in semi-structural analysis. We here

use our main result in Section 3.2 to show that the influential Local Projection Instrumental

Variable estimation procedure is equivalent to estimating a VAR with the instrument ordered

first, irrespective of the underlying structural model.

An instrumental variable (IV) is defined as an observed variable zt that is contempora-

neously correlated only with the shock of interest ε1,t, but not with other shocks that affect

the macro aggregate yt of interest (Stock, 2008; Stock & Watson, 2012; Mertens & Ravn,

2013).21 More precisely, given Assumption 10, the IV exclusion restrictions are that

Cov(zt, εj,s | {zτ , wτ}−∞<τ<t) 6= 0 if and only if both j = 1 and t = s. (3.15)

Stock & Watson (2018, p. 926) refer to this assumption as “LP-IV⊥,” and it is routinely

made in theoretical and applied work, as reviewed by Ramey (2016) and Stock & Watson

(2018). The assumption requires that, once we control for all lagged data, the instrument is

not contaminated by other structural shocks or by lags of the shock of interest.

Without loss of generality, we can use projection notation to phrase the IV exclusion

restrictions (3.15) as follows.

Assumption 11.

zt = cz +∞∑`=1

(Ψ`zt−` + Λ`wt−`) + αε1,t + vt, (3.16)

where α 6= 0, cz,Ψ` ∈ R, Λ` ∈ R1×nw , vti.i.d.∼ N(0, σ2

v), and vt is independent of εt at all leads

and lags. The lag polynomial 1 −∑∞

`=1 Ψ`L` is assumed to have all roots outside the unit

circle, and {Λ`}` is absolutely summable.

21We focus on the case of a single IV. If multiple IVs for the same shock are available, Plagborg-Møller &Wolf (2019a) show that (i) the model is testable, and (ii) all the identifying power of the IVs is preserved bycollapsing them to a certain (single) linear combination.

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Crucially, the assumption allows the IV to be contaminated by the independent measurement

error vt. In some applications, we may know by construction of the IV that the lag coefficients

Ψ` and Λ` are all zero (so zt satisfies assumption “LP-IV” of Stock & Watson, 2018, p. 924,

without controls); obviously, such additional information will not present any difficulties for

any of the arguments that follow.

The Local Projection Instrumental Variable (LP-IV) approach estimates the impulse

responses to the first shock using a two-stage least squares version of LP. Loosely, Mertens

(2015), Jorda et al. (2015, 2019), Leduc & Wilson (2017), Ramey & Zubairy (2018), and

Stock & Watson (2018) propose to estimate the LP equation (3.1) using zt as an IV for xt.

To describe the two-stage least-squares estimand in detail, define Wt ≡ (zt, w′t)′ and consider

the “reduced-form” IV projection

yt+h = µRF ,h + βRF ,hzt +∞∑`=1

δ′RF ,h,`Wt−` + ξRF ,h,t (3.17)

for any h ≥ 0. Consider also the “first-stage” IV projection22

xt = µFS + βFSzt +∞∑`=1

δ′FS ,`Wt−` + ξFS ,t. (3.18)

Notice that the first stage does not depend on the horizon h. As in standard cross-sectional

two-stage least-squares estimation, the LP-IV estimand is then given by the ratio βLPIV ,h ≡

βRF ,h/βFS of reduced-form to first-stage coefficients (e.g. Angrist & Pischke, 2009, p. 122).23

Stock & Watson (2018) show that, under Assumptions 10 and 11, the LP-IV estimand

βLPIV ,h correctly identifies the relative impulse response Θnr+2,1,h/Θnr+1,1,0. Importantly,

this holds whether or not the shock of interest ε1,t is invertible in the sense of (3.11).

22As always, the coefficients and residuals in (3.17)–(3.18) should be interpreted as linear projections.23In the over-identified case with multiple IVs, the IV estimand can no longer be written as this simple

ratio; we focus on a single IV as in most of the applied literature.

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We now use our main result from Section 3.2.1 to show that the LP-IV impulse responses

can equivalently be estimated from a recursive VAR that orders the IV first. As in Section 3.2,

this result is nonparametric and assumes nothing about the underlying structural model or

about the IV zt.

Corollary 1. Let Assumption 9 hold for the expanded data vector Wt ≡ (zt, w′t)′ in place of

wt. Assume also that βFS 6= 0, cf. (3.18). Consider a recursively ordered SVAR(∞) in the

variables (zt, w′t)′, where the instrument is ordered first (the ordering of the other variables

does not matter). Let θy,h be the SVAR-implied impulse response at horizon h of yt with

respect to the first shock. Let θx,0 be the SVAR-implied impact impulse response of xt with

respect to the first shock.

Then θy,h/θx,0 = βLPIV ,h.

Proof. Let zt ≡ αε1,t + vt and a ≡√E(z2

t ) =√α2 + σ2

v . Proposition 7 states that θy,h =

a× βRF ,h for all h, and θx,0 = a× βFS . The claim follows.

This nonparametric result implies that, given the structural Assumptions 10 and 11, valid

identification of relative structural impulse responses can equivalently be achieved through

LP-IV or through an “internal instrument” recursive SVAR with the IV ordered first.24

Importantly, under Assumptions 10 and 11, these equivalent estimation strategies are valid

even when the shock of interest ε1,t is not invertible (Stock & Watson, 2018). Intuitively,

although adding the IV zt to the VAR does not render the shock ε1,t invertible, the only

reason that the shock may be non-invertible with respect to the expanded information set

{zτ , wτ}−∞<τ≤t is the presence of the measurement error vt in the IV equation (3.16).25 But

24Plagborg-Møller & Wolf (2019a) show that point identification of absolute impulse responses – and thusvariance decompositions – can be achieved under a further recoverability assumption that is mathematicallyand substantively weaker than assuming invertibility.

25Note that, even though Assumption 11 allows zt to be correlated with lags of wt, non-invertibility of ε1,tis entirely consistent with Theorem 1 of Stock & Watson (2018). That theorem states that if the shock isnon-invertible, then it is possible to construct an example of an IV zt satisfying E(ztεj,t) = 0 for all j 6= 1and E(ztεj,t−` | {wτ}τ<t) 6= 0 for some j and ` ≥ 1 (so zt does not satisfy Assumption 11).

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this measurement error merely leads to attenuation bias in the estimated impulse responses,

and the bias (in percentage terms) is the same at all response horizons and for all response

variables. Thus, it does not contaminate estimation of relative impulse responses.

IV identification is therefore an example of a setting where SVAR analysis works even

though invertibility fails (including the partial invertibility notion of Forni et al., 2019, and

Miranda-Agrippino & Ricco, 2018a). The “internal instrument” recursive SVAR(∞) pro-

cedure estimates the right relative impulse responses despite the fact that no invertible

structural VAR model generally exists under our assumptions. Our result implies that it is

valid to include an externally identified shock in a SVAR even if the shock is measured with

(independent) error, as long as the noisily measured shock is ordered first.26

Unlike the non-invertibility-robust procedure of ordering the IV first in a VAR, the pop-

ular SVAR-IV (also known as proxy-SVAR) procedure (Stock, 2008; Stock & Watson, 2012;

Mertens & Ravn, 2013) is only valid under invertibility. This procedure uses an SVAR to

identify the shock of interest as

ε1,t ≡1√

Var(z†t )× z†t ,

where z†t is computed as a linear combination of the reduced-form residuals ut from a VAR

in wt alone (i.e., excluding the IV from the VAR):

z†t ≡ E(zt | ut) = E(zt | {wτ}−∞<τ≤t).

If Assumptions 10 and 11 and the invertibility condition (3.11) hold, then SVAR-IV is

valid. In fact, in this case SVAR-IV removes any attenuation bias, thus correctly identifying

26Romer & Romer (2004) and Barakchian & Crowe (2013) include an externally identified monetary shockin a SVAR, but they order it last, which assumes additional exclusion restrictions. Kilian (2006), Ramey(2011), Miranda-Agrippino (2017), and Jarocinski & Karadi (2019), among others, mention the strategy ofordering an IV first in a SVAR, but these papers do not consider the non-invertible case.

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absolute (not just relative) impulse responses.27 However, in the general non-invertible case,

SVAR-IV mis-identifies the shock as ε1,t 6= ε1,t.28 Plagborg-Møller & Wolf (2019a, Appendix

B.4) characterize the bias of SVAR-IV under non-invertibility and show that the invertibility

assumption can be tested using the IV.

To summarize, the relative impulse responses obtained from the LP-IV procedure of Stock

& Watson (2018) are nonparametrically identical to the relative impulse responses from a

recursive SVAR with the IV ordered first (an “internal instrument” approach). Assuming an

SVMA model and the IV exclusion restrictions, these procedures correctly identify relative

structural impulse responses, irrespective of the invertibility of the shock of interest. This

allows researchers to exploit VAR estimation techniques – with their associated bias-variance

properties discussed in Section 3.3 – while relying on the same invertibility-robust identifying

restrictions as the popular two-stage least squares implementation of LP-IV. In contrast, the

SVAR-IV procedure of Stock & Watson (2012) and Mertens & Ravn (2013) (an “external

instrument” approach) requires invertibility.29

3.4.4 Estimands in non-linear models

Our main result in Section 3.2.1 implies that linear local projections are exactly as “ro-

bust to non-linearities” as VAR methods, in population. We now show that the common

LP/VAR estimand can be given a mathematically well-defined “best linear approximation”

interpretation when the true underlying structural DGP is in fact non-linear.

27Consistent with our analytical results, Carriero et al. (2015) observe in a calibrated simulation study that,under invertibility, SVAR-IV correctly identifies absolute impulse response functions, while direct projectionson the IV suffer from attenuation bias.

28The VARX approach of Paul (2018) is equivalent with SVAR-IV under Assumption 9.29SVAR-IV does have one advantage over LP-IV (and thus also over the “internal instruments” VAR

approach): Provided the shock is invertible, SVAR-IV does not require zt to only be correlated with laggedshocks through observed lagged variables as in Assumption 11, cf. Stock & Watson (2018, sec. 2.1).

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Assume that the structural DGP has the nonparametric causal structure

wt = g(εt, εt−1, εt−2, . . . ), (3.19)

where g(·) is any non-linear function that yields a well-defined covariance stationary process

{wt}, and {εt} is an nε-dimensional i.i.d. process with Cov(εt) = Inε . The number of

structural shocks εt may exceed the number of variables in wt.

We show formally in Appendix C.3 that we can represent the process (3.19) as the linear

Structural Vector Moving Average model

wt = µ∗ +∞∑`=0

Θ∗`εt−` +∞∑`=0

Ψ∗`ζt−`,

where ζt is an nw-dimensional white noise process that is uncorrelated at all leads and lags

with the structural shocks εt. The argument exploits the Wold decomposition of the residual

of wt after projecting on the structural shocks. Hence, the linear SVMA model (3.9) in

Assumption 10 should not be thought of as restrictive, provided we do not restrict the

number of “shocks” relative to the number of variables.

The linear SVMA impulse responses Θ∗` corresponding to the structural shocks εt have a

“best linear approximation” interpretation. Specifically,

(Θ∗0,Θ∗1, . . . ) ∈ argmin

(Θ0,Θ1,... )

E

[(g(εt, εt−1, . . . )−

∑∞`=0 Θ`εt−`

)2]. (3.20)

Thus, if a second-moment LP/VAR identification scheme is known to correctly identify the

impulse responses in a linear SVMA model (3.9), and there is doubt about whether the true

underlying DGP is in fact linear, the population estimand of the identification procedure

can be given a formal “best linear approximation” interpretation. This is analogous to the

“best linear predictor” property of Ordinary Least Squares in cross-sectional regression. In

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contrast, identification approaches that depart from standard linear projections – such as

identification through higher moments or through heteroskedasticity – may not have a clear

interpretation under functional form misspecification.

Of course, in some applications, the non-linearities of the true underlying DGP may be of

interest per se. In such cases, non-linear VAR or LP estimators can be applied, for example

by adding interaction or polynomial terms, regime switching, stochastic volatility, etc. Such

issues are outside the scope of this paper, which deals exclusively with linear estimators.

3.5 Empirical application

We finally illustrate our theoretical equivalence results by empirically estimating the dynamic

response of corporate bond spreads to a monetary policy shock. We adopt the specification

of Gertler & Karadi (2015), who, using high-frequency financial data, obtain an external

instrument for monetary policy shocks.30 Because of possible non-invertibility (Ramey, 2016;

Plagborg-Møller & Wolf, 2019a), we do not consider the external SVAR-IV estimator, but

instead implement direct projections on the IV through (i) local projections and (ii) an

“internal instrument” recursive VAR, following the logic of Corollary 1. In both cases, our

vector of macro control variables exactly follows Gertler & Karadi (2015); it includes output

growth (log growth rate of industrial production), inflation (log growth rate of CPI inflation),

the 1-year government bond rate, and the Excess Bond Premium of Gilchrist & Zakrajsek

(2012) as a measure of the non-default-related corporate bond spread. The data is monthly

and spans January 1990 to June 2012.31

30The external IV zt is constructed from changes in 3-month-ahead futures prices written on the FederalFunds Rate, where the changes are measured over short time windows around Federal Open Market Com-mittee monetary policy announcement times. See Gertler & Karadi (2015) for details on the construction ofthe IV and a discussion of the exclusion restriction.

31The data were retrieved from: https://www.aeaweb.org/articles?id=10.1257/mac.20130329

122

https://www.aeaweb.org/articles?id=10.1257/mac.20130329


Figure 3.2: Response of bond spread to monetary shock: VAR and LP estimates

0 5 10 15 20

-1

-0.5

0

0.5

1

1.5

2

2.5

0 5 10 15 20

-1

-0.5

0

0.5

1

1.5

2

2.5

Note: Estimated impulse response function of the Excess Bond Premium to a monetary policyshock, normalized to increase the 1-year bond rate by 100 basis points on impact. Left panel: laglength p = 4. Right panel: p = 12. The horizontal line marks the horizon p after which the VAR(p)and LP(p) estimates may diverge substantially.

Figure 3.2 shows that LP-IV and “internal instrument” VAR impulse response estimates

agree at short horizons, but diverge at longer horizons, consistent with Proposition 8. The

figure shows point estimates of the response of the Excess Bond Premium to the monetary

policy shock, for different projection techniques and different lag lengths. For all specifica-

tions, the Excess Bond Premium initially increases after a contractionary monetary policy

shock, consistent with the results in Gertler & Karadi (2015). The left panel shows results for

LP(4) and VAR(4) estimates. Up until horizon h = 4, the estimated impulse responses are

closely aligned. At longer horizons, the iterated VAR structure enforces a smooth return to

0, while direct local projections give more erratic impulse responses. The right panel shows

an analogous picture for LP(12) and VAR(12) estimates: The estimated impulse responses

agree closely until horizon h = 12, but they diverge at longer horizons.

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These results provide a concrete empirical illustration of our earlier claim that LP and

VAR estimates are closely tied together at short horizons, not just in population but also in

sample. The larger the lag length used for estimation, the more impulse response horizons

will exhibit agreement between LP and VAR estimates. As this exercise is merely meant to

illustrate our theoretical results, we refrain from conducting formal statistical tests of the

relative finite-sample efficiency of the different estimation methods.

3.6 Conclusion

We demonstrated a general nonparametric equivalence of local projection and VAR impulse

response function estimands. This result has several implications for empirical practice:

1. VAR and local projection estimators of impulse responses should not be regarded as

conceptually distinct methods – in population, they estimate the same thing, as long

as we control flexibly for lagged data.

2. Efficient finite-sample estimation requires navigating a bias-variance trade-off. Low-

order VAR and local projection estimators resolve this trade-off differently, and several

other recently proposed methods also lie on the continuum of possible dimension re-

duction approaches. Neither low-order VARs nor low-order local projections should be

treated as having special status generally.

3. The bias-variance trade-off is equivalent to the well-known trade-off between direct and

iterated forecasts. Thus, the finite-sample mean-square error ranking of different im-

pulse response estimation methods depends on smoothness/sparsity properties of the

autocovariance function of the data. The forecasting literature offers extensive guid-

ance on the bias-variance trade-off (see references in Section 3.3). No single estimation

method dominates for all empirically relevant data generating processes.

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4. At short impulse response horizons, the various estimation methods are likely to ap-

proximately agree, but at longer horizons the bias-variance trade-off is unavoidable.

A VAR estimator with large lag length will give similar results as a local projection,

except at very long horizons.

5. It is a useful diagnostic to check if different estimation methods reach similar con-

clusions. If estimated impulse responses from VARs and local projections differ sub-

stantially at longer horizons, it must mean that the sample partial autocorrelations

at long lags are not small. This possibly calls into question the validity of the VAR

approximation to the distribution of the data, depending on the standard errors.

6. Structural identification is logically distinct from the dimension reduction choices that

must be made for estimation purposes. It may be counterproductive to follow stan-

dard practice in assuming a finite-order SVAR model whenever the discussion turns

to structural identification, as this conflates the population identification analysis and

the dimension reduction technique of using a low-order VAR estimator.

7. Any structural estimation method that works for SVARs can be implemented with local

projections, and vice versa. For example, if a paper already relies on local projections

for parts of the analysis, then an additional sign restriction identification exercise, say,

can also be implemented in a local projection fashion.

8. If an instrument/proxy for the shock of interest is available, structural impulse re-

sponses can be consistently estimated by ordering the instrument first in a recursive

VAR (an “internal instrument” approach), even if the shock of interest is non-invertible.

In contrast, the popular SVAR-IV estimator (an “external instrument” approach) is

only consistent under invertibility.

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9. Linear local projections are exactly as “robust to non-linearities” in the underlying

data generating process as linear VARs.

We stress that this paper has focused entirely on identification and estimation of impulse

responses using linear methods. Identification of other objects, such as variance or historical

decompositions, is more involved, as for example shown in Plagborg-Møller & Wolf (2019a).

Our work points to several promising areas for future research. First, it would be use-

ful to adapt the results in the present paper to non-linear estimators, such as regressions

with interactions or polynomial terms. Second, future research could consider data with

near-unit roots or cointegration. Third, we only discussed the population properties of IV

estimators, and thus ignored weak IV issues. Fourth, it would be interesting to generalize

our LP-IV equivalence result to settings with multiple instruments/proxies. Finally, we have

deliberately avoided questions related to inference.

126

Appendix A

Appendix for Chapter 1

A.1 Model details

This appendix provides additional details on the rich class of structural models underlying

my exact demand equivalence results. In Appendix A.1.1 I outline the full model and

offer a formal definition of equilibrium transition paths. Appendix A.1.2 then discusses the

particular parametric model variants used for illustration of approximate equivalence results.

A.1.1 The benchmark model

A.1.1.1 Full model outline

Recall that the model is populated by households, firms, and the government. Whenever

there is no risk of confusion, I replace the full decision problems of agents by simple con-

ditions characterizing their actual optimal behavior. I do so because many of the problems

considered here (in particular for price-setting entities) are notationally involved, but at the

same time extremely well-known and so require no repetition.

127

A. Appendix for Chapter 1

Households. The household consumption-savings problem was described in detail in Sec-

tion 1.2.1. For the general estimated HANK model of Section 1.4, the only change is that I

allow for a borrowing wedge; that is, the liquid interest rate satisfies

ib(bh) =

ib if bh ≥ 0

ib + κb if bh < 0

It remains to specify the problem of a wage-setting union k. A union sets wages and labor

to maximize weighted average utility of its members, taking as given optimal consumption-

savings behavior of each individual member household, exactly as in Auclert et al. (2018).

Following the same steps as those authors, it can be shown that optimal union behavior is

summarized by a standard non-linear wage-NKPC:1

πwt (1 + πwt ) =εwθw`ht

[ ∫ 1

0

{− u`(cit, cit−1, `

ht )−

εw − 1

εw(1− τ`)wteit{uc(cit, cit−1, `

ht )

+βEt[uc−1(cit+1, cit, `ht+1)]}

}di]

+ βπwt+1(1 + πwt+1) (A.1)

where 1 + πwt = wtwt−1× 1

1+πt, εw is the elasticity of substitution between different kinds of

labor, and θw denotes the Rotemberg adjustment cost. Given prices (πππ,w) as well as a

consumption path c, (A.1) provides a simple restriction on total labor supply `h.2 Note

that, without idiosyncratic labor productivity risk and so common consumption cit = ct, the

derived wage-NKPC (A.1) is to first order identical to the standard specification in Erceg

et al. (2000). An extension to partially indexed wages, as in Smets & Wouters (2007) or

Justiniano et al. (2010), is straightforward and omitted in the interest of notational simplicity.

1For notational simplicity, in the derivation of this wage-NKPC assume that βi = β for all i. Thegeneralization to heterogeneous βi’s is conceptually straightforward, but notationally cumbersome.

2For θw →∞, equation (A.1) is vacuous, so then I instead simply assume that `h = `f .

128


Together, the consumption-savings problem and (A.1) characterize optimal household

and union behavior. I assume that the solutions to each problem exist and are unique, and

summarize the solution in terms of aggregate consumption, saving and union labor supply

functions c(sh, εεε), bh(sh, εεε), and `h(su), where sh = (ib,πππ,w, `, τττ e,d) and su = (πππ,w, c). In

particular, the union problem gives

ˆPEε ≡ `h(πππ, w, c(sh;εεε))− ¯h

For my theoretical equivalence results, I will impose the high-level assumption that all of

those infinite-dimensional vector functions are at least once differentiable in their arguments.

Firms. I first study the problem of each of the three types of firms in isolation. I assume

that all firms discount at the common rate 1 + rbt ≡1+ibt−1

1+πt.3

1. Intermediate Goods Producers. The problem of intermediate goods producer j is to

max{dIjt,yjt,`jt,kjt,ijt,ujt,b

fjt}

E0

[∞∑t=0

(t−1∏q=0

1

1 + rbq

)dIjt

]

such that

dIjt = pIt yjt − wt`jt︸︷︷︸πjt

−ξjt × 1ijt 6=0 − (1− 1ijt<0 × ϕ)ijt − φ(kjt, kjt−1, ijt, ijt−1)

−bfjt +1 + ibt−1

1 + πtbfjt−1

yjt = y(ejt, ujtkjt−1, `jt)

ijt = kjt − [1− δ(ujt)]kjt−1

3Along a perfect foresight transition path, discounting at 1 + rbt is equivalent to discounting at the(common) stochastic discount factor of all households with strictly positive asset holdings. If firm savingand borrowing in the liquid asset is not constrained, then this choice of discount rate is needed to preventarbitrarily large desired saving or borrowing.

129


−bfjt ≤ Γ(kjt−1, kjt, πjt)

dIjt ≥ d

Adjustment costs have a convex and continuously differentiable part φ, a firm-specific

fixed adjustment cost ξjt (distributed with cdf F (ξ) over support R+), and may feature

partial irreversibility, with ϕ ∈ [0, 1]. Firms can vary capital utilization, with higher

utilization leading to faster depreciation, i.e. δ′(•) > 0. The solution to the firm problem

gives optimal production y(•), labor demand `f (•), investment i(•), intermediate goods

producer dividends dI(•), capital utilization rates u(•) and liquid corporate bond savings

bf (•) as a function of nominal returns ib, inflation πππ, wages w, and the intermediate

goods price pI .

2. Retailers. A unit continuum of retailers purchases the intermediate good at price pIt ,

costlessly differentiates it, and sells it on to a final goods aggregator. Price setting is

subject to a Rotemberg adjustment cost. As usual, optimal retailer behavior gives rise

to a standard NKPC as a joint restriction on the paths of inflation and the intermediate

goods price. In log-linearized form:

ˆπt =εpθp

εp − 1

εp︸︷︷︸κp

× ˆpIt + β ˆπt+1

where εp denotes the substitutability between different kinds of retail goods, and θp de-

notes the Rotemberg adjustment cost. In an equivalent (to first-order) Calvo formulation,

the slope of the NKPC instead is given as

κp =(1− 1

1+rφp)(1− φp)φp

130


where 1− φp is the probability of a price re-set. A further extension to partially indexed

prices, as in Smets & Wouters (2007) or Justiniano et al. (2010), is straightforward and

omitted in the interest of notational simplicity. Total dividend payments of retailers are

dRt = (1− pIt )yt

3. Aggregators. Aggregators purchase retail goods and aggregate them to the composite final

good. They make zero profits.

Total dividend payments by the corporate sector are given as

dt = dIt + dRt

With some algebra, it is straightforward to show that in fact

dt = yt − wt`t − it

Using the restriction on the intermediate goods price implied by optimal retailer behavior,

aggregate dividends can be obtained solely as a function of sf = (ib,w,πππ).

We can now summarize the aggregate firm sector simply through a set of optimal produc-

tion, labor hiring, investment, dividend pay-out and bond demand functions, y = y(sf ;εεε),

`f = `f (sf ;εεε), i = i(sf ;εεε), d = d(sf ;εεε) and bf = bf (sf ;εεε), as well as a restriction on the

aggregate path of inflation, πππ = πππ(sf ;εεε), where sf = (ib,πππ,w). As before, I will assume that

these aggregate firm sector-level functions are at least once differentiable in their arguments.

Government. The fiscal authority was discussed in detail in Section 1.2.1, with an example

of a concrete fiscal rule given in Appendix A.1.2.2. It remains to describe central bank

behavior. In line with standard empirical practice I assume that the nominal rate on bonds

131


ib is set according to the conventional Taylor rule

îbt = ρmîbt−1 + (1− ρm)

(φπ ˆπt + φy ˆyt + φdy ˆyt−1

)

A generalization to feature a notion of potential output, as in Justiniano et al. (2010), is

straightforward.

Market-Clearing. Equating liquid asset demand from households and intermediate goods

producers, as well as liquid asset supply from the government, we get

bht + bft = bt

Equating labor demand and supply:

`ft = `ht

Finally, aggregating all household, firm and government budget constraints, we obtain the

aggregate output market-clearing condition4

ct + it + gt = yt

A.1.1.2 Equilibrium definition

All results in this paper rely on the following equilibrium definition.

Definition 2. Given initial distributions µh0 = µh and µf0 = µf of households and intermedi-

ate goods producers over their idiosyncratic state spaces, an initial real wage w−1 = w, price

4So as to not excessively clutter market-clearing conditions with various adjustment cost terms, I assumethat adjustment costs are ex-post rebated lump-sum back to the agents facing the adjustment costs. Ofcourse, all subsequent equivalence results are unaffected by this rebating. An alternative interpretation isthat adjustment costs are perceived utility costs, as in Auclert et al. (2018).

132


level p−1, and real government debt b−1 = b, as well as exogenous shock paths {εt}∞t=0, a recur-

sive competitive equilibrium is a sequence of aggregate quantities {ct, `ht , `ft , b

ht , b

ft , bt, yt, it, dt,

kt, gt, τt}∞t=0 and prices {πt, ibt , wt}∞t=0 such that:

1. Household Optimization. Given prices and government rebates, the paths of aggregate

consumption c = c(sh;εεε), labor supply `h = `h(su), and asset holdings bh = bh(sh;εεε) are

consistent with optimal household and wage union behavior.

2. Firm Optimization. Given prices, the paths of aggregate production y = y(sf ;εεε), invest-

ment i = i(sf ;εεε), capital k, labor demand `f = `f (sf ;εεε), dividends d = d(sf ;εεε) and asset

holdings bf = bf (sf ;εεε) are consistent with optimal firm behavior. Furthermore, the path

of inflation is consistent with optimal retailer behavior.

3. Government. The liquid nominal rate is set in accordance with the monetary authority’s

Taylor rule. The government spending, rebate, and debt issuance paths are jointly consis-

tent with the government’s budget constraint, its exogenous laws of motion for spending

and discretionary rebates, and its financing rule.

4. Market Clearing. The goods market clears,

ct + it + gt = yt

the bond market clears,

bht + bft = bt

and the labor market clears,

`ht = `ft

for all t = 0, 1, 2, . . ..

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A.1.2 Parametric special cases

The quantitative illustrations and accuracy checks in Sections 1.2 and 1.4 largely rely on two

particular structural models: the simple spender-saver RBC model and the estimated HANK

model. This section provides further details on both models and discusses my preferred

parameterizations.

A.1.2.1 The spender-saver RBC model

The simple model is a special case of the rich benchmark model of Section 1.2.1. For

convenience, I here explicitly state the equations characterizing the model equilibrium.

Model Sketch. A mass λ ∈ (0, 1) of households are spenders, indexed by h. They in-

elastically supply labor and receive lump-sum transfers τht; since βh = 0, their consumption

satisfies

cht = wt ¯+ τht (A.2)

The residual fraction of households are savers, indexed by r. Since there are no adjustment

costs or portfolio restrictions, I can characterize the consumption-savings problem of savers

as a simple one-asset problem with exogenous dividend receipts:

max{crt,bt}

∞∑t=0

βt log(crt)

such that

crt + bt = (1 + rt)bt−1 + dt + wt ¯+ τrt

Optimal saver behavior is characterized by the Euler equation

c−1rt = β(1 + rt+1)c−1

rt+1 (A.3)

134


A single representative firm chooses investment to maximize the present value of dividend

payments to savers, discounted at the real rate faced by savers (and so their stochastic

discount factor, to first order). Its problem is

max{dt,yt,kt,`t,it}

∞∑t=0

(t−1∏s=0

1

1 + rs

)dt

such that

dt = yt − it − wt`t (A.4)

yt = kαt−1`1−αt (A.5)

it = kt (A.6)

where the final relation uses full depreciation of the capital good. From now on I use that,

in equilibrium, `t = ¯. Optimal firm investment is characterized by the relation

1 + rt = αkα−1t

¯1−α (A.7)

and wages satisfy

wt = (1− α)kαt−1¯−α (A.8)

The government consumes an exogenously determined amount of the final good,

gt = εgt (A.9)

and sets rebates to spenders as

τht = τh +1

λετt (A.10)

135


The scaling factor 1λ

is chosen to ensure that the amount of stimulus rebate given to spenders

overall is independent of the mass of spenders in the economy. Expenditure is financed fully

through contemporaneous lump-sum taxation on savers, so

λτht + (1− λ)τrt + gt = 0 (A.11)

bt = 0 (A.12)

Note that, in the notation of Section 1.2.1, τht = 1λτxt and τrt = 1

1−λ τet. Finally, aggregate

market-clearing dictates that

yt = ct + it + gt (A.13)

where

ct = λcht + (1− λ)crt (A.14)

A recursive competitive equilibrium for aggregate prices and quantities {ct, cht, crt, rt, wt, dt,

yt, kt, it, gt, τht, τrt, bt} is fully characterized by the relations (A.2) - (A.14).

Without loss of generality, and to simplify the algebra, I normalize

¯=[(αβ)

α1−α − (αβ)

11−α

]−1

which ensures that c = 1 and y = 1/(1 − αβ). I furthermore, and also for notational

simplicity, assume that steady-state rebates τh are such that steady-state consumption of

spenders and savers are equalized.

Log-linear Solution. It is straightforward to characterize the (log-linear) solution of this

model in closed form. Log-linearizing (A.13), and using (A.9), we get

(1− αβ)ˆct + αβ ˆkt +1

yεgt = αˆkt−1 (A.15)

136


Log-linearizing (A.3) and plugging into (A.7), we get

ˆcrt+1 − ˆcrt = (α− 1)ˆkt (A.16)

Expressing saver consumption in terms of aggregate and spender consumption using (A.14),

solving for spender consumption in terms of the rebate shock and capital using (A.2) and

(A.8), and plugging into (A.16), we get

1

1− λˆct+1 −

λ

1− λ(αˆkt +

1

λετt+1)− 1

1− λˆct +

λ

1− λ(αˆkt−1 +

1

λετt) = (α− 1)ˆkt (A.17)

All other equilibrium objects are immediately determined from the remaining equilibrium

relations, so the equilibrium is fully characterized by (A.15) and (A.17). Plugging (A.15)

into (A.17) to eliminate consumption, we get the single equation

1

1− λ

[α

1− αβˆkt −

αβ

1− αβˆkt+1 − εgt+1

]− λ

1− λ

(αˆkt +

1

λετt+1

)− 1

1− λ

[α

1− αβˆkt−1 −

αβ

1− αβˆkt − εgt

]+

λ

1− λ

(αˆkt−1 +

1

λετt

)= (α− 1)ˆkt (A.18)

I solve the model exploiting the well-known equivalence between perfect foresight and first-

order perturbation solutions. I thus treat (A.18) as a second-order expectational difference

equation (replacing all variables dated t+ 1 by their expectation), and find its unique stable

solution. To this end conjecture that

ˆkt = θkˆkt−1 + ωd(εgt + ετt) (A.19)

137


Plugging in and matching coefficients, we find that the guess is confirmed,5 with

θk = α

ωd = − 1− αβ1− λ(1− αβ)

Plugging this back into (A.15), we get

ˆct = αˆkt−1 +αβ

1− λ(1− αβ)× (εgt + ετt)− εgt (A.20)

as claimed.

Parameterization. For the graphical illustration in Figure 1.1 I use standard parameter

values: β = 0.99, α = 1/3 and λ = 0.3.

A.1.2.2 The estimated HANK model

The analysis in Section 1.4 builds on a rich estimated one-asset HANK model, featuring a

consumption-savings problem under imperfect insurance embedded into an otherwise stan-

dard medium-scale DSGE environment. This section provides details on the model, the

solution algorithm, my approach to likelihood-based estimation, and the final parameteriza-

tion used to generate the results in Section 1.4 (as well as the simpler check in Section 1.2.4).

Model Outline. The model is an extension of the rich baseline environment outlined

in Section 1.2.1, violating Assumption 2 (households and government borrow at different

interest rates) and Assumption 3 (strong wealth effects and imperfectly rigid wages).

5It is straightforward to verify existence and uniqueness of the equilibrium following the arguments inBlanchard & Kahn (1980) or Sims (2000).

138


Households have separable preferences over consumption and labor,

u(c, `) =c1−γ − 1

1− γ− χ `

1+ 1ϕ

1 + 1ϕ

,

and discount the future at rate β. The log-linearized wage-NKPC then takes the form

ˆπwt = κw ×[

1

ϕˆt − ( ˆwt − γ ˆc∗t )

]+ β ˆπwt+1 (A.21)

where κw is a function of model parameters and c∗t satisfies

c∗t ≡[∫ 1

0

eitc−γit di

]− 1γ

(A.22)

Results are unchanged if I instead use the average marginal utility of aggregate consumption

−γ ˆct in the union wage target (as in Hagedorn et al., 2019). I furthermore slightly generalize

the model of Section 1.2.1 to allow for stochastic death with probability ξ. All households

receive identical lump-sum transfers τt but are heterogeneous in dividend payment receipts.

In particular, I assume that the model is populated by different illiquid wealth “types”, who

each receive an exogenous (and time-invariant) endowment of illiquid shares.

The intermediate goods production block – in particular the production function y(•),

the investment adjustment cost function φ(•), and the capacity utilization depreciation rate

δ(•) – is set up exactly as in Justiniano et al. (2010). Relative to the model outlined in

Section 1.2.1, I then add structural shocks to output and investment productivity, monetary

policy, price mark-ups and wage mark-ups to complement the already included impatience

and government spending shocks. All shocks are modeled as in Justiniano et al. (2010), so

I omit details. Finally, for purposes of the model estimation, I assume that

τet = −(1− ρτ )× bt−1 (A.23)

139


The endogenous part of transfers is cut in response to increases in bt. For plots of approximate

equivalence results, I let transfer shocks be financed using this rule, and then assume that

government spending shocks are financed using the same (potentially scaled) intertemporal

tax profile, consistent with Assumption 2. The partial equilibrium financing paths of the two

shocks will thus always be multiples of each other; without a borrowing wedge, they would

be identical, at least in partial equilibrium. Since households spend most of the rebate

immediately, results are very similar if I instead simply use the rule (A.23) for all shocks.

Steady-State Calibration. Solving for the deterministic steady-state of the model re-

quires specification of several parameters. On the household side, I need to set income risk

and share endowment processes, specify preferences, and choose liquid borrowing limits as

well as the substitutability between different kinds of labor. On the firm side, I need to spec-

ify production and investment technologies, as well as the substitutability between different

kinds of goods. Finally, on the government side, I need to set taxes, transfers, and total

bond supply. Government spending is then backed out residually. My preferred parameter

values and associated calibration targets are displayed in Table A.1.

The first block shows parameter choices on the household side. For income risk, I adopt

the 33-state specification of Kaplan et al. (2018), ported to discrete time. For share en-

dowment, I split the illiquid wealth distribution from the 2016 SCF into four bins (< 15,

15−50, 50−85, and > 85 percentiles), and then exactly match wealth in the four bins κw by

allowing for four permanent illiquid wealth types with mass pd. I set the average return on

(liquid) assets in line with standard calibrations of business-cycle models. The discount and

death rates are then disciplined through targets on the total amount of liquid wealth as well

as average household age. Households can borrow up to one time average quarterly labor

earnings (which in turn are normalized to 1), and the borrowing wedge is set to discipline

the fraction of households with negative liquid wealth. All remaining parameters are set

140


Steady-State Parameter Values, HANK Model

Parameter Description Value Target Model Data

Households

ρe, σe Income Risk - Kaplan et al. - -

κd, pd Div. Endowment - Ill. Wealth Shares - -

β Discount Rate 0.97 B/Y 1.04 1.04

rb Average Return 0.01 Annual Rate 0.04 0.04

ξ Death Rate 1/180 Average Age 45 45

γ Preference Curvature 1 Standard

ϕ Labor Supply Ela. 1 Standard

εw Labor Subs. 10 Standard

b Borrowing Limit -1 Kaplan et al.

κb Borr. Wedge (yr) 0.06 Fraction b < 0 0.15 0.15

Firms

α Capital Share 0.2 Justiniano et al.

δ Depreciation 0.016 Total Wealth/Y 10.64 10.64

εp Goods Subs. 16.67 Profit Share 0.06 0.06

Government

τ` Labor Tax 0.3 Avg. Labor Tax 0.30 0.30

τ/Y Transfer Share 0.05 Transfer Share 0.05 0.05

B/Y Liquid Wealth Supply 1.04 Gov’t Debt/Y 1.04 1.04

Table A.1: HANK model, steady-state calibration.

in line with conventional practice. The second block shows parameter choices on the firm

side. I discipline the Cobb-Douglas production function y = kα`1−α by setting α in line

with Justiniano et al. (2010), identify goods substitutability by targeting the profit share,

and finally back out the depreciation rate from my target of total wealth (and so corporate

sector valuation) in the economy as a whole.6 The third block informs the fiscal side of the

6More conventional higher values of α change impulse responses, but do not break demand equivalence.Similarly, the results also remain accurate with the low value of α entertained in Auclert & Rognlie (2018).

141


model. The average government tax take, transfers, and debt issuance are all set in line with

direct empirical evidence.

Importantly, with household self-insurance severely limited, the average MPC in the

economy is high, around 30% out of an unexpected 500$ income gain. As a result, the model

can replicate the large (yet gradual) empirically observed consumption response to income

tax rebates, as argued previously in Auclert et al. (2018).

Dynamics: Computational Details. I solve the model using a variant of the popular

Reiter method (Reiter, 2009). In particular, I use a discrete-time variant of the methods

developed in Ahn et al. (2017) to reduce the dimensionality of the state space. Without

dimensionality reduction, the number of idiosyncratic household-level states is too large to

allow likelihood-based estimation. With dimensionality reduction, the number of states is

reduced to around 300, making estimation feasible.

Dynamics: Estimation. With two exceptions, I estimate the remaining model parame-

ters (which exclusively govern dynamics around the deterministic steady state) using stan-

dard likelihood methods, as in An & Schorfheide (2007). The estimation procedure then

sticks as closely as possible to Justiniano et al. (2010): I consider the same set of macro ob-

servables (over the same time period), and impose identical priors whenever possible.7 As a

result, the estimation exercise does not really take advantage of the additional opportunities

afforded by micro data; instead, it is merely a slightly more disciplined approach to arrive

at a plausible parameterization for the non-household block of the model.

The first exception is the transfer adjustment parameter ρτ ; since I do not include data

on government debt, this parameter would likely be poorly identified. I thus simply set ρτ =

0.85, in line with the VAR evidence documented in Galı et al. (2007) and Appendix A.2.3.

7For the discussion of data construction I thus refer the interested reader to their appendix. I thankBrian Livingston for help in assembling the data.

142


Second, as it is central to my approximate equivalence results, I directly discipline the degree

of wage stickiness from micro data. Exploiting the standard first-order equivalence of Calvo

price re-sets and Rotemberg adjustment costs, it is easy to show that the slope parameter

of the wage-NKPC (A.21) can be equivalently written as

κw =(1− 1

1+rφw)(1− φw)

φw(εw1ϕ

+ 1)

where 1− φw is the probability of wage adjustment in the quarter. I set the wage stickiness

parameter consistent with the micro evidence in Grigsby et al. (2019) and Beraja et al. (2019),

giving φw = 0.6 – price re-sets every 2.5 quarters. Direct estimation of this parameter would

instead suggest a much larger value, consistent with the findings of Justiniano et al. (2010)

and other estimated New Keynesian models.

The results of the estimation are displayed in Table A.2. Since they are not relevant for

my purposes here, I omit estimates of shock persistence and volatility; some brief remarks

on those follow at the end. I find the posterior mode using the csminwel routine provided

by Chris Sims; for accuracy of the demand equivalence approximation beyond the mode

parameterization of the model, see the discussion in Appendix A.4.1.7.8

On the whole, the results are quite consistent with the parameter estimates in Justiniano

et al. (2010). Relative to their rich framework, the two central changes in my model are,

first, the introduction of uninsurable income risk, and second, the absence of habit formation.

The first change ties consumption and income more closely together, while the second leads

to less endogenous persistence and worsens the Barro-King puzzle (Barro & King, 1984).

Jointly, these changes dampen the importance of impatience shocks as a driving force of

consumption fluctuations, but also give a somewhat smaller role for investment efficiency

shocks as a source of cyclical fluctuations. These findings are consistent with the intuition in

8The optimization routine is available at http://sims.princeton.edu/yftp/optimize/.

143

http://sims.princeton.edu/yftp/optimize/


Dynamics Parameter Values, HANK Model

Prior Posterior

Parameter Description Density Mean Std Mode

φp Price Calvo Parameter B 0.66 0.10 0.87

ζ Capacity Utilization G 5.00 1.00 3.71

κ Investment Adjustment Cost G 4.00 1.00 2.45

ρm Taylor Rule Persistence B 0.60 0.20 0.86

φπ Taylor Rule Inflation N 1.70 0.30 2.08

φy Taylor Rule Output N 0.13 0.05 0.08

φdy Taylor Rule Output Growth N 0.13 0.03 0.06

Table A.2: HANK model, parameters governing dynamics, estimated using conventionallikelihood-based methods. For the priors, N stands for Normal, B for Beta and G for Gamma.

Werning (2016) and the estimation results on the no-habit model in Justiniano et al. (2010).

Ultimately, given the similarity in model environment and data sources, the similarity of

the resulting parameter estimates should not come as a surprise. A more serious estimation

exercise on the effects of micro heterogeneity on macro fluctuations would also leverage the

advantages afforded by time series of richer micro data, and is left for future work.

Simplified model. The simplified HANK model considered for the accuracy check in

Section 1.2.4 is identical to the estimated model except for one change: I set κb = 0 and b = 0.

As a result, households and government face identical interest rates, and any inaccuracy in

my approximations is exclusively due to short-run wealth effects in labor supply.

144


A.2 Empirical appendix

This appendix provides additional details on the empirical results needed to implement my

two-step methodology. Appendix A.2.1 discusses estimates of the direct partial equilibrium

consumption response to income tax rebates, Appendix A.2.2 does the same for investment

tax credit, and Appendix A.2.3 offers supplemental information on the VAR-based identifi-

cation of government spending shocks. Finally, in Appendix A.2.4, I briefly discuss how to

account for joint estimation uncertainty in micro and macro estimators.

A.2.1 Direct response: micro consumption elasticities

Proposition 3 shows that, with truly exogenous cross-sectional heterogeneity in shock expo-

sure, micro difference-in-differences regressions estimate direct partial equilibrium responses.

In the empirical analysis of Johnson et al. (2006) and Parker et al. (2013), matters are slightly

more subtle – all households are exposed to the shock, but exposure differs over time for

exogenous reasons. Building on Kaplan & Violante (2014), this appendix discusses how to

interpret their regression estimands. Parker et al. estimate a differenced version of (1.14):

∆cit = time fixed effects + controls + β0ESPit + β1ESPit−1 + uit (A.24)

where ESPit is the dollar amount of the rebate receipt at time t. To establish that the regres-

sion estimands are interpretable as MPC0,0 and MPC1,0 −MPC0,0, respectively, consider

again the structural model of Section 1.2.1, and suppose – roughly in line with the actual

policy experiment (see Kaplan & Violante, 2014) – that a randomly selected fraction ω of

households receive a lump-sum rebate at t = 0 (ετi0 = 1), and that the remaining households

145


receive the same rebate at t = 1 (ετi1 = 1). The model analogue of regression (A.24) is then

∆cit = δ∆t + β0ετit + β1ετit−1 + uit, t = 0, 1 (A.25)

Now suppose additionally that receipt of the rebate is a surprise for all households; in par-

ticular, it is a surprise at t = 1 for households who receive the delayed check.9 We can then

follow exactly the same steps as in the proof of Proposition 3 to show that, to first order,

β0 = MPC0,0

β1 = MPC1,0 −MPC0,0

Of course, as emphasized by Kaplan & Violante (2014), it may be dubious to assume

that the delayed check was a surprise to all households. If instead the delayed check was

perfectly anticipated, then the regression estimands are β0 = MPC0,0 −MPC0,1 and β1 =

MPC1,0−MPC1,1, where MPCt,1 ≡∫ 1

0∂cit∂τ1di is the response of consumption at t to a rebate

received at t = 1, but anticipated at t = 0. Encouragingly, at least in my estimated HANK

model, MPC0,0 and MPC1,1 are quite similar, and MPC0,1 is relatively small (similar to

Auclert et al. (2018)). Thus, even if the rebate was partially anticipated, the approximation

underlying my estimate of the direct response in Figure 1.4 is likely to be accurate.

My analysis in Section 1.3.2 relies on the estimates of Parker et al. (2013). Since their

lagged spending estimates are not significant, I base my direct spending path on the signif-

icant impact spending response in their Table 3, consistent with the headline presentation

of their results in the introduction. My conclusions are, however, quite similar if the impact

and delayed spending responses are evaluated at the point estimates in their Table 5.

9However, note that I still assume that the aggregate perfect foresight transition path is perfectly antic-ipated by all households; in that case, aggregate general equilibrium feedback is differenced out. I discussbelow what happens if the transition path and all individual rebates are anticipated by households.

146


A.2.2 Direct response: micro investment elasticities

Koby & Wolf (2020) generalize the static analysis of Zwick & Mahon (2017) and estimate

dynamic projection regressions of the form

ijt+h = αj + δt + βqh × zn(j),t + ujt (A.26)

where zn(j),t is the size of the bonus depreciation investment stimulus for industry n(j) of firm

j. We estimate this regression on a quarterly Compustat sample from spanning the years

1993–2017; the sample period in particular features the two bonus depreciation episodes

of 2001-2004 and 2008-2010, exactly as in Zwick & Mahon (2017). We then give sufficient

conditions under which the estimands {βqs} are interpretable as the direct partial equilibrium

response of investment to a one-time bonus depreciation stimulus. I briefly repeat the main

insights here.

First, firms must not be subject to financial frictions. In the presence of financial fric-

tions, the indicator zn(j),t does not remain a sufficient statistic summarizing the effects of a

given bonus depreciation policy. Second, all meaningful capital adjustment costs must be

internal to the firm, while the aggregate supply of capital goods must be perfectly flexible.10

Reassuringly, this assumption is consistent with the findings in House & Shapiro (2008),

Edgerton (2010) and House et al. (2017), who all conclude that the supply of new capital

goods is very elastic. Third, general equilibrium feedback associated with the investment

demand stimulus should not co-vary with exposure to the stimulus itself. This assumption

is satisfied if low- and high-depreciation firms do not systematically vary in their cyclicality

(also see the discussion Zwick & Mahon, 2017).

10This restriction is necessary because (A.26) differences out aggregate capital price effects.

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Given the estimated partial path {iPEqt }3t=0, I recover the full partial equilibrium invest-

ment response by fitting a single Gaussian basis function, exactly as in Barnichon & Matthes

(2018). It then remains to construct the corresponding output path yPEq , which requires pa-

rameter choices (α, ν, δ). I have experimented with a wide range of parameter values, and

found result to be robust; for example, with α = 0.3 and ν = 0.8 (a set-up closer to standard

heterogeneous-firm model calibrations), the investment counterfactual barely changes.

A.2.3 The missing intercept: VAR estimation

My analysis of the transmission of transitory government spending shocks closely follows

the important contributions of Perotti (2007) and Ramey (2011), both in terms of data

and in terms of model specification. I construct the government forecasts errors exactly as

Ramey (2011). I then treat these forecast errors as a valid external instrument for structural

government spending shocks, as formalized in Assumption 4. Following Plagborg-Møller

& Wolf (2019b), I study their transmission by ordering them first in a recursive VAR. In

addition to the forecast error variable, the VAR contains measures of government spending,

consumption, output, investment, taxes, and hours worked; in an expanded version, I also

include total government debt.

Data. My benchmark VAR consists of the log real per capita quantities of total govern-

ment spending, total output (GDP), total (non-durable, durable and services) consumption,

private fixed investment, total hours worked, a measure of the federal average marginal tax

rate (Alexander & Seater, 2009),11 and a measure of private business compensation. All

variables are defined and measured as in Ramey (2011). To study the effects of the spending

shock on debt issuance, I construct a log per capita measure of total federal debt, deflating

11The tax measure of Barro & Redlick (2011) includes state income taxes; given my focus on federalexpenditure, I regard the Alexander & Seater series as more suitable for my purposes.

148


the nominal debt series in the St. Louis Fed’s FRED database (data series: GFDEBTN).

In a further robustness check, I replace professional forecaster errors with Greenbook defense

spending forecast errors, closely following the analysis in Drautzburg (2016).12

Estimation Details. I specify both the benchmark and the extended VAR in levels, with

a quadratic time trend and four lags. The lag length selection is informed by standard

information criteria, and is also consistent with the recommendation of Ramey (2016) in the

postscript to her handbook chapter. For estimation of the model, I use a uniform-normal-

inverse-Wishart distribution over the orthogonal reduced-form parameterization (Arias et al.,

2018). Throughout, I display confidence bands constructed through 10,000 draws from the

model’s posterior.

Benchmark Results. Figure A.1 shows the impulse responses of government spending,

output, consumption, investment and taxes in the benchmark VAR, as well as the impulse

response of total government debt in the expanded VAR.

As in most existing structural VAR work, I construct 16th and 84th percentile confidence

bands; the output and debt responses, however, remain significant at the more conventional

95 per cent level. In line with most of the previous literature I find a significant positive

output response (corresponding to around a unit multiplier), and a flat impulse response for

consumption. Total debt rises immediately and significantly, suggesting that the government

spending expansion is debt-financed. In fact, I also find a delayed and persistent increase in

labor income taxes. However, the tax response is somewhat sensitive to details of the model

specification, and sometimes not significant.

12For demand matching I need to re-scale public and private demand shocks to be in common dollar(and not percentage) terms. This is easily done using information on the GDP shares of consumption,investment, and government consumption plus investment. I take those data from FRED, and then simplycompute averages for the different shares across the VAR sample period.

149


Figure A.1: Benchmark Government Spending Shock, VAR IRFs

Note: Impulse responses after a one standard deviation innovation to the forecast error. The dashedlines correspond to 16th and 84th percentile confidence bands, constructed using 10,000 draws fromthe posterior distribution of the reduced-form VAR parameters.

Robustness. My central results – the 1-1 increase in output, the limited crowding-out

of private expenditure, and the persistent rise in debt – are robust to various changes in

model specification. First, I have experimented with different sub-samples. Starting earlier

(1971Q1) means that I need to link forecasts on real federal spending (available after 1981) to

earlier forecasts of military spending, as in Ramey (2011). Depending on the set of included

controls, the undershooting of consumption and investment is, in this earlier sample, usually

more pronounced (similar to Ramey, 2011). However, the undershooting then goes hand-

in-hand with a similar undershooting of spending itself, invalidating the required demand

matching.13 Continuing the sample to 2016Q4 means that I need to stop controlling for

13Note, however, that – unlike the impact co-movement of fiscal spending and output – the dynamicunder-shooting of consumption and output is not statistically significant at the 95 per cent level. It is alsosomewhat dependent on the set of controls; for example, with most controls dropped, I instead find (againlargely insignificant) over-shooting.

150


taxes, as my available measures only continue until 2009. Results in this expanded sample

suggest that crowding-in is slightly stronger, consistent with standard intuition on zero lower

bound constraints. The results are, however, not particularly robust, similar to the findings

in Ramey & Zubairy (2018) and Debortoli et al. (2019).14 Second, replacing my benchmark

measure of government spending forecast errors with Greenbook defense spending forecast

errors leaves my results almost completely unchanged. This suggests that either (i) the

benchmark VAR itself is largely picking up the response to military spending forecast errors

or (ii) multipliers are invariant to the spending type (similar to Gechert (2015)). Third,

removing individual controls in the benchmark specification does not materially impact the

results. This is consistent with the intuition in Plagborg-Møller & Wolf (2019b) – my es-

timands are projection coefficients which do not depend on the set of included controls, at

least in terms of population estimands. Fourth, changes in the number of lags do not affect

the overall flavor of my results. And fifth, frequentist inference (or a flat prior) gives almost

identical impulse response estimates to those displayed here.

Alternative identification. Following Blanchard & Perotti (2002), I consider a second

approach to the analysis of government spending shock propagation. I estimate the same

benchmark VAR as before, but now consider the dynamic propagation of an innovation

to the equation for government spending gt itself, rather than for its forecast error. This

identification scheme is identical to the original approach of Blanchard & Perotti (2002),

except for the fact that I now control implicitly for past government spending forecast errors.

Similar to Caldara & Kamps (2017), I find that this alternative identification scheme iden-

tifies a government spending shock with a more persistent response of government spending

14I have also allowed the aggregate effects of spending shocks to be heterogeneous across expansions andrecessions, through a local projection implementation identical to Ramey & Zubairy (2018). Similar to thoseauthors, I find no evidence of such state dependence.

151


itself. Qualitatively, the responses of other macroeconomic aggregates – in particular output,

consumption and investment – look similar to those for my benchmark identification.15

Importantly, because both sets of impulse responses are identified in the same reduced-

form VAR, I can easily account for joint uncertainty by drawing from the posterior of that

reduced-form VAR, rotating forecast residuals in line with either my benchmark or the

Blanchard-Perotti identification scheme, and then finding the best fit to net demand paths

following (1.23). Detailed impulse response plots for this alternative identification scheme

are available upon request.

A.2.4 Joint Uncertainty

I throughout ignore estimation uncertainty for microeconomic difference-in-differences es-

timators. This approach is in line with standard empirical practice, which largely takes

microeconomic point estimates of household MPCs and investment price elasticities at face

value (e.g. Kaplan & Violante, 2014; Auclert et al., 2018; Koby & Wolf, 2020). In princi-

ple, however, it is straightforward to account for joint estimation uncertainty: Under my

identifying assumptions, microeconomic and macroeconomic estimation uncertainty are in-

dependent, so sampling uncertainty for the micro and macro estimators is independent. Joint

standard errors can thus be straightforwardly constructed from the individual standard errors

of the micro and macro estimators.

I only provide a sketch of the argument here. To ease the notional burden, I consider a

simple static model; the generalization to the dynamic case is conceptually straightforward,

15Also similar to Caldara & Kamps (2017), I find that additionally controlling for professional forecasterrors has quite limited effects. Even in a pure Blanchard-Perotti recursive VAR (without the error control)I find an approximately unit fiscal spending multiplier and little response of private spending.

152


but more notationally involved. I assume that consumption of household i satisfies

cit = βτετt + βgεgt + σvνt︸︷︷︸macro shocks

+ βPEτ ξit + σζζit︸︷︷︸micro shocks

where the macro shocks are distributed (ετt, εgt, νt)′ iid∼ N(0, I3), and the micro shocks are

distributed (ξit, ζit)′ iid∼ N(0, I2), independently of the macro shocks and any individual char-

acteristics. It is then straightforward to see that a simple OLS estimator for (1.14) satisfies

βPEτ =1N

∑Ni=1(cit − cit)(ετit − ετit)1N

∑Ni=1(ετit − ετit)2

= βPEτ +1N

∑Ni=1(ξit − ξit)σζ(ζit − ζit)

1N

∑Ni=1(ξit − ξit)2

where the averages are taken over the N households i. Crucially, any macroeconomic uncer-

tainty is differenced out. Similarly, by the projection arguments in Plagborg-Møller & Wolf

(2019b), a recursive VAR estimator gives

βg =1T

∑Tt=1(ct − ct)(εgt − εgt)

1T

∑Tt=1(εgt − εgt)2

= βg +1T

∑Tt=1 βτ (εgt − εgt)(ετt − ετt)

1T


+1T

∑Tt=1 σν(νt − νt)(ετt − ετt)

1T


where now all averages are taken over time t, in a total sample with length T . Importantly,

by the proof of Proposition 3, micro shocks have no aggregate effects, so all estimation

uncertainty for βg is driven by macroeconomic uncertainty.

Since sampling uncertainty for the micro and macro estimators is independent, their

covariance is zero, so construction of joint uncertainty bands is conceptually trivial. Under

less stringent assumptions, it is always possible to construct conservative bounds using the

methods developed in Cocci & Plagborg-Møller (2019).

153


A.3 Proofs and auxiliary lemmas

A.3.1 Proof of Lemma 1

From the specification of the household and firm problems in Appendix A.1.2.1, it is im-

mediate that there exist differentiable functions c(r,w,d, τττ e;εεε), y(r), i(r) and d(r) that

fully characterize optimal firm and household behavior. But by (A.8) and (A.4) we can also

obtain w = w(r), so the expression (1.9) is well-defined.

Next, since g = g(εεε) by (A.9), we can conclude that (1.9) is necessary for any perfect

foresight transition equilibrium. Since τττ e = τττ e(εεε) by (A.11) it is similarly immediate that

(1.10) is necessary. To show sufficiency, note that (A.2), (A.3) as well as (A.4), (A.5) and

(A.7) hold by optimal household and firm behavior, respectively, and that all other equations

simply residually determine remaining model variables. Thus, if an interest rate path r and

a saver transfer path τττ e are such that (1.9) and (1.10) hold, then they are in fact part of a

perfect foresight equilibrium. By existence and uniqueness of the perturbation solution (see

Appendix A.1.2.1), and by equivalence of perfect foresight transition paths and perturbation

solutions (Boppart et al., 2018), we know that this transition path exists and is unique.

A.3.2 Proof of Proposition 1

By differentiability of the consumption, investment and output supply functions, a perfect

foresight equilibrium is, to first order, a solution to the linear system of equations

∂c

∂εεε× εεε+

∂c

∂r× r +

∂c

∂τττ e× τττ e +

∂g

∂εεε× εεε =

(∂y

∂r− ∂i

∂r

)× r (A.27)

τττ e =∂τττ e∂εεε× εεε (A.28)

154


The existence of a unique perturbation solution (see the discussion in Appendix A.1.2.1) in

conjunction with Lemma 1 implies that this equation also has a unique bounded solution for

(r, τττ r). Thus there exists a unique linear map H such that

r

τττ e

= H×

∂c∂εεε× εεε+ ∂g

∂εεε× εεε

∂τττe∂εεε× εεε

(A.29)

where H is the left inverse of ∂y∂r− ∂i

∂r− ∂c

∂r− ∂c∂τττr

0 ∂τττr∂εεε

Since there exists a unique bounded solution, this left inverse is unique. Thus, in response

to a generic shock εεεs, the response path of consumption satisfies

cε =∂c

∂εεε× εεε︸︷︷︸

cPEε

+

(∂c∂r

∂c∂τeτeτe

)×H×

∂c∂εεε× εεε+ ∂g

∂εεε× εεε

∂τττe∂εεε× εεε

︸︷︷︸

≡D×( ∂c∂εεε×εεε+∂g∂εεε×εεε)

(A.30)

The definition of the “demand multiplier” map D uses my assumptions on the government

financing rule – both policy experiments can be (and in fact are) financed using identical

paths of lump-sum saver taxes.16 General equilibrium feedback is thus identical, giving (1.5),

and (1.6) follows.

A.3.3 Auxiliary Lemma for Proposition 2

Lemma 2. Consider the structural model of Section 1.2.1. A perfect foresight equilibrium

is a sequence of nominal interest rates {ibt}t≥0, aggregate output {yt}t≥0, wages {wt}t≥0 and

16Of course, given Ricardian equivalence, this does not really matter. It only matters that the presentvalue of the implied tax burdens on savers is the same, which is ensured by the intertemporal governmentbudget constraint (and since government and savers borrow and save at a common rate).

155


the endogenous part of tax rebates {τet}t≥0 such that

c(sh(x);εεε) + i(sf (x);εεε) + g(εεε) = y(sf (x);εεε)

`h(su(x;εεε)) = `f (sf (x);εεε)

y(sf (x);εεε) = y

τττ e(sf (x;εεε);εεε) = τττ e

where the consumption, production, investment, labor supply and labor demand functions

c(•), y(•), i(•), `h(•) and `f (•) are derived from optimal firm, household and union behavior,

and xt = (ibt , yt, wt, τet).

To prove Lemma 2 I proceed in two steps. First, I show that all relevant inputs to the

household and firm problems can be obtained as functions only of x and εεε. Second, I show

sufficiency of the four equations in the statement of the result.

1. Given (ib,y), the Taylor rule of the monetary authority allows us to back out the path of

inflation πππ. Thus all inputs to the firm problem are known,17 so indeed sf = sf (x). We

thus obtain y, i and `f . Setting ` = `f and since τττ e ∈ x, all inputs to the household prob-

lem are known, so indeed sh = sh(x). We can thus also solve for the path of consumption,

so that indeed su = su(x;εεε), and we finally recover union labor supply.

2. Optimal household, firm and government behavior is assured by assumption. It thus re-

mains to check that (i) all markets clear (ii) that the input path of output is consistent

with firm production, and (iii) that the rebate path is consistent with the government

budget constraint. Output and labor market-clearing are ensured by the first two equa-

tions in the statement of the lemma, and asset market-clearing then follows from Walras’

17Note that the path of the intermediate goods price pI is obtained from the problem of retailers.

156


law. The third set of equations in the lemma statement then ensures consistency in ag-

gregate production, while the fourth set – which uses that the only relevant quantities

for the government budget constraint are (r,w, `) – ensures that the government budget

constraint holds period-by-period.

Together, 1. - 2. establish sufficiency of the conditions in the statement of Lemma 2.

Necessity is immediate, completing the argument.


By Lemma 2, a perfect foresight equilibrium is, to first order, a solution to the system of

linear equations

(∂c

∂x× x +

∂c

∂εεε× εεε)

+

(∂i

∂x× x +

∂i

∂εεε× εεε)

+∂g


(∂y

∂x× x +

∂y

∂εεε× εεε)

(∂`h

∂x× x +

∂`h

∂εεε× εεε)

=

(∂`f

∂x× x +

∂`f

∂εεε× εεε)

(∂y

∂x× x +

∂y

∂εεε× εεε)

= J2 × x(∂τττ e∂x× x +

∂τττ e∂εεε× εεε)

= J4 × x

where Ji denotes the infinite-dimensional generalization of the selection matrix selecting the

ith entry of a vector xt. Assuming equilibrium existence and uniqueness,18 there exists a

18Existence and uniqueness of a bounded transition path for representative-agent models can be shown asusual. For the heterogeneous-agent models, I have verified existence and uniqueness for particular numericalexamples, using the conditions of Blanchard & Kahn (1980).

157


unique linear map H such that

x = H︸︷︷︸GE adjustment

×

∂c∂εεε

+ ∂i∂εεε

+ ∂g∂εεε− ∂y

∂εεε

∂`h

∂εεε− ∂`f

∂εεε

∂y∂εεε

∂τττe∂εεε

× εεε

︸︷︷︸direct shock response

where H is a left inverse of

∂y∂x− ∂c

∂x− ∂i

∂x

∂`f

∂x− ∂`h

∂x

J2 − ∂y∂x

J4 − ∂τττe∂x

The assumed existence and uniqueness of the equilibrium ensures that this left inverse is

unique. Now consider consumption demand and government spending shocks. To reduce

unnecessary clutter, I use the notation ∂∂εεεs

(rather than the generic ∂∂εεε

) to denote derivatives

for a shock path where only entries of shock s are non-zero. By the arguments in the proof

of Lemma 2, we know that ∂i∂εεεd

= ∂y∂εεεd

= ∂`f

∂εεεd= 0, and similarly that ∂i

∂εεεg= ∂y

∂εεεg= ∂`f

∂εεεg= 0. We

also know that ∂`h

∂εεεg= 0. I now distinguish two cases.

(i) Suppose that Assumption 3 holds. Then we can also conclude that ∂`h

∂εεεd= 0. The two

direct shock responses are then

∂c∂εεεd

0

0

∂τττe∂εεεd

× εεεd =

cPEd

0

0

τττPEed

158


and

∂g∂εεεg

0

0

∂τττe∂εεεg

× εεεg =

gg

0

0

τττPEeg

By Assumption 2, we know that there exists a matrix T such that τττPEed = T × cPEd ,

τττPEeg = T × gg, and τττPEed = τττPEeg if cPEd = gg. Thus, in response to consumption demand

and government spending shocks, the response path of consumption satisfies

cd =∂c

∂εεεd× εεεd︸︷︷︸

cPEd

+∂c

∂x×H×

cPEd

0

0

τττPEed

︸︷︷︸

D×cPEd

and

cg = 0 +∂c

∂x×H×

gg

0

0

τττPEeg

︸︷︷︸

D×gg

respectively, where D is a common demand multiplier. This establishes that

cGEτ = cGEg

and so (1.13) follows from simple addition.

159


(ii) Without Assumption 3, the two direct shock responses are

∂c∂εεεd

∂`h

∂εεεd

0

∂τττe∂εεεd

× εεεd =

cPEd

ˆPEd

0

τττPEed

and

∂g∂εεεg

0

0

∂τττe∂εεεg

× εεεg =

gg

0

0

τττPEeg

The response paths of consumption now satisfy

cd =∂c


cPEd

+∂c

∂x×H×

cPEd

ˆPEd

0

τττPEed

and

cg = 0 +∂c

∂x×H×

gg

0

0

τττPEeg

160


Combining the two:

cd = cPEd + cg +∂c

∂x×H×

0

ˆPEd

0

0

︸︷︷︸

error(ˆPEd

)

In particular, the third term is immediately seen to be the general equilibrium response

of consumption to a leisure shock leading to a desired union labor supply adjustment

of ˆPEd , as claimed.


The proof proceeds in three steps. First, I show that aggregate impulse responses to the

heterogeneous shocks {εdi0} are identical to impulse responses to the common aggregate

shock εd0 ≡∫ 1

0εdi0. Second, I prove that cdi − cd = (ξdi0 − 1) × cPEd + ζζζ i, where

∫ 1

0(ξdi0 −

1)ζζζ idi = 0. And third, I exploit standard properties of fixed-effects regression to complete

the argument. As in the proof of Proposition 2, I use the notation ∂∂εεεs

to denote derivatives

for a shock path where only entries of shock s are non-zero.

1. We study impulse responses to the shock path εεεd ≡ e1, where e1 = (1, 0, 0, . . .)′. The

direct partial equilibrium response of consumption to the shock is

cPEd ≡∫ 1

0

∂ci∂εεεd× ξdi0 × εεεddi

161


where ci(•) is the consumption function of individual i, defined analogously to the aggre-

gate consumption function c(•). Since∫ 1

0ξdi0di = 1 and since ξdi0 is assigned randomly

across households (and so does not correlate with ∂ci∂εεεd× εεεd at any t), we have that

cPEd =

∫ 1

0

∂ci∂εεεd0

× εεεddi×[1 +

∫ 1

0

(ξdi0 − 1)di

]=

∫ 1

0

∂ci∂εεεd× εεεddi

The direct partial equilibrium response of aggregate consumption is thus identical to the

response in an economy where all individuals i face the common shock εεεd. The same

argument applies to the desired partial equilibrium contraction in labor supply, ˆPEd . But

if direct partial equilibrium responses are the same, then general equilibrium adjustment

is the same, and so all aggregates are the same.

2. Consumption of household i along the transition path satisfies

cid =∂ci∂x× x +

∂ci∂εεεd× ξdi0 × εεεd

where x was defined in Lemma 2. We thus get

cid − cd = (ξdi0 − 1)× ∂c


cPEd

+

(∂ci∂x− ∂c

∂x

)× x + ξdi0

(∂ci∂εεεd− ∂c

∂εεεd

)× εεεd︸︷︷︸

≡ζζζi

Note that, since by definition we have∫ 1

0∂ci∂xdi = ∂c

∂xand

∫ 1

0∂ci∂εεεddi = ∂c

∂εεεd, the residual term

ζζζ i must satisfy∫ 1

0(ξdi0 − 1)ζζζ idi = 0.

3. By the standard properties of fixed-effects regression, we can re-write regression (1.14) as

cit+h − ct+h = βdh × (ξit − 1)εdt + uit+h − ut+h (A.31)

162


By standard projection results, the estimand βββd satisfies

βββd =

∫ 1

0

[(ξdi0 − 1)cPEd + ζζζ i

](ξdi0 − 1)di∫ 1

0(ξdi0 − 1)2di

= cPEd

where I have used the fact that Var(ξdit) > 0.


By definition of yg, we know that

ygh = Cov(yt+h, εgt)

ygh is thus the estimand of a local projection on εgt. (1.20) then follows immediately by

Corollary 1 in Plagborg-Møller & Wolf (2019b).19

A.3.7 Auxiliary Lemma for Proposition 5

Lemma 3. Consider the structural model of Section 1.2.1. Under Assumptions 5 to 8,

all firm sector price inputs sf can be derived as functions only of the path of aggregate

consumption c. Sequences of consumption c and shocks εεε are part of a perfect foresight

equilibrium if and only if

c + i(sf (c);εεε) + g(εεε) = y(sf (c);εεε) (A.32)

19Strictly speaking, it remains to verify their Assumption 1, ensuring that the process (zt, y′t)′ is not

stochastically singular. It is straightforward to augment the model of Section 1.2.1 with more structuralshocks or measurement errors to ensure that this is the case for any vector of macroeconomic observables yt.

163


where the production and investment functions y(•), i(•) are derived from optimal firm

behavior.

To prove Lemma 3 I as before proceed in two steps. First, I show that all relevant inputs

to the firm problem can be obtained as functions only of c and εεε. Second, I show sufficiency

of the aggregate market-clearing equation.

1. By Assumption 6, the household block admits aggregation to a single representative

household with period felicity function u(c, c−1, `). Given c, the Euler equation of the

representative household allows us to back out the path of real interest rates r. Given

r, the Fisher equation and the Taylor rule of the monetary authority (by Assumption 8)

allow us to recover the path of aggregate inflation πππ, and so by the NKPC of retailers

we recover pI . Next, given Assumption 7, the wage-NKPC allows us to recover the path

of real wages w. Together with εεε we thus have all inputs to the firm problem, and in

particular indeed sf = sf (c), as claimed.

2. Optimal firm and government behavior is assured by construction. Next, since the Euler

equation and wage-NKPC hold, the only missing condition for household optimality is the

lifetime budget constraint. But by assumption the aggregate market-clearing condition

(A.32) holds at all times, so the household lifetime budget constraint must hold.

Together, 1. - 2. establish sufficiency of the conditions in the statement of Lemma 3.

Necessity is immediate, completing the argument.


By Lemma 3, a perfect foresight equilibrium is, to first order, a solution to the system of

linear equations

c +∂i

∂c× c +

∂i

∂εεε× εεε+

∂g


∂y

∂c× c +

∂y

∂εεε× εεε

164


As before, we thus in general have

c = H×(∂i

∂εεε× εεε− ∂y

∂εεε× εεε+

∂g

∂εεε× εεε)

for a unique linear map H. Now again use the notation ∂∂εεεs

to denote derivatives for a

shock path where only entries of shock s are non-zero. In response to investment tax and

government spending shocks, the response path of investment satisfies

iq =∂i

∂εεεq× εεεq︸︷︷︸

iPEq

+∂i

∂c×H×

(∂i

∂εεεq× εεεq −

∂y

∂εεεq× εεεq

)︸︷︷︸

Di×(iPEq −yPEq )

and

ig = 0 +∂i

∂c×H×

(∂g

∂εεε× εεεg

)︸︷︷︸

Di×gg

respectively. This establishes (1.21). The equations for output are exactly analogous.

A.3.9 Proof of Corollary 2

It is straightforward to show that a generalization of Lemma 2 holds for the system

e(sh(x);εεε) + i(sf (x);εεε) + g(εεε) = y(sf (x);εεε)

`h(su(x;εεε)) = `f (sf (x);εεε)

y(sf (x);εεε) = y

τττ(sf (x);εεε) = τττ

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where e is now the aggregated optimal household expenditure function for durable and non-

durable consumption. Applying the same steps as in the proof of Proposition 2 to this new

system, the result follows.

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A.4 Additional results

This appendix collects supplemental results. I discuss (i) robustness checks for accuracy of

the demand equivalence approximation, (ii) a generalization of my methodology to correct for

wealth effects in labor supply, (iii) a model variant with strong general equilibrium crowding-

in, (iv) a re-parameterization of the estimated HANK model that more closely matches my

empirically estimated output and consumption impulse responses to fiscal shocks, and (v) a

generalization of the equivalence result away from the model’s deterministic steady state.

A.4.1 Approximation accuracy

This section provides details for the extensions and robustness checks referenced in Sec-

tions 1.2, 1.4 and 1.5. I discuss (i) a model variant without unions and with weak wealth

effects in labor supply, (ii) a two-asset HANK model, (iii) inaccuracies for persistent demand

shocks, (iv) a model with durables, (v) useful (valued or productive) government spending,

(vi) multi-good economies, (vii) random draws for all parameters governing dynamics in the

estimated HANK model, (viii) approximate equivalence under imperfect matching of private

and public excess demand paths and (ix) approximate investment demand equivalence.

A.4.1.1 Weak wealth effects in labor supply

Empirical evidence suggests weak – but non-zero – short-term wealth effects associated

with (small) unexpected income gains (Cesarini et al., 2017; Fagereng et al., 2018). My

benchmark structural model – which features preferences with strong wealth effects, but

sticky-wage unions – cannot directly speak to these weak short-term wealth effects, as micro-

level difference-in-differences regressions invariably difference out the effects of direct labor

adjustments (recall the proof of Proposition 3). In this section I thus instead consider a model

167


with non-standard household preferences and without sticky-wage unions. Importantly, the

model is designed to be consistent with the empirically documented weakness and cross-

sectional homogeneity in short-run wealth effects in labor supply.

Model. I consider the estimated HANK model of Section 1.4, but with three changes.

First, the model is now populated by a double unit continuum of households – a unit con-

tinuum of families f ∈ [0, 1], and a unit continuum of households i ∈ [0, 1] for each f . Each

family is a replica of the unit continuum of households in the benchmark model, but shock

exposures may be heterogeneous across families. I will explain the purpose of this artificial

construction momentarily. Second, there are no unions – each household decides on its own

labor supply. Third, I change household preferences. Similar to Jaimovich & Rebelo (2009)

and Galı et al. (2012), I assume that

uft(cift, ìft) =c1−γift − 1

1− γ− χθift

`1+ 1

ϕ

ift

1 + 1ϕ

where the preference shifter θift satisfies20

θift = xγft × c−γift

The variable xft is central. To jointly ensure arbitrarily weak short-run wealth effects in labor

supply, homogeneous wealth effects in the cross section of households, and direct earnings

responses showing up in cross-sectional regressions, I assume that

xft = x1−ωft−1 × c

ωft

20Households do not internalize the effect of their consumption on the shifter.

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This preference specification is the simplest design with all three desired properties. First,

by varying the parameter ω, I can control the strength of short-term wealth effects, exactly

as in Galı et al. (2012). With ω = 0 wealth effects are 0, and so Assumption 3 is satisfied.

Second, solving for optimal household labor supply decisions, we get

χ`1ϕ

ift = wtx−γft (A.33)

If all “families” are equally affected by the shock, then everyone’s labor supply is identical,

giving the desired homogeneity. Thus, for the first two requirements, the family construction

is not necessary – we could simply replace cft by ct, giving the natural heterogeneous-agent

analogue of the preferences in Galı et al. (2012). But third, with heterogeneous family-level

shock exposures, cross-sectional regressions as in Proposition 3 will pick up direct earnings

responses.21 In particular, let `h = `h(w, c) denote the mapping from wages and family

consumption into family labor supply induced by (A.33). The micro regression estimand in

(1.14) then satisfies

cPEτ =

(I − ∂c

∂`× ∂`h

∂c

)−1

×(∂c

∂τττ· dτττ)

(A.34)

For my accuracy checks, I simply match this regression estimand with an identical expansion

in aggregate government spending.

Parameterization. All parameters related to the sticky-wage block of the model are now

irrelevant; the only new model parameter is ω. To ensure consistency with empirical evidence,

I set ω = 0.05. As in Cesarini et al. (2017), this specification results in a peak partial

equilibrium labor supply response of around 4$ for every 100$ response in consumption.

Results. Results are displayed in Figure A.2.

21I could have used a similar family construction for the union model. Without changes in preferences,however, this model would be inconsistent with empirical evidence on the weakness of wealth effects.

169


Figure A.2: Approximate Demand Equivalence, Weak Wealth Effects

Note: Impulse response decompositions and demand equivalence approximation for the HANKmodel with weak wealth effects. The direct response and the indirect general equilibrium feedbackare computed following Definition 1.

Estimated wealth effects are weak, so Assumption 3 is nearly satisfied, and the approxi-

mation is again highly accurate, with a maximal error of around 4 per cent. In addition to

general equilibrium feedback associated with the labor supply contraction itself, the slight

over-statement displayed in the right panel now also reflects a second, more subtle effect:

Since the government spending expansion gεg only replicates the direct consumption response

net of earnings changes, its present value is lower than that of the corresponding income tax

rebate, and so the associated tax burden is lower. Matching ∂c∂τττ· dτττ (rather than the true

direct response displayed in (A.34)) would instead make the approximation as accurate as

in the benchmark model.

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A.4.1.2 A two-asset HANK model

This section provides further details on the two-asset HANK model of Section 1.4.3. I sketch

the expanded model (in particular the new consumption-savings problem), discuss the model

parameterization, and finally display and interpret the results of my accuracy check.

Model sketch. Households invest in an illiquid asset with nominal return ih and a liquid

asset with return ih − κ1,b + 1bh<0κ2,b. The household consumption-savings problem then is

max{cit,bhit,ahit}

E0

[∞∑t=0

βtiζt(εεεv)u(cit, cit−1, ìt)

]

such that

cit+bhit+a

hit = (1−τ`)wteitìt+

1 + iht−1 − κ1,b + 1bhit−1<0κ2,b

1 + πtbhit−1+

1 + iht−1

1 + πtahit−1+φa(a

hit, a

hit−1)+τit

and

bhit ≥ b, ahit ≥ a

where φa(•, •) is the adjustment cost function for illiquid asset holdings. Similar to Kaplan

et al. (2018), I assume that

φa(a′, a) =

χ1

χ2

×

(|a′ − 1+ih

1+πa|

χ0 + 1+ih

1+πa

)χ2

×

(χ0 +

a′ − 1+ih

1+π

a

)

Returns in the economy are determined as follows. Both liquid and illiquid assets are

issued by a mutual fund, which in turn owns all government debt and all claims to corporate

profits in the economy. Let ωt ≡ bht + aht denote total funds managed by the mutual fund.

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Returns earned by the mutual fund imt then satisfy

ωt ×1 + imt−1

1 + πt= bt

1 + ibt−1

1 + πt+ (dt + vt)

where vt denotes the value of the corporate sector, which by arbitrage satisfies

1 + ibt−1

1 + πt=vt + dtvt−1

except possibly at t = 0. I assume that the mutual fund is competitive, and faces inter-

mediation costs κ1,b to make assets liquid and κ1,b + κ2,b to lend liquid assets. It follows

immediately that we must have iht = imt .

The rest of the economy is unchanged; in particular, firms still discount at1+ibt−1

1+πt, which

in the absence of aggregate risk is equivalent to discounting at1+imt−1

1+πt=

1+iht−1

1+πt. The only

change to the equilibrium Definition 2 is the new asset market-clearing condition:

bht + bft + aht = bt + vt

Parameterization. For simplicity, I keep all parameters governing dynamics identical to

the estimated 1-asset HANK model, and only re-calibrate the steady state of the model.

Table A.3 displays all parameters from the re-calibrated 2-asset model that are different

from those displayed in Table A.1 for the benchmark 1-asset model.

I choose the parameters of the adjustment cost function to ensure a reasonable fit to the

liquid-illiquid wealth distribution in U.S. data (Kaplan et al., 2018). To provide a stringent

test of the demand equivalence approximation, I set the wedge between returns on household

deposits and government debt to be an (arguably implausible) 1 per cent per quarter. With

smaller return gaps, the approximation would improve.

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Steady-State Parameter Values, 2-Asset HANK Model

Parameter Description Value Target Model Data

Households

χ0 Adj. Cost Parameter 0.25 Kaplan et al.

χ1 Adj. Cost Parameter 6.13 Fraction b = 0 0.29 0.30

χ2 Adj. Cost Parameter 2.03 A/Y 11.21 10.64

β Discount Rate 0.98 B/Y 1.29 1.04

rh Return 0.0125 Kaplan et al.

κ1,b Liquid Wedge 0.01 Upper Bound

κ2,b Borrowing Wedge 0.03 Fraction b < 0 0.09 0.15

b Borrowing Constraint -1 Kaplan et al.

Firms

δ Depreciation 0.025 Firm Valuation

Table A.3: 2-asset HANK model, steady-state calibration.

As is typical for two-asset models, the average household MPC is lower than in a liquid-

wealth calibration of a one-asset model, now at around 14 per cent. Intuitively, this is so

because households have more vehicles to self-insure.

Approximation accuracy. Results are displayed in Figure A.3. Two features stand out.

First, the model now features stronger general equilibrium crowding-out. Relative to the

simpler one-asset HANK model, this model features (i) smaller average MPCs and (ii) no

mechanical redistribution effects related to heterogeneous dividend exposure.22 Both changes

tend to dampen general equilibrium amplification. Second, even though both wealth effects in

labor and (implausibly large) heterogeneity in borrowing and lending rates lead the demand

equivalence to over-state the response of aggregate consumption, the approximation remains

reasonable, with a maximal error around 7 per cent of the peak consumption response.

22All returns are received by the mutual fund and passed on to households, whereas before householdswere directly exposed to (mildly countercyclical) dividend payments.

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Figure A.3: Approximate Demand Equivalence, 2-Asset HANK Model

Note: Impulse response decompositions and demand equivalence approximation for the two-assetHANK model. The direct response and the indirect general equilibrium feedback are computedfollowing Definition 1.

A.4.1.3 Persistent demand shocks

Figure A.4 shows that, even in the rigid-wage model of Justiniano et al. (2010), the de-

mand equivalence approximation deteriorates for very persistent demand shocks. This is

not surprising: Persistent shocks induce – through pronounced long-term wealth effects –

a persistent decline in desired labor supply. Since wages are not sticky forever, the de-

cline in desired labor supply ultimately feeds into a decline in actual hours worked, so the

approximation error is larger at long horizons.

A.4.1.4 Durables

I extend the household consumption-savings problem to feature durable and non-durable

consumption:

max{cit,dhit,bhit}

E0

[∞∑t=0

βtiζt(εεεv)u(cit, dhit, ìt)

](A.35)

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Figure A.4: Approximate Demand Equivalence, Justiniano et al. (2010)

Note: Impulse response decompositions and demand equivalence approximation for persistent andhump-shaped demand shocks in the model of Justiniano et al. (2010), solved at the posterior modeand with a fraction λ → 0 of spenders. The direct response and the indirect general equilibriumfeedback are computed following Definition 1.

such that

cit + dhit + bhit = (1− τ`)wteitìt +1 + ibt−1(bhit−1)

1 + πtbhit−1 + (1− δ)dhit−1 + τit + dit + φd(d

hit−1, d

hit)

and

bit ≥ b− (1− θ)dit

where φd(•) is the durables adjustment cost function, 1− θ is the share of durable holdings

that can be collateralized, and – in a slight abuse of notation – I only use the superscript h

to distinguish between household durables consumption dhit and dividend receipts dit. Note

that this specification allows for all of the bells and whistles considered in quantitative

studies of durable and non-durable consumption (Barsky et al., 2007; Berger & Vavra, 2015):

175


Households have potentially non-separable preferences over c and dh, adjustments in durables

may incur additional costs, and households can borrow against their durable goods holdings.

Crucially, I assume that the common final good yt can be costlessly turned into the

durable good dht , so that the aggregate resource constraint becomes

yt = ct + dht − (1− δ)dht︸︷︷︸et

+it + gt

where et is aggregate household expenditure. The equilibrium definition in Appendix A.1.1

thus generalizes straightforwardly, with aggregate household expenditure replacing non-

durable consumption expenditure. Defining a PE-GE decomposition for total household

expenditure as in Definition 1, the demand equivalence result then still applies, now for the

aggregated household expenditure path e:

Corollary 2. Consider the structural model of Section 1.2.1, extended to feature durable

goods, as in Problem (A.35). Suppose that, for each one-time shock {τ, g, v}, the equilibrium

transition path exists and is unique. Then, under Assumptions 1 to 3, the response of

consumption to a generic consumption demand shock d (either impatience v or tax rebate τ)

and to a government spending shock g with gg = ePEd satisfy, to first order,

ed = ePEd︸︷︷︸PE response

+ eg︸︷︷︸= GE feedback

(A.36)

A.4.1.5 Useful government spending

In the benchmark model of Section 1.2.1 as well as the estimated HANK model in Section 1.4,

government spending is useless – it is neither valued by households, nor does it have any

productive benefits. Some previous work has instead allowed for such benefits of government

spending. To gauge the extent to which such benefits threaten my approximations, I in

176


this section review related empirical evidence and provide model-based intuition as well as

quantitative accuracy checks.

Valued Spending. It is immediate that all equivalence results go through unchanged if

government expenditure enters household utility in an additively separable fashion, i.e. the

per-period felicity function can be represented as

u(cit, cit−1, ìt, gt) = u(cit, cit−1, ìt) + v(gt)

More interestingly, exact equivalence also holds with particular kinds of (popular) non-

separable preferences. For example, suppose that preferences take the form

u(cit, ìt, gt) = log(cνitg1−νt )− χ `

1+ 1ϕ

it

1 + 1ϕ

Log preferences are popular in the business-cycle literature (and used in my own HANK

model), while an inner Cobb-Douglas aggregator is popular in the trade literature (Fajgel-

baum et al., 2018). It is straightforward to see that the marginal utility of consumption is

then unaffected by changes in government expenditure, so consumption decisions are again

unaffected, and demand equivalence survives under the same assumptions as before.

With other types of non-separabilities, exact equivalence does not survive. For example,

Leeper et al. (2017) assume that households have conventional preferences over a synthetic

consumption aggregate c∗it, where

c∗it = cit + αGgt

There is little direct empirical evidence on the magnitude or even sign of αG. Since pri-

vate and public consumption co-move in post-war aggregate data, standard likelihood-based

estimation exercises with representative-agent models usually call for a negative coefficient

177


(Leeper et al., 2017). Models with high average household MPCs instead endogenously

tie private and public spending together, and so would likely require little or no comple-

mentarity (Galı et al., 2007). Nevertheless, and to threaten the accuracy of the demand

equivalence approximation as much as possible, I consider a variant of my benchmark model

with αG = −0.24, exactly as in Leeper et al. (2017). Results are displayed in Figure A.5.

Figure A.5: Approximate Demand Equivalence, Valued G

Note: Impulse response decompositions and demand equivalence approximation in the estimatedHANK model, augmented to feature complementarities in private and public consumption, asin Leeper et al. (2017). The direct response and the indirect general equilibrium feedback arecomputed following Definition 1.

Since private and public consumption are complements in household preferences, private

consumption is directly stimulated by an increase in public spending, and so the demand

equivalence approximation over-states (black line). However, given an estimate of αG, it

is straightforward to correct for this inaccuracy: Ignoring for simplicity the presence of

potentially binding borrowing constraints, the wedge in household Euler equations associated

with additional government spending is equal to αG× (gt− gt+1). Thus, for every additional

dollar of government spending, private partial equilibrium demand increases by −αG dollars

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(ignoring the permanent level shift due to unchanged lifetime income). This suggests a simple

fix: Instead of matching cPEτ and gg, researchers should match cPEτ and a scaled spending

response (1 +αG)× gg. The third purple line in Figure A.5 shows that, with this alternative

approach, the demand equivalence approximation is again nearly exact.

Productive Spending. If government spending has productive benefits, then the aggre-

gate effects of private and public spending should differ. Consistent with this intuition,

empirical estimates of public investment multipliers are usually larger than those of public

spending (Leduc & Wilson, 2013; Gechert, 2015). These results caution against the use of

public investment multipliers for the demand equivalence approximation. Reassuringly, my

empirical estimates are almost identical for overall and for pure military spending forecast

errors, suggesting that my analysis is not picking up the effect of public investment spending.

For completeness, I also illustrate this conclusion through a structural analysis. In Boehm

(2016) and Leeper et al. (2010), government expenditure on investment goods is productive in

the sense that the aggregate stock of government “capital” kgt directly affects the production

capabilities of intermediate goods producers:

yjt = y(ejt, kgt , ujtkjt−1, `jt)

where kgt = (1− δ)kgt−1 +gt. Analogously, I consider a variant of my estimated HANK model

with a production function of the form

yjt = (kgt )αg(ujtkjt−1)α`1−α

jt

I set the output elasticity to αg = 0.2, large enough to generate substantial asymmetry in

multipliers (and larger than usual estimates in this literature, e.g. αg = 0.05 in Leeper et al.

(2010)). Results are displayed in Figure A.6.

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Figure A.6: Approximate Demand Equivalence, Productive G

Note: Impulse response decompositions and demand equivalence approximation in the estimatedHANK model, augmented to feature productive benefits of public spending, as in Leeper et al.(2010). The direct response and the indirect general equilibrium feedback are computed followingDefinition 1.

Unsurprisingly, the approximation is accurate on impact (where the demand pressure of

the shock dominates), but deteriorates over time, as higher government spending gradually

expands the productive capacity of the economy. These results are entirely consistent with

Leduc & Wilson (2013), who empirically document “an initial effect due nominal rigidities

and a subsequent medium-term productivity effect.”

A.4.1.6 Multi-good economies

Heterogeneity in consumption baskets for private and public consumption can break the

demand equivalence result, at least as long as factors of production are imperfectly mobile

across sectors or production functions are sector-specific. In such a segmented economy, rel-

ative prices will respond to spending shocks (Ramey & Shapiro, 1998), and so the demand

equivalence approximation will fail. Previous work has also emphasized that heterogene-

180


ity in the factor incidence of private and public demand shocks may matter for aggregate

transmission (Alonso, 2017; Baqaee, 2015), and that the effects of government spending on

consumption goods may differ from those on investment goods (Boehm, 2016).

With the notable exception of productive long-lived investments, evidence on asymmetry

in government spending multipliers by the type of spending is relatively scarce (Gechert,

2015; Ramey, 2016). I complement this evidence with a less direct, model-based approach: I

study the accuracy of the demand equivalence approximation in a series of structural models,

rich enough to allow for the mechanisms reviewed above and disciplined to be consistent with

empirical evidence on their likely strength.

Encompassing Model. I consider a generalized variant of my benchmark model of Sec-

tion 1.2. The model deviates from this benchmark framework in the following ways. First,

it features three goods – two consumption goods and an investment good. Households have

preferences over a consumption basket cit, which is given as a mix of the two individual

consumption goods:

cit = cνi1tc1−νi2t

I let the ideal price index of the consumption bundle be the numeraire of my economy,

and denote the relative prices of two consumption goods by q1t and q2t. Investment is only

possible using the economy’s investment good, whose real relative price is denoted qIt. The

government purchases each of the three goods, with potentially different aggregate spending

multipliers for each.

Second, total household labor supply `ht is an aggregator of labor supply for each of the

three goods in the economy:

`ht ≡[`ϕ+µϕ

1t + `ϕ+µϕ

2t + `ϕ+µϕ

It

] ϕϕ+µ

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µ = 0 corresponds to perfect labor mobility across the sectors, while µ = 1 corresponds

to perfect immobility, with all labor types entering separately into my particular choice of

household utility functions. For each type of labor, labor supply is intermediated by a unit

continuum of sticky-wage unions. Optimal union behavior then gives the three log-linearized

wage-NKPCs; under my choice of household preferences, they take the form

ˆwmt =β

1 + βˆwmt+1 − κw

[ˆwmt −

(1− µϕ

ˆht +

µ

ϕˆmt

)− γ ˆc∗t

]− 1

1 + βˆπt +

β

1 + βˆπt+1 +

1

1 + βˆwmt−1

for m = 1, 2, I, and where as before c∗t is the virtual consumption aggregate defined in (A.22).

Note that, with µ = 0, wages in all sectors are at all times equalized. Overall, household i

then receives eitwt`t worth of labor earnings, where wt is the aggregated wage index.

Third, there are separate production sectors for each of the three goods. Briefly, I simply

repeat the production sector of the estimated 1-asset HANK model three times, but with

good-specific final prices qmt and potentially heterogeneous capital shares αm. All three

sectors then purchase capital goods at price qIt, hire labor at cost wmt, and sell their own

good at real price qmt.

For all subsequent results, I build on the parameterization of the estimated HANK model

of Section 1.4, but with one notable difference: I materially lower the degree of nominal price

rigidities. In the model, the probability of price re-sets governs relative price movements after

a demand shock for a specific good. I have included measures of relative prices in my VARs

and find little response, similar to Nakamura & Steinsson (2014); however, Ramey & Shapiro

(1998) show that, after large government spending shocks that move output by almost 4 per

182


cent, relative prices move by 2.5 per cent.23 To be conservative, I choose a model calibration

with φp = 0.6, giving relative price responses consistent with this evidence.

In the remainder of this section I study the quality of the demand equivalence approx-

imation for different types of government purchases. In particular, I consider two model

variants: (i) a two-good model with a consumption good and an investment good, similar

to Boehm (2016), and (ii) a model with two consumption goods and a separate investment

good, and with heterogeneous factor shares. Both models implicitly allow for the relative

price effects emphasized in Ramey & Shapiro (1998).

Investment Goods. To study the effects of multiplier heterogeneity for investment and

consumption goods, I consider a special case of the above economy with perfect factor mobil-

ity across the two consumption sectors (allowing aggregation to a single composite consump-

tion good), but imperfect capital and labor mobility across the composite consumption and

investment good sectors (µ = 1). The government consumes both goods, and the overall size

of the composite investment sector is calibrated to correspond to 20% of aggregate output in

steady state. In keeping with Boehm (2016), I assume homogeneous production technologies

across the two sectors, i.e. α1 = α2 = αI ≡ α.

Results are displayed in Figure A.7. The right panel shows that the approximation is

still accurate for government purchases of consumption goods. Perhaps more surprisingly, it

remains accurate for government purchases of investment goods.24

My results are inconsistent with Boehm (2016). The differences between the two analyses

can be traced back to three model features. First, Boehm’s model features flexible wages,

while in my model wages are sticky. Wage flexibility turns out to be crucial to his mechanism:

When the government buys the consumption good, its relative price rises, consumption is

23For my VAR analysis, I follow Ramey & Shapiro (1998) and – in a VAR with military spending forecasterrors – include a measure of the relative price of manufacturing goods.

24Correspondingly, I also find that the output responses for both types of government spending are almostidentical.

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Figure A.7: Approximate Demand Equivalence, Investment v Consumption

Note: Impulse response decompositions and demand equivalence approximation in the two-sectorHANK model. The direct response and the indirect general equilibrium feedback are computedfollowing Definition 1.

crowded out, and households work harder. When it buys the investment good, in contrast,

relative prices barely change (because of the high intertemporal elasticity of investment

demand), consumption is not crowded out, and labor supply does not respond. Nominal

wage rigidity breaks this mechanism. Second, my shocks are much more transitory than

his. As a result, in the presence of short-lived wage rigidity, the labor supply channel is

particularly dampened. Third, his model features an extremely large intertemporal price

elasticity of firm investment demand. Consistent with both my macro estimation as well as

the micro evidence of Zwick & Mahon (2017), my model features much stronger adjustment

costs, and so investment is not as easily crowded out. This increases the strength of aggregate

demand effects after government purchases of investment goods.

Consumption Basket Heterogeneity. I analyze relative price effects and heterogeneous

factor incidence using a full three-sector version of my extended economy. I set µ = 1,

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ν = 0.5, labor shares for each of the three production sectors are exactly as in Alonso (2017,

Table 3), and finally the fraction of labor in each of the three sectors is set so that their

relative sizes are also data-consistent (again following Alonso (2017)). As before, I restrict

factors to be imperfectly mobile across the three sectors. Results are displayed in Figure A.8.

Figure A.8: Approximate Demand Equivalence, Heterogeneous Factor Incidence

Note: Impulse response decompositions and demand equivalence approximation in the three-sectorHANK model. The direct response and the indirect general equilibrium feedback are computedfollowing Definition 1.

In the data, the network-adjusted labor share of the average government consumption

good exceeds that of the average consumption good. I thus in the right panel of Figure A.8

show the demand equivalence approximation for government purchases of the second (labor-

intensive) consumption good. The approximation error is clearly visible, and goes in the

expected direction: Since the MPC out of labor income is higher than that out of capital

income, the approximation using the second consumption good over-states; similarly, for the

first good, it under-states (not shown). However, and consistent with the conclusions in

Alonso (2017) and Baqaee (2015), these incidence effects are not particularly strong. The

intuition is simple: In the data, the average consumption good has a labor share of around

185


of 0.4, while the network-adjusted labor share of government consumption is around 0.65.

Assuming an (extreme) average quarterly MPC out of labor income of around 0.5, and an

MPC out of any residual income of 0.05, the resulting second-round demand difference from

spending on the two goods would be around 11 cents for every dollar of spending.25

Finally, for completeness, I have re-computed the demand equivalence approximations in

a model with imperfect factor mobility, but homogeneous production functions, leaving only

the inaccuracies associated with relative price movements. I find that the demand equivalence

approximation is then almost as accurate as in my benchmark model, suggesting that almost

all of the inaccuracy in Figure A.8 is driven by the factor incidence mechanism.

A.4.1.7 Random parameter draws

The accuracy displayed in Figure 1.6 is not at all special to the mode parameterization of the

estimated HANK model, but a generic feature of standard business-cycle models with at least

moderate wage and price stickiness. To illustrate this point, I proceed as follows: Rather

than fixing the dynamics parameter values as in Table A.2, I randomly draw their values

from uninformative uniform distributions over wide supports, as displayed in Table A.4.26

For each parameter draw, I compute the maximal demand equivalence error relative to the

true model-implied peak consumption response. This procedure is repeated for 10,000 draws

from the joint uniform distributions in Table A.4.

I find that the approximation accuracy is largely orthogonal to all parameters except for

the price stickiness φp. Fixing φp at the posterior mode and merely randomly drawing all

other parameters, I find that 95 per cent of draws give a maximal prediction error below 3.3

per cent. For φp = 0.1, and fixing all other parameters at the posterior mode, the prediction

25Arguably, this is an upper bound for the likely size of the effect, since heterogeneity in MPCs by skillimplies the opposite conclusion: Government expenditure is concentrated on relatively high-skilled labor(Baqaee, 2015); if MPCs out of skilled labor are smaller, then the gap displayed in Figure A.8 shrinks.

26This approach to documenting a generic property of a family of quantitative models closely follows theanalysis in Canova & Paustian (2011).

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Dynamics Parameter Values, Random Draws

Parameter Description Lower Bound Upper Bound

φp Price Calvo Parameter 0.1 0.99

ζ Capacity Utilization 0.1 10

κ Investment Adjustment Cost 0.1 10

ρm Taylor Rule Persistence 0.01 0.99

φπ Taylor Rule Inflation 1 5

φy Taylor Rule Output 0 1

φdy Taylor Rule Output Growth 0 1

Table A.4: Supports for uniform parameter draws in the HANK model.

error increases to almost 9 per cent; intuitively, output is now not demand-determined, but

given my calibrated moderate degree of wage rigidity, shifts in desired household labor supply

still have rather limited aggregate effects.27

A.4.1.8 Imperfect demand matching

The excess demand paths in Figure 1.4 and Figure 1.8 are matched well, but of course not

perfectly. To gauge the distortions associated with moderate mis-matching, I again consider

the estimated HANK model of Section 1.4.1, but now do not assume perfectly matched

excess demand paths; instead, I construct the demand equivalence approximation for an

inaccurately matched government spending path gg with

ggt = (1 + νt) × cPEτt (A.37)

where νt ∼ N(0, σ2ν). I set σ2

ν to get average errors identical in size to those displayed in

Figure 1.4; this gives σ2ν = 0.123.

27Conversely, with rigid prices and flexible wages, labor is still demand-determined, so labor supply shiftsonly move wage relative to dividend income. In my model, these incidence effects turn out to be relativelysmall, so the demand equivalence approximation is also accurate with flexible wages and rigid prices.

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I then construct the demand equivalence approximation for 10,000 draws of the error

sequence ννν, and for each compute the maximal prediction error relative to the peak true

consumption response. I find that 95 per cent of prediction errors lie below 8.7 per cent, so

the approximation remains accurate.28 The intuition is simple: Since the model only features

relatively moderate general equilibrium amplification, prediction errors for consumption can

only be large if the demand path perturbation itself is substantial. The errors in demand

matching, however, are by construction small, and thus so are the overall approximation

errors.29 To illustrate, Figure A.9 shows the quality of the demand equivalence approximation

for one particular draw of the error sequence ννν.

Figure A.9: Approximate Demand Equivalence, Imperfect Matching

Note: Impulse response decompositions and demand equivalence approximation in the estimatedHANK model, with imperfect demand matching, following (A.37). The direct response and theindirect general equilibrium feedback are computed following Definition 1.

28Most of the large approximation errors come from draws in which the ννν’s are so far from 0 that demandmatching is clearly violated, so the results displayed here are actually an upper bound on likely inaccuracies.

29Consistent with this intuition, the average error is even smaller when I repeat the same exercise inthe model of Appendix A.4.4, which features general equilibrium amplification matched to my empiricalestimates on government spending shock transmission.

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A.4.1.9 Approximate investment demand equivalence

Consider again the estimated HANK model of Section 1.4. In keeping with my empirical in-

vestment application, I enrich the model to feature an investment tax credit shock that yields

the same partial equilibrium investment response as that estimated in Section 1.5.2. I then

construct an investment demand equivalence approximation for the aggregate investment

response via the decomposition in (1.21). Results are displayed in Figure A.10.

Figure A.10: Approximate Investment Demand Equivalence, HANK Model

Note: Impulse response decompositions and investment demand equivalence approximation in theestimated HANK model, with details on the parameterization in Appendix A.1.2.2. The investmenttax credit path is matched to replicate the direct investment response estimated in Section 1.5.2.The direct response and the indirect general equilibrium feedback are computed following thenatural generalization of Definition 1.

The demand equivalence approximation remains reasonably accurate (in particular at

short horizons), with the maximal error over all horizons equal to around 10 per cent of

the true impact impulse response of investment. Intuitively, the equivalence approximation

over-states the actual investment response since each of Assumptions 6 to 8 tends to weaken

the extent of general equilibrium crowding-out. First, without Assumption 6, short-term

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increases in labor earnings (and cuts in dividend pay-outs) lead to excess consumption de-

mand pressure. Second, without Assumption 7, wages need to increase more sharply to

induce additional labor supply. And third, without Assumption 8, the monetary authority

more aggressively leans against the investment demand stimulus. Nevertheless, all three

effects are small, and so the additional degree of general equilibrium crowding-out in the

estimated model is relatively modest. The investment demand equivalence approximation

thus promises to be informative even if the underlying assumptions do not hold exactly.

A.4.2 Correcting for wealth effects in labor supply

Instead of ignoring the labor supply error term in Proposition 2, a simple alternative is to

first estimate the direct partial equilibrium labor supply response ˆPEd from micro data and

then estimate the aggregate consumption effects of an equivalent household “leisure” shock.

Generalized Methodology. Let cψ denote the impulse response of aggregate consump-

tion to a leisure shock – a labor wedge εεεψ that changes desired household labor supply by

ˆPEd . Using the equilibrium construction of Lemma 2, it is straightforward to see that such

a shock has no other direct partial equilibrium effect. It is thus immediate that, under the

assumptions of Proposition 2 (but without imposing Assumption 3), we have

cd = cPEd + cg + cψ

In practice, cψ is presumably not available, since there is no good evidence (to the best of

my knowledge) on the aggregate effects of pure shocks to the labor wedge. Instead, the best

related evidence is on changes in labor income taxes (Mertens & Ravn, 2013). Estimates of

the consumption response to labor income tax changes are likely to be informative about

cψ, but have two problems. First, to translate the size of the tax change into a partial

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equilibrium change in labor supply ˆPEd , we need an estimate of the Frisch elasticity of

labor supply. Second, the tax may generate revenue, which could be used to finance greater

government spending or reduce future tax burdens.

Results. Direct micro estimates (Cesarini et al., 2017; Fagereng et al., 2018) suggest that,

for every 100$ consumption spending response to a one-off unexpected income receipt, total

labor income very briefly dips by around 4$. For the income tax rebate studied by Parker

et al. (2013), partial equilibrium consumption spending increased by around 1.5%. Assuming

that consumption spending roughly equals labor income, the direct labor supply response

ˆPEd thus equals around 0.06% on impact, and very little thereafter.

With a unit Frisch elasticity, a labor supply drop of this magnitude would correspond

to a transitory labor income tax increase of 0.06 percentage points. According to the point

estimates of Mertens & Ravn, such a transitory tax hike in turn induces a general equilibrium

drop of consumption of around 0.07%. Abstracting from the effects of future tax adjustments

associated with the tax hike today, we would thus subtract around 0.07% from the benchmark

estimates of the impact consumption response in Figure 1.5 – a hardly relevant adjustment.

A.4.3 General equilibrium amplification

The equivalence result in Proposition 2 asserts that general equilibrium effects are tied

together across shocks, but is silent on the strength of this common general equilibrium

feedback. In this section I give two extreme examples, one with full general equilibrium

crowding-out, and one with strong general equilibrium amplification.

The first example is a variant of the rich benchmark model, restricted to feature flexible

prices and wages, labor-only production, and household preferences as in Greenwood et al.

(1988). In this model, an income tax rebate does not move aggregate output, consumption,

or labor. The argument is well-known and straightforward: Given a rebate path τττ , consider

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an interest rate path r such that, at (τττ , r) and facing steady-state wages forever, households

are willing to consume steady-state consumption c forever. But then the output and labor

markets clear by construction, and so we have indeed found an equilibrium. Thus, in this

model, interest rate feedback fully crowds out partial equilibrium consumption demand.

The second example is quantitative. I consider the benchmark estimated HANK model

of Section 1.4, but set the household borrowing wedge to zero and further assume that

preferences are as in Greenwood et al. (1988).

Figure A.11: Demand Equivalence, GHH-HANK

Note: Impulse response decompositions after equally large, one-off tax rebate and governmentspending shocks in the HANK model with GHH preferences. The direct response and the indirectgeneral equilibrium feedback are computed following Definition 1.

Given strong complementarities in consumption and labor supply, the extra production

induced by the demand shock will lead to yet more consumption demand, setting in motion

a strong general equilibrium feedback cycle (see Auclert & Rognlie, 2017, for an analytical

characterization). Results are displayed in Figure A.11.

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A.4.4 Impulse response matching

The estimated model of Section 1.4 predicts some general equilibrium crowding-in of con-

sumption following a transitory tax rebate. Minor parameter changes are however enough to

ensure close agreement between model-implied and empirically estimated government spend-

ing impulse responses. In particular, it is enough to slightly lower the degree of nominal price

rigidity (to φp = 0.65) and to make monetary policy somewhat more aggressive (increasing

the output response to φy = φdy = 0.15). Figure A.12 provides an illustration.

Figure A.12: Impulse Response Matching, Public Spending Shock

Note: Impulse responses to a transitory expansion in government spending. Empirical estimates(grey) exactly as in Section 1.3.2. Model-implied impulse responses (orange) in estimated HANKmodel, but with φp = 0.65 and φy = φdy = 0.15. The government spending path exactly matchesthe model-implied household spending response to a transitory rebate, as in Figure 1.4.

The two panels show the consumption and output responses to a transitory increase

in government spending. In both plots, the grey lines are the empirical estimates of Sec-

tion 1.3.2, and the orange lines are model-implied analogues. Clearly, with the proposed

parameter changes, the impulse responses align. The underlying model is thus a promising

laboratory for the structural analysis of tax rebates or other consumption demand shifters.

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A.4.5 Demand equivalence along transition paths

All equivalence results presented in this paper were stated for transition paths starting at the

deterministic steady state. However, it is immediate from the proofs of Propositions 2 and 5

that nothing in my logic hinges on the starting point. Intuitively, the crucial restriction in

my arguments is that they are valid to first order, but not that they only apply to particular

expansion points. All results can thus equivalently be interpreted as applying to first-order

perturbation solutions around a given (deterministic) transition path.

For example, initial states µh0 , µf0 , w−1 and p−1 could be such that the economy is in a

deep recession or brisk expansion. My equivalence results would then apply to deviations

from the unshocked transition path of the economy back to steady state. These deviations

need not agree with impulse responses at steady state, but they remain tied together across

different kinds of demand shocks.

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A.5 Application: income redistribution

Both applications in the main text are semi-structural: I recover spending impulses from mi-

cro data, and then use demand equivalence to map micro estimates into general equilibrium

counterfactuals. For some interesting shocks, however, micro data are not rich enough to es-

timate the required direct spending responses. Appealingly, however, construction of direct

spending responses only requires researchers to specify one block of the economy. Given this

partial equilibrium block, my methodology can again be used to provide the mapping into

full general equilibrium counterfactuals.

I illustrate this insight with an application to a simple redistributive, budget-neutral

stimulus policy: The government imposes a lump-sum tax on the richest 10 per cent (in

terms of liquid wealth holdings) of households, and uses the proceeds to finance a lump-sum

rebate to the poorest 10 per cent.

Direct response. Jappelli & Pistaferri (2014) document that, because poor households on

average have higher MPCs than rich households, a redistributive policy of this sort should

stimulate short-term demand. However, as pointed out in Auclert & Rognlie (2018), all

households spend their income at some point in time, so the demand stimulus today is

necessarily offset by a demand contraction in the future. Since estimates of heterogeneity in

dynamic iMPCs across the household wealth distribution are hard to obtain, I instead use

the partial equilibrium consumption-savings problem (1.1) – parameterized exactly as in my

estimated HANK model – to construct the partial equilibrium consumption demand path

associated with the budget-neutral redistributive policy.

The solid green in the top right panel of Figure A.13 shows the estimated direct con-

sumption response. Consistent with the empirical estimates of Jappelli & Pistaferri (2014),

consumption sharply increases on impact. Since the taxed rich households behave almost

195


exactly in line with the permanent income hypothesis, their consumption decreases slightly

but persistently, so overall consumption demand decreases slightly but persistently over time.

Figure A.13: Redistribution Shock, Impulse Responses

Note: Output and consumption responses to a redistribution shock, with the partial equilibriumnet output response path matched to a linear combination of government spending shocks. Theconsumption response is computed in line with Proposition 2. The plot also shows the requireddemand matching as well as the implied labor tax response (cf. Assumption 2). The dashed linesagain correspond to 16th and 84th percentile confidence bands.

The missing intercept. I match the implied partial equilibrium excess demand path

through a combination of expansionary and contractionary government spending shocks,

similar to the bonus depreciation application in Section 1.5.2. The top left panel shows that

the partial equilibrium excess demand path is matched reasonably well, if with substantial

uncertainty at higher horizons. The bottom right panel shows that taxes – which in theory

196


need not respond, since the implied partial equilibrium excess demand path has zero net

present value – only respond very little, so Assumption 2 is reasonable.

Macro counterfactuals. The top right panel computes the general equilibrium consump-

tion counterfactual implied by the demand equivalence decomposition (1.13). Importantly,

while the direct consumption response was derived from my partial equilibrium consumption-

savings block, all general equilibrium feedback is estimated semi-structurally. Consistent

with the results in the rest of this paper, I find limited general equilibrium feedback, so con-

sumption rises significantly (if briefly) following the redistributive shock. The bottom left

panel shows that this general equilibrium increase in consumption is accommodated through

an (imprecisely estimated) increase in aggregate output.

197

Appendix B


B.1 Identified sets

Throughout the paper I informally refer to identified sets of SVARs and impulse response

functions. A proper definition of identified sets requires a formal treatment of identifying

information. Following Rubio-Ramırez et al. (2010), I allow identifying information to take

the form of linear restrictions on transformations of the structural parameter space into

q × nx matrices, where q > 0. Denote the transformation by f(·). Linear restrictions on the

transformation can then be represented via q×q matrices Zj and Sj, with j = 1, . . . , nx. Here

the Zj allow us to impose exact linear restrictions on f(·) through the requirement Zjf(·)ej =

0, and the Sj allow us to impose linear sign restrictions through the requirement Sjf(·)ej ≥ 0.

In the SVAR literature, most identifying restrictions take the form of restrictions on impulse

response functions. Formally, the impulse response of variable i to shock j at horizon h is

defined recursively as the (i, j)th element of the matrix

IRFh =

A−1

0 if h = 0∑h`=1A

−10 AÌRFh−` if h > 0

198

B. Appendix for Chapter 2

Transformation functions f(·) are then typically of the following form:

f({Aj}) =

. . .

IRFh

. . .

where {Aj} collects the structural VAR matrices (A0, A1, . . .). The matrices Zj and Sj simply

consist of 0’s, 1’s and −1’s, placed so as to ensure that the desired zero and sign restrictions

are imposed. Simple examples of such matrices are provided, for example, in Rubio-Ramırez

et al. (2010) or Arias et al. (2019). Finally, covariance restrictions and outside information

are appended by sign normalizations on A0: We require the diagonal elements of A0 to be

non-negative, which just means that a unit positive change in the ith structural shock is

interpreted as a one standard deviation positive innovation to the ith variable in the VAR.

Definition 3. Consider the reduced-form VAR ({Bj},Σu), and let b(Σu) denote an invertible

nx× nx matrix such that b(Σu)b(Σu)′ = Σu. Let R be an identifying restriction, defined by a

transformation function f(·) and a set of restriction matrices Zj, Sj, j = 1, . . . , n. Then the

identified set of rotation matrices with respect to the basis b(Σu), QbR, is defined as the set of

orthogonal rotation matrices Q ∈ O(nx) such that A0 ≡ Q × b(Σu)−1 is consistent with the

reduced-form covariance matrix, the normalization rule and the identifying restriction R:

QbR ≡ {Q | Q ∈ O(nx), A0 ≡ Qb(Σu)−1, A−1

0 A−1′

0 = Σu, diag(A0) > 0, and

Zjf({Aj})ej = 0, Sjf(A)ej ≥ 0 for 1 ≤ j ≤ nx}

Identified sets can be empty, have a single member, or they can have multiple members.

Generically, sign restrictions are set-identifying, and the identified set of rotation matrices

199


has strictly positive Haar measure (independent of basis). It is now straightforward to define

the identified set of impulse responses:1

Definition 4. Consider the reduced-form VAR ({Bj},Σu). Let R be an identifying restric-

tion, defined by a transformation f(·) and a set of restriction matrices Zj, Sj, j = 1, . . . , nx.

Then the identified set of impulse responses for variable i in response to shock j at horizon

h, ISi,j,h, is defined as the set of impulse responses generated by some rotation matrix Q in

the identified set of SVARs:

ISi,j,h ≡ {a ∈ R | a = IRFi,j,h({Aj}({Bj},Σu, Q)), Q ∈ QbR}

The upper bound of the identified set of impulse responses for variable i in response to shock

j at horizon h is defined as follows:

ISi,j,h ≡ supQ∈QbR

IRFi,j,h({Aj}({Bj},Σu, Q))

The lower bound ISi,j,h is defined analogously.

The bounds can be obtained using the closed-form expressions provided in Gafarov et al.

(2018). Strictly speaking, an exclusive focus on bounds is, of course, only justified if the

identified set is convex. In almost all applications considered in this paper, this is easy to

establish using Lemma 5.1 in Giacomini & Kitagawa (2016). The sole exception is the case

of multiple simultaneously identified structural shocks, considered in Section B.4.3; even in

that case, numerical explorations suggest convexity of the identified set.

1It is trivial to show that identified set and in particular boundaries are independent of the chosen basisb(Σu). No further reference to the basis is thus needed.

200


B.2 Supplementary results on invertibility

This appendix complements the discussion of Section 2.2.3. First, I prove Proposition 6.

Second, I discuss in more detail why the R2m in the structural model of Smets & Wouters

(2007) is high. And third I conclude that, by near-invertibility, we can, without much loss

of generality, restrict attention to the first entry (P0) of the matrix polynomial P (L).

Shock Weights & Non-Invertibility. I will prove a slightly generalized version of Propo-

sition 6, in fact asserting that

|P`(k, j)| ≤√R2`,j −R2

`−1,j

where R2`,j = 1 − Var (εj,t | {xτ}−∞<τ≤t+`). At ` = 0, this statement is identical to that of

Proposition 6 (since R2−1,j is trivially 0).

Proof. By Lemma B.1 and Lemma B.2 in the Online Appendix,

R20,j = Var (E (εj,t | {xτ}−∞<τ≤t))

= Cov (ut, εj,t)′Σ−1

u Cov (ut, εj,t)

= M ′•,j,0Σ−1

u M•,j,0

Similarly, it can be established that

R2`,j = R2

`−1,j +M ′•,j,`Σ

−1u M•,j,` ∀` ≥ 2

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Next, we know from Lemma B.1 in the Online Appendix and the definition of the VAR

structural shocks et that

et = Q× Σ−1/2u ×

∞∑`=0

M`εt−` ≡∞∑`=0

P`εt−`

where Q ∈ O(nx) is an orthogonal matrix. Summing over all identified shocks k at some

given horizon `, the total squared weights on shock j are given as the (j, j)th element of

M ′`Σ−1u M`

But this is just R2`,j − R2

`−1,j. Any individual squared weight is thus bounded above by the

difference in R2’s.

To assign this maximal weight to a single identified shock, it suffices to have the jth

column of P0 be proportional to the standard basis vector ej. We can ensure this by setting

qj, the jth row of Q, proportional to M ′•,j,0Σ

−1/2u (with normalization to ensure unit length),

and the other nx − 1 rows orthogonal to qj and to each other, again with unit length.

By linking shock weights to a quantitative measure of the degree of invertibility, Propo-

sition 6 formalizes the notion that SVAR inference can succeed if and only if the R20,j is

sufficiently close to 1.

Near-Invertibility in Smets & Wouters (2007). I find that, in a trivariate VAR in

(yt, πt, it) induced by the structural model of Smets & Wouters (2007), the R20,m for monetary

policy shocks is robustly close to 1. It is equal to 0.8702 at my benchmark parameterization,

and it remains high for most draws from the model’s posterior, as well as for much shorter

VAR(p) representations (see Section B.3 in the Online Appendix).

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To understand why the R20,m is so robustly high, it will prove useful to re-write it in a

slightly more interpretable fashion. Following Forni et al. (2019) and Plagborg-Møller &

Wolf (2019a), we can equivalently re-write the R20,j as follows:

R20,j = Cov(ut, εj,t)

′ × Var(ut)−1 × Cov(ut, εj,t) (B.1)

In words, the R20,j is large if and only if the true structural shock εj,t is responsible for a lot

of the variation in the reduced-form VAR forecasting error ut made by the econometrician.

Equivalently, the system is invertible for shock εj,t if and only if

Var(ut)− Cov(ut, εj,t) Cov(ut, εj,t)′︸︷︷︸

≡Varj(ut)

(B.2)

is reduced-rank. To interpret (B.1) and (B.2), suppose first that the econometrician only

observes a single macro aggregate. In that case, the R20,j for a shock j would be high if and

only if that shock accounts for almost all forecasting uncertainty in that single aggregate xt.

Formally, if shock j drives all of the forecasting uncertainty in xt, then the unconditional

variance Var(ut) is equal to the variance induced only by shock j, Varj(ut). Thus, by (B.2),

we have invertibility. Of course, since monetary policy shocks are arguably unimportant for

the evolution of most macro aggregates, the R20,m in univariate ARs is invariably small.

Matters are much more subtle in the multivariate case, however. Here, macro data can be

very (in fact even fully) informative about a quite unimportant shock, if that shock induces

an atypical covariance pattern, in the sense that the induced conditional variance Varj(ut)

is far from being proportional to Var(ut). For example, if unconditionally two entries in ut

co-move, but conditional on shock j they diverge, then intuitively the data should be quite

informative about j; formally, the residual covariance matrix should become nearly singular.

203


This logic explains the robustly high R20,m in Smets & Wouters (2007). If the econome-

trician observes a negative forecast error in nominal interest rates and a positive forecast

error in inflation, then – because only monetary policy shocks induce such conditional co-

movement – she concludes that a contractionary monetary policy shock is reasonably likely

to have occurred. Consistent with this intuition, the R20,m is already large in a bivariate VAR

in (πt, it) (it is equal to 0.7800, so the maximal weight is 0.8832), but remains stubbornly

low in VARs that omit either macro aggregate, in particular the policy rate.2

Another way to see this is to study monetary policy shock R2’s in a variant of the Smets

& Wouters model with forward guidance shocks, as done in Plagborg-Møller & Wolf (2019a).

Figure B.1: R2 for Forward Guidance Shock

Note: I take the benchmark model of Smets & Wouters (2007), but delay the monetary policyshock by two periods. The figure shows the implied R2

`,FG with the conventional small set of macroobservables: output, inflation, and interest rates.

Figure B.1 (which is taken straight from that paper) shows various R2’s for a forward

guidance shock two quarters ahead. In addition to the R20,FG relevant for SVAR inference, I

additionally consider the implied R2’s using current, past and some future values of macro

aggregates – that is, the more general R2`,j terms of Proposition 6. The x-axis in Figure B.1

2For example, in a four-variable VAR in consumption, investment, output and inflation, the R20,m at my

benchmark parameterization of Smets & Wouters (2007) is equal to 0.0934.

204


varies the number of periods ` that the econometrician is allowed to look into the future.

At horizon 0, the R2 is very low, for two reasons. First, forward guidance shocks do not

drive much of the variation in aggregate data (just like conventional policy shocks), and

second, they move inflation and interest rates in the same direction (unlike conventional

policy shocks). As a result, any econometrician observing the conventional small set of macro

aggregates hit by a forward guidance shock would presumably conclude that a demand shock

must have occurred. Two periods out, however, when the shock materializes, interest rates

and inflation are untied, so the econometrician realizes that the economy was presumably

hit by a forward guidance shock – accordingly, the R2 jumps to the high level familiar for

conventional monetary policy shocks.

Weight Decay. Since the structural model of Smets & Wouters (2007) does not feature

news shocks, the incremental information in future realizations of macro aggregates – that

is, R2`,j − R2

`−1,j – is generically small. For example, for monetary policy shocks, we have

R20,m = 0.8702 and R2

∞,m = 0.8763. Consistent with this intuition, the coefficients in the

higher-order entries of the lag polynomial P (L) in (2.9) are generically small.

Figure B.3 in Section B.3 of the Online Appendix provides an illustration for dynamic

shock weights under the identification scheme of Uhlig (2005), studied in Section 2.3. Given

the displayed extremely fast decay of weights, it is nearly sufficient to look at the coefficients

in the first entry P0, as done for most of the plots displayed in Sections 2.3 to 2.5.

B.3 Shock volatility and Bayesian posteriors

I first provide a formalization of the limit results displayed in panel (b) of Figure 2.3.

Proposition 9. In the sign-identified three-equation model, impose the uniform Haar prior

µ over the set of orthogonal matrices O(nx). Denote by λ = λ(σm) the induced posterior

205


probability of a negative output response as a function of the monetary policy shock volatility

σm. Then, if φy > 0,

limσm→0

λ(σ) = 0

limσm→∞

λ(σ) = 1

Proof. The boundaries of the identified set of rotation vectors p are defined as the solution

to the following optimization problems:

max\minp∈S(3)

1

1 + φy + φπκ

(σdpmd + φπσ

spms − σmpmm)

such that

κσdpmd − (1 + φy)σspms − κσmpmm ≤ 0

(φy + φπκ)σdpmd − φπσspms + σmpmm ≥ 0

where S(n) denotes the n-dimensional unit sphere. Now let σd, σs → 0, and fix σm.3 It is

then easy to see that

limσd,σs→0

ISy,m,0 = 0

The Haar prior assigns strictly positive mass to the sign-identified set, since pmm ∈ [0, 1]

and the weights on the demand and supply shocks are unrestricted. But the upper bound is

only attained for a weight of 0 on the monetary policy shock, which maps into a measure-0

subset of the unit sphere under the Haar prior.

3With σm → ∞, the upper bound would remain finite and strictly positive, but the weight on themonetary policy shock corresponding to any strictly positive output response would also be vanishinglysmall, so the argument is unchanged.

206


Next, as σm → 0, the identifying restrictions simplify to

κσdpmd − (1 + φy)σspms ≤ 0

(φy + φπκ)σdpmd − φπσspms ≥ 0

Thus pmd and pms must both be of the same sign. Re-arranging the inequalities, we see that,

if φy > 0, then pmd, pms must both be positive and, whenever pmd, pms > 0, we require that

pmdpms∈[

φπφy + φπκ

,1 + φyκ

]× σs

σd

where the interval has strictly positive length precisely because φy > 0. With the relative

demand and supply weights in this interval, and pmm unrestricted, we see again that the

posterior mass of the identified set is strictly positive. Since pmd, pms ≥ 0, we conclude that

the lower bound of the identified set is 0, attained with a weight of 1 on the monetary policy

shock. Again this maps into a measure-0 subset of the unit sphere under the uniform Haar

prior.

Proposition 9 provides a formal rationalization of the results in Paustian (2007) and

Canova & Paustian (2011): For more volatile structural shocks, the Haar-implied posterior

probability of a correctly signed impulse response increases.

However, it is also straightforward to use the constructive logic of the proof to find

alternative measures µ that, for any finite set of variances σ = (σd, σs, σm), give arbitrary

posterior probabilities for either positive or negative output responses. By the discussion in

Section 2.3.1, the identified set of the static three-equation model always contains strictly

positive and strictly negative output responses. For example, for a given set of variances

σ = (σd, σs, σm), let P+ denote the set of weight vectors that give a strictly positive output

response. For any alternative set of volatilities σ = (σd, σs, σm), and for any p ∈ P+, let

207


p∗i = pi× σi

σi, i ∈ (d, s,m), and p ≡ p∗

||p∗|| . All weight vectors p in the thus defined set P+ give

strictly positive output responses in the rescaled model. It remains to simply pick a measure

µ that assigns arbitrarily large prior probability to P+.4

4Since my results rely on statements about relative shock volatilities, it is unsurprising that the conclusionsof Section 2.3 apply just as well to structural models estimated on different sample periods. For example,in a variant of Smets & Wouters (2007) estimated on pre-Great Moderation samples, all shocks are morevolatile, but again demand and supply shocks are more prominent than monetary policy shocks, so the shapeof posterior distributions over identified sets is largely unchanged.

208

Appendix C


C.1 Equivalence result with finite lag length

We here prove Proposition 8 from Section 3.2.3. We proceed mostly as in the proof of

Proposition 7. As a first step, the Frisch-Waugh theorem implies that

βh(p) =Cov(yt+h, xt(p))

E(xt(p)2). (C.1)

We now introduce the notation Covp(·, ·), which denotes covariances of the data {wt} as

implied by the (counterfactual) stationary “fitted” SVAR(p) model

A(L; p)wt = B(p)ηt, ηt ∼WN (0, I), (C.2)

i.e., where ηt is truly white noise (unlike the residuals from the VAR(p) projection on the

actual data). For example Covp(yt, xt−1) denotes the covariance of yt and xt−1 that would

obtain if wt = (r′t, xt, yt, q′t)′ were generated by the model (C.2) with parameters A(L; p)

and B(p) obtained from the projection on the actual data, as defined in Section 3.2.3. We

209

C. Appendix for Chapter 3

similarly define any covariances that involve ηt. Note that stationarity of the VAR model

(C.2) follows from Brockwell & Davis (1991, Remark 2, pp. 424–425).

It follows from the argument in Brockwell & Davis (1991, p. 240) that Covp(wt, wt−h) =

Cov(wt, wt−h) for all h ≤ p (see also Brockwell & Davis, 1991, Remark 2, pp. 424–425 for the

multivariate generalization of the key step in the argument). In words, the autocovariances

implied by the “fitted” SVAR(p) model (C.2) agree with the autocovariances of the actual

data out to lag p, although generally not after lag p.

Under the counterfactual model (C.2), we have the moving average representation wt =

C(L; p)B(p)ηt, and thus

θh(p) = Cnr+2,•,h(p)B•,nr+1(p) = Covp(yt+h, ηx,t), (C.3)

where ηx,t is the (nr + 1)-th element of ηt. Since B(p) is lower triangular by definition, it is

straight-forward to show from (C.2) that

Bnr+1,nr+1(p)ηx,t = xt − Ep(xt | rt, {wτ}t−p≤τ<t) = xt − E(xt | rt, {wτ}t−p≤τ<t) = xt(p),

(C.4)

where Ep(· | ·) denotes linear projection under the inner product Covp(·, ·), the second

equality follows from the above-mentioned equivalence of Covp(·, ·) and Cov(·, ·) out to lag

p, and the last equality follows by definition. Since Covp(ηx,t, ηx,t) = 1, equation (C.4) implies

Bnr+1,nr+1(p)2 = Covp(xt(p), xt(p)) = E(xt(p)2),

210


where the last equality again uses the equivalence of Covp(·, ·) and Cov(·, ·) out to lag p.

Putting together (C.3), (C.4), and the above equation, we have shown that

θh(p) =1√

E(xt(p)2)× Covp(yt+h, xt(p)).

Under the stated assumption that xt(p) = xt(p − h), the covariance on the right-hand side

above depends only on autocovariances of the data wt at lags ` = 0, 1, 2, . . . , p. Hence, we

can again appeal to the equivalence of Covp(·, ·) with the covariance function of the actual

data, and the expression (C.1) yields the desired conclusion.

C.2 Long-run identification using local projections

We show that the LP-based long-run identification approach in Example 2 is valid. Define

the Wold innovations ut ≡ wt − E(wt | {wτ}−∞<τ<t) and Wold decomposition

wt = χ+ C(L)ut, C(L) ≡ I2 +∞∑`=1

C`L`. (C.5)

Since both structural shocks are assumed to be invertible, there exists a 2×2 matrix B such

that εt = But. Comparing (3.9) and (C.5), we then have Θ(1)B = C(1). Let e1 ≡ (1, 0)′.

Note that the Blanchard & Quah assumption e′1Θ(1) = (Θ1,1(1), 0) implies that

e′1C(1) = e′1Θ(1)B = Θ1,1(1)e′1B,

and therefore

e′1C(1)ut = Θ1,1(1)× e′1But = Θ1,1(1)× ε1,t.

211


By the result in Section 3.2.2, the claim in Example 2 follows if we show that

limH→∞

β′H = e′1C(1). (C.6)

Define Σu ≡ Var(ut). Applying the Frisch-Waugh theorem to the projection (3.13), and

using w1,t = ∆gdpt, we find

β′H = Cov(gdpt+H − gdpt−1, ut)Σ−1u = Cov

(H∑`=0

w1,t+`, ut

)Σ−1u =

H∑`=0

Cov(w1,t+`, ut)Σ−1u .

(C.7)

On the other hand, the Wold decomposition (C.5) implies (recall the fact that ut is white

noise)∞∑`=0

Cov(wt+`, ut)Σ−1u =

∞∑`=0

C` = C(1). (C.8)

Comparing (C.7) and (C.8), we get the desired result (C.6).

C.3 Best linear approximation under non-linearity

Here we give the technical details behind the “best linear approximation” interpretation of

a non-linear model, cf. Section 3.4.4. Assume the nonparametric model (3.19), and that

{wt} is covariance stationary and purely nondeterministic. Let the linear projection of wt on

the orthonormal basis (εt, εt−1, εt−2, . . . ) be denoted∑∞

`=0 Θ∗`εt−`, with projection residual

vt. Assume vt is either identically zero or purely non-deterministic. Then it has a Wold

decomposition

vt = µ∗ +∞∑`=0

Ψ∗`ζt−`,

where {ζt} is nw-dimensional white noise with Cov(ζt) = Inw . Since vt is a function of

{ετ}τ≤t, and {εt} is i.i.d., we have Cov(εt+`, vt) = 0nε×nw for all ` ≥ 1. Moreover, since vt is

212


a residual from a projection onto {ετ}τ≤t, we also have Cov(εt+`, vt) = 0nε×nw for all ` ≤ 0.

By the Wold decomposition theorem, ζt lies in the closed linear span of {vτ}τ≤t, so we must

have Cov(εt+`, ζt) = 0nε×nw for all ` ∈ Z. Finally, the best linear approximation property

(3.20) is a standard consequence of linear projection. We have thus verified all claims made

in Section 3.4.4.

213

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Essays on Identification in Macroeconomics · 2020. 7. 13. · Abstract This dissertation consists of three independent chapters on questions of identi cation and causal e ect estimation