A Structural Investigation of Quantitative EasingPreliminary results, comments are very welcome. Latest version:

http://gregorboehl.com/live/qe_bs.pdf

Gregor Boehl∗, Felix Strobel

Institute for Monetary and Financial Stability, Goethe University Frankfurt

April 16, 2019

Abstract

Did the measures of unconventional monetary policy conducted during the Great Reces-sion induce real effects? We seek to answer this question by using an estimated mediumscale DSGE model featuring a banking sector, financial frictions and the endogenouslybinding effective lower bound on interest rates (ELB) to decompose the US dynamics.We conclude that from 2009 to 2015 the overall quantitative easing (QE) measures in-creased output about 0.5 percent. While the effects of measures of liquidity provisionwere negligible, both the bond and MBS purchases had positive impact on consumption.However, other than the MBS purchases, government bond purchases had negative effectson investment and a negative net contribution to GDP. This can be explained by thefact that, whereas QE stimulates asset prices in the short-run, the persistent reductionof excess spreads can lower banks’ net worth, the loan supply, and hence real economicactivity in the mid-run. Forecasts suggest that, through the link between banks balancesheet and investment, shutting down the QE program will have a strong recessionary”hangover” effect, leading to a net-decrease in output for the years after the end of QE.

Keywords: Quantitative Easing, Liquidity Facilities, Zero Lower Bound, BayesianEstimationJEL: E63, C63, E58, E32, C62

1 Introduction

In 2008, the financial sector in the US collapsed, causing a sharp drop in asset pricesand credit supply and resulting in what is known today as the Great Recession. Asa response the Federal Reserve reduced the Federal Fund Rate to zero and engaged inmeasures of quantitative easing (QE), thereby drastically extending the size of its balancesheet. Early on with the unfold of the crisis, it also assumed its role as a lender of last

?We are grateful to Alexander Meyer-Gohde and participants of several conferences and seminars fordiscussions and helpful comments on the contents of this paper.∗Corresponding authorEmail addresses: boehl@econ.uni-frankfurt.de, strobel@econ.uni-frankfurt.de

resort and injected large amounts of liquidity directly into the banking sector. Whilethis initial response was undertaken to counter the immediate effects of the crisis, laterprograms had the purpose of stimulating the economy in a low-growth environment. Assuch, it additionally carried out large-scale purchases of government debt and MortgageBacked Securities lasting until 2014.

The effects of these measures are subject to a controversial debate. Critics’ assess-ments range from ”a drop in the bucket” (Cochrane, 2011) to ”[not having] fulfilled itsstated objective of stimulating the economy” (Taylor, 2014). On the other side, propo-nents firmly stated that ”evidence suggests that forward guidance and security purchases[...] did help spur consumption and business spending, lower the unemployment rate, andstave off disinflationary pressures” (Yellen, 2017). In an attempt to defend his policies,Ben Bernanke famously quipped that ”the problem with QE is, it works in practice, butit doesn’t work in theory”.

While there is an emerging consensus that QE measures had at least some impacton financial market indicators,1 evidence for the effects on the real economy is stillsparse. We seek to fill this gap by employing a DSGE model, which incorporates severalQE measures and takes account of the effective lower bound on nominal interest rates(ELB), as a lens to look at the data. The model is a hybrid of the models by Boehland Strobel (2019) – in turn a variant of Smets and Wouters (2007) – and Gertler andKaradi (2013). It thereby features all the bells and whistles to be suitable for estimation,has properties allowing ELB durations of an extended period of time, and it includes thefinancial frictions needed to allow for effects of QE on the real economy. Quantitativeeasing in our model affects different parts of the balance sheet of the financial sector,capital holdings, bond holdings, and its liabilities. The effects are passed through to thereal economy via the supply of loans to non-financial firms. Following the estimation, weanalyse the drivers of macroeconomic dynamics in the aftermath of the Great Recessionand quantitatively assess the effects of QE on the US economy.

We find that QE had a relevant impact on the US economy. According to our es-timates, in its peak, QE raised aggregate output and labor hours by roughly 0.7%,consumption by 0.6%, and aggregate investment by almost 4%. The impact on inflationwas very small. However, while we find that QE had a stimulating impact, this stimuluswas small in comparison to the sharp and persistent drop in output and labor below thepre-crisis trend. Furthermore, our analysis suggests that the stimulating effects of QEmight be followed by a drag on the real economy once the balance sheet of the centralbank shrinks. In our model this drag can be attributed to the persistent decline in ex-cess spreads on capital assets and government bonds, which weakens the balance sheetof financial intermediaries and in turn reduces investment and output in the long run.When decomposing the effects of QE into the effects of three separate policy measures– purchases of capital assets and bonds, as well as emergency liquidity injections intothe banking sector – we find that the purchases of capital assets had the strongest effectin stimulating the economy. The purchases of bonds on the other hand, while slightlyincreasing consumption, crowded out investment and had a negative net-effect on outputand labor. The effects of liquidity injections on asset prices and interest rate spreads we

1See, e.g., Swanson (2017), Del’Ariccia et al. (2017), Borio and Zabai (2018), and Kuttner (2018).For a critique see Greenlaw et al. (2018).

2

found to be sizable, but they die out quickly. Hence their long term effect on the realeconomy is negligible.

The financial sector in our model builds on Gertler and Karadi (2011) and Gertlerand Karadi (2013). In recent years, this general framework has often been adapted andextended to capture various aspects of the global financial crisis.2 Within this model, theprivate securities purchased by the central bank as part of its QE programs are in essenceclaims on the productive capital stock, whereas the large bulk of non-government bondassets purchased by the Fed during the last decade were mortgage backed securities.Despite this discrepancy, we model the link between financial and real economy as inGertler and Karadi (2013) since on the one hand it captures the key link between thebalance sheet policy of the central bank and the decline in risk premia and lending rates inthe economy, and on the other hand it facilitates an easy comparison to theoretical resultsin the literature. The effects of the asset purchases by the central bank are thereforelargely similar to the effects in Gertler and Karadi (2013): they initially stimulate theeconomy, but a few years after the implementation of the policy measures the economydrops below the steady state. However, given the additional rigidities and our estimatedparameter values, these QE effects are somewhat smaller than in the calibrated model.

There are several other papers that analyze the effects of QE. Chen et al. (2012)find that the effects of government bond purchases in a model with segmented bondmarkets are rather small, resulting in a 0.5 % increase in GDP and very little effect oninflation, whereas Del Negro et al. (2017) find in a calibrated model that in response toa shock that raises the convenience yield, liquidity injections can have a sizable effect.Accordingly, our results are more in line with the modest effects of QE implied by Chenet al. (2012). Curdia and Woodford (2011) point out, that in a model, in which firmsfinance investment with long-term debt, the effect of QE depends on which type of assetsthe central bank buys. Bond purchases are neutral in their framework, whereas purchasesof assets from markets which are in distress can alleviate the pressure on the financialsector and stimulate the economy. In contrast, government bond purchases do haveeffects in our framework. Nonetheless, we confirm the assertion that purchases of capitalassets are more effective. Both, bond and capital purchases, however, lower investment inthe long run according to our analysis. This is consistent with a different line of thoughtsuggested by Brunnermeier and Koby (2018), who warn of potentially contractionaryeffects of QE. They argue that the persistently decreased net margins of the financialsector can induce a reduction of real activity. Our quantitative result of a hangover afterthe initial stimulus of QE is consistent with their theoretical argument. The design ofthis study is closely related to Boehl and Strobel (2019), where we focus on the effects offorward guidance in a medium-scale model without a financial sector. We document thatthe most parameter estimates are largely unaffected by the introduction of a financialsector in Appendix C.

This paper is also linked to empirical work on the effects of QE. Using a BayesianVAR, Hesse et al. (2017) investigate on the effects of announcements of asset purchases.They find that these announcements had rather moderate effects. Shocks normalizedto 1% of GDP, increase GDP and CPI by roughly 0.2% for both the US and the UK.Additionally they find that the earlier purchase programs had slightly stronger effects in

2Some examples are Gertler et al. (2012), Gertler and Kiyotaki (2015), Kirchner and van Wijnbergen(2016), Rannenberg (2016), Meeks et al. (2017)

3

the US and in the UK than the later programs, as they had a larger surprise effect onmarket participants. For the UK, VAR studies from Kapetanios et al. (2012) find largereffects of QE. Their results suggest peak effects of 1.5% on GDP and 1.25% on annualizedCPI. For Japan Schenkelberg and Watzka (2013) find rather small effects of QE on thereal economy. All considered, our finding of significant positive macroeconomic effectsare broadly in line with the VAR literature on this topic. The literature on the effectsof QE on financial variables focusses on the announcement effects of QE in daily orhigh-frequency data. Gilchrist and Zakrajsek (2013) find that announcements of largescale asset purchases (LSAPs) led to a significant reduction in corporate credit risk, asmeasured by CDS indexes. Swanson (2017) separately identifies the effects of forwardguidance and LSAP announcments on financial market variables and finds that forwardguidance is more successful at moving short-term Treasury yields, whereas LSAP had alarger effect on long-term yields. The finding that QE had important effects on financialconditions is corroborated by Borio and Zabai (2016) and Kuttner (2018). In contrastGreenlaw et al. (2018) exploit the movement of financial variables on FOMC dates andother days of large bond yield movements and conclude that the role of QE for financialmarkets is rather small. In our paper, we use quarterly data instead of high-frequencydata and realized policy measures instead of announcements. Although not directlycomparable for this reason, our results that LSAPs had indeed a sizable impact on riskpremia speaks to the findings of the bulk of this literature.

Apart from our results regarding the effects of monetary policy measures, our findingssuggest that the dynamics of macroeconomic and financial variables were mainly drivenby risk premium shocks. While Beraja et al. (2016) point out that the effects and theimportance of different shocks for the US business cycle and in particular for labor marketvariables depend on the degree of disaggregation of the data, the important role of riskpremium shocks for the post-Great Recession era finding appears to be a robust result instructural estimations on US-wide aggregate data (see, e.g. results in Gust et al. (2017),Kulish et al. (2017), Linde et al. (2017), and Boehl and Strobel (2019)). In contrastto Christiano et al. (2015), who analyze the Great Recession within a framework witha detailed modelling of the labor market, we abstain in this paper from including afine-grained labor market in our model in order to facilitate an easier comparison withtheoretical results in the literature on the effects of QE on the business cycle.

To our knowledge, the analysis of the real effects of QE from the perspective of anestimated, structural model model is a novel contribution to the literature. In order toovercome the challenges associated with the estimation of a model at the effective lowerbound, we employ the methods suggested by Boehl (2019). The linearized model is solvedwith the effective lower bound as an endogenous occasionally binding constraint. Thelikelihood is evaluated with an ensemble-type Kalman filter (the IPA smoother, which is ahybrid between Particle Filter and Iterative Kalman Filter), and full Bayesian estimationof the parameters using the Affine Invariant MCMC Ensemble sampler (Goodman andWeare, 2010). The IPA smoother is also used to extract the distribution of economicshocks for our quantitative analysis.

Section 2 presents the model in use. Section 3 includes a brief description of thedata, as well as of the solution, filtering and estimation methodology, and presents anddiscusses the estimation results. Section 4 entails our analysis of the drivers of the USeconomy, as well as a detailed discussion of the effects of QE measures. Lastly, Section5 concludes.

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2 Model

For our analysis, we extend a standard medium scale model as in Smets and Wouters(2007) with separable preferences as in Boehl and Strobel (2019) and financial intermedi-ates as in Gertler and Karadi (2011), which extend loans to nonfinancial firms and holdgovernment bonds. This section gives a brief description of the model.

Time is discrete, and one period in the model represents one quarter. The modelfeatures households, banks, intermediate good producers, capital good producers, retail-ers, labor unions, a fiscal and a monetary authority. Households consume, supply labor,and save in the form of bank deposits. Labor is differentiated by a union with monopolypower that faces nominal rigidities. The firm sector consists of three types of firms.Intermediate good producers employ labor and capital to produce their goods. Each pe-riod, after producing their output, they sell their used capital stock to the capital goodsproducers. The latter repair it, and invest in new capital. At the end of the period theyre-sell the capital to the intermediate good producers, which use it for production in thenext period. The intermediate good producers finance their purchases of capital withloans from the banks. Intermediate goods are purchased by retailers, which repackagethem, and sell them with a markup as final goods to households, the capital producers,and to the government. Banks hold loans and government bonds on the asset side oftheir balance sheets. On the liability side are deposits and the banks net worth. Thegovernment consumes final goods, collects taxes, and issues government bonds. Mon-etary policy takes the form of a Taylor rule. The model includes habit formation inconsumption, convex investment adjustment costs, and nominal rigidities as in Calvo(1983) in both, price and wage setting to enhance the empirical plausibility of the modeldynamics,

2.1 Households

There is a continuum of households with a unit mass. As in Gertler and Karadi(2011) a constant fraction f of each household’s members works as banker, whereas theother fraction (1 − f) consists of workers who supply labor to the intermediate goodproducers. While workers receive their wage income every period, bankers reinvest theirgains in asset holdings of the bank over several periods, and contribute to the householdsincome only when exiting the banking sector, bringing home the accumulated profits. Toensure that both fractions of the household face the same consumption stream, perfectconsumption insurance within the household is assumed. Households’ expected lifetimeutility is as follows

Ut = E0

∞∑t=0

βt(

(Ct(k)− hCt−1)1−σc − 1

1− σc− ν log(1− Lt(k))

)where parameters β, h, σc, and ν are, respectively, the discount factor the degree ofexternal habit formation in consumption, the coefficient of relative risk aversion, and aweight on the disutility of labor. Ct(k) is consumption and Lt(k) are the hours worked.Habit formation is with respect to past aggregate consumption, Ct−1 The optimizinghousehold k earns nominal wages Wt from its labor supply Lt(k) and receives income fromone-period bank deposits, Dt(k), that yield an interest rate, Rt. As for all stock variables,we use the end-of-priod notation, so that Dt(k) denotes the household’s deposits at the

5

end of period t. The return on deposits is subject to a disturbance term, vd,t which followsan AR(1) process. This term drives a wedge between the interest rate controlled by thecentral bank and the required return on assets. Smets and Wouters (2007) label theseshocks risk-premium shocks. In our context, this shock to the deposit rate can as wellbe interpreted as variations in the confidence in the banking system. Households spendtheir funds on consumption Ct(k), and save in new deposits. The budget constraint ofconsumers reads

PtCt(k) +Dt(k)

vd,tRt= Dt−1(k) +WtLt(k)− Tt + Pt. (1)

Here, Pt denotes the price level at time and Tt are lump sum taxes raised by thegovernment to finance government spending, and Pt are profits of monopolistic firmsand banks that accrue to the households.

This formulation of the utility function – separability in consumption and leisure –stands in contrast to the non-separable formulation in Smets and Wouters (2007). AsBoehl and Strobel (2019) argue, the non-separable formulation has unfavorable effectson the size of the determinacy region at the ELB. While the same model with separablepreferences can, given an appropriately strong recessionary shock, easily stay at theELB for 25+ periods (which are of course fully anticipated on impact), in the modelwith non-separable preferences larger shocks quickly lead to non-existence of an rationalexpectation equilibrium (without additional central bank activity).

2.2 Firm sectors

The model contains three types of firms. Intermediate goods are produced by perfectlycompetitive firms, which use capital and labor as inputs for production. Monopolisticallycompetitive retailers buy a continuum of intermediate goods, and assemble them into afinal good. Nominal frictions as in Calvo (1983) make the retailers optimization problemdynamic. Additionally, a capital producing sector buys up capital from the intermediategood producer, repairs it, and builds new capital, which it sells to the intermediate goodsector again. Investment in new capital is subject to investment adjustment costs.

2.2.1 Intermediate good producers

Intermediate good producers are in monopolistic competition, employ labor and cap-ital services from households, and set their prices as markups over the marginal cost.Firm i produces according to the Cobb-Douglas Function

Ym,t(i) = eztKt(i)α(γtLt(i)

)1−α − γtΦ. (2)

Here, Ym,t(i) are intermediate goods, zt is a TFP-shock, Kt(i) is the effective capital usedin production defined as Kt(i) = Ut(i)Kt−1(i), and parameter α is the output elasticitywith respect to effective capital. γt represents the labor-augmenting growth rate in theeconomy, and Φ is the fixed cost of production.

At the end of each period the intermediate good producer sells the capital stock thatit used for production to the capital producer which repairs the capital, and purchasesthe capital stock that it is going to use in the next period from the capital producer.To finance the purchase of the new capital at the price Qt per unit, it issues a claim for

6

each unit of capital it acquires to banks, which trade at the same price. The interestrate the firm has to pay on the loan from the bank is Rk,t. Under the assumption thatthe competitive firms make zero profits, the interest rate on their debt will just equal therealized ex-post return on capital. Furthermore, we assume that the firm incurs costsof capital utilization that are proportional to the amount of capital used, a(Ut)Kt. Theperiod profit function of firm i is therefore

Pm,t(i)Yt(i)−WtLt(i)−RKt Kt(i)− aUt(i)Kt(i).

Hence each period the firm determines its optimal capital purchase by maximizing

Et[βΛt,t+1(−Rk,t+1QtKt(i)+Pm,t+1(i)Ym,t+1(i)−Wt+1Lt+1(i)−a(Ut)Kt(i)+(1−δ)Qt+1Kt(i))]

with respect to Kt(i). As all firms make the same decisions, we can drop the index i. Inoptimum the ex-post return then is as follows

Rk,t+1 =Pm,t+1α

Ym,t+1

Kt+ (1− δ)Qt+1

Qt. (3)

Additionally, the choices for optimal labor input and optimal capital utilization yield thefirst order conditions

Wt = Pm,t(1− α)Ym,tLt

. (4)

a′(Ut)Kt = αPm,tYm,tUt

(5)

2.2.2 Capital good producers

Capital evolves according to the following law of motion

Kt = (1− δ)Kt−1 + vi,t

(1− S

(ItIt−1

))It, (6)

where δ is the depreciation rate and the function S(·), indicates a cost of adjusting thelevel of investment. In steady state it holds that S = 0, S′ = 0, and S′′ > 0. and vi,tfollows an AR(1) process. While Smets and Wouters (2007) interpret vi,t as an investmentspecific technology disturbance, Justiniano et al. (2011) stress that this shock can as wellbe viewed as a reduced-form way of capturing financial frictions, as it drives a wedgebetween aggregate savings and aggregate investment.The capital good producer’s role in the model is to isolate the investment decision thatbecomes dynamic through the introduction of convex investment adjustment costs, whichis a necessary feature to generate variation in the price of capital. Capital good producersbuy the used capital, restore it and produce new capital goods. Since capital producersbuy and sell at the same price, the profit they make is determined by the differencebetween the quantities sold and bought, i.e. investment. Capital producers bear theresource costs associated with changes in investment. They choose the optimal amount

7

of investment to maximize

E0

∞∑t=0

βtΛ0,t

Qt

(1− S

(It(k)

It−1(k)

))vi,t − 1

It.

The first order condition of the capital producer reads

1 = Qtvi,t

(1− S

(ItIt−1

)− S′

(ItIt−1

)ItIt−1

)+Et

Qt+1vi,t+1S

′(It+1

It

)(It+1

It

)2

2.2.3 Final good producers

Final good producers buy the goods produced by the intermediate good producersand sell them to final good producers. They act under monopolisitic competition. Eachperiod, retailers firms face a constant probability of being able to optimally adjust theirprices, ζp. Those firms, which cannot optimally adjust their prices in a given period,index their prices to a weighted average of last periods inflation and steady state inflation.These assumptions give rise to the following maximization problem for firm i

maxPt(i)

Et

∞∑s=0

(βζp)s Λt,t+s

Πt,t+s

[Pt(i)Π

sl=1(Π

ıpt+l−1Π1−ıp)−MCt+s

]Yt+s(i) (7)

s.t.Yt+s(i)

Yt+s= G

′−1

(Pt(i)Π

sl=1(Π

ıpt+l−1Π1−ıp)

Pt+sτt+s

). (8)

Pt(i) is the price set by firm i, Πt,t+s is the accumulated change in the aggregate pricelevel between periods t and t+s, Λt,t+s is the stochastic discount factor of the firm, Yt+sis the demand by final good firms for intermediate goods. Parameter ıp is the degree ofprice indexation. Function G governs how the relative price of firm i affects the amountsof goods it can sell. We make the same assumptions on G as Smets and Wouters (2007).3

Furthermore, τt+s ≡∫ 1

0G′(Yt+s(i)Yt+s

)Yt+s(i)Yt+s

di. The aggregate price index is in this case

given by

Pt = [(1− ζp)(P ∗t )G′−1

[P ∗t τtPt

]+ ζpΠ

ıpt−1Π(1−ıp)Pt−1G

′−1

[Π

ıpt−1Π(1−ıp))Pt−1τt

Pt

](9)

where P ∗t is the optimal price in period t. The price markup set by final goods producers,vp,t, is time-varying and subject to markup shocks, vp,t, which follow an AR(1)-process.

3That is that G′ > 0, G′′ < 0, and G(1) = 1. As shown by Kimball (1995), the assumptions on Gimply that the demand for a good is decreasing in its relative price, and that the elasticity of demand fora good increases with its relative price, which in turn implies a higher persistence of aggregate inflationdynamics.

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2.2.4 Retailers

Retailers act under perfect competition. They buy the goods from final good pro-ducers, bundle them in final goods. and sell them to the public. Their maximizationproblem reads

maxYt,Yt(i)

PtYt − Pt(i)Yt(i) (10)

s.t.

[∫ 1

0

G

(Yt(i)

Yt, λp,t

)di

]= 1. (11)

2.2.5 Unions and Labor Packers

The supply of labor to intermediate good firms is organized by unions and labor pack-ers. Households supply labor to a labor union, which differentiates the labor services andsets wages. Unions act in monopolistic competition with each other and set their wages,Wt(i), as a markup over the average marginal rate of substitution between consumptionand leisure of households. We assume that the wage setting process, in the same way asprice setting, is subject to a Calvo type friction, and that unions, which cannot adjusttheir wages in a given period, index their last wage to a weighted average of last periodsinflation and steady state inflation. Labor packers buy the labor services from unions,bundle them, and provide them to intermediate good firms at the wage Wt. Thus, themaximization problem of labor packers is

maxLt,Lt(i)

WtLt −Wt(i)Lt(i) (12)

s.t.

[∫ 1

0

Gw

(Lt(i)

Lt, λw,t

)di

]= 1, (13)

where Lt is labor provided by labor packers to intermediate good firms, Lt(i) denotesthe labor services sold by unions to labor packers, and Gw is the labor aggregator onwhich we make the same assumptions as on G. The wage markup, λw,t, is time-varyingand subject to wage markup shocks, vw,t, which follows an AR(1)-process, vw,t.

Labor unions observe the average marginal rate of substitution between consumptionand leisure of all households,

MRSt =ν(Ct − hCt−1)σ

(1− Lt), (14)

and charge a markup on top of it. Their maximization problem is

maxWt(i)

Et

∞∑s=0

(βζw)sΛt,t+sΠt,t+s

[Wt(i)Π

sl=1(Πıw

t+l−1Π1−ıw)−MRSt+s]Lt+s(i) (15)

s.t.Lt+s(i)

Lt+s= G

′−1w

(Wt(i)Π

sl=1(Πıw

t+l−1Π1−ıw)

Wt+sτwt+s

). (16)

Wt(i) is the wage set by union i, parameter ıw is the degree of wage indexation, and

9

τwt+s ≡∫ 1

0G′w

(Lt+s(i)Lt+s

)Lt+s(i)Lt+s

di. The aggregate wage index is in this case given by

Wt = [(1−ζw)(W ∗t )G′−1w

[W ∗t τ

wt

Wt

]+ζwΠıw

t−1Π(1−ıw))Wt−1G′−1w

[Πıwt−1Π(1−ıw))Wt−1τ

wt

Wt

],

(17)where W ∗t is the optimal wage set by labor unions in period t.

2.3 Banks

Banks finance their operations by creating deposits, Dt, which are held by households,and by their net worth, Nt. They use their funds to extend loans to intermediate goodproducers for acquiring capital, Kt, and for the purchases of government bonds, Bt attheir market price Qbt . To distinguish the capital assets and bonds that are held by banksfrom those which are held by the central bank, we denote them with Kb,t and Bb,tThebalance sheet of bank j is given by

QtKb,t(j) +QbtBb,t(j) = Nt(j) +Dt(j). (18)

The banks retain the earnings, generated by the return on their assets purchased in theprevious period, and add it to their current net worth. Thus, the law of motion for thenet worth of a bank is given by

Nt(j) = Rk,tQt−1Kb,t−1(j) +Rb,tQbt−1Bb,t−1(j)− vd,tRt−1Dt−1(j). (19)

Note that while the interest rate on deposits raised in period t− 1, is determined in thesame period, the return of the risky capital assets and risky government bonds purchasedin period t−1 is determined only after the realization of shocks at the beginning of periodt. Substituting the balance sheet into the law of motion for net worth yields

Nt(j) = (Rk,t−vd,tRt−1)Qt−1Kb,t−1(j)+(Rb,t−vd,tRt−1)Qbt−1Bb,t−1(j)+vd,tRt−1Nt−1(j).(20)

Bankers continue accumulating their net worth until they exit the business. Each period,each banker faces a lottery, which determines, regardless of the history of the banker,whether he exits his business or stays in the sector. Bankers exit the business with anexogenous probability, 1− θ, or continue their operations with probability θ. The drawsof this lottery are i.i.d.. When a banker leaves the sector, he adds his terminal wealthVt(j) to the wealth of its household. Therefore, bankers seek to maximize the expecteddiscounted terminal value of their wealth

Vt(j) = max Et

∞∑i=0

(1− θ)θiβi+1Λt,t+1+iNt+1+i(j)

= max Et [βΛt,t+1(1− θ)Nt+1(j) + θVt+1(j)] .

As banks operate under perfect competition, in the case of perfect capital markets therisk adjusted return on loans and government bonds would equal the return on deposits.However, bankers face an endogenous limit on the amount of funds that householdsare willing to supply as deposits. Following Gertler and Karadi (2011), we assume that

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bankers can divert a fraction of their assets and transfer it to their respective households.However, if they do so, their depositors will choose to withdraw their remaining fundsand force the bank into bankruptcy. To avoid this scenario, households will keep theirdeposits at a bank only as long as the bank’s continuation value is higher or equal to theamount that the bank can divert. Formally, the incentive constraint of the bank reads

Vt(j) ≥ λQtKb,t(j) + λbQbtBb,t(j), (21)

where λ, is the fraction of loans that the bank can divert, and λb is the fraction ofgovernment bonds it can divert. We calibrate λb to be smaller than λ. This is motivatedby the fact, that, in general, the collateral value of government bonds is higher than thatof loans.4 The reason is that loans to private firms are less standardized than governmentbonds contracts. Additionally, information on the credit-worthiness of the governmentis publicly available, while the credit-worthiness of private firms is often only known tothe bank and the firm, and not easy to assess for depositors, making it easier for banksto misrepresent the value of loans.The initial guess for the form of the value function is

Vt(j) = νk,t(j)QtKb,t(j) + νb,t(j)QbtBb,t(j) + νn,tNt(j), (22)

where νk,t, νb,t and νdjt are time varying coefficients. Maximizing (22) with respect toloans and bonds, subject to (21) yields the following first order conditions for loans,bonds, and µt(j), the Lagrangian multiplier on the incentive constraint

νk,t(j) = λµt(j)

1 + µt(j), (23)

νb,t(j) = λbµt(j)

1 + µt(j), (24)

νk,t(j)QtKb,t(j) + νb,t(j)QbtBb,t(j) + νn,t(j)Nt(j) = λQtKb,t(j) + λbQ

btBb,t(j). (25)

Given that the incentive constraint binds5, a bank’s supply of loans can be written as

QtKb,t(j) =νb,t(j)− λbλ− νk,t(j)

QbtBb,t(j) +νn,t(j)

λ− νk,t(j)Nt(j). (26)

As (27) shows, the supply of loans decreases with an increase in λ, which regulates thetightness of the incentive constraint with respect to capital, and increases with an increasein λb, which makes the holding of bonds more costly in terms of a tighter constraint.Plugging the demand for loans into (23), and combining the result with (24) and (25)

4This is in the vein of Meeks et al. (2014), who in their setting use the same approach to distinguishbetween the collateral values of loans and asset backed securities.

5The constraint binds in the neighborhood of the steady state. For convenience, we make the as-sumption that it is binding throughout all experiments.

11

one can write the terminal value of the banker as a function of its net worth6

Vt(j) = (1 + µt(j))νn,t(j)Nt(j). (27)

A higher continuation value, Vt(j), is associated with a higher shadow value of holdingan additional marginal unit of assets, or put differently, with a higher shadow value ofmarginally relaxing the incentive constraint. Defining the stochastic discount factor ofthe bank to be

Ωt(j) ≡ Λt−1,t((1− θ) + θ(1 + µt(j))νn,t(j)), (28)

plugging (28) into the Bellman equation, and using the law of motion for net worth, onecan then write the value function as

Vt(j) = Et[βΛt,t+1(1− θ)Nt+1(j) + θVt+1(j)]

= Et[βΩt+1(j)((Rk,t+1 − vd,tRt)QtKt(j) + (Rb,t+1 − vd,tRt)QbtBt(j) + vd,tRtNt−1(j))],

and verify the initial guess for the value function as

νk,t(j) = βEtΩt+1(j)(Rk,t+1 − vd,tRt), (29)

νb,t(j) = βEtΩt+1(j)(Rb,t+1 − vd,tRt), (30)

νn,t(j) = βEtΩt+1(j)vd,tRt. (31)

To facilitate aggregation of financial variables, we assume that banks share the samestructure to the extent that they derive the same respective values from holding loans andbonds, and from raising deposits (i.e., ∀j : νk,t(j) = νk,t, νb,t(j) = νb,t, νn,t(j) = νn,t).Furthermore, we assume that all banks have the same ratio of capital assets to governmentbonds, ςt ≡ QtKt

QbtBt, in their portfolio. As an implication, the leverage ratio of banks does

not depend on the conditions that are specific to individual institutes, and all banksshare the same weighted leverage ratio7

φt ≡νn,t(1 + ςt)

(λ− νk,t)(1 + λbλ ςt)

=QtKb,t +QbtBb,t

Nt. (32)

Note that the lower divertability of government bonds relative to capital assets, allowsthe bank to increase its leverage ratio, compared to a scenario in which banks only holdcapital assets. The aggregate balance sheet constraint reads

QtKb,t +QbtBb,t = Dt +Nt. (33)

The net worth of the fraction of bankers that survive period t−1 and continue operatingin the banking sector, θ, can be written as

Not = θ[Rk,tQt−1Kb,t−1 +Rb,tQ

bt−1Bb,t−1 − vd,tRt−1Dt−1

]. (34)

A fraction (1− θ) of bankers leaves the business. There is a continuum of bankers, and

6Detailed derivations are delegated to the appendix.7Details are delegated to the appendix.

12

the draws out of the lottery, which determines whether a banker stays in business orexits the sector, are iid. Hence, by the law of large numbers, it follows that the shareof assets that leaves the sector is a fraction (1 − θ) of the total assets. At the sametime, new bankers enter the sector. New bankers are endowed with ”start-up funding”by their households. The initial endowment of the new bankers is proportionate to theassets that leave the sector. The net worth of the new bankers, Nn,t, can be written as

Nn,t = ω[Qt−1Kb,t−1 +Qbt−1Bb,t−1

], (35)

where ω is calibrated to ensure that the size of the banking sector is independent of theturnover of bankers. Aggregate net worth, Nt, is then the sum of the net worth of oldand new bankers

Nt = Not +Nn,t. (36)

2.4 Policy and market clearing

The policy makers in our model are the central bank as conductor of interest ratepolicy and QE, and the government sector.

2.4.1 Monetary Policy

The central bank in this model has various policies at its disposal. The control of theshort term nominal interest rate, large-scale purchases of government bond and privateassets, and liquidity injections to the banking sector.The short-term nominal interest rates follows a Taylor rule that takes account of theoccasionally binding nature of the effective lower bound

Rt = max

1, Rρt−1

(R

(Πt

Π

)φπ (YtY

)φy ( YtYt−1

)φdy)(1−ρ)

evr,t

.

ρ is the interest smoothing parameter, and φπ, φy, and φdy are the coefficients, whichgovern the strength of the response of interest rate policy to deviations of inflation,output and output growth from their target level. vr,t represents a shock to the Taylorrule and follows an AR(1)-process.Additionally, the central bank purchases capital assets and government bonds. The shareof central bank capital assets and government bonds of the economy-wide stock of capitaland bonds can be written as

XK,t =Kt −Kb,t

Kt,

XB,t =Bt −Bb,t

Bt.

We model these purchases as exogenous and following AR(1) processes. Lastly, thecentral can lend to banks in the form of emergency liquidity facilities, which are modelledas well as exogenous and following an AR(1) process. We assume that emergency liquidityis lent at a zero interest rate and as in Gertler and Kiyotaki (2010) they relax the incentiveconstraint of commercial banks. The extended incentive constraint of commercial banks

13

then readsVt ≥ λQtKb,t + λbQ

btBb,t − λcblCBLt, (37)

where CBLt are central bank liquidity injections, and λcbl governs the degree to whichthe liquidity injections relax the constraint. As a consequence, central bank liquidityaffects the supply of bank loans to firms

QtKb,t =νb,t − λbλ− νk,t

QbtBb,t +λcbl

λ− νk,tCBLt +

νn,tλ− νk,t

Nt. (38)

Throughout all our exercises we assume that the central bank purchases assets from allbanks equally and provides the same amount of liquidity to all banks.

2.4.2 Fiscal policy and market clearing

Government spending, Gt, is exogenous and follows an AR(1) process

Gt = Gegt , (39)

and gt = ρggt−1 + εgt , (40)

where G is the steady state government consumption, ρg is the autocorrelation of govern-ment consumption, and εgt is a shock to government spending. The government financesits expenditures, by issuing government bonds, which are bought by banks and the cen-tral bank, as well as by raising lump sum taxes, Tt. Taxes follow a simple feedback rule,such that they are sensitive to the level of public debt,

Tt = T + κb(Bt−1 −B), (41)

where T and B are the steady state levels of tax revenue and government debt, respec-tively. κb is set to ensure that the real value of debt grows a rate smaller than the grossreal rate on government debt. As shown by Bohn (1998), this rule is a sufficient conditionto guarantee the solvency of the government. To allow for the calibration of a realisticaverage maturity of government debt, bonds are modeled as consols with geometricallydecaying coupon payments, as in Woodford (1998) and Woodford (2001). A bond issuedin period t at the price of Qbt , pays out a coupon of rc in period t+ 1, a coupon of ρcrcin period t + 2, a coupon of ρ2

crc in t + 2, and so on. Setting the decay factor ρc equalto zero captures the case of a one-period bond in which the entire payoff of the bond isdue in period t + 1. Setting ρc = 1 delivers the case of a perpetual bond. The averagematurity of a bond of this type is 1/(1 − βρc). For investors, this payoff structure isequivalent to receiving the coupon rc and a fraction, ρc, of a similarly structured bond inperiod t + 1. The beginning-of-period debt of the government can thus be summarizedas (rc + ρcQ

bt)Bt−1.

Rb,t =rc + ρcQ

bt

Qbt−1

. (42)

The flow budget constraint of the government reads

Gt +Rb,tQbt−1Bt−1 = QbtBt + Tt (43)

14

Finally, the aggregate resource constraint in real terms reads

Yt = Ct + It +Gt + a(Ut)Kt−1,

where the last term on the right hand side of the equation marks the resource costs ofadjusting the utilization of installed capital.

2.5 Linearized Equilibrium conditions

This subsection briefly presents the linearized equilibrium conditions. Small lettersdenote the log-deviation of the corresponding variable from its steady state value.

2.5.1 Non-financial part of the economy

ct =h/γ

(1 + h/γ)ct−1 +

1

1 + h/γEt[ct+1]− (1− h/γ)

(1 + h/γ)σc(rt − Et[πt+1] + vd,t) (44)

Equation (44) is the aggregated Euler equation for consumption. The presence ofhabit formation justifies the presence of lagged consumption in the equation.

it =1

1 + β[it−1] +

β

1 + βEt[it+1] +

1

(1 + β)γ2S′′qkt (45)

where β = βγ(1−σc). Equation (45) is the linearized first order condition for investment.The dynamics of investment are governed by Tobins q. S′′ is the steady state valueof the second derivative of the investment adjustment cost function. The accumulationequation of physical capital reads

kt = (1− δ)/γkt−1 + (1− (1− δ)/γ)it + (1− (1− δ)/γ)(1 + β)γ2S′′vi,t. (46)

The marginal cost of the firms and the marginal product of capital are given by (47) and(48), while (49) is the equation for the return on capital.

mct = wt − zt + α(lt − kt) (47)

mpkt = wt − kt + lt (48)

Rkrk,t = MC ∗MPK(mct + yt − kt−1) + (1− δ)qt −Rkqt−1. (49)

The relation between physical capital and effective capital is given by (50). Here, param-eter ψ is the elasticity of the capital utilization adjustment cost function and normalizedto be between zero and one.

kt =1− ψψ

rkt + kt−1. (50)

(51) is the aggregate production function, and (52) is the aggregate resource constraint.

15

yt = Φ(αkt + (1− α)lt + zt) (51)

yt =G

Ygt +

C

Yct +

I

Yit +

RkK

Y

1− ψψ

rkt (52)

πt =β

1 + ıpβEtπt+1 +

ıp

1 + ıpβπt−1 +

(1− ζpβ)(1− ζp)(1 + βıp)ζp((Φ− 1)εp + 1)

(wt − zt + αlt − αkt)

(53)

Equation (53) is the New Keynesian Phillips curve. The last term in parenthesis corre-sponds to the marginal cost of production. As we employ the Kimball aggregator, thesensitivity of inflation to fluctuations in marginal cost is affected by the market powerof firms, represented by the steady state price markup, (Φ− 1).8 Furthermore, the cur-vature of the Kimball aggregator, εp, affects the adjustment of prices to marginal cost,since a higher εp implies a higher degree of strategic complementarity in price settingand dampens the price adjustment to shocks.

wt =1

1 + βγ(wt−1 + ıwπt−1) +

βγ

1 + βγEt[wt+1 + πt+1]− 1 + ıwβγ

1 + βγπt

+(1− ζwβγ)(1− ζw)

(1 + βγ)ζw((λw − 1)εw + 1)(wht − wt)

(54)

Equation (54) is the Wage Phillips curve. wht is the wage that would prevail in theabsence of market power by unions. Therefore, (wt−wht ) is the wage markup. Analogousto equation (53), the terms λw and εw represent the steady state wage markup andthe curvature of the Kimball aggregator for labor services. The efficient wage in (55),equals the average marginal rate of substitution between consumption and leisure of allhouseholds.

wht =σc

(1− h)(ct − hct−1) +

L

1− Llt (55)

The relationship between the real and the nominal interest rate is given by the Fisherequation.

rrt = rt − Et[πt+1]. (56)

8Note, that in equilibrium, the fixed cost parameter is related to the steady state price markup by azero profit condition.

16

2.5.2 The financial sector

The financial sector features the linearized law of motion for net worth

ntN =θ

γ[RkKrk,t + (Rk −R)K(qt−1 + kb,t−1) +RbQ

bBrb,t + (Rb −R)QbB(qbt−1 + bb,t−1)]

+θ

γ(RNnt−1 −RD(rrt−1 + vd,t−1)] + nn,tNn

(57)

and the linearized equation for the net worth that bankers are endowed with that newlyenter the business.

nn,tNn =ω

γ[K(qt−1 + kb,t−1) +B(qbt−1 + bb,t−1)] (58)

The linearized first order conditions for the banking sector yield the shadow value ofholding capital assets as a function of the spread between the loan rate and the rate ondeposits, (59) and the shadow value of holding bonds as function of the spread betweenthe bond return and the deposit rate, (60). The shadow value of another unit of networth depends on the rate on deposits (61). Equation (62) shows that in optimum theseshadow values are in balance.

νk,tνk = βΩ[(Rk −R)Ωt+1 + rk,t+1Rk − (rrt + vd,t)R], (59)

νb,tνb = βΩ[(Rb −R)Ωt+1 + rb,t+1Rb − (rrt + vd,t)R], (60)

νn,t = Ωt+1 + rrt + vd,t, (61)

νk,t = νb,t, (62)

The linearized stochastic discount factor of the banker is

ΩtΩ = θνn(1 + µ)(Ωt+1 + rrt + vd,t + νk,tνk

λk − νk)− (rrt−1 + vd,t−1)Ω. (63)

Furthermore, the loan supply of the financial intermediaries to non-financial firms reads

(q+kb,t−νk,t)K =(νb − λb)(qbt + bb,t) + νb,tνb

λk − νkQbB+

(νn,t + nt)νnN

λk − νk+

λCBLλk − νk

cblt. (64)

2.5.3 Policy and exogenous processes

The fiscal sector can be summarized by the linearized budget constraint, which alreadyentails the tax rule (65), and the linearized return on long-term bonds, (66).

RbQbB

γ(rb,t + qbt−1 + bt− 1) ∗Ggt − κbbt−1

B

γ= QbB(qbt + bt), (65)

Rb(rb + qbt−1) = ρcqbt . (66)

17

Equation (67) is the linearized Taylor rule in terms of the net interest rate set by thecentral bank.

rt = max0, ρrt−1 + (1− ρ)(φππt + φyyt + φdy(yt − yt−1)) + vrt, (67)

When the economy is away from the effective lower bound, the stochastic process ertrepresents a regular interest rate shock. When the nominal interest rate is at zero, vrtserves as a forward guidance shock as is explained in section 4. The asset purchases bythe central bank can be written as

xk,t = kt − kb,t,

xb,t = bt − bb,t.

Finally, the stochastic drivers of our model are the following ten processes:

vd,t =ρdvd,t−1 + εdt , (68)

zt =ρzzt−1 + εzt , (69)

gt =ρggt−1 + εgt , (70)

vr,t =ρrvr,t−1 + εrt , (71)

vi,t =ρivi,t−1 + εit, (72)

vp,t =ρpvp,t−1 + εpt , (73)

vw,t =ρwvw,t−1 + εwt , (74)

xk,t =ρx,kxk,t−1 + εkt , (75)

xb,t =ρx,bxb,t−1 + εbt , (76)

xCBL,t =ρCBLxCBL,t−1 + εCBLt . (77)

where the last three processes are the unconventional policy tools employed by the central

bank. It holds that εktiid∼ N(0, σ2

k) for all k = d, z, g, r, i, p, w, k, b, CBL.

3 Estimation and Methodology

This section first gives a very brief idea of the solution, filtering and estimationmethodology. We then describe the data and finally present and discuss our parameterestimates.

3.1 Solution method

We make use of the methodology presented in Boehl (2019) which is implementedin the pydsge package written in Python.9 All packages used to solve and estimate themodel in this paper can be downloaded from https://github.com/gboehl/pydsge.

9Python can provide speed benchmarks that are en-par with compiled languages such as Fortranwhile comprising the advantages of a high-level programming language. We want to promote free andopen software and advocate the avoidance of proprietary languages. Open source alternatives providevery efficient and highly flexible environments. Although such technology transitions can be expensive,we are convinced that the long-run payoff are excessive.

18

The paper first shows how to cast the piecewise linear model in the form

N

[xt

vt−1

]+ c max

b

[xt

vt−1

], r

= Et

[xt+1

vt

], (78)

where vt is an (n × 1)-vector that contains all the (latent) state variables and xt is an(m × 1)-vector containing all forward looking variables. N is a matrix of dimensions(n+m)× (n+m). r is the maximum value the constrained variable, r, which is in ourcase the nominal interest rate, can take. b is a row vector of dimension 1x(n+m) andthe vector (n+m)x1 vector c, contains the coefficients of the constrained variable. Thisspecification of the system has the advantage that the rational expectations solution ofsystem state v in period t+ s, depending on vt−1 while for any k and l, where k is thenumber of periods that the system spends at the ELB and the transition of l periodstowards the ELB, can be expressed in closed form as

Ls(l, k,vt−1) =Nmaxs−l,0 (N + cb)minl,s

S(l, k,vt−1) (79)

+ (I−N)−1(I−Nmaxs−l,0)cr (80)

=

[xt+1+s

vt+s.

](81)

where

S(l, k,vt−1) =

[xt

vt−1

]: QNk (N + cb)

l

[xt

vt−1

]= −Q(I−N)−1(I−Nk)cr

(82)

and Q =[I −Ω

]for xt = Ωvt−1 representing the linear rational expectations solu-

tion of the unconstrained system as given by Blanchard and Kahn (1980). Finding theequilibrium values of (l, k) has to be done numerically. To this purpose, the above repre-sentation comes in handy for computational reasons since it can be avoided to simulatethe whole equilibrium path when iterating over (l, k). For further details such as theequilibrium conditions and the numerical procedure we refer to Boehl (2019). Let usdenote the resulting transition function as

vt = fL(vt−1), (83)

which is linear for the region where the ELB does not bind and (increasingly) nonlinearwhen it binds. Given that the duration of how long the ELB is expected to bind isdetermined endogenously, this corresponds to the so called perfect foresight solution.It is accurate as long as shocks are moderate since the Jensen inequality is ignored.We argue that the approximation error is limited, especially compared to observationerrors and errors related to model misspecification.10 We further argue that such anapproximation is necessary to solve a large scale model within reasonable computationtime. For the model presented here, the implementation in pydsge is able to find thesolution of about 80.000 particles draws per second (given one processor).

10It is hard to test for the extent of this approximation error since solving a model with large stateaccurately at a global domain is extremely difficult.

19

3.2 Filtering and Estimation Method

Bayesian filtering – i.e. the extraction of the distribution of states in each periodgiven the series of observables – is done using the IPA smoother that is also introduced inBoehl (2019). IPA smoother stands short for iterative path-adjusting transposed-ensembleRTS-Smoother and is a combination of the particle filter, Kalman filter technology anditerative method. Similar to the particle filter, a set of points (the ensemble) is sentthrough the transition function in the prediction step of the filter. However, instead ofre-sampling (particle filter), the filter makes use of statistical linearization to update theparticles after each new observation. This ensemble update allows to efficiently approx-imate the state distribution of large-scale nonlinear systems with only a few hundredparticles while only requiring the assumption that states at each point in time are ap-proximately Gaussian distributed. The second step is ensemble smoothing similar to theRauch-Tung-Striebel smoother (Rauch et al., 1965). As the last step, iterative meth-ods (iterative path-adjustment) are used to guarantee that the estimates fully respectthe nonlinearity of the transition function. The IPA smoother is implemented in theeconsieve package.11

This approach stands in contrast to the methods used for instance in Fratto andUhlig (2014); Linde et al. (2017); Kulish et al. (2017) which are either linear or parameterdependent like the Unscented Kalman Filter (UKF). The approximation of the likelihoodof a nonlinear model with a linear filter is bound to be problematic per se. Likewise, whenusing parametric filters like the Sigma-Point Kalman filters (e.g. the UKF) the optimalparametrization of the filter depends on the values of the model’s parameters. Thus,for wide regions in the model’s parameter space, the parametrization of the filter maynot be optimal and hence it may deliver biased estimates. Hence, using the same set ofparameters for the filter might approximate the likelihood of different model parametersdifferently well. Both types of filters – linear and parametric – are unable to produce exactestimates of the distribution of exogenous variables that allow to precisely reproduce theoriginal time series of the data. This does not hold for the filter introduced in Guerrieriand Iacoviello (2017), which enables to exactly reproduce the data but can not be seenas a Bayesian estimator as it does not take into account the probability distribution ofthe states.

For the Bayesian estimation the Affine Invariant MCMC Ensemble sampler is used asproposed by Goodman and Weare (2010). This relatively new method hast the advantagethat the sampling can be efficiently parallelized: instead of having a small number ofchains that are each dependent on its own state as in the Metropolis algorithm, theensemble sampler uses a large number of chains that communicate after each iterationand exchange states and regions of the parameter space with high likelihood. The methodhas been extensively applied in particular in the field of astrophysics. Prior to the MCMCsampler, the posterior mode is obtained using the simplex method developed by Nelderand Mead (1965).

3.3 Data and Filtered Series

For the estimation – and later for the filtering of shock processes – we use time seriesof 10 observables. Our observables are real GDP growth, real consumption growth,

11See https://github.com/gboehl/econsieve.

20

2000 2004 2008 2012 2016GDP

2

1

0

1

2000 2004 2008 2012 2016Cons

3

2

1

0

1

2000 2004 2008 2012 2016Inv

7.5

5.0

2.5

0.0

2.5

2000 2004 2008 2012 2016Lab

0

5

10 IPASdata

Figure 1: Data series with simulated observables based on IPA-smoothened shocks. Note: medians over500 simulations. Each simulation is based on the means of IPA-smoothened shocks given a draw fromthe posterior parameter distribution.

investment growth, labor hours, the log change of the GDP deflator, real wage growth,the federal funds rate, the central bank holdings of government bonds, its holdings ofmortgage back securities, and aggregated liquidity facilities by the central banks. Thedetails of the construction of these variables as well as data sources are provided inAppendix A. We employ a sample from 1998Q1 to 2018Q2 in our quantitative analysis.This is in line with the sample chosen in Boehl and Strobel (2019) but shorter than inGust et al. (2017), Linde et al. (2017) or Fratto and Uhlig (2014). The reason for thatis that we want the estimation to capture idiosyncrasies of the episode where QE waseffective. Our special focus hier lies in particular on changes the slope of the PhillipsCurve and the persistence of endogenous and exogenous variables. Using a longer samplehence would have borne the risk of misspecification. On the other hand, using a shortersample yields less data points for the estimation procedure and might let priors dominatethe posterior distribution for parameters that are not well identified. Correspondingly,we reduce the parameter space where possible and opt for the shorter sample but willkeep the last critique point in the back of our minds.

We assume an observation error of√.1 ≈ 32% of each time series standard deviation

apart from the federal funds rate, where we only√.001 ≈ 3.2% of the standard deviation.

We opt to make use of observation errors not only because we think they are a realisticfeature that enriches the model, but also because this allows more leeway for the smoother

21

2000 2004 2008 2012 2016Infl

0.00

0.25

0.50

0.75

1.00

2000 2004 2008 2012 2016Wage

2

0

2

2000 2004 2008 2012 2016FFR

0.0

0.5

1.0

1.5

2000 2004 2008 2012 2016CBL

0

2

4

6

8

2000 2004 2008 2012 2016CB_Loans

0

2

4

6

8

10

2000 2004 2008 2012 2016CB_Bonds

2.5

5.0

7.5

10.0

12.5

Figure 2: Data series with simulated observables based on IPA-smoothened shocks. Note: medians over500 simulations. Each simulation is based on the means of IPA-smoothened shocks given a draw fromthe posterior parameter distribution.

to find more probable shocks. The filtered and smoothed observables are drawn in Figure1 and 2. The orange line depicts the data series while the blue line depicts the filteredand smoothed series. As the figure shows, our IPA-smoothened series matches the dataseries of the observables very well, although our choice for the observation error is quitebroad.

22

3.4 Parameter estimates

We fix several parameters in our estimation. This in particular concerns the meanvalues of most of our observables. For the observables given in growth rates, these meansgovern the slope of the steady state relative to the levels of the variables. For example,assuming a high trend of output implies a relatively lower level of the percentage steadystate deviation after the Great Recession. Likewise, assuming a very low trend of outputwill imply that output recovered quickly after the financial crises. It is clear that fromthe probabilistic perspective of the model, a recession is far more likely if it is short thanif it is long. Hence, an estimation of this trend parameter will always prefer a rather lowtrend. But this would not be in line with the common understanding that the recessionwas both very strong, quite long, and followed by a period of weak growth. In order toaccount for that feature, this can not be left to be evaluated to the estimation but wemust calibrate these parameters. Note that, while the trend of most observables is fixed,their level (relative to the steady state) is endogenously determined by the filter giventhe initial distribution of states.

We rechecked robustness of our results to other values of the observable means and –somewhat surprisingly – found almost no effects. As a note of caution, while our resultsare robust to changes in the calibration of the output trend, we find that alterations interms of parameters included in the estimation, or other choices made in the estimationcan have an impact on the estimated posterior distributions of some parameters. Whilemost crucial parameters seem to be identified, it seems that the parameter distributionis polymodal and it is hard to ex-ante discriminate the different modi.

The target rate of inflation is set to 0.5, which corresponds to an inflation target ofabout 2% annually. While the common prior would be 0.62, we opt for the lower value toaccount for temporary deviations in the inflation target. The mean of the observed laborseries and of the common growth trend are set to their average values of the pre-crisisdecade, i.e. 6.5415 and 0.344, respectively. Prior to the crisis, central bank holdingsof government bonds relative to GDP barely move at all. Here we set the mean of thisobservable to 5.65 percent of GDP. Holdings of Private Assets and the Liquidity Measuresare near zero before the crisis which is reflected by their steady state values.

Additionally to the means of observables, we fix the steady state markup in the labormarket, λw to 1.1 and the curvature parameters of the Kimball aggregator for both,labor and goods markets, ε, to 10. As the value of ψ does not seem to matter much forour results, we fix it at 0.79. Lastly, from pre-studies we observe that the values of αand h are close to invariant across the estimations that we have conducted, and close tothe values found in Boehl and Strobel (2019). Therefore we set α = 0.19 and h = 0.72,which corresponds on the one hand to our results and on the other hand are close toconventional values for these parameters.

Table 1 shows our choice of priors and our posterior estimates. Except for the uniformand the inverse gamma distribution, the parameters of the priors correspond to the meanand standard deviation of the distributions. The priors for the parameters that pertain tothe real economy are chosen in line with Smets and Wouters (2007, SW).12 In addition, weestimate the steady state leverage of financial intermediaries, LEV , their survival rate

12In contrast to Gust et al. (2017), the computational efficiency of our approach allows us to use thesame standard deviations as in SW, instead of choosing narrow priors.

23

distribution mean/alpha sd/beta mean sd mc error hpd 2.5 hpd 97.5

tpr beta gamma 0.250 0.10 0.156 0.052 0.000 0.050 0.247sig c normal 1.500 0.37 1.093 0.191 0.001 0.660 1.438S′′ normal 4.000 1.50 7.524 1.265 0.007 4.963 9.948i p beta 0.500 0.15 0.340 0.084 0.000 0.174 0.492i w beta 0.500 0.15 0.445 0.142 0.001 0.167 0.733zeta p beta 0.500 0.10 0.805 0.044 0.000 0.727 0.887zeta w beta 0.500 0.10 0.680 0.058 0.000 0.555 0.783Phi p normal 1.250 0.12 1.305 0.119 0.001 1.062 1.516phi pi normal 1.700 0.25 1.828 0.246 0.001 1.307 2.302phi y normal 0.125 0.05 0.160 0.027 0.000 0.104 0.211

phi dy normal 0.125 0.05 0.207 0.045 0.000 0.127 0.295LEV normal 3.000 1.00 1.802 0.457 0.003 1.131 2.616kap t normal 0.100 0.10 0.146 0.051 0.000 0.037 0.235theta beta 0.950 0.05 0.908 0.074 0.000 0.762 0.994lamb cbl uniform 0.000 10.00 2.694 0.842 0.006 0.995 4.134rho u beta 0.500 0.20 0.766 0.051 0.000 0.667 0.862rho beta 0.700 0.20 0.784 0.040 0.000 0.697 0.849rho r beta 0.700 0.20 0.488 0.092 0.002 0.332 0.683rho g beta 0.500 0.20 0.838 0.101 0.000 0.661 0.983rho i beta 0.500 0.20 0.816 0.069 0.000 0.692 0.944

rho z beta 0.500 0.20 0.583 0.193 0.002 0.214 0.888rho p beta 0.700 0.20 0.260 0.053 0.000 0.158 0.363rho w beta 0.700 0.20 0.455 0.094 0.000 0.314 0.660rho cbl beta 0.500 0.20 0.555 0.064 0.000 0.439 0.670rho qe b beta 0.500 0.20 0.863 0.041 0.000 0.778 0.945rho qe k beta 0.500 0.20 0.921 0.033 0.000 0.857 0.980sig g inv gamma 0.100 2.00 0.380 0.066 0.000 0.272 0.495sig b inv gamma 0.100 2.00 1.104 0.400 0.002 0.427 1.803sig z inv gamma 0.100 2.00 0.339 0.256 0.001 0.051 0.873sig r inv gamma 0.100 2.00 0.215 0.063 0.001 0.118 0.349

sig i inv gamma 0.100 2.00 0.268 0.071 0.000 0.125 0.398sig p inv gamma 0.100 2.00 0.397 0.107 0.000 0.200 0.605sig w inv gamma 0.100 2.00 0.948 0.189 0.002 0.564 1.259sig cbl inv gamma 10.000 1.00 2.080 0.301 0.001 1.585 2.670sig qe b inv gamma 10.000 1.00 1.844 0.218 0.001 1.424 2.285sig qe k inv gamma 10.000 1.00 1.779 0.247 0.002 1.361 2.315

Table 1: Estimation results

θ, the sensitivity of the incentive constraint to liquidity injections λCBL, the feedbackcoefficient for government debt in the tax rule, κb and the shock processes for the QEmeasures. For the prior of LEV we choose a normal distribution centered around 3, andwith a standard deviation of 1. A larger leverage LEV implies a more fragile financial

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system and with it a larger amplification of financial shocks and QE shocks. As weare not aware of prior estimates of this parameter in the context of a structural model,we opt for choosing a rather wide prior.13 For θ, we choose a beta distribution for theprior, reflecting its nature of a probability that is bound between 0 and 1. The standarddeviation of 0.05 makes this prior rather wide enough to allow for significant departuresfrom the mean, 0.95, which corresponds to an expected time horizon of the bankers of5 years. A higher θ is associated with a higher stochastic discount factor of financialintermediaries and therefore shapes the persistence of the effects of QE shocks. For λCBLwe set a wide uniform prior from 0 to 10. The shape and width of the prior reflectour agnostic approach to the effect of central bank liquidities injections. For λCBL = 0,liquidity injections have no effect at all. The prior for κb is again normal with a meanof 0.1 and a standard deviation of 0.1 which can again be understood to be a ratheruninformative prior. For the persistence parameters of the shock processes of the policymeasures, we apply the same priors as for the other AR(1) processes. Lastly, we choosethe priors for the standard deviations of the QE shocks such as to allow for movementsin the policy measures on a magnitude such as we see them in the respective observables.The reader can find further technical details on the estimation procedure in AppendixB.

Where applicable, our estimates are largely in line with common findings in the liter-ature. The posterior mean of tpr beta, the time preference rate, is close to the estimateby SW. We estimate σc close to log utility. The mean of the investment adjustment costparameter, S′′ is found to be rather high in our sample. It is noteworthy that the inflationPhillips Curve is estimated to be very flat with an estimated Calvo parameter ζ = 0.805,a price indexation parameter of ip and a fixed cost parameter, Φp = 1.305 that in ourmodel matches the steady state markup of firms, and which enters the Phillips curve dueto the use of the Kimball aggregator. The values for the wage Phillips curve parametersare roughly in line with the estimates by SW, but imply a steeper wage Phillips Curvethan estimated in Linde et al. (2017). The values are also higher than the correspondingestimates in Boehl and Strobel (2019), where we use virtually the same methodology buta model without featuring financial frictions. The estimated feedback coefficients as wellas the interest rate smoothing parameter, ρ, in the Taylor rule also are in the range ofconventionally found values.

Our estimations show that the data favors small values for our parameters pertainingto the financial sector, LEV and θ. It can be suspected that this is the fact becausethese value govern the sensitivity of the economy with respect to the QE measures,whereas the model seems to prefer smaller effects. In particular, the posterior mean forthe steady state leverage, LEV = 1.802 is below our expectations and our prior mean.This value is far smaller than the leverage ratio of banks but rather conform to whatGertler and Karadi (2011) reason to be a plausible value for non-financial corporations.The posterior mean of θ implies an average time-horizon of ten quarters for the banker.For the sensitivity of the incentive constraint to liquidity injections, λCBL = 2.694, theposterior mean is at the lower end of our prior. The estimated persistence parameters ofthe shock processes imply that technology shocks and government spending shocks areless persistent than the often-found values close to a unit root (see, e.g. SW, Linde et al.,

13Villa (2016) estimates a version of the model by Gertler and Karadi (2011) on US data, but calibratesLEV and θ to fixed values.

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2017). On the other hand, our value for the persistence of the monetary shock, ρr, isrelatively high. We find that this difference is due to the estimation on a shorter sample.For an estimation of the model on a sample starting in 1966, we again find a relativelysmall value for ρr. At the zero lower bound, higher values of ρr imply a stronger ability ofthe central bank to change the expected path of future interest rate, and hence strongereffects of forward guidance. The persistence of the QE shocks is rather high for thepurchases and rather low for the liquidity injections, which conforms to a first eyeballingof the observed series.

4 Empirical Analysis

We start this section with a general decomposition of the dynamics during the sampleperiod. We then zoom in specifically on the effects of the different measures of quan-titative easing. Lastly, we compare the magnitude of the effects of QE with those ofconventional interest rate and forward guidance policies. The methodology for each ofthese exercises is similar: we start by drawing 500 parameter sets from the posterior dis-tribution (more precisely: from the converged Markov chains). For each of these drawswe use the IPA smoother to infer on the distribution of shock innovations given the seriesof observables and uncertainty about the measures and initial states. In this regard, thestrategy is similar to the procedure used to obtain the simulated series in Figures 1 and2 from the previous section. For the decompositions and counterfactuals below we thenswitch of one or several of these series or parts of them off and show the median over theresulting draws. When possible in terms of a clear display, we also add 68% intervals.

4.1 A decomposition of the dynamics

Let us start with the discussion of the decomposition of the business cycle dynamicsin the Great Recession and its aftermath. We identify variations in the risk premiumto be the dominant driver of most of the observables.14 As the shock series depictedin Figure 3 to 5 convey, the sharp drop in output and its components, and well as inlabor hours, inflation and wages was triggered by a sharp increase in the risk premiumon safe assets. Variations in the risk premium dominate the other shocks in explainingthe macroeconomic dynamics in the years following the financial crisis. This findingcoincides with results by Boehl and Strobel (2019) and Gust et al. (2017). This increasein the risk premium forces the nominal interest rate endogenously to the effective lowerbound. Spiking sharply at the end of 2008 – coinciding with the collapse of LehmanBrothers – the risk premium stays persistently high until the end of the sample. It iseasy to see that without this driver, the Great Recession would at best have been mildand the nominal rate would not have reached the effective lower bound at all.

As one can see, the effects of the different monetary policy shocks (interest rateshocks - pink, aggregated QE shocks - grey) had an effect on the real economy, but itwas not very large compared to the effects of non-policy shocks and, in particular, the

14At the step of filtering and smoothing, we ran some trail runs with several different financial shocks,such as the capital quality shock in Gertler and Karadi (2011) and a shock to the incentive constraintas in Dedola et al. (2013). For the time being we found that the risk-premium shock together with theinvestment specific shock, which has been interpreted as a financial shock as well (see, e.g. Justinianoet al., 2011, Gust et al., 2017, Kulish et al., 2017) delivers the most robust results.

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risk premium shock. We analyse the consequences of the convetional and unconventionalmonetary policy shocks in more detail in the next subsections.

Next to the risk-premium shock, the investment specific technology shock, shown inyellow, played a significant role for economic activity. It caused additional pressure onoutput, investment and labor hours during the drop in 2009. For the inflation dynamicshowever only the markup shocks had notable impact next to the exogenous part of therisk premium. The wage markup shock is also an important driver of wages in the post-crisis time. The government spending process is mostly negative throughout the sample,with the exception of the years around 2010. These years correspond to the years ofthe American Recovery and Reinvestment Act (ARRA), which is well captured by oursmoothed series. Although the effect of government spending on output is very small,the fiscal contraction crowds in investment at the end of the ELB period. Around theonset of the ELB and the ARRA program, it also manages to crowd in consumption.The measured technology shocks are quite small and hardly matter for business cycledynamics.

4.2 The Effects of the QE Measures

In this section, we focus on the contribution of the QE measures to the dynamics ofthe US economy and discuss the diverse effects of the different single policy measures in

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isolation. Figure 7 shows the development of the three policy measures in this modelover time. The emergency liquidity measures, which encompassed several programs suchas the Term Auction Facility Program (TAF), the Term Asset-Backed Securities LoanFacility (TALF), the primary dealer and other broker-dealer credit, and several more,took place around the height of the financial crisis in 2008/2009 and were quickly reducedin the following years. Instead, the Federal Reserve turned to programs of large-scaleasset purchases. The first of these programs, dubbed QE1, which lasted from December2008 to March 2010, included net purchases of $1.25 trillion in mortgage-backed securities(MBS) and $175 billion in agency securities. After a brief pause, it was followed byQE2, a program, which centered around the net-purchases of $600 billion in long-termgovernment Treasury securities. Again after a brief pause, the Fed started what is nowknown as Operation Twist, a portfolio shift of the size of $600 billion from short andmedium-term Treasuries with a maturity up to three years, to long-term Treasurieswith maturities of six years and above. This effect of Operation Twist is not explicitlyincorporated in our analysis, as our data series of government bonds pools Treasuriesof all maturities. Lastly, QE3 started in September 2012 and lasted until December2014. It increased the balance sheet of the Fed through net-purchases of $95 billion eachmonths - $45 billion in long-term bonds and $40 billion in MBS.15 As was shown in theforegoing subsection, despite the unprecedented size of the Fed’s balance sheet expansion,the contribution of QE to business cycle dynamics was rather small compared to the non-policy-based drivers. However, a closer look at the effects of QE in isolation reveals thatthey were of relevance, and that the different tools affected the real economy differently.

Figures 6 and 7 show the total effect of unconventional monetary policy measuresdecomposed in the effects of the each tools separately. Green marks the effect of MBSpurchases, which in our model are equivalent to purchases of private assets, yellow are theeffects of government bond purchases, and blue are the effects of the liquidity injections.Liquidity injections took place at the onset of the crisis. As one can see, they significantlyincreased the price of capital assets, q, as well as of bonds, qb, and lowered the excessreturn on assets (premk in the lower right panel of the figure is the excess return oncapital). The decrease in the excess returns on the banks assets decreased their respectiveshadow values for the bank. Additionally, the relaxation of the financial constraint of thebanking sector that was triggered by liquidity injections was associated with an economy-wide portfolio shift from bond holdings to capital, triggering an increase in investmentthat peaked at 1%. The increase of investment in turn caused an increase in the demandfor labor associated with an increase of labor hours and wages, the latter correspondingto the increase in the marginal product of labor due to an increased capital stock. Thisresulted in a small increase in consumption. The overall increase of output peaked at0.25%. Given that the peak injections reached 8% of GDP, the peak output multiplierfor liquidity injections is only at roughly 0.031. Additionally, the emergency liquidityfacilities were relatively short-lived, and had themselves no strong long-lasting effects onthe real economy. This contrasts with the findings by Del Negro et al. (2017) who findlarge effects by liquidity injections on the real economy. One of the potential reason forthis difference is that the persistence of liquidity injections in their model is tied to thecalibration of a very high value for the persistence parameter of the liquidity shock.

15The figures regarding QE1-QE3 are taken from Williamson (2017).

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The largest effects on the non-financial sector was caused by the central bank pur-chases of capital assets. The Fed started its purchase program of private securities inthe last quarter of 2008, raising its holdings up to roughly 9,7% of GDP in 2015, when itslowly started to reduce its position. The securities purchases consisted largely of mort-gage backed securities (MBS), but included the smaller and more shortly-lived programof the commercial paper funding facility (CPFF) as well. The effects of private securitypurchases were in many ways qualitatively similar to the effects of liquidity injections.Figure 8 depicts the theoretical impulse response of selected variables given the estimatesfor the parameters to an increase in capital purchases by 11 % of GDP. The purchaseslowered the interest rate spread for capital assets and raised the price of capital, q. Byarbitrage they had the same effect on the prices and spreads of bonds as well, just toa lesser extent16. After the initial capital gains, the reduced excess premium on hold-ing assets, reduced the shadow value of banks for holding assets on their balance sheet.Likewise, the effect on the net worth of banks was ambiguous. Initially, the increase inasset prices raised net worth, but after a while, the reduction in the accumulation of newnet worth due to the decrease in the interest rate spreads prevails.

Additionally, an increase in the central bank holdings of capital is associated with adecline in the holding of capital assets by financial intermediaries, which further reducedtheir profits. As in the theoretical impulse respnses, the overall effect of capital purchaseon the real economy in the US were largely positive. They boosted aggregate invest-ment up to 3.5% and had a lasting positive impact on the accumulation of productivecapital. As a consequence, we see in Figure 6 that capital purchases contributed thelion’s share to the stimulus of output, which was raised by up to 0.75%. Again, as withliquidity injections, the capital purchases caused an increase in the demand for labor,increasing labor hours and real wages simultaneously. Additionally, they had the effectof increasing inflation. However, this effect was very small. Nonetheless, it signals thatdespite increasing the productive capacity, the overall effect of this form of QE acted likea demand shock to the US economy. Furthermore, the increase in inflation reduced thereal rate on deposits (rr in the figure) at the ELB, supporting the increase in households’consumption expenditures. While purchases on capital assets supported the economy fora prolonged time, these effects died out and turned negative at the end of the sample.The reason is that the slow reduction of private security holdings by the Fed is associatedwith an end of the stimulating effects, while the persistently lower spreads on assets heldby the banking sector weaken the sectors balance sheet and reduce the supply of loans.

Lastly, the effects of government bond purchases played out differently. The Fedreduced its bond holdings during the crisis when it shifted its balance sheet towardsliquidity facilities. In 2010 it embarked in large-scale purchases of long-term governmentbonds, which it later extended in the QE3 program. Overall the bond holdings of the Fedreached close to 14% of GDP.17 On the financial variables, bond purchases had effectsthat were similar to the ones of capital purchases. However, the variables’ reaction tobond purchases is smaller. As can be seen in the theoretical impulse responses in Figure

16The comovement of asset prices and spreads in a model with financial frictions is a known finding.We therefore abstain from showing the second set of assets and spreads due to space constraints.

17We anchor our mean of the observable of government bonds at 5.65% of GDP. The plotted seriesin Figure 7 shows what we account for as the exogenous increases, decreases of bond holdings from thislevel.

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9, the response of asset prices is smaller and dies out more quickly. The reaction ofnet worth is dominated by the reduction of bank’s profits that stems from the persistentdecline in spreads as well as from a reduction of their holdings of bonds, which mirrors theexpansion of bond holdings on the Fed’s balance sheet. The lower estimated persistenceof the bond-related QE shock compared to the capital-related QE shock (ρx,b = 0.863vs. ρx,k = 0.921) contributes to the quicker dying out of the support of asset prices. Themore detrimental effects on the financial sector are associated with negative consequencesfor the loan supply and crowding-out of investment. Therefore, the positive contribution

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of bond purchases to investment and output quickly disappears after the Fed startsexpanding its position. As a consequence, investment, output and labor were raised in theyears until 2013 and thereafter reduced by bond purchases. Quantitatively, the reductionwas more pronounced. On wages and inflation, the initial positive push persisted untilthe end of the sample.

Overall, our analysis suggests that the stimulating effects of asset purchases of bothkinds are followed by a hangover period. The implications for a future in which the Fedunwinds its position of asset holdings and shrinks its balance sheet are depicted in Figures

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10 and 11. Here we show the overall impact of all three QE tools. For the periods after2018Q2 we draw the projection. Our forecasts clearly show that the negative long-termconsequences of QE have the potential to be of the same size as the stimulus effects in theshort-run. In particular, investment is projected to fall by more than the peak increasedue to QE. Additionally the slump in investment is more persistent than the stimuluscaused by QE. In this scenario, the trough for banks’ net worth is projected to be reachedin 2019, when the excess spreads on capital assets return to their initial level. The troughfor aggregate output, investment, and labor hours follows briefly thereafter. In summary,the detrimental effects of QE that are projected match or exceed the positive effects intheir size. Exceptions are consumption and inflation. However, for both variables theresponses are rather weak. Our result of a contractionary effect of QE in the long-runis in line with arguments made by Brunnermeier and Koby (2018) that there exists areversal rate at which accommodative monetary policy reverses its effect and becomescontractionary. In their setting the reversal becomes more likely, the longer QE lasts.As in their analysis, in our model this is due to the mismatch of the rather short-livedcapital gains and the persistent decline of excess spreads.

Nonetheless, as a note of caution, this result of a looming hangover effect should betaken with a grain of salt. The way in which the financial sector is modelled is crucial

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for the effects of QE, which reach the real economy via the supply of loans. In the modelstructure by Gertler and Kiyotaki (2010), Gertler and Karadi (2011) and Gertler andKaradi (2013) that we adopt, the possibility of a hangover is hardwired into the modelas the decrease in the profitability of banks, which is linked to the decline in the returnon capital, and therefore the dynamics of the capital stock, is more persistent than theincrease in asset prices, which is tied to investment dynamics. Given our parameterestimates, the hangover is quite pronounced. Possibly, this result will not survive in ananalysis in which QE affects the financial sector through other channels. For instance,the model does not incorporate the possibility of defaults by non-financial agents on theirobligations to banks. In this example, lower interest rates on loans would help to preventdefaults and thereby stabilize the banking sector. At the current step of our analysis, weconsider this to be a preliminary result.

4.3 Comparing the effects of QE and interest rate policy

Figures 12 to 13 show the effects of interest rate policy in yellow and the combinedQE measures in blue. The joint effect of all monetary policy tools on the real economyis non negligible in its size. The Fed Funds Rate hit the effective lower bound in the lastquarter of 2008. Interest rate shocks were mostly expansionary before, but also duringthe ELB period, and turned contractionary at the end of the sample. At the onset of thecrisis, when reaching the effective lower bound, the Fed decreased its interest rate quickerthan implied by the Taylor rule. This can also be interpreted by the Fed following theTaylor rule but not accounting for interest rate smoothing.

During the ELB period, these shocks capture changes in the expectations of interestrate policy. Hence, we interpret these shocks as triggering expectations by the forwardguidance announcement of the Federal Open Market Committee, which started a policyof announcing the future path of federal fund rate as expected by the Fed. Throughoutthe ELB period, forward guidance shocks also were slightly expansionary, though at aconsiderably lower level. This suggests a rather dovish stance of the Fed on the impliedduration of the ELB. The graphs reveal that while the traditional interest rate shockshad an effect prior to the ELB period, the forward guidance shocks at the effective lowerbound hardly had an impact on the real economy. However, the effects of expansionaryinterest rate shocks prior to the crisis persisted for a long time for several variables.For instance, whereas the increase of investment due to interest rate shocks died out atthe onset of the ELB, the aftereffects on the capital stock decreased only slowly overtime. As a consequence, effects on consumption and wages lasted for a long time. Atthe time of the exit, the interest rate shocks are found to be restrictive, signalling thatrates were raised quicker than expected after the exit from the ELB. This is perceived asa recessionary shock. This role of interest rate policy for the US economy is broadly inline with the results of Boehl and Strobel (2019), in which we discuss the role of forwardguidance in more detail.

During the ELB, QE proved to be more consequential for the dynamics of the realeconomy as well as for financial variables. The stimulus for output and consumptionprovided by QE at the ELB is only slightly smaller than the magnitude of the effects ofinterest rate shocks in the pre-crisis decade. Starting with the end of the ELB, interestpolicy resumes to be of consequence for the macroeconomy. As the interest rate shocks arerestrictive, they reinforce the recessionary effects that come with the hangover followingexpansionary QE, which were described in the foregoing section. Our results contradict

38

2000 2004 2008 2012 2016y

1

0

1

2totalQEeps_r

2000 2004 2008 2012 2016c

0.0

0.5

1.0

1.5

2000 2004 2008 2012 2016i

4

2

0

2

4

2000 2004 2008 2012 2016l

1

0

1

2

2000 2004 2008 2012 2016r

0.4

0.2

0.0

0.2

2000 2004 2008 2012 2016rr

0.6

0.4

0.2

0.0

0.2

2000 2004 2008 2012 2016Pi

0.00

0.05

0.10

2000 2004 2008 2012 2016w

0.0

0.5

1.0

Figure 12: Decomposition of the net-effect of unconventional monetary policy measures into the contri-bution of QE and of forward guidance (νr). Note: medians over 500 simulations. Each simulation isbased on the means of IPA-smoothened shocks given a draw from the posterior parameter distribution.

39

2000 2004 2008 2012 2016kb

0.0

0.5

1.0

1.5

2.0 totalQEeps_r

2000 2004 2008 2012 2016n

5

0

5

2000 2004 2008 2012 2016q

1

0

1

2

3

4

2000 2004 2008 2012 2016premk

2.0

1.5

1.0

0.5

0.0

Figure 13: Decomposition of the net-effect of unconventional monetary policy measures into the contri-bution of QE and of forward guidance (νr). Note: medians over 500 simulations. Each simulation isbased on the means of IPA-smoothened shocks given a draw from the posterior parameter distribution.

Greenlaw et al. (2018), which make the argument that forward guidance shocks were amore effective policy tool than quantitative easing programs implemented by the Fed.To the contrary we find that at the ELB quantitative easing has far stronger effects thanforward guidance.

5 Conclusion

In this paper, we develop a medium scale DSGE model with a financial sector and useit to analyse the effects of several measures of unconventional monetary policy. For thispurpose, we solve and estimate the model with an occasionally binding effective lowerbound on nominal interest rates.

We find the effects of quantitative easing to be relatively small. This holds in partic-ular in comparison to the effects of non-policy shocks and especially exogenous variationsin the risk premium. Nonetheless, the effects are economically non-negligible. We findthat the most effective tool for stimulating real activity in the short-run are central bankpurchases of claims on the private capital stock. According to our analysis, central bankliquidity injections, while providing temporary relief for the financial sector, had thesmallest and most short-lived effect on the real economy. In contrast, government bond

40

purchases, while lowering the real rate on safe deposits did not support economic growthbut crowded out investment and decreased GDP.

We document that the overall stimulating effects of QE are followed by an impliedeconomic slump that we dub the hangover effect of QE. In our model this is due tothe declining profitability of the financial sector that results from the decrease in excessspreads on assets purchased by the central bank. As a consequence, the weakened finan-cial sector contracts its loan supply in the long run. This mechanism is loosely relatedto the argument of the reversal rate made by Brunnermeier and Koby (2018).

Other sceptics of QE have argued that concerns surrounding the unwinding of QEcreated uncertainty and contributed to the slow recovery of the US economy following theGreat Recession (see, e.g. Taylor, 2014). This and other aspects such as the signallingcontent of monetary policy decisions are not captured in our analysis. Investigatinghow the effects of QE in the US as implied by a nonlinearly estimated model hinge onthe modelling of the incorporated financial sector or a more detailed modelling of labormarket dynamics is a promising route for future research.

References

Beraja, M., Hurst, E., Ospina, J., 2016. The Aggregate Implications of Regional Business Cycles.Working Paper 21956. National Bureau of Economic Research.

Blanchard, O.J., Kahn, C.M., 1980. The solution of linear difference models under rational expectations.Econometrica: Journal of the Econometric Society , 1305–1311.

Boehl, G., 2019. Efficient Solution, Filtering and Estimation of Models with OBCs. Technical Report.URL: https://gregorboehl.com.

Boehl, G., Strobel, F., 2019. The Great Recession and the Zero Lower Bound. Technical Report.unpublished manuscript. URL: https://gregorboehl.com.

Bohn, H., 1998. The behaviour of us public debt and deficits. The Quarterly Journal of Economics 117,949–963.

Borio, C., Zabai, A., 2016. Unconventional monetary policies: a re-appraisal. BIS Working Papers 570.Bank for International Settlements. URL: https://ideas.repec.org/p/bis/biswps/570.html.

Borio, C., Zabai, A., 2018. Unconventional monetary policies: a re-appraisal, in: Research Handbookon Central Banking. Edward Elgar Publishing. Chapters. chapter 20, pp. 398–444.

Brunnermeier, M.K., Koby, Y., 2018. The Reversal Interest Rate. NBER Working Papers 25406. NationalBureau of Economic Research, Inc. URL: https://ideas.repec.org/p/nbr/nberwo/25406.html.

Calvo, G.A., 1983. Staggered prices in a utility-maximizing framework. Journal of Monetary Economics12, 383–398.

Chen, H., Curdia, V., Ferrero, A., 2012. The macroeconomic effects of large-scale asset purchase pro-grammes*. The Economic Journal 122, F289–F315. doi:10.1111/j.1468-0297.2012.02549.x.

Christiano, L.J., Eichenbaum, M.S., Trabandt, M., 2015. Understanding the great recession. AmericanEconomic Journal: Macroeconomics 7, 110–67. URL: http://www.aeaweb.org/articles?id=10.1257/mac.20140104, doi:10.1257/mac.20140104.

Cochrane, J., 2011. Inside the Black Box: Hamilton, Wu and QE2. Technical Report.Curdia, V., Woodford, M., 2011. The central-bank balance sheet as an instrument of monetary policy.

Journal of Monetary Economics 58, 54–79.Dedola, L., Karadi, P., Lombardo, G., 2013. Global implications of national unconventional poli-

cies. Journal of Monetary Economics 60, 66–85. URL: https://ideas.repec.org/a/eee/moneco/

v60y2013i1p66-85.html, doi:10.1016/j.jmoneco.2012.12.Del Negro, M., Eggertsson, G., Ferrero, A., Kiyotaki, N., 2017. The Great Escape? A Quantitative

Evaluation of the Fed’s Liquidity Facilities. American Economic Review 107, 824–857.Del’Ariccia, G., Laeven, L., Suarez, G.A., 2017. Bank leverage and monetary policy’s risk-taking channel:

Evidence from the united states. The Journal of Finance 72, 613–654. doi:10.1111/jofi.12467.Fratto, C., Uhlig, H., 2014. Accounting for post-crisis inflation and employment: A retro analysis.

Technical Report. National Bureau of Economic Research.

41

Gertler, M., Karadi, P., 2011. A model of unconventional monetary policy. Journal of Monetary Eco-nomics 58, 17–34.

Gertler, M., Karadi, P., 2013. QE 1 vs. 2 vs. 3. . . : A Framework for Analyzing Large-Scale AssetPurchases as a Monetary Policy Tool. International Journal of Central Banking 9, 5–53.

Gertler, M., Kiyotaki, N., 2010. Financial intermediation and credit policy in business cycle analysis,in: Friedman, B.M., Woodford, M. (Eds.), Handbook of Monetary Economics. Elsevier. volume 3.chapter 11, pp. 547–599.

Gertler, M., Kiyotaki, N., 2015. Banking, liquidity, and bank runs in an infinite horizon economy.American Economic Review 105, 2011–43. URL: http://www.aeaweb.org/articles?id=10.1257/

aer.20130665, doi:10.1257/aer.20130665.Gertler, M., Kiyotaki, N., Queralto, A., 2012. Financial crises, bank risk exposure and government

financial policy. Journal of Monetary Economics 59, 17–34.Gilchrist, S., Zakrajsek, E., 2013. The impact of the federal reserve’s large-scale asset purchase programs

on corporate credit risk. Journal of Money, Credit and Banking 45, 29–57. doi:10.1111/jmcb.12070.Goodman, J., Weare, J., 2010. Ensemble samplers with affine invariance. Communications in applied

mathematics and computational science 5, 65–80.Greenlaw, D., Hamilton, J.D., Harris, E., West, K.D., 2018. A Skeptical View of the Impact of the

Fed’s Balance Sheet. Working Paper 24687. National Bureau of Economic Research. URL: http:

//www.nber.org/papers/w24687, doi:10.3386/w24687.Guerrieri, L., Iacoviello, M., 2017. Collateral constraints and macroeconomic asymmetries. Journal of

Monetary Economics 90, 28–49. URL: https://ideas.repec.org/a/eee/moneco/v90y2017icp28-49.html, doi:10.1016/j.jmoneco.2017.06.

Gust, C., Herbst, E., Lopez-Salido, D., Smith, M.E., 2017. The empirical implications of the interest-ratelower bound. American Economic Review 107, 1971–2006.

Hesse, H., Hofmann, B., Weber, J., 2017. The macroeconomic effects of asset purchases revisited. BISWorking Papers. Bank for International Settlements.

Justiniano, A., Primiceri, G., Tambalotti, A., 2011. Investment Shocks and the Relative Price of Invest-ment. Review of Economic Dynamics 14, 101–121. URL: https://ideas.repec.org/a/red/issued/09-248.html, doi:10.1016/j.red.2010.07.003.

Kapetanios, G., Mumtaz, H., Stevens, I., Theodoridis, K., 2012. Assessing the economy-wide effects ofquantitative easing*. The Economic Journal 122, F316–F347. doi:10.1111/j.1468-0297.2012.02555.x.

Kimball, M.S., 1995. The Quantitative Analytics of the Basic Neomonetarist Model. NBER WorkingPapers 5046. National Bureau of Economic Research, Inc.

Kirchner, M., van Wijnbergen, S., 2016. Fiscal deficits, financial fragility, and the effectiveness ofgovernment policies. Journal of Monetary Economics 80, 51–68.

Kulish, M., Morley, J., Robinson, T., 2017. Estimating dsge models with zero interest rate policy. Journalof Monetary Economics 88, 35 – 49. URL: http://www.sciencedirect.com/science/article/pii/S0304393217300442, doi:https://doi.org/10.1016/j.jmoneco.2017.05.003.

Kuttner, K.N., 2018. Outside the box: Unconventional monetary policy in the great recession andbeyond. Journal of Economic Perspectives 32, 121–46. URL: http://www.aeaweb.org/articles?id=10.1257/jep.32.4.121, doi:10.1257/jep.32.4.121.

Linde, J., Maih, J., Wouters, R., 2017. Estimation of Operational Macromodels at the Zero LowerBound. Technical Report. manuscript.

Meeks, R., Nelson, B., Alessandri, P., 2017. Shadow banks and macroeconomic instability. Journal ofMoney, Credit and Banking 49, 1483–1516.

Nelder, J.A., Mead, R., 1965. A simplex method for function minimization. The computer journal 7,308–313.

Rannenberg, A., 2016. Bank leverage cycles and the external finance premium. Journal of Money, Creditand Banking 48, 1569–1612. doi:10.1111/jmcb.12359.

Rauch, H.E., Striebel, C., Tung, F., 1965. Maximum likelihood estimates of linear dynamic systems.AIAA journal 3, 1445–1450.

Schenkelberg, H., Watzka, S., 2013. Real effects of quantitative easing at the zero lower bound: Structuralvar-based evidence from japan. Journal of International Money and Finance 33, 327–357.

Smets, F., Wouters, R., 2007. Shocks and frictions in us business cycles: A bayesian dsge approach.American Economic Review 97, 586–606.

Swanson, E.T., 2017. Measuring the Effects of Federal Reserve Forward Guidance and Asset Purchaseson Financial Markets. Working Paper 23311. National Bureau of Economic Research. doi:10.3386/w23311.

42

Taylor, J.B., 2014. After unconventional monetary policy. Testimony before the Joint Economic Com-mittee of Congress at the Hearing on “Unwinding Quantitative Easing: How the Fed Should PromoteStable Prices, Economic Growth and Job Creation” URL: https://web.stanford.edu/~johntayl/

2014_pdfs/JEC_March2014_Written_Testimony.pdf.Villa, S., 2016. Financial frictions in the euro area and the united states: A bayesian assessment.

Macroeconomic Dynamics 20, 1313–1340. doi:10.1017/S1365100514000881.Williamson, S., 2017. Quantitative Easing: How Well Does This Tool Work? Working Paper 2015

Third Quarter. Federal Reserve Bank of St. Louis. URL: https://www.stlouisfed.org/~/media/

publications/regional-economist/2017/third_quarter_2017/qe_lead.pdf.Woodford, M., 1998. Public debt and the price level. mimeo .Woodford, M., 2001. Fiscal requirements of price stability. Journal of Money, Credit and Banking 33,

669–728.Yellen, J.L., 2017. A challenging decade and a question for the future. 2017 Herbert Stein Memo-

rial Lecture at National Economicsts Club” URL: https://www.federalreserve.gov/newsevents/

speech/files/yellen20171020a.pdf.

Appendix A Data

Our measurement equation contains ten variables:

• GDP: ln(GDP/GDPCTPI/CNP16OV)*100

• CONS: ln(PCEC/GDPCTPI/CNP16OV)*100

• INV: ln(FPI/GDPCTPI/CNP16OV)*100

• LAB: ln((AWHNONAG*CE16OV)/CNP16OV)

• INFL: GDPCTPI/4

• WAGE: ln(COMPNFB/A006RD3Q086SBEA*100)

• FFR: FEDFUNDS/4

• CB Bonds: TREAST/GDP

• CB Loans: (MBST+WCPFF)/GDP

• CB Liquidity: (WACBS + FEDDT + TERAUCT + OTHLT)/GDP

For GDP, CONS, INV and WAGE we use the log changes in our measurement equations.Data sources:

• GDP: GDP - Gross Domestic Product, Billions of Dollars, Quarterly, SeasonallyAdjusted Annual Rate, FRED

• GDPCTPI: Gross Domestic Product: Chain-type Price Index, Index 2009=100,Quarterly, Seasonally Adjusted, FRED

• PCEC: Personal Consumption Expenditures, Billions of Dollars, Quarterly, Sea-sonally Adjusted Annual Rate, FRED

• FPI: Fixed Private Investment, Billions of Dollars, Quarterly, Seasonally AdjustedAnnual Rate, FRED

43

• AWHNONAG: Average Weekly Hours of Production and Nonsupervisory Employ-ees: Total private, Hours, Quarterly, Seasonally Adjusted, FRED

• CE16OV: Civilian Employment Level, Index Q12009=100, FRED

• CNP16OV NBD20090101: Civilian Noninstitutional Population, Index Q1 2009=100,Quarterly, Seasonally Adjusted, FRED

• COMPNFB, Nonfarm Business Sector: Compensation Per Hour, Index 2009=100,Quarterly, Seasonally Adjusted, FRED

• FEDFUNDS: Effective Federal Funds Rate, Percent, FRED

• TREAST: U.S. Treasury securities held by the Federal Reserve: All Maturities,Millions of Dollars, Quarterly, Not Seasonally Adjusted, FRED

• MBST: Mortgage-backed securities held by the Federal Reserve: All Maturities,Millions of Dollars, Quarterly, Not Seasonally Adjusted, FRED

• WCPFF: Net Portfolio Holdings of Commercial Paper Funding Facility LLC, Bil-lions of Dollars, Quarterly, Not Seasonally Adjusted, FRED

• WACBS: Central Bank Liquidity Swaps, Millions of Dollars, Quarterly, Not Sea-sonally Adjusted, FRED

• FEDDT: Federal agency debt securities held by the Federal Reserve: All Maturities,Millions of Dollars, Quarterly, Not Seasonally Adjusted, FRED

• TERAUCT: Term auction credit held by the Federal Reserve: All Maturities, Mil-lions of Dollars, Quarterly, Not Seasonally Adjusted, FRED

• OTHLT: Other loans held by the Federal Reserve: All Maturities, Millions ofDollars, Quarterly, Not Seasonally Adjusted, FRED

Appendix B Details on the estimation

Figures B.14 to B.19 show the traces and final distribution after burn in of 100 chainsrunning 4000 iterations. 3000 of these are discarded as burn-in.

44

0.1 0.2 0.30

2

4

6

8

Freq

uenc

y

tpr_beta

0 1000 2000 3000 4000

0.1

0.2

0.3

Sam

ple

valu

e

tpr_beta

0.6 0.8 1.0 1.2 1.4 1.6 1.80.0

0.5

1.0

1.5

2.0

Freq

uenc

y

sig_c

0 1000 2000 3000 40000.50

0.75

1.00

1.25

1.50

Sam

ple

valu

e

sig_c

4 6 8 10 120.0

0.1

0.2

0.3

Freq

uenc

y

phiss

0 1000 2000 3000 40004

6

8

10

Sam

ple

valu

e

phiss

0.1 0.2 0.3 0.4 0.5 0.60

1

2

3

4

Freq

uenc

y

i_p

0 1000 2000 3000 4000

0.2

0.4

Sam

ple

valu

e

i_p

0.0 0.2 0.4 0.6 0.80

1

2

3

Freq

uenc

y

i_w

0 1000 2000 3000 40000.0

0.2

0.4

0.6

0.8

Sam

ple

valu

e

i_w

0.60 0.65 0.70 0.75 0.80 0.85 0.900

2

4

6

8

Freq

uenc

y

zeta_p

0 1000 2000 3000 4000

0.7

0.8

0.9

Sam

ple

valu

e

zeta_p

Figure B.14: Traces of 100 chains (means, 66% and 95% confidence intervals).45

0.5 0.6 0.7 0.80

2

4

6

8

Freq

uenc

y

zeta_w

0 1000 2000 3000 40000.5

0.6

0.7

0.8

Sam

ple

valu

e

zeta_w

1.0 1.2 1.4 1.6 1.80

1

2

3

Freq

uenc

y

Phi_p

0 1000 2000 3000 40001.0

1.2

1.4

1.6

1.8

Sam

ple

valu

e

Phi_p

1.00 1.25 1.50 1.75 2.00 2.25 2.500.0

0.5

1.0

1.5

Freq

uenc

y

phi_pi

0 1000 2000 3000 40001.0

1.5

2.0

2.5

Sam

ple

valu

e

phi_pi

0.10 0.15 0.20 0.250

5

10

15

Freq

uenc

y

phi_y

0 1000 2000 3000 4000

0.10

0.15

0.20

0.25

Sam

ple

valu

e

phi_y

0.05 0.10 0.15 0.20 0.25 0.30 0.350

2

4

6

8

Freq

uenc

y

phi_dy

0 1000 2000 3000 4000

0.1

0.2

0.3

0.4

Sam

ple

valu

e

phi_dy

1 2 3 40.0

0.5

1.0

Freq

uenc

y

LEV

0 1000 2000 3000 4000

1

2

3

Sam

ple

valu

e

LEV

Figure B.15: Traces of 100 chains (means, 66% and 95% confidence intervals).46

0.0 0.1 0.2 0.3 0.40

2

4

6

8

Freq

uenc

y

kap_t

0 1000 2000 3000 40000.0

0.1

0.2

Sam

ple

valu

e

kap_t

0.4 0.5 0.6 0.7 0.8 0.9 1.00

5

10

15

Freq

uenc

y

theta

0 1000 2000 3000 4000

0.8

1.0

Sam

ple

valu

e

theta

0 1 2 3 4 50.0

0.2

0.4

Freq

uenc

y

lamb_cbl

0 1000 2000 3000 40000

2

4

Sam

ple

valu

e

lamb_cbl

0.65 0.70 0.75 0.80 0.85 0.900

2

4

6

8

Freq

uenc

y

rho_u

0 1000 2000 3000 4000

0.7

0.8

0.9

Sam

ple

valu

e

rho_u

0.6 0.7 0.8 0.90.0

2.5

5.0

7.5

10.0

Freq

uenc

y

rho

0 1000 2000 3000 4000

0.7

0.8

0.9

Sam

ple

valu

e

rho

0.3 0.4 0.5 0.6 0.7 0.80

1

2

3

4

Freq

uenc

y

rho_r

0 1000 2000 3000 40000.2

0.4

0.6

Sam

ple

valu

e

rho_r

Figure B.16: Traces of 100 chains (means, 66% and 95% confidence intervals).47

0.4 0.5 0.6 0.7 0.8 0.9 1.00

2

4

6

Freq

uenc

y

rho_g

0 1000 2000 3000 4000

0.6

0.8

1.0

Sam

ple

valu

e

rho_g

0.6 0.7 0.8 0.9 1.00

2

4

6

Freq

uenc

y

rho_i

0 1000 2000 3000 40000.6

0.7

0.8

0.9

1.0

Sam

ple

valu

e

rho_i

0.2 0.4 0.6 0.80.0

0.5

1.0

1.5

2.0

Freq

uenc

y

rho_z

0 1000 2000 3000 40000.00

0.25

0.50

0.75

1.00

Sam

ple

valu

e

rho_z

0.1 0.2 0.3 0.40

2

4

6

8

Freq

uenc

y

rho_p

0 1000 2000 3000 4000

0.2

0.3

0.4

Sam

ple

valu

e

rho_p

0.2 0.3 0.4 0.5 0.6 0.7 0.80

2

4

Freq

uenc

y

rho_w

0 1000 2000 3000 40000.2

0.4

0.6

0.8

Sam

ple

valu

e

rho_w

0.4 0.5 0.6 0.70

2

4

6

Freq

uenc

y

rho_cbl

0 1000 2000 3000 4000

0.4

0.5

0.6

0.7

Sam

ple

valu

e

rho_cbl

Figure B.17: Traces of 100 chains (means, 66% and 95% confidence intervals).48

0.75 0.80 0.85 0.90 0.95 1.000.0

2.5

5.0

7.5

10.0

Freq

uenc

y

rho_qe_b

0 1000 2000 3000 4000

0.8

0.9

1.0

Sam

ple

valu

e

rho_qe_b

0.80 0.85 0.90 0.95 1.000

5

10

Freq

uenc

y

rho_qe_k

0 1000 2000 3000 4000

0.7

0.8

0.9

1.0

Sam

ple

valu

e

rho_qe_k

0.2 0.3 0.4 0.5 0.6 0.70

2

4

6

Freq

uenc

y

sig_g

0 1000 2000 3000 4000

0.2

0.3

0.4

0.5

0.6

Sam

ple

valu

e

sig_g

0.5 1.0 1.5 2.00.00

0.25

0.50

0.75

1.00

Freq

uenc

y

sig_b

0 1000 2000 3000 40000.0

0.5

1.0

1.5

2.0

Sam

ple

valu

e

sig_b

0.0 0.2 0.4 0.6 0.8 1.0 1.20

2

4

6

8

Freq

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y

sig_z

0 1000 2000 3000 4000

0.0

0.5

1.0

Sam

ple

valu

e

sig_z

0.10 0.15 0.20 0.25 0.30 0.35 0.400

2

4

6

8

Freq

uenc

y

sig_r

0 1000 2000 3000 4000

0.2

0.4

Sam

ple

valu

e

sig_r

Figure B.18: Traces of 100 chains (means, 66% and 95% confidence intervals).49

0.1 0.2 0.3 0.4 0.50

2

4

6

8

Freq

uenc

y

sig_i

0 1000 2000 3000 4000

0.2

0.4

Sam

ple

valu

e

sig_i

0.2 0.4 0.6 0.80

1

2

3

4

Freq

uenc

y

sig_p

0 1000 2000 3000 4000

0.2

0.4

0.6

Sam

ple

valu

e

sig_p

0.6 0.8 1.0 1.2 1.4 1.60.0

0.5

1.0

1.5

2.0

Freq

uenc

y

sig_w

0 1000 2000 3000 4000

0.50

0.75

1.00

1.25

1.50

Sam

ple

valu

e

sig_w

1.5 2.0 2.5 3.00.0

0.5

1.0

Freq

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y

sig_cbl

0 1000 2000 3000 4000

1.5

2.0

2.5

3.0

Sam

ple

valu

e

sig_cbl

1.25 1.50 1.75 2.00 2.25 2.50 2.750.0

0.5

1.0

1.5

2.0

Freq

uenc

y

sig_qe_b

0 1000 2000 3000 4000

1.5

2.0

2.5

Sam

ple

valu

e

sig_qe_b

1.25 1.50 1.75 2.00 2.25 2.50 2.750.0

0.5

1.0

1.5

2.0

Freq

uenc

y

sig_qe_k

0 1000 2000 3000 40001

2

3

4

Sam

ple

valu

e

sig_qe_k

Figure B.19: Traces of 100 chains (means, 66% and 95% confidence intervals).50

Appendix C Comparison of Parameter Estimates to a model without fi-nancial sector

(TBD)

distribution mean/alpha sd/beta mean sd mc error hpd 2.5 hpd 97.5

tpr beta gamma 0.250 0.10 0.362 0.105 0.001 0.164 0.581lamb uniform 0.000 1.00 0.019 0.018 0.000 0.000 0.057sig c normal 1.500 0.37 1.163 0.205 0.001 0.809 1.605phiss normal 4.000 1.50 7.650 1.174 0.010 5.580 10.085i p beta 0.500 0.15 0.203 0.084 0.000 0.066 0.380i w beta 0.500 0.15 0.526 0.148 0.001 0.241 0.802zeta p beta 0.500 0.10 0.841 0.036 0.000 0.773 0.914zeta w beta 0.500 0.10 0.638 0.065 0.000 0.515 0.757Phi p normal 1.250 0.12 1.912 0.099 0.000 1.716 2.111psi beta 0.500 0.15 0.825 0.081 0.000 0.659 0.960

phi pi normal 1.700 0.25 1.724 0.268 0.004 1.264 2.270phi y normal 0.125 0.05 0.204 0.034 0.000 0.142 0.271phi dy normal 0.125 0.05 0.179 0.056 0.001 0.074 0.296rho beta 0.700 0.20 0.864 0.030 0.001 0.803 0.917rho r beta 0.700 0.20 0.329 0.061 0.001 0.199 0.430rho g beta 0.500 0.20 0.621 0.249 0.002 0.237 0.955rho i beta 0.500 0.20 0.784 0.110 0.001 0.553 0.949rho z beta 0.500 0.20 0.673 0.202 0.001 0.239 0.920rho u beta 0.500 0.20 0.882 0.030 0.001 0.824 0.940rho p beta 0.700 0.20 0.453 0.103 0.000 0.230 0.625

rho w beta 0.700 0.20 0.536 0.092 0.001 0.368 0.719sig g inv gamma 0.100 2.00 0.590 0.214 0.001 0.290 1.028sig u inv gamma 0.100 2.00 0.605 0.148 0.004 0.329 0.863sig z inv gamma 0.100 2.00 0.261 0.175 0.001 0.048 0.607sig r inv gamma 0.100 2.00 0.351 0.089 0.001 0.207 0.535sig i inv gamma 0.100 2.00 0.256 0.144 0.001 0.099 0.503sig p inv gamma 0.100 2.00 0.190 0.090 0.000 0.084 0.414sig w inv gamma 0.100 2.00 0.646 0.139 0.003 0.365 0.898

Table C.2: Estimation results from Boehl and Strobel (2019) (same model without financial frictions).

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