Reaching Inflation Stability

Antonio Moreno ∗

October 21, 2002

Abstract

Inflation volatility has significantly declined over the last 20 years in the U.S. Tofind out why, I follow a structural approach. I estimate a complete New Keynesianmodel that imposes cross-equation restrictions on the time series of inflation, theoutput gap and the interest rate. I perform the analysis with three measures ofinflation: Consumer Price Index (CPI), Gross Domestic Product Deflator (GDPD)and Personal Consumption Expenditure Deflator (PCE). While CPI and PCE in-flation volatility fell because the internal propagation mechanism changed, GDPDinflation volatility decreased because of the smaller shocks. The shift towards amore aggressive monetary policy can explain a large percentage of the decrease inCPI and PCE inflation volatility.

JEL Classification: C32, C62, E32, E52.Keywords: Volatility, New Keynesian Model, Monetary Policy Rule

∗Columbia University. Department of Economics, 420 W.118th Street, New York, NY 10027. Email:am716@columbia.edu. I thank Jean Boivin, Seonghoon Cho, Tim Cogley, Martijn Cremers, Marc Gian-noni, Ching-Chih Lu, Frederic S. Mishkin Edmund S. Phelps and Luis Rivera-Batiz for useful commentsand suggestions. I am especially grateful to Geert Bekaert for stalwart guidance and support. Financialsupport from the Fundacion Ramon Areces is gratefully acknowledged.

1 Introduction

Inflation volatility has declined over the last 20 years in the U.S. This fact constitutes a

major macroeconomic development since both consumers and producers obtain welfare

gains in a less uncertain inflationary environment. As policy makers of all around the

world strive to tame inflation fluctuations, it is of critical importance to understand better

the causes behind big inflation episodes and the factors which cause inflation volatility

to fall. In this paper we attempt to identify the driving factors which led to our current

stable inflationary economy.

The top panel of Figure 1 graphs the historical inflation series since 1957. We present

three measures for inflation: Consumer Price Index (CPI), Personal Consumption De-

flator (PCE) and GDP deflator. The figure shows the steady increase of inflation since

the mid-60’s up to the beginning of the 80’s. One difference among the series is that the

GDP deflator peaks at the end of 1974, after the first oil shock, whereas the CPI and

PCE measures peak in the beginning of the 80’s. The bottom panel of Figure 1 shows

rolling standard deviations for the three inflation measures. It also shows an important

increase since the mid 60s followed by a steady decline since the early 80’s. Table 1

lists the sample statistics of the inflation series for different sample periods. All the

inflation series have experienced a significant volatility drop since the end of 1980. A

difference across the series is that GDP deflator inflation, unlike CPI and GDP deflator,

did not experience a large decrease in first order autocorrelations. Table 1 shows that

these empirical facts are reinforced if we remove the observations included in the period

1978-1983. The overall picture of lower volatility that emerges from Table 1 and Figure 1

gives raise to the central question in this paper: What led to a lower inflation volatility?

The approach followed in this paper to answer this question is based upon two building

blocks. First, we formulate a monetary New Keynesian model of the macroeconomy which

comprises aggregate supply, aggregate demand and monetary policy rule equations with

endogenous persistence. The introduction of a model has the advantage that it allows to

identify specific propagation mechanisms of aggregate fluctuations. The New Keynesian

model seems adequate for our exercise, as it implies macroeconomic dynamics which

represent a good approximation to those observed in the data, as shown by Rotemberg

and Woodford (1998) and others. In fact, our model estimates yield standard deviations

and autocorrelation patterns which are broadly consistent with the data.

1

Second, we develop a counterfactual analysis in order to determine the driving forces

behind the shift towards a more stable macroeconomic environment. This methodology is

particularly useful for our task, as it makes the private agents and the monetary authority

confront shocks of different sample periods. Hence, it reveals how would the macro

variables have fared under different macroeconomic conditions (shocks), or alternative

behavior of the private sector and the Fed. In this way, we can determine what factors

were critical in reducing macroeconomic volatility.

We impose model’s implied cross-equation constraints in estimation. We estimate

the model across sample periods with three measures of inflation: CPI, PCE and GDP

deflator. We find that while CPI and PCE inflation volatility fell because the internal

propagation mechanism changed, GDPD inflation volatility decreased because of smaller

shocks. The shift towards a more aggressive monetary policy can explain a large per-

centage of the decrease in CPI and PCE inflation volatility.

Our study also highlights the importance of the interaction between the private sector

pricing behavior and the Central Bank reaction function. We show that as agents put

more weight on expected inflation in the AS equation, as historically estimated for the

U.S. economy in the paper, two effects associated with the more activist Fed of the 80’s

and 90’s take place: On the one hand, an increase in the Fed’s degree of activism has

a lower impact on inflation volatility under a less sluggish price setting. On the other

hand, the more aggressive reaction of the Fed to expected inflation has a more stabilizing

effect on output volatility under a more forward looking price setting.

The literature on the drop of inflation volatility is quite recent. Cogley and Sargent

(2002) and Ahmed, Levin, and Wilson (2002) use empirical models to explain this decline.

The first paper shows a contemporaneous decline of the persistence of CPI inflation

with an increase in the Fed’s degree of responsiveness to expected inflation. Our paper

also reports this contemporaneous interaction but in the context of a structural model.

Ahmed, Levin, and Wilson (2002) develop counterfactual analysis with structural VARs

and find that good monetary policy and practices were responsible for the decline of CPI

inflation volatility. In our study we analyze the differences across inflation measures and

find that the lower shocks are instrumental in the drop of the GDP deflator volatility.

Two closely related papers are Stock and Watson (2002) and Boivin and Giannoni

(2002). They also use a monetary model within the New Keynesian class and perform

2

counterfactual analysis. These two papers have a different focus, however. The first one

analyzes systematically the decline in output volatility whereas the second one studies the

factors behind the smaller impact of the monetary policy shocks in the 80’s and 90’s. In

our paper, unlike Stock and Watson (2002) we estimate all the parameters of the model.

We show that the parameter on expected inflation in the AS equation has changed across

subsamples, especially for the CPI and PCE inflation measures. Boivin and Giannoni

(2002) find that the reduced effect of the monetary policy shock is associated with a

stronger systematic reaction of the Fed to inflation. In our paper, we show that, for

parameter values that match the response of output to the monetary policy shock, the

shift towards more stabilizing monetary policy in the early 80’s induced the decline of

CPI and PCE inflation volatility.

This paper proceeds as follows. Section 2 lays out the model of the economy. Section

3 discusses the Rational Expectations solution associated with the model. Section 4

describes the data and the estimation procedure. In Section 5 we perform the break date

tests in order to determine the subsample periods. In Section 6 we show our main results.

Section 7 concludes.

2 A New Keynesian Model of the Economy

This section lays out a simple linear Rational Expectations model of the macroeconomy

which has been employed in recent studies of monetary policy such as Rotemberg and

Woodford (1998). The model comprises Aggregate Supply (AS), Aggregate Demand (IS)

and monetary policy equations. The derivations of each of the equations are consigned

to the Appendix.

The aggregate supply equation is a generalization of the contracting equation pro-

posed by Fuhrer and Moore (1995):

πt = αAS + δEtπt+1 + (1− δ)πt−1 + λ(yt + yt−1) + εASt (1)

πt is inflation between t− 1 and t, αAS is a constant and yt stands for inflation and the

output gap between t− 1 and t. εASt is the aggregate supply structural shock, assumed

to be independently and identically distributed with homoskedastic variance σ2AS. It can

3

be interpreted as a cost push shock that deviates real wages from their equilibrium value

or simply as a pricing error. Et is the Rational Expectations operator conditional on the

information set at time t, which comprises πt, yt, rt (the nominal interest rate at time t)

and all the lags of these variables. Equation 1 shows that δ grows as the private sector

puts more weight on expected inflation. A virtue of this pricing specification is that it

captures the persistence present in the inflation rate. The key feature that imparts the

empirical inflation persistence to this AS equation is the real wage contracting between

firms and workers. As Fuhrer and Moore (1995) show, this AS relation captures the

empirical properties of U.S. inflation dynamics quite accurately.

The IS or demand equation is based on representative agent intertemporal utility

maximization with external habit persistence proposed by Fuhrer (2000) :

yt = αIS + µEtyt+1 + (1− µ)yt−1 − φ(rt − Etπt+1) + εIS,t (2)

where αIS is a constant and εISt is the IS shock, assumed to be independently and iden-

tically distributed with homoskedastic variance σ2IS. In our specification, it is the habit

formation specification in the utility function which allows for the endogenous persistence

of the output gap. The monetary policy channel in the IS equation is the contemporane-

ous output gap dependence on the ex ante real rate of interest which arises in standard

Euler equations derived by lifetime utility maximization. The monetary transmission

mechanism depends negatively on the inverse of the elasticity of substitution, σ and, for

σ > 1, on the parameter that indexes habit persistence, h, since φ = 1σ(1+h)−h

. Appendix

A.2 shows that εISt can be seen as the deviations of the real interest rate from its long

run steady state.

We close the model with the monetary policy rule formulation proposed by Clarida,

Galı, and Gertler (2000)

rt = αMP + ρrt−1 + (1− ρ) [βEtπt+1 + γyt] + εMPt (3)

αMP is a constant and εMPt is the monetary policy shock, assumed to be independently

and identically distributed with homoskedastic variance σ2MP . The policy rule has two

well differentiated parts. On the one hand the monetary authority smooths interest

rates, placing a weight of ρ in the past interest rate. On the other hand, it reacts to high

expected inflation and to deviations of output from its natural rate. The parameter β

4

measures how aggressively the Central Bank fights expected inflation, whereas γ describes

its reaction to output gap fluctuations. In section 6 we show what Central Bank objective

function gives rise to this monetary policy rule. We assume that the Federal funds rate

is the monetary policy instrument, as much of the previous literature.

3 Rational Expectations Equilibrium

3.1 Model Equilibrium and Implications

In this section we follow the framework laid out in Cho and Moreno (2002) to derive the

Rational Expectations equilibrium of the model. Our macroeconomic system of equations

(1), (2) and (3) can be expressed in matrix form as follows:

1 −λ 0

0 1 φ

0 − (1− ρ) γ 1

πt

yt

rt

=

αAS

αIS

αMP

+

δ 0 0

φ µ 0

(1− ρ)β 0 0

Et

πt+1

yt+1

rt+1

+

1− δ λ 0

0 1− µ 0

0 0 ρ

πt−1

yt−1

rt−1

+

εASt

εISt

εMPt

In more compact notation:

B11Xt = α + A11EtXt+1 + B12Xt−1 + εt, εt ∼ (0, D) (4)

where Xt = (πt yt rt)′, B11, A11 and B12 are the coefficient matrices of structural pa-

rameters, and α is a vector of constants. εt is the vector of structural errors, D is the

diagonal variance matrix and 0 denotes a 3 × 1 vector of zeros.1 Following a standard

Undetermined Coefficients approach, a Rational Expectations equilibrium to the system

in (4) can be written as the following reduced form:

Xt+1 = c + ΩXt + Γεt+1 (5)

1In what follows, 0 will denote a matrix, vector or scalar of the appropriate dimension.

5

where c is a 3×1 vector of constants and Ω and Γ are 3×3 matrices. To see this, substitute

equation (5) into equation (4) and rearrange by applying Rational Expectations. Then:

(B11 − A11Ω)Xt = α + A11c + B12Xt−1 + εt (6)

Since the 3 structural equations are linearly independent, (B11 − A11Ω) is nonsingular.

Then, pre-multiplying by (B11−A11Ω)−1 on both sides in equation (6) and matching the

coefficient matrices of Xt−1 and εt, we obtain:

Ω = (B11 − A11Ω)−1B12 (7)

Γ = (B11 − A11Ω)−1 (8)

c = (B11 − A11Ω− A11)−1α (9)

Therefore, equation (5) with Ω, Γ and c satisfying equations (7), (8) and (9) is a solution

to equation (4). Once we solve for Ω as a function of A11, B11 and B12, Γ and c can be

easily calculated. Notice that the implied reduced form of our structural model is simply

a VAR of order 1 with highly nonlinear parameter restrictions. There is a linear relation

between the structural errors, εt and the reduced form Rational Expectations errors (vt),

through Γ,

vt = Γεt (10)

The Rational Expectations equilibrium also yields a simple linear relation between Ω and

Γ through B12, which captures the dependence of the system on the lagged predetermined

variables.

Ω = ΓB12 (11)

3.2 Characterization of the Rational Expectations Equilibrium

I will utilize two methods in order to determine the Rational Expectations equilibrium

to our system. First, I will use the generalized Schur matrix decomposition method

(QZ) developed by Klein (2000) and outlined by McCallum (1999) in order to obtain

the Rational Expectations equilibrium of our system. The QZ method is particularly

useful when the matrix A11 is singular, which is the case in our model, and it allows us

to determine whether there exists a stationary real-valued solution. In Appendix B.1 we

describe the derivation of the Rational Expectations Solution through the QZ method.

6

For Ω satisfying (4) to be admissible as a solution, it must be real-valued and exhibit

stationary dynamics. Because Ω is a nonlinear function of the structural parameters in

B11, A11 and B12, there could potentially be multiple equilibria. Since in this case the QZ

method is silent about determinacy of equilibrium, we will use the recursive method de-

veloped by Cho and Moreno (2002), who solve the model forward recursively and propose

an alternative recursive selection criterion that is stationary and real-valued by construc-

tion. The recursive method is described in Appendix B.2. In it, agents coordinate in an

equilibrium which yields a unique vector of self-fulfilling expectations. This equilibrium

imposes a transversality condition that distant future expectations converge to a zero

vector. The remaining expectations are discarded, since agents deem them incapable

of being satisfied.2 Hence, I will use the QZ and recursive methods jointly in order to

determine the solution to our macroeconomic system.

4 Data and Estimation

We use quarterly data which spans the period between the second quarter of 1957 and the

fourth quarter of 2001. We estimate with three different measures of inflation: CPI, PCE

and GDP deflator. The Federal funds rate is the monetary policy instrument. We use

the average of the Federal funds rate over the previous quarter. Our results are similar if

we use the 3 month T-Bill rate. We use output detrended quadratically, although results

are similar if we employ a linear trend. The data is annualized and in percentages. CPI

and Federal funds rate data were obtained from Datastream, real GDP and GDP deflator

were obtained from the National Income and Product Accounts (NIPA) and the PCE

deflator was collected from the Federal Reserve of St. Luis website.

We estimate the structural parameters using Full Information Maximum Likelihood

(FIML) by assuming normality of the structural errors. Our FIML estimation procedure

allows us to obtain the structural parameters and the VAR reduced form in one stage,

affording a higher efficiency than two-stage instrumental variables techniques. It seems

2If the solution falls into the indeterminacy region, the recursive method ignores the possibility ofthe sunspot shocks discussed in Farmer and Guo (1994). In this case, our equilibrium can be seen as asunspot equilibrium without sunspots. As it will be shown below, we obtain multiple equilibrium in thefirst subsample under the CPI and PCE specifications. Lubik and Schorfheide (2002) allow for sunspotshocks in their estimation and cannot reject the existence of a sunspot equilibrium without sunspots inthe pre-Volcker period.

7

adequate to estimate the whole model jointly, given the simultaneity between the private

sector and the Central Bank behavior, as explained by Leeper and Zha (2000).

The log likelihood function can be written as:

ln L(θ|XT , XT−1, ..., X1) =T∑

t=2

[−3

2ln 2π − 1

2ln |Σ| − 1

2(Xt − ΩXt−1)

′Σ−1(Xt − ΩXt−1)]

(12)

where Xt = Xt−EXt, θ = (δ, λ, µ, φ, ρ, β, γ, σ2AS, σ2

IS, σ2MP ) and Σ = ΓDΓ′. Ω and Γ can

be calculated by the QZ method or the recursive method. Note that we maximize the

likelihood function with respect to the structural parameters in θ, not the reduced form

ones in Ω or Γ. Given the structural parameters, the matrices Ω and Γ must be calculated

at each iteration. This requires checking whether there is a unique, real-valued stationary

solution at each iteration. Whenever there are multiple solutions at the i-th iteration, we

apply the recursive method to select one solution. We choose the initial parameters from

the values used in the literature. In order to check for robustness of our estimates we

set up different initial conditions, randomizing around the obtained parameter estimates

five times. In all of the cases convergence to the same parameter estimates was attained.

We also found that the estimates obtained through our recursive method converge to the

c, Ω and Γ matrices obtained through the QZ method.

5 Dating the Structural Break in the U.S. Economy

Our strategy is to determine the contribution of changes in the economy’s shocks and the

model’s parameters to the drop in volatility. To this end, we perform a structural break

date test, which detects the most likely break date in all the coefficients of an unrestricted

VAR. The idea is that variations in these coefficients reflect changes in the underlying

structural model parameters. Bernanke, Gertler, and Watson (1997) and Clarida, Galı,

and Gertler (1999), among others, have shown evidence of parameter instability across

sample periods. Since it is a fundamental change in the propagation mechanism of the

economy that we are trying to identify, we break up the whole sample period in two

subsamples separated by the date of the break.

We use the Sup-Wald test derived by Bai, Lumsdaine, and Stock (1998), which de-

tects the most likely structural break date in the reduced form coefficients of a vector

8

autoregression. In Table 2 we report the Sup-Wald test associated with unconstrained

VARs of orders one to five using the CPI inflation rate. Except for the VAR(1), the

beginning of the 4th quarter of 1980 is identified as the most likely break date for the pa-

rameters of the reduced form relation (in the case of the VAR(1), the break date selected

is the third quarter of 1980). Since the Schwarz criterion selects the VAR(3) as the order

which provides the best fit to the data, we set the beginning of the fourth quarter of

1980, one year after Paul Volcker became Federal Reserve chairman, as our break date.

Figure 2 graphs the time series of the Wald statistics for the VAR(3). This break date

is robust across inflation and output gap measures and significant at the 1% level.3 The

90% confidence interval is very tight, including only three quarters. This date coincides

with the largest increase, between two quarters, in the average Federal funds rate during

the whole sample: From 9.83% in the 3rd quarter of 1980 to 15.85% in the 4th. The first

sample finishes on the 3rd quarter of 1980 and the second starts the 4th quarter of 1980.4

While there appears to be a clear break date in the relation among our three macroe-

conomic variables, it seems plausible that more than one structural break has occurred

in the joint fluctuations of inflation, the output gap and the Federal funds rate over the

complete sample period. Stock and Watson (2002), for instance, perform a battery of

univariate and multivariate tests and conclude that the most likely break date test for the

majority of the macroeconomic series is around 1984.5 In order to gauge the robustness of

our break date, we perform the following experiment: We estimate unconstrained VARs

for the two subsamples separated by the original break date. Then, with the residuals

of these vector autoregressions, we run the Sup-Wald test for both samples. If no other

clear structural break dates existed, no obvious break dates should arise in this exercise,

since the sample splitting would make the unconstrained parameters approximately sta-

ble across samples. Table 3 shows the break date statistics for unconstrained VARs for

the two subsamples and Figure 3 graphs the time series of Wald statistics.6 While the

years 1974 and 1986 appear as significant candidates for break dates across subsamples,

3The associated asymptotic critical values can be found in Bekaert, Harvey, and Lumsdaine (2002).4Empirical results are similar if we start the sample outside the 90% confidence interval.5One major difference, however, is that when testing for an unknown break date in a multivariate

context, they restrict their attention to the break in the GDP. They perform the Sup-Wald test on VARswith different components of the GDP, but do not include inflation or the Federal funds rate. In ourcase, the highly nonlinear behavior of the Federal funds rate at the beginning of the 80’s seems to beinstrumental in provoking the break.

6Note that we trim the initial and final 15% of the sample when running the Sup-Wald test. AsMaddala and Kim (1998) point out, it is customary to do so in order to rule out breaks around the ends.

9

these breaks do not seem to be as clear as in the original case, since the exact quarter

differs for each VAR order. Given this finding and the fact that two relatively large

subsamples will be available for estimation, we will proceed with our analysis assuming

that there was a single structural break in the 4th quarter of 1980.

6 Results

In this section we present our main findings. Firstly, we report the U.S. FIML structural

parameter estimates for the three different inflation specifications. Secondly, we analyze

the properties of the implied Rational Expectations equilibrium and the model’s goodness

of fit. Then we proceed to explain the drop in inflation volatility and the impact of

monetary policy. Finally, we provide an interpretation of the monetary policy reaction

parameters in terms of the preferences of the government and the AS and IS parameters.

6.1 Parameter Estimates

Table 4 reports the U.S. FIML estimates with the with the three inflation measures:

CPI, PCE and GDP deflator In order to accommodate the documented change in the

deterministic trend growth of output (see, for instance, Orphanides and Porter (1998))

we allow in estimation for separate quadratic trends across subsamples, just as in Ireland

(2001).7

The estimates in Table 4 have all the right sign and are statistically significant except

φ, the coefficient on the real rate in the IS equation and λ, the Phillips curve parameter

in the AS equation. Estrella and Fuhrer (1999), Smets (2000) and Ireland (2001) also

obtained small and insignificant estimates for these two parameters.8 In the AS equation,

agents put more weight on expected inflation than in past inflation in both periods

(except under the first period PCE inflation specification), whereas in the IS equation

7Notice that the break date reported in the previous section is the same for all VAR orders whetherwe use a segmented trend for output or not. While the segmented trend reduces the volatility of theoutput gap, the differences across subperiods are approximately the same. The original ratio of thevolatilities (first period divided by second period) is 1.05, whereas for the segmented trend is 1.07.

8Nelson and Nikolov (2002) perform a survey of studies which estimate this IS coefficient, φ. Bayesianand Minimum Distance methods yield larger values for φ than estimates obtained through MaximumLikelihood or Instrumental Variables estimators.

10

they put about the same weight on the expected and past output gap across periods.

The implied habit persistence parameter, h, is around 1 and significant across inflation

measures and for both samples. Fuhrer (2000) found it to be 0.80. The implied estimates

of the elasticity of substitution range from 15 to 150, but are not significantly different

from zero across periods and inflation periods. Finally, the estimates of the monetary

policy reaction function reflect the smoothing behavior of the Fed, as the persistence

coefficient, ρ, is of large magnitude. They also show that the Fed reacted more strongly

to future inflation in the second period, although not significantly, and that it acted in a

countercyclical fashion, as γ has positive signs in all cases .

Three major stylized facts emerge from Table 4. First, the three standard deviations

of the structural AS and IS shocks are lower in the second period, especially the one

corresponding to the IS shock. Interestingly, Blanchard and Simon (2001) and Ahmed,

Levin, and Wilson (2002) also report decreases in their output equation innovations of a

very similar magnitude.9 Stock and Watson (2002) also present evidence that structural

shocks have been milder since 1984. Second, the probability distribution of the Fed’s

reaction to expected inflation shifted to the right in the second period, but the difference

across estimates is not statistically significant. On this respect, the evidence is mixed

across studies. Clarida, Galı, and Gertler (1999), with single equation GMM estimation

and both Lubik and Schorfheide (2002) and Cogley and Sargent (2002), with a Bayesian

MLE approach in a system framework, find significant increases in the Fed reaction

to inflation. On the other hand, Sims (1999), with a regime switching model for the

reaction function and Ireland (2001) and Cho and Moreno (2002), through frequentist

MLE in a system framework, do not find a significant increase. Third, private agents put

significantly more weight on expected inflation in the AS equation in the second period.

This is specially the case in the estimations with CPI and PCE inflation. Less attention

has been paid to the third fact, however. The exception is Boivin and Giannoni (2002),

who also report an increase in this parameter.10

9Cogley and Sargent (2002) also report a 40% decrease in variance in their unemployment equation.10Woodford (2002) develops an AS specification with persistence within a Calvo (1983) pricing frame-

work. In his model, the endogenous persistence arises due to the existence of price setters who do notadjust optimally. These price setters implement the following indexation rule: log pt =logpt−1 + ϕπt−1.As a result, ϕ reflect the degree of indexation with respect to past inflation. Our baseline estimatesimply that ϕ was 0.81 in the first period, and 0.61 in the second. Notice also that except for the firstperiod estimates with PCE inflation, our estimates are consistent with the implied upper bound of 0.5for the backward looking term of this AS specification.

11

6.2 Implied Equilibrium and Model’s Goodness of Fit

Table 5 reports the generalized eigenvalues associated with the Rational Expectations

equilibrium in both subsamples for the three data specifications. Whereas the second

period equilibrium is unique in all cases, the first period estimates give rise to multiple

equilibria under CPI and PCE inflation, as there are more than three eigenvalues less than

unity. Under Ricardian fiscal policy, multiple equilibria can arise due to the violation of

the “Taylor principle”, whereby the Fed does not stabilize inflation fluctuations (β < 1).11

Then, for the first subsample, we select the solution implied by the recursive method,

which selects the equilibrium associated with the three smallest eigenvalues.

We will measure the model’s goodness of fit across two dimensions: measuring its

ability to match the sample volatilities and vector autocorrelations and performing a

standard likelihood ratio test. Table 6 compares, across sample periods, the volatilities

of the variables found in the data with the volatilities implied by the New Keynesian

model estimates of the baseline specification.12 All the volatilities are matched with

precision except in the case of the interest rate volatility of the second period. This

seems to be due to the highly non-linear behavior of the federal funds rate during the

beginning of the 80’s under the Volcker disinflation. However, Cogley and Sargent (2002)

allow for stochastic volatility in their time-varying parameter model and does not seem

to affect the rejection of the time variance in the coefficients of their empirical model.

Figure 4 compares the sample autocorrelation functions with the vector autocorrela-

tions implied by our structural model.13 The sample inflation autocorrelation functions

across sample periods fall within the model’s autocorrelation confidence bands,14 except

11Cho and Moreno (2002) show that indeterminacy can also arise under negative values of λ. Dupor(2001) derives the condition λ(β − 1) > 1 for determinacy of equilibrium in a New Keynesian modelwithout endogenous persistence.

12Since the structural model is nested in a VAR(1) system, all the elements of the implied variance-covariance matrix of the model (V(Xt)) can be easily computed from the Rational Expectations modelsolution in (5) as: vec(V(Xt)) = (I−Ω⊗Ω)−1(Γ⊗Γ)vec(D), where I is the identity matrix of dimension9× 9.

13The model’s autocorrelation functions of the macro variables can be simply computed as the ra-tios of the diagonal elements of the covariance matrix between Xt and Xt−i implied by the model(Cov(Xt, Xt−i)) to the diagonal elements of V(Xt). Cov(Xt, Xt−i) can be calculated as Cov(Xt, Xt−i) =Ωi ·V(Xt) ∀i = 1, . . . ,K. For instance, the first order autocorrelation of inflation can be computed asthe ratio of the (1,1) element of the matrix Cov(Xt, Xt−1) to the (1,1) element of the matrix V(Xt).

14The empirical autocorrelation confidence bands were constructed through the following Monte Carloprocedure: First, we construct the standard errors of Ω and Γ through the delta-method. Then, we per-form random draws from the normal distributions of D,Γ and Ω, discarding the non-stationary solutions.

12

for distant autocorrelations in the first period. The model output gap autocorrelation

seems to overpredict its sample counterpart in both periods. Finally, the model matches

the interest rate autocorrelation function across sample periods quite closely. The cross-

correlations are not as precisely matched as the autocorrelations. This seems to be due

to the fact that two important cross-coefficients (φ and λ) are not precisely estimated.

As Table 7 shows, our model is statistically rejected according to a standard likelihood

ratio test against an unrestricted VAR(1). This fact is perhaps not surprising since a

likelihood ratio test is a very difficult test to pass for any dynamic stochastic model of

this kind.15 This is also true in our model given its parsimonious VAR(1) form. Cho and

Moreno (2002) perform a small sample study of the likelihood ratio test of this model and

find that when exogenous correlation is added to the model, the model is only marginally

rejected looking at the small sample. They show that one important source of rejection

of the model is the presence of a price puzzle in the unrestricted VAR, an anomaly found

in many empirical studies. Despite this fact, they show that the model implies impulse

response functions that are broadly consistent with those present in the data.

6.3 Explaining the Drop in Inflation Volatility

In this subsection we attempt to determine the sources of the increased stability in the

inflation rate in the context of our New Keynesian macro model. To this end, we develop

a counterfactual analysis given the parameter estimates obtained in both sample periods.

We also develop a sensitivity analysis to provide a better understanding of our results.

We first quantify the contribution of both shocks and the model’s propagation mechanism

to the decline in inflation volatility. Then we focus our attention in the role of monetary

policy.

Finally, we compute the empirical variances, covariances and autocorrelation functions. Replicating thisprocedure 1,000 times yields the empirical distributions of the model autocorrelation functions. Theconfidence bands correspond to the 2.5% and 97.5% quantiles of the distributions.

15Fuhrer (2000) does not reject a structural consumption equation with habit persistence. However,in his methodology, the expectation terms are approximated with several lags.

13

6.3.1 Shocks or Propagation?

In this section we study the role of exogenous shocks and internal propagation in the

documented inflation volatility drop. Table 8 compares the standard deviations of in-

flation and the output gap for all the sample combinations of shocks and propagation.

Let Di be the matrix of structural shocks in period i and Φj the matrix of propagation

coefficients of period j. Then, for instance, σk(Di, Φj), where k = π, y, r and i, j = 1, 2,

denotes the standard deviation of the variable k implied by the system including the

shocks of sample i and the propagation of sample j. There are 5 possible counterfactual

comparisons that can be carried out:

1. If σk(D1, Φ1) > σk(D1, Φ2), the changes in propagation contribute to a lower volatil-

ity of variable k.

2. If σk(D1, Φ1) > σk(D2, Φ1), the changes in shocks contribute to a lower volatility

of variable k.

3. If σk(D2, Φ2) < σk(D1, Φ2), the changes in shocks contribute to a lower volatility

of variable k.

4. If σk(D2, Φ2) < σk(D2, Φ1), the changes in propagation contribute to a lower volatil-

ity of variable k.

5. If σk(D1, Φ2) < σk(D2, Φ1), changes in propagation are more important than changes

in shocks in explaining a lower volatility of variable k. To see this, suppose that the

four previous inequalities hold. In that case, both shocks and volatility contributed

to lower volatility. To determine which factor was more influential we compare the

volatilities implied by the more stabilizing propagation and the lower shocks with

the destabilizing propagation and the larger shocks.

Whereas the first and second inequalities describe how, given an initial subsample,

changes in propagation or shocks would affect the volatilities, the third and fourth in-

equalities reflect the changes in volatilities that would be brought about by returning

to past scenarios of shocks or propagation. The fifth comparison allows us to gauge the

overall importance of shocks relative to propagation in the (relevant, as we will see below)

case that in a given subperiod shocks were milder and both Central Bank and private

sector behavior became more stabilizing.

14

A comparison between σπ(D1, Φ1) and σπ(D1, Φ2) for the three inflation specifica-

tions reveals that the changes in propagation in the second period contributed to the

decrease of inflation volatility. This was triggered by the more forward looking agents

in the AS equation in the 2nd period and the more aggressive reaction of the Fed to

expected inflation. This result is confirmed by the fact that σπ(D2, Φ2) < σπ(D2, Φ1).

The lower shocks also contributed to lower inflation volatility as σπ(D2, Φ1) < σπ(D1, Φ1)

and σπ(D1, Φ2) > σπ(D2, Φ2). Therefore, the question remains as to whether the lower

shocks were more influential than the more stabilizing propagation in reducing inflation

volatility. The direction of the fifth inequality above gives the answer. While for CPI

and PCE inflation σπ(D2, Φ1) > σπ(D1, Φ2), for the GDP deflator the direction of this

inequality is reversed. This implies that for the first two specifications, the lower infla-

tion volatility has been more influenced by changes in propagation rather than by lower

shocks. However, the GDP deflator seems to be more influenced by lower shocks or better

macroeconomic conditions.

Table 9 performs a sensitivity analysis to determine the robustness of our previous

finding. We fix the Phillips curve parameter value, λ, and the coefficient on the real rate, φ

across sample periods, estimate the rest of the model parameters and compare σπ(D2, Φ1)

with σπ(D1, Φ2). The results remain intact for the three inflation specifications.

One could argue, however, that our results may be driven by the structure of the

model and that they may not be present in the data. We contemplate this possibility and

estimate an unrestricted VAR(1) across sample periods. Then we perform an analogous

counterfactual analysis. Table 10 reports σπ(D2, Φ1) and σπ(D1, Φ2) for the three data

specifications. For the CPI and PCE inflation measures, the VAR counterfactuals confirm

the larger influence of the propagation mechanism. For the GDP eflator, propagation and

shocks seem to have the same influence, since σπ(D2, Φ1) is almost the same as σπ(D1, Φ2).

While this implies that our model may be understating the influence of propagation on

the drop of the GDP deflator volatility, it confirms that shocks were more influential in

the decline of GDP Deflator volatility than in that of the CPI and PCE.16

Why do we obtain different results with the GDP Deflator? In order to answer this

question, we recognize the three main differences among the GDP Deflator, the CPI

and the PCE index: First, the CPI and the PCE include the price of imported goods,

16Interestingly, Stock and Watson (2002) also find that GDP deflator inflation volatility decreased dueto shocks. In their study, however, they fix the AS and IS study in all the estimations.

15

whereas the GDP Deflator does not. Second, the GDP deflator includes the price of

goods purchased by investors, the government, and by foreign buyers of domestic goods,

whereas the CPI and PCE do not. Finally, the CPI is a fixed price index whereas

GDP deflator and the PCE account for the change in the domestic production and

consumption, respectively. We do not believe that our finding is related with differences

in the formulas of the different indexes (third reason), since the CPI is a fixed price index

and the PCE is not, and we obtain similar results for both. It seems that divergences in

indexes compositions are driving the different results. In particular, the GDP deflator,

which is the broadest index in scope, includes the prices of goods purchased by firms,

government, investors and foreign buyers, which may be more sensitive to shocks than

the prices of consumption goods.

As for the output gap volatility, the decrease that we are explaining in our sample is

fairly small. Nevertheless, the key factor underlying this small output gap volatility drop

is to be found in the smaller shocks, since σy(D2, Φ1) < σy(D1, Φ2) in our three data

specifications. The changes in propagation did not contribute to this lower volatility,

but actually the opposite, as σy(D1, Φ1) < σy(D1, Φ2) and σy(D2, Φ1) < σy(D2, Φ2). The

decrease in output gap volatility is induced by the significant decrease in the IS shock,

which falls to about half in the three specifications. This result is consistent with Simon

(2000), Blanchard and Simon (2001), and Ahmed, Levin, and Wilson (2002), who, with

different methodologies, also find that the key factor behind the drop of output volatility

was the smaller shocks of the 80’s and 90’s.

6.3.2 Did Monetary Policy Matter?

We now turn to our attention to the influence of monetary policy in the documented

drop of inflation volatility. It is important to recognize first how monetary policy op-

erates in our structural model. There are two transmission mechanisms in our model:

First, monetary policy affects the real interest rate. The real rate enters the IS equation

influencing the output gap, which, in turn, affects inflation through the excess demand

term in the AS equation. Second, monetary policy affects inflation through expectational

effects. To illustrate this second channel, suppose that there is a supply shock to which

the Fed reacts through its policy rule. If the Fed can influence inflation through the first

channel, the private agents will form their expectations accordingly and inflation and

16

output expectations will be affected by the Fed’s actions. In the absence of first channel

(when φ or/and λ are zero), the second mechanism will not be active since private agents

realize that the Fed cannot do anything to tame inflation fluctuations.

One feature of our baseline maximum likelihood estimates in Table 4 is that both

φ and λ are very small and imprecisely estimated. This results in a small impact of

monetary policy. To see this, Figure 5 compares the model’s response of the output

gap to the monetary policy shock to that of an identified monetary policy shock in

an empirical VAR(1).17 As can be seen in the figure, for our baseline estimates, the

model cannot match the impulse response present in the data. This impulse response

is especially relevant for our primary transmission mechanism as monetary policy tames

the inflation rate by reducing the output gap. In order to solve this problem, we fix the

parameters which govern the monetary transmission mechanism at higher values so as to

approximately match this impulse response. and we estimate the rest of the parameters

by our maximum likelihood procedure.

Table 11 lists in percentage terms the influence on inflation of the shift towards a more

aggressive monetary policy. The first row of each of the data specifications corresponds

to our baseline estimates. The second and third row list the impact of monetary policy

associated with the estimates obtained when we calibrate the coefficient φ so as to match

the impulse response function, as described above. It is interesting to observe that as the

parameters which govern the monetary transmission mechanism (φ and λ) increase, the

estimated Fed’s response to expected inflation increases. This fact reflects the importance

of the expectational effects previously explained: As monetary policy becomes more

effective under larger values of φ and λ, expectations adjust accordingly so that the Fed

has more incentives to stabilize expected inflation.

When we adjust the estimation to match the impulse response function, the impact

of monetary policy grows considerably. In the case of the CPI, whereas our baseline

estimates implied that monetary policy was responsible of 30% of the decrease in inflation

volatility, under the alternative estimation specification, this percentage increases to 51%

and 71%. For the PCE, it increases to 63% and 53% from the original 42% and for the

GDP deflator it goes up to 35% from 19%. Therefore, our results confer monetary policy

17We identify the monetary policy shock in the VAR(1) as follows: Let wt be the vector of errorterms of the unconstrained VAR(1) and εt the structural shocks. Under the null of the model, Γεt = wt.Therefore Γ−1wt yields the structural error shocks.

17

an instrumental role in the decline of inflation volatility.

6.3.3 Conditional effects of the more aggressive reaction function

One advantage of our study is that it allows us to gauge the influence of monetary

policy changes conditional on the changes in the private sector behavior. As our baseline

estimates show, in the second period agents became more forward looking in the AS

equation. We proceed then to measure the influence of the monetary policy switch

conditional on increases in the forward looking pricing behavior.

The top panel of Figure 6 shows the conditional gain in inflation volatility of switching

to a more responsive monetary policy. This gain is calculated as the difference of inflation

volatility between a passive monetary policy regime (σ2π(βS)) and an activist regime

(σ2π(βL)). When the agents are less forward looking in the AS equation this gain is large.

However, as the price setting becomes more forward looking this gain dissipates. The

reason is that under very forward looking agents in the AS equation, monetary policy

ceases to have relevance in the future path of inflation, as future inflation depends less

on current inflation.

The bottom panel of Figure 6 shows the conditional gain in output volatility of

switching to a more responsive monetary policy. It shows that for a more sluggish price

setting behavior (when δ is around 0.5) switching to a more responsive monetary policy

is not without high costs. Then, as the price setting becomes more forward looking these

losses turn into gains. Figure 7 shows why this is the case. Panel A of Figure 7 shows

that an increase in β exacerbates the recessionary effect of the AS shock, whereas it

dampens the fluctuations on the output gap provoked by the IS and the monetary policy

shocks. Panel B of Figure 7 shows that increases in δ are associated with a diminishing

real effect of the AS shock and a larger effect of both the IS and the monetary policy

shocks on output. The smaller effect of the AS shock and the larger impact of the IS

shock are caused by the Fed’s smaller interest rate reaction implied by the higher value

of δ. The impact of the monetary policy shocks is amplified since the lower response

of inflation under higher values of δ makes the Central Bank reduce the interest rate

by a lesser amount, amplifying the recessionary effects of the monetary policy shock.18

18Notice that increases in β, under high values of δ, can reduce the output gap volatility also in theabsence of monetary policy shocks.

18

Therefore, under a more forward looking price setting, the more activist Fed stabilized

the (magnified) impact of IS and monetary policy shocks.

6.4 Interpreting the Fed reaction Function

While an increase in β tells us that the Fed reacted more strongly to expected inflation

in the 80’s and 90’s, it would be interesting to determine to what extent this increase

was due to the change in the actual preferences of the Fed against expected inflation.

It could be that the Fed is reacting more strongly because of a modification in other

structural parameters of the economy. In order to answer this question, we formulate the

optimization problem of the Federal Reserve which gives rise to our interest rate reaction

function:

minit1

2(ψλπ(Etπt+1 − π)2 + y2

t + λii2t + λ∆(it − it−1)

2) (13)

λπ, λi and λ∆ are the objective function weights on expected inflation, interest rate

variation and interest rate changes, respectively. As a result, this objective function

implies that the Fed is trying to stabilize expected excess of inflation and the current

output gap. Additionally, the Fed is trying to avoid excessive interest rate variation as

well as deviations of the current interest rate from its past value. Since this objective

function does not contain expected future terms beyond the next period inflation forecast,

this optimization problem falls under the category of discretion. In other words, the Fed

reoptimizes every period the objective function in (13) in order to choose its instrument,

the Federal funds rate.19 The implied interest rate rule becomes, in mean deviation form:

it =λ∆

λi + λ∆

it−1 − ψλπ

λi + λ∆

· ∂Etπt+1

∂itEtπt+1 − 1

λi + λ∆

· ∂yt

∂ityt (14)

Matching these coefficients with those of the Clarida, Galı, and Gertler (2000) rule, which

19Svensson (2001) criticizes forecast-based instrument rules of this kind on the grounds of time in-consistency. Our goal in this paper is not to derive a formal optimal monetary policy plan, but ratherprovide a deeper interpretation of the estimated parameters of our interest rate rule which seems tocapture the short term interest rate dynamics quite accurately.

19

we estimated, yields:

β = −ψλπ

λi

∂Etπt+1

∂it(15)

γ = − 1

λi

∂yt

∂it(16)

ρ =λ∆

λi + λ∆

(17)

The partial derivatives in the coefficients reflect the constraints imposed by the AS and IS

equation in the optimization program of the monetary authority. Equations (15) and (16)

show that it is optimal to react more strongly to expected inflation and the output gap

as these two variables become more sensitive to changes in the interest rate, respectively.

Equation (17) shows that the optimal ρ grows in tandem with the tendency of the Fed

towards smoothing the interest rate. One difficulty to compute these partial derivatives

in our setting is the endogeneity of it in our model. However, given our policy rule, we

can distinguish an exogenous part (εMPt) as a source of interest rate fluctuations. In

Appendix A.3 we derive the partial derivatives proxying changes in it with changes in its

exogenous part. Hence, we can identify λπ, λi and λ∆ uniquely by setting ψ to 0.99 and

using our parameter estimates to obtain the left hand side of (15)-(17) and the partial

derivatives.

Table 12 lists the implied monetary policy weights on the expected inflation, inter-

est rate and interest rate difference terms for both periods under our objective function

specification. For the three inflation specifications there was an increase in λπ, implying

a larger concern of the Fed towards deviation of expected inflation from its target. Ad-

ditionally, while the term λi is smaller in the second period, revealing less concern for

interest rate variance.

We finally turn our attention to the optimal changes in the Fed’s response to inflation

as a function of the price setting parameter δ. This is a relevant issue, since the estimation

yielded a significant increase in δ across sample periods. Since our setup includes two

equations, (15) and (16), to determine a unique optimal relation between β and δ, we

will assume in our analysis that the Fed does extreme inflation targeting, i.e. γ = 0. In

Appendix C.2, we show how to obtain numerical approximations in order to answer this

question using equation (15). These approximations can be computed via the Implicit

Function theorem, as the partial derivative depends on β and δ. As shown in this

20

Appendix, the sign relation between the Fed’s response to inflation and the price setting

parameter can be expressed as:

∂β

∂δ= − k · ∂f(β,δ,·)

∂δ

1 + k · ∂f(β,δ,·)∂β

f(β, δ, ·) =∂Etπt+1

∂it, k > 0 (18)

where β is the implicit function of β. The top panel of Figure 8 graphs f(β, δ, ·) as

a function of β. The remaining AS and IS parameters are fixed at their first period

estimated values, whereas the policy parameters are fixed at their second period values

(γ is fixed at zero), since our break date seems to be mostly induced by monetary policy

changes.20 The top Panel of Figure 8 shows that, for realistic parameter values (β > 0.2),

f(θ) is an increasing function of β. The bottom panel of Figure 8 shows that f(θ) is a

decreasing function of δ for values of δ smaller than 0.6, but then it becomes an increasing

function. This implies that, approximately, for historical estimated values:

∂β

∂δ> 0 δ < 0.6

∂β

∂δ< 0 δ > 0, 6

What is the mechanism at work in the optimal relation between β and δ? When

the price setting exhibits a high degree of inertia, increases in δ make expected inflation

respond more strongly to increases in the interest rate, as expected inflation depends

highly on current inflation. Then, at low values of δ, it is optimal for the Fed to react more

strongly to expected inflation as δ increases. However, when the price setting is highly

forward looking (for values of δ above 0.6), current inflation is not so instrumental in

determining future inflation, so that interest rate changes are not so effective in reducing

future inflation. Therefore, our results suggest that it is not optimal for the Fed to

become more responsive to expected inflation when agents become more forward looking

in the AS equation beyond a given cutoff value for δ. Panel B of Figure 8 makes this

point clearer. It shows that for small Phillips curve parameters (λ), the cutoff value is

smaller. This is due to the fact that, under a small λ, interest rate increases have a lower

impact on inflation. Consequently, it is then optimal to stop increasing the reaction to

expected inflation earlier as δ increases.

20Results are similar if we fix all the remaining parameters at their 1st or 2nd period values.

21

7 Conclusion

In the context of a New Keynesian model, we showed, on the one hand, that CPI and

PCE inflation volatility declined because the internal propagation mechanism changed

in the 80’s and 90’s. On the other hand, we showed that the smaller shocks had more

importance on the decrease of GDP deflator inflation volatility. We also showed that for

meaningful parameter values for the monetary transmission mechanism, the shift towards

a more aggressive monetary policy was instrumental in lowering CPI and PCE inflation

volatility

This paper raises a number of questions for future research, but perhaps the most

pressing one is related to the contemporaneous increase in the Fed’s responsiveness to

inflation and the private sector’s forward looking behavior in the AS equation. As Wood-

ford (2002) observes, variations in agents’ price setting behavior are exogenous in stan-

dard AS specifications with endogenous persistence. An exception is Dotsey, King, and

Wolman (1999), who derive a state-dependent pricing specification. The present paper

underscores the need to model and estimate the explicit links between the price setting

behavior and the monetary authority degree of activism.

22

Appendix

A A New Keynesian model for the US Economy

A.1 AS Equation

We assume that the log of the price level at time t, pt, is an average of the current and

past logs of nominal wages, wt and wt−1, respectively, so that:

pt =1

2(wt + wt−1) (19)

Private agents negotiate real wage contracts with the firms in a staggered fashion,21 so

that current real wages depend on past and expected real wages in the following way:

wt − pt = δEt(wt+1 − pt+1) + (1− δ)(wt−1 − pt−1) + λ(yt + yt−1) (20)

Et is the Rational Expectations operator conditional on the information set at time t.

Therefore, δ, which is between 0 and 1, grows as agents put more weight on the expected

prevailing real wages at time t + 1. Substituting wt − pt into the definition of the price

level in 19 yields the AS equation:

πt = αAS + δEtπt+1 + (1− δ)πt−1 + λ(yt + yt−1) + εASt (21)

where εASt is an added pricing shock.

A.2 IS Equation

The IS equation describes the demand side of the economy. It is derived by a represen-

tative agent model where the utility function given by

U(Ct) =( Ct

Ht)1−σ − 1

1− σ(22)

21See Phelps (1978) and Taylor (1980) for pioneer work on supply specifications based on wage con-tracting models.

23

Ct is the consumption level, Ht is the external level of habit and σ is the inverse of

the elasticity of substitution. The habit level is external in the sense that the consumer

does not consider it as an argument to maximize his utility function. We assume that

Ht = Cht−1 ex post, where h (> 0) measures how strong the habit level is. It is the habit

specification that will introduce persistence in the IS equation. The budget constraint

that the agent faces is

Ct + Bt ≤ Pt−1

Pt

Bt−1Rt + Wt (23)

This constraint implies that agents’ consumption at time t, Ct, plus the value of his asset

holdings, Bt, cannot exceed his endowment each period, which comes from an exogenous

labor income, Wt, and the real value of the asset holdings that he had at the beginning

of the period, Pt−1

PtBt−1, multiplied by the nominal gross return on those assets, Rt.

The agent is infinitely lived and maximizes his lifetime stream of utility, subject to

the dynamic budget constraint in (23). The Euler equation is given by

1 = Et[ψU ′(Ct+1)

U ′(Ct)

Pt

Pt+1

Rt] (24)

where ψ is the subjective time discount factor and Pt is the price level at time t. By

assuming joint lognormality of consumption and inflation, the following expression can

be derived:

ct = αC + µEtct+1 + (1− µ)ct−1 − φ(rt − Etπt+1) (25)

where αC =−ln(ψ)− 1

2Vt(σct+1+πt+1)

σ(1+h)−h, ct is the log of consumption at time t, Vt is the con-

ditional variance operator at time t, µ = σσ(1+h)−h

and φ = 1σ(1+h)−h

. As can be seen

in equation (25), the monetary transmission mechanism is a function of the inverse of

the elasticity of substitution of consumption across periods, σ, and the habit persistence

parameter, h.

From the market clearing condition, Y ∗t = Ct +Gt, where Y ∗

t is the aggregate demand

and Gt denotes the remaining demand components: investment, government expenditures

and net exports. Taking logs, ct = y∗t + zt, where y∗t is the log of GDP at time t and

zt =log(Y ∗t −Gt

Y ∗t). Let y∗t = yT

t + yt, where yTt denotes the potential output or trend

component of y∗t and yt is the output gap. Then, equation (25) can be rewritten as:

yt = αC + µEtyt+1 + (1− µ)yt−1 − φ(rt − Etπt+1) + gt (26)

24

where gt = −(zt +yTt )+µEt(zt+1 +yT

t+1)+(1−µ)(zt−1 +yTt−1). Note that yt rises with Gt.

Finally, define αg = Egt, where E is the unconditional expectation operator, in order to

rewrite the IS equation as

yt = αIS + µEtyt+1 + (1− µ)yt−1 − φ(rt − Etπt+1) + εIS,t (27)

where αIS = αC + αg and εISt = gt − αg. Since we do not model Gt explicitly, εIS,t is

assumed to be independent and identically distributed with homoskedastic variance σ2IS.

Equation (27) can also be expressed as

yt = µEtyt+1 + (1− µ)yt−1 − φ(rt − Etπt+1 − rrt) (28)

where rrt = rr +εISt

φ, is the real rate of interest and rr = αIS

φ. rr represents the long run

equilibrium real rate of interest and εISt can then bee seen as shocks to the real rate of

interest.

A.3 Monetary Policy Rule

The instrument of the monetary authority, the Federal funds rate, is set according to the

following reaction function:

rt = ρrt−1 + (1− ρ)r∗t + εMPt (29)

r∗t = r∗ + β(Etπt+1 − π) + γyt (30)

π is the long run equilibrium level of inflation and r∗ is the desired nominal interest rate.

There are two parts to the equation and εMPt is the monetary policy shock. The lagged

interest rate captures the well known tendency of the Federal Reserve towards smoothing

interest rates, whereas r∗t represents the “Taylor rule” whereby the monetary authority

reacts to deviations of expected inflation from its target and to deviations of output from

its natural level. Hence, the monetary policy equation becomes:

rt = αMP + ρrt−1 + (1− ρ) [βEtπt+1 + γyt] + εMPt (31)

where αMP = (1− ρ)(r ∗ −βpi)

25

B Rational Expectations Equilibrium

B.1 The QZ Method

In this appendix we derive the Rational Expectations solution of our New Keynesian

model using the generalized (QZ) Decomposition. For ease of exposition we reproduce

equations (4)and (5) in mean deviation, so that Xt = Xt − EXt:

B11Xt = A11EtXt+1 + B12Xt−1 + εt (32)

Xt+1 = ΩXt + Γεt+1 (33)

We further assume, for the sake of generality, that the error terms are serially correlated,

i.e. εt = Fεt−1 + wt. Our goal is to solve for Ω and Γ, since they completely determine

the equilibrium dynamics of our system.

Substituting equation (33) into equation (32) we can obtain

A11[Ω(ΩXt−1 + Γεt) + ΓFεt] = B11[ΩXt−1 + Γεt] + B12Xt−1 + εt (34)

Collecting the Xt−1 and εt terms yields, respectively:

A11Ω2 = B11Ω + B12 (35)

A11ΩΓ + A11ΓF = B11Γ + I (36)

Expressing equation (35) in matrix companion form:

[A11 0

0 I

][Ω2

Ω

]=

[B11 B12

I 0

][Ω

I

](37)

Define the 2n × 2n matrices A =

[A11 0

0 I

]and B =

[B11 −B12

I 0

]where n is the

number of endogenous variables. Then the generalized Schur decomposition guarantees

the existence of invertible matrices Q and Z such that QAZ = S and QBZ = T , with

S and T triangular. The ratios tiisii

are the generalized eigenvalues of the matrix pencil

26

B − λA, where λ ∈ C is a given generalized eigenvalue. Premultiply (37) by Q, define

H = Z−1 and apply the QZ decomposition to obtain:

[S11 0

S21 S22

][H11 H12

H21 H22

][Ω2

Ω

]=

[T11 0

T21 T22

] [H11 H12

H21 H22

][Ω

I

](38)

where the submatrices Sij, Hij and Tij are of dimension n × n. The first row can be

written as:

S11(H11Ω + H12)Ω = T11(H11Ω + H12) (39)

Equation (39) is satisfied for an Ω such that:

Ω = −H−111 H12 = −H−1

11 (−H11Z12Z−122 ) = Z12Z

−122 (40)

Finally, once we solve for Ω, Γ is obtained directly from equation (36):

vec(Γ) = [I + F ′ ⊗ (A11Ω−B11)−1A11]

−1vec[(A11Ω−B11)−1] (41)

B.2 The Recursive Method

We can characterize the stationarity, uniqueness and real-valuedness of the equilibrium

of our system as follows: If all the eigenvalues, tiisii

, are less than unity in absolute value,

then Ω is stationary. If the number of stable generalized eigenvalues is the same as that

of the predetermined variables (3 in our model, the lagged endogenous variables), then

there exists a unique solution. If there are more than 3 stable generalized eigenvalues,

then we have multiple solutions. Conversely, if there are less than 3 stable eigenvalues,

there is no stable solution. Finally, Ω is real-valued if (a) each one of its eigenvalues

is real-valued, or (b) for every complex eigenvalue of Ω, the complex conjugate is also

an eigenvalue of Ω. Unfortunately, in the case of multiple stationary solutions, there

seems to be no agreement about the selection of a solution among the candidates.22 In

this case, we use the recursive method developed by Cho and Moreno (2002), who solve

the model forward recursively and propose an alternative simple selection criterion that

22Blanchard and Kahn (1980) suggest the choice of the 3 smallest eigenvalues and McCallum (1999)suggests the choice that would yield Ω = 0 if it were the case that B12 = 0. Uhlig (1997) observesthat McCallum’s criterion is difficult to implement but it often coincides with Blanchard and Kahn’scriterion.

27

is bubble-free and real-valued by construction. The idea is to construct sequences of

convergent matrices, Ck, Ωk, Γk, k = 1, 2, 3, ... such that:

Xt = CkEtXt+k+1 + ΩkXt−1 + Γkεt (42)

We characterize the solution that is fully recursive as follows. We check first whether

Ω∗ ≡ limk→∞

Ωk and Γ∗ ≡ limk→∞

Γk exist, and Ω∗ is the same as one of the solutions obtained

through the QZ method. For the limit to equation (42) to be a bubble-free solution,

limk→∞

CkEtXt+k+1 must be a zero vector. Then the solution must be of the form:

Xt = Ω∗Xt−1 + Γ∗εt (43)

Finally, we check whether limk→∞

CkEtXt+k+1 = limk→∞

CkΩ∗k = 0 using equation (43).

C The Fed Reaction Function Coefficients

C.1 Computing the Partial Derivatives

Equations (15) and (16) show that the optimal β and γ coefficients in our reaction

function depend on partial derivatives with respect to interest rate changes. In order to

compute these partial derivatives we recognize endogenous and exogenous parts in the

interest rate in our model:

it = it + it (44)

where it = ρit−1 + (1 − ρ)(βEtπt+1 + γyt) and it = εMPt . Therefore, it constitutes the

exogenous part. In this setting, we can proxy the partial derivative terms involving

changes in it by changes in it applying vector differentiation rules to our model solution.

Since next period expectations can be expressed as EtXt+1 = ΩXt, using the model

solution:

EtXt+1 = Ω(ΩXt−1 + Γεt) (45)

Therefore ∂EtXt+1

∂εt= ΩΓ, so that ∂Etπt+1

∂itis simply the (1,3) element of the product matrix

ΩΓ, i.e.:∂Etπt+1

∂it= Ω11Γ13 + Ω12Γ23 + Ω13Γ33 (46)

28

In order to obtain the term ∂yt

∂itin equation (16) we follow the procedure describe above

and we use the IS equation to obtain:

∂yt

∂it= Γ13(µΩ21 + φΩ11) + Γ23(µΩ22 + φΩ12) + Γ33(µΩ23 + φ(1− Ω13)) (47)

C.2 Optimal Fed behavior

In this final Appendix we derive the optimal reaction of the Fed to expected inflation

as a function of changes in the behavior of the private sector in the AS equation. We

reproduce equation (15) for ease of exposition:

β = −ψλπ

λi

∂Etπt+1

∂it(48)

In the previous subsection, we computed an approximation to the term ∂Etπt+1

∂it. As

can be seen in equation (46), this term depends on the reduced form elements of the

Rational Expectations solution which, in turn, also depend on β. Therefore, we can use

the Implicit Function theorem in equation (48) to determine how the Fed would change

its reaction to expected inflation when the private sector puts more weight on expected

inflation; i.e. we can compute the sign of ∂β∂δ

.

Let us first introduce some additional notation:

f(θ) =∂Etπt+1

∂itδ, λ, µ, φ, β, γ, ρ ∈ θ (49)

Then, we can rewrite (48) as:

β = β + k · f(θ) = 0 k =ψλπ

λi

> 0 (50)

Using the Implicit Function theorem:

∂β

∂δ= − k · ∂f(θ)

∂δ

1 + k · ∂f(θ)∂β

(51)

In order to obtain the sign of the partial derivatives, we can compute numerically vectors

of the Ωij and Γij terms in (46) as a function of a given θ, i.e. δ and β, holding the

29

remaining parameters constant. In this way, we can construct f(θ) in (46) as a function

of δ and β, identify the sign of the terms ∂f∂δ

and ∂f∂β

and, in turn, the sign of the derivative∂β∂δ

. The top and bottom panels of Figure 8 graph the functions involved in (51).

30

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33

Table 1: Descriptive statistics

1957:2Q-1980:3Q 1980:4Q-2001:2Q 1957:2Q-1977:4Q 1983:1Q-2001:2Q

CPIπ 4.7383 3.4923 3.8939 3.0934

(0.7243) (0.3424) (0.6083) (0.2636)σπ 3.7771 2.0876 3.0329 1.5440

(0.4687) (0.3671) (0.4248) (0.1966)ρπ 0.8215 0.6075 0.7444 0.4610

(0.0654) (0.1105) (0.0856) (0.0975)

PCEπ 1.8552 1.3849 1.5840 1.1878

(0.2582) (0.1321) (0.2341) (0.1197)σπ 1.3048 0.7014 1.1193 0.5515

(0.1486) (0.1131) (0.1794) (0.0601)ρπ 0.9112 0.7161 0.8669 0.6234

(0.0443) (0.0671) (0.0904) (0.0262)

GDPDπ 4.3584 3.1967 3.8594 2.5841

(0.5601) (0.3812) (0.5481) (0.1972)σπ 2.7266 1.8127 2.4958 0.8215

(0.5601) (0.3812) (0.3768) (0.0923)ρπ 0.9983 0.9203 0.9874 0.9484

(0.0360) (0.0332) (0.0507) (0.0389)

This Table shows the descriptive statistics of CPI inflation (CPI), PCE inflation (PCE) and GDP deflatorinflation (GDPD). π stands for the average, σπ is the standard deviation and ρπ is the first orderautocorrelation. These statistics and their respective standard errors (in parentheses) were computedusing generalized method of moments (GMM) estimation. The weighting matrix is constructed 3 Newey-West lags. The following system of equations was estimated for each inflation measure:

e1t = πt − π

e2t = (πt − π)2 − σ2π

e3t = (πt − ¯π)(πt−1 − ¯π)− ρπ(πt−1 − ¯π)

where ¯π is the sample mean of inflation. e1t, e2t and e3t are the disturbances so that et = e1t, e2t, e3tand E[et] = 0. There are three parameters to be estimated and three orthogonality conditions, so thatthe system is exactly identified.

34

Table 2: Sup-Wald Break Date Statistics

Sample Period VAR Sup-Wald Break Date 90% Confidence Interval

1957:2Q-2001:4Q 1 68.35 1980:3Q 1980:2Q-1980:4Q

1957:2Q-2001:4Q 2 115.42 1980:4Q 1980:3Q-1981:1Q

1957:2Q-2001:4Q 3 115.02 1980:4Q 1980:3Q-1981:1Q

1957:2Q-2001:4Q 4 146.70 1980:4Q 1980:3Q-1981:1Q

1957:2Q-2001:4Q 5 163.56 1980:4Q 1980:3Q-1981:1Q

This Table lists the Sup-Wald values of the break date test derived by Bai, Lumsdaine, and Stock (1998).The test detects the most likely break date of a break in all of the parameters of unconstrained VARsof orders 1 to 5. The Table shows the results of the test using the CPI, quadratically detrended outputgap and the Federal funds rate.

35

Table 3: Sup-Wald Break Date Statistics (Robustness Test)

Panel A: 1st subsample

Sample Period VAR Sup-Wald Break Date 90% Confidence Interval

1957:2Q-1980:3Q 1 23.01 1974:4Q 1974:2Q-1975:2Q

1957:2Q-1980:3Q 2 33.67 1974:2Q 1974:1Q-1974:3Q

1957:2Q-1980:3Q 3 46.30 1974:2Q 1974:1Q-1974:3Q

1957:2Q-1980:3Q 4 54.83 1974:4Q 1974:3Q-1975:1Q

1957:2Q-1980:3Q 5 81.36 1975:1Q 1974:4Q-1975:2Q

Panel B: 2nd subsample

Sample Period VAR Sup-Wald Break Date 90% Confidence Interval

1980:4Q-2001:4Q 1 33.62 1986:3Q 1986:2Q-1986:4Q

1980:4Q-2001:4Q 2 52.75 1985:4Q 1985:3Q-1986:1Q

1980:4Q-2001:4Q 3 90.64 1986:1Q 1985:4Q-1986:2Q

1980:4Q-2001:4Q 4 128.04 1986:2Q 1986:1Q-1986:3Q

1980:4Q-2001:4Q 5 174.00 1987:01Q 1986:4Q-1987:2Q

Panel A lists the Sup-Wald values of the break date test derived by Bai, Lumsdaine, and Stock (1998)applied to a VAR(3) of the first subsample residuals implied by the vector autoregression of CPI inflation,the output gap and the Federal funds rate. Panel B lists the values associated with the analogous exercisefor the second subsample.

36

Table 4: FIML estimates

CPI PCE GDP Defl.1st P. 2nd P. 1st P. 2nd P. 1st P. 2nd P.

δ 0.5482 0.6188 0.4702 0.5769 0.5377 0.5595(0.0182) (0.0294) (0.0496) (0.0239) (0.0151) (0.0174)

λ 0.0072 0.0009 0.0045 0.0003 0.0013 0.0008(0.0104) (0.0032) (0.0033) (0.0005) (0.0032) (0.0009)

µ 0.5180 0.4959 0.4699 0.4795 0.5007 0.4820(0.0213) (0.0219) (0.0510) (0.0355) (0.0220) (0.0344)

φ 0.0146 0.0032 0.0339 0.0058 0.0098 0.0064(0.0126) (0.0035) (0.0261) (0.0053) (0.0052) (0.0058)

ρ 0.7740 0.8784 0.7889 0.8772 0.8395 0.8480(0.0509) (0.0305) (0.0539) (0.0297) (0.0212) (0.0401)

β 0.9825 1.9652 2.2442 3.4490 0.8279 1.6445(0.1400) (0.5890) (0.3634) (1.3847) (0.2080) (0.4497)

γ 0.6992 0.5686 0.8001 0.6865 1.1827 0.5720(0.2448) (0.3676) (0.3142) (0.3508) (0.1389) (0.2791)

σAS 1.2034 1.0015 0.3100 0.2587 0.7880 0.4603(0.0984) (0.0865) (0.0273) (0.0214) (0.0603) (0.0389)

σIS 0.7149 0.3794 0.6842 0.3879 0.5878 0.3877(0.0566) (0.0301) (0.0683) (0.0344) (0.0457) (0.0356)

σMP 0.7551 0.7322 0.7466 0.7288 0.8239 0.7226(0.0559) (0.0564) (0.0578) (0.0573) (0.0585) (0.0549)

This Table shows the FIML parameter estimates of the structural New Keynesian macro model for ourthree data specifications: with CPI, PCE and GDP deflator. Output is detrended quadratically and theFederal funds rate is used as interest rate. 1st P. stands for the 1st period estimation and 2nd P. standsfor the second period estimation. he first subsample spans the period 1957:2Q-1980:3Q and the secondsubsample covers the period 1980:4Q-2001:4Q. The model’s equations in demeaned form are:

πt = δEtπt+1 + (1− δ)πt−1 + λ(yt + yt−1) + εASt

yt = µEtyt+1 + (1− µ)yt−1 − φ(rt − Etπt+1) + εIS,t

rt = ρrt−1 + (1− ρ) [βEtπt+1 + γyt] + εMPt

37

Table 5: Generalized Eigenvalues: Baseline Specifications

Panel A: CPI1st period 2nd period

ξ1 0.7484 0.6158ξ2 0.8319-0.1245i 0.9130-0.0314iξ3 0.8319+0.1245i 0.9130+0.0314iξ4 0.9978 1.0118ξ5 1.1234 1.0579ξ6 ∞ ∞

Panel B: PCE1st period 2nd period

ξ1 0.8338 0.7328ξ2 0.8735-0.1444i 0.9143-0.0464iξ3 0.8735+0.1444i 0.9143+0.0464iξ4 1.2378-0.0472i 1.0110ξ5 1.2378+0.0472i 1.1246ξ6 ∞ ∞

Panel C: GDP Deflator1st period 2nd period

ξ1 0.8430 0.7822ξ2 0.8644-0.0991i 0.8990-0.0429iξ3 0.8644+0.0991i 0.8990+0.0429iξ4 0.9955 1.0117ξ5 1.1326 1.1194ξ6 ∞ ∞

This Table reports the generalized eigenvalues which determine the stability of the structural macromodel under the three data specifications.

38

Table 6: Standard Deviations

Panel A: CPI1st period 2nd period

Sample Model Sample Modelσπ 3.7730 3.6374 2.2309 2.0454σy 2.6774 2.2807 2.4975 2.2033σr 2.9495 2.7841 3.3178 2.1951

Panel B: PCE1st period 2nd period

Sample Model Sample Modelσπ 1.3072 1.4395 0.7546 0.6532σy 2.6774 2.5708 2.4975 2.2181σr 2.9495 2.7832 3.3178 2.0958

Panel C: GDP Deflator1st period 2nd period

Sample Model Sample Modelσπ 2.8553 2.8285 1.8273 1.3218σy 2.6774 2.0710 2.4975 2.1593σr 2.9495 2.6571 3.3178 2.0863

This Table reports the sample standard deviations along with the ones implied by the structural modelin (4) across sample periods and data specifications.

Table 7: Likelihood Ratio Statistics

CPI PCE GDPD1st P. 2nd P. 1st P. 2nd P. 1st P. 2nd P.

48.5963 29.6927 19.4468 24.0107 35.8363 29.7965(0.0000) (0.0000) (0.0016) (0.0002) (0.0000) (0.0000)

This Table reports the likelihood ratio statistics implied by the estimates structural model in (4) acrosssample periods and data specifications. The corresponding probability values appear below in parenthe-ses.

39

Table 8: Counterfactual Standard Deviations

Panel A: CPID1,Φ1 D1,Φ2 D2,Φ1 D2,Φ2

σπ 3.6374 2.4772 2.9769 2.0454

σy 2.2807 4.0479 1.2802 2.2033

σr 2.7841 3.1404 2.1197 2.1951

Panel B: PCED1,Φ1 D1,Φ2 D2,Φ1 D2,Φ2

σπ 1.4395 0.7900 1.1489 0.6532

σy 2.5708 3.748 1.6492 2.2181

σr 2.7832 2.9573 1.9725 2.0958

Panel C: GDP DeflatorD1,Φ1 D1,Φ2 D2,Φ1 D2,Φ2

σπ 2.8285 2.2481 1.6555 1.3218

σy 2.0710 3.2490 1.3911 2.1593

σr 2.6571 3.0049 1.8580 2.0863

This Table reports the sample standard deviations implied by all the combinations of volatilities andpropagation. The volatilities in column Di,Φj , i = 1, 2, j = 1, 2, are those associated with the i − th

period structural error standard deviations and the j − th period propagation coefficients.

40

Table 9: Counterfactuals Sensitivity Analysis

Panel A: CPIAS,IS Mon. Pol. Std. Dev.

λ φ β1 γ1 β2 γ2 σπ(1,2) σπ(2,1)0.010 0.025 1.02 0.66 2.94 0.56 2.81 3.130.010 0.050 1.04 0.64 3.59 0.68 2.75 3.200.020 0.050 1.09 0.58 5.32 0.58 3.35 3.40

Panel B: PCEAS,IS Mon. Pol. Std. Dev.

λ φ β1 γ1 β2 γ2 σπ(1,2) σπ(2,1)0.002 0.01 2.25 0.83 4.14 0.57 0.94 1.180.002 0.025 2.24 0.84 4.40 0.64 0.99 1.160.001 0.025 2.25 0.87 3.91 0.70 0.79 1.14

Panel C: GDP DeflatorAS,IS Mon. Pol. Std. Dev.

λ φ β1 γ1 β2 γ2 σπ(1,2) σπ(2,1)0.005 0.010 0.90 1.06 1.86 0.47 2.62 1.900.010 0.010 0.98 1.01 1.81 0.35 3.79 2.030.010 0.025 0.93 1.06 1.99 0.53 2.68 1.77

This Table reports a robustness analysis of the counterfactuals fixing λ and φ across periods. βi and γi

represent the estimates of β and γ in period i. Panel A lists the analysis with CPI inflation, Panel Bwith PCE inflation and Panel C with the GDP Deflator. σπ(1,2) is the volatility of inflation under theshocks of the first period and propagation of the second. σπ(2,1) is the volatility of inflation under theshocks of the second period and propagation of the first.

Table 10: Empirical VAR(1) Counterfactuals

CPI PCE GDPDσπ(1,2) σπ(2,1) σπ(1,2) σπ(2,1) σπ(1,2) σπ(2,1)2.4816 2.78764 0.7952 0.8921 1.9027 1.9085

This Table reports counterfactual analysis based on the estimates of a VAR(1) which includes inflation,the output gap and the Federal funds rate. Results with the three measures of inflation are presented.σπ(1,2) is the volatility of inflation under the shocks of the first period and propagation of the second.σπ(2,1) is the volatility of inflation under the shocks of the second period and propagation of the first.

41

Table 11: Monetary Policy Influence in the Drop of Inflation Volatility

Panel A: CPI1st P. 2nd P. Frac. Var.

λ1 φ1 β1 γ1 λ2 φ2 β2 γ2

0.007 0.015 0.98 0.70 0.001 0.003 1.97 0.57 30.280.010 0.075 1.07 0.73 0.005 0.015 2.36 0.57 50.920.015 0.085 1.11 0.71 0.010 0.020 2.73 0.44 70.78

Panel B: PCE1st P. 2nd P. Frac. Var.

λ1 φ1 β1 γ1 λ2 φ2 β2 γ2

0.0045 0.034 2.24 0.80 0.0003 0.006 3.45 0.69 42.480.0042 0.075 2.23 0.85 0.0010 0.015 3.90 0.68 53.290.0041 0.085 2.23 0.85 0.0020 0.020 4.34 0.61 63.06

Panel C: GDP Deflator1st P. 2nd P. Frac. Var.

λ1 φ1 β1 γ1 λ2 φ2 β2 γ2

0.0013 0.010 0.83 1.18 0.0008 0.006 1.65 0.57 19.010.0050 0.030 0.93 1.08 0.0008 0.006 1.65 0.57 35.380.0050 0.025 0.94 1.63 0.0008 0.006 1.65 0.57 25.59

This Table reports the contribution of the change in monetary policy to the decline of inflation volatilityacross inflation measures. It lists the parameter values associated with this contribution for the first (1stP.) and second period (2nd P.) This contribution (Frac. Var.) is computed as: σ2

π(1,1)−σ2π(1,2∗)

σ2π(1,1)−σ2

π(2,2)

where σ2π(i, j) is the model’s implied inflation variance under first period shocks and second period

propagation. σ2π(1, 2∗) is the implied variance under first period shocks and first period propagation

except for the second period reaction function coefficients (β and γ).

42

Table 12: Implied weights of the U.S. Fed in its Objective Function

Panel A: CPI1st period 2nd period

λπ 1.2937 6.1078λi 0.1039 0.0599λ∆ 0.3545 0.4334

Panel B: PCE1st period 2nd period

λπ 8.3550 14.2925λi 0.1517 0.0364λ∆ 0.9921 0.2384

Panel C: GDP Deflator1st period 2nd period

λπ 1.9415 2.3054λi 0.0461 0.0378λ∆ 0.2531 0.2469

This Table lists the preference parameters in the objective function of the Federal Reserve in (13) acrosssample periods. λπ is the weight on expected inflation, λi is the weight on the interest rate and λ∆ isthe weight on the interest rate differences.

43

1962 1967 1972 1977 1982 1987 1992 1997 2002−2

0

2

4

6

8

10

12

14

16Inflation Rates

CPIPCEGDPD

1962 1967 1972 1977 1982 1987 1992 19970

0.5

1

1.5

2

2.5

3

3.5

4

4.5Rolling Standard Deviations of Inflation Rates

CPIPCEGDPD

Figure 1: Inflation Series and their Rolling Standard Deviations

The top figure graphs the historical series of the CPI, PCE and GDP deflator inflation rates from 1957:2Qto 2001:2Q. The bottom panel graphs the corresponding rolling standard deviation of the inflation rates.The rolling standard deviations are constructed using a forward looking 20 quarter window.

44

63:4Q 68:4Q 73:4Q 78:4Q 83:4Q 88:4Q 93:4Q 98:4Q0

20

40

60

80

100

120Sup−Wald: 1980:4Q

Figure 2: Series of Wald Statistics: All parameters break for a VAR(3)

This Figure shows the values of the time series of the Sup-Wald statistic for a break in all the parametersof an unconstrained VAR(3) developed by Bai, Lumsdaine, and Stock (1998). The variables in the VARare CPI inflation, quadratically detrended output and the Federal funds rate. The sample period is1957:2Q-2001:4Q. The initial and final 15% of the sample are trimmed.

87:4Q 92:4Q 97:4Q0

20

40

60

80

1002nd subsample

63:4Q 68:4Q 73:4Q 78:4Q0

20

40

60

801st subsample

Sup−Wald: 1974:2Q

Sup−Wald: 1986:1Q

Figure 3: Sup-Wald robustness test

These two figures graph the time series of the Sup-Wald statistics for a break in all the parameters ofan unconstrained VAR(3) of the residuals associated with the vector autoregression of CPI inflation,the output gap and the Federal funds rate. The first subsample spans the period 1957:2Q-1980:3Q andthe second subsample covers the period 1980:4Q-2001:4Q. The initial and final 15% of the samples aretrimmed.

45

Panel A: 1st sample

−0.5

0

0.5

1Inf and Lag Inf

−0.5

0

0.5

1Inf and Lag Output

−0.5

0

0.5

1Inf and Lag Int Rate

−0.5

0

0.5

1Output and Lag Inf

−0.5

0

0.5

1Output and Lag Output

−1

−0.5

0

0.5

1Output and Lag Int Rate

0 5 10 15 20−0.5

0

0.5

1Int Rate and Lag Inf

0 5 10 15 20−0.5

0

0.5

1Int Rate and Lag Output

0 5 10 15 20−0.5

0

0.5

1Int Rate and Lag Int Rate

samplemodel 2.5%97.5%

Panel B: 2nd sample

−0.5

0

0.5

1Inf and Lag Inf

−0.5

0

0.5

1Inf and Lag Output

−0.5

0

0.5

1Inf and Lag Int Rate

−0.5

0

0.5

1Output and Lag Inf

−0.5

0

0.5

1Output and Lag Output

−0.5

0

0.5

1Output and Lag Int Rate

0 5 10 15 20−0.5

0

0.5

1 Int Rate and Lag Inf

0 5 10 15 20−0.5

0

0.5

1Int Rate and Lag Output

0 5 10 15 20−0.5

0

0.5

1Int Rate and Lag Int Rate

samplemodel2.5%97.5%

Figure 4: Model and Sample autocorrelation Functions

This figure graphs the implied model’s autocorrelations (solid thick lines) together with the sampleautocorrelations (dashed lines) using the CPI data specification. The 95% confidence intervals lie withinthe solid thin lines. The first sample period is 1957:2Q-1980:3Q and the second is 1980:4Q-2001:4Q.

46

Panel A: CPI

0 10 20 30 40 50−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2Output Response to MP shock (1stP., Baseline Est.)

0 10 20 30 40 50−0.6

−0.4

−0.2

0

0.2Output Response to MP shock (1stP., φ=0.085,λ=0.01)

0 10 20 30 40 50−0.25

−0.2

−0.15

−0.1

−0.05

0Output Response to MP shock (2ndP., Baseline Est.)

0 10 20 30 40 50−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1Output Response to MP shock (2ndP., φ=0.015,λ=0.005)

MODELVAR(1)

Panel B: PCE

0 10 20 30 40 50−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2Output Response to MP shock (1st P., Baseline Est.)

0 10 20 30 40 50−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2Output Response to MP shock (1stP., φ=0.075,λ=0.004)

0 10 20 30 40 50−0.2

−0.15

−0.1

−0.05

0

0.05Output Response to MP shock (2nd P., Baseline Est.)

0 10 20 30 40 50−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1Output Response to MP shock (2ndP., φ=0.015,λ=0.001)

MODELVAR(1)

Figure 5: Output Response to the Monetary Policy Shock

47

Panel C: GDPD

0 10 20 30 40 50−0.4

−0.35

−0.3

−0.25

−0.2

−0.15

Output Response to MP shock (1stP. Baseline Est.)

0 10 20 30 40 50−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

Output Response to MP shock (1stP. φ=0.03, λ=0.005)

0 10 20 30 40 50−0.14

−0.12

−0.1

−0.08

−0.06

−0.04

−0.02

0 Output Response to MP shock (2ndP. Baseline Est.)

0 10 20 30 40 50−0.14

−0.12

−0.1

−0.08

−0.06

−0.04

−0.02

0 Output Response to MP shock (2ndP. Baseline Est.)

MODELVAR(1)

Figure 5: (Continued) Output Response to the Monetary Policy Shock

This Figure compares the model’s output gap impulse response function to an identified monetary policyshock to that of a VAR(1) for the three inflation measures. The solid line represents the model’s response,whereas the dashed line is the empirical VAR response. Footnote 17 explains the identification of themonetary policy shock in the VAR(1)

48

Panel A: Inflation Volatility

0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 10

2

4

6

8

10

12

14β

S = 0.98; β

L =1.50

βS = 0.98; β

L= 1.96σπ

2(βS)−σπ

2(βL)

δ

Panel B: Output Volatilty

0 5 10 15 20 25 30 35 40 45 50−1.6

−1.4

−1.2

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

βS=0.98; β

L=1.50

βS=0.98; β

L=1.96

σy2(β

S)−σ

y2(β

L)

δ

Figure 6: Gains Implied by a More Responsive Reaction Function

This top figure graphs the gains of a shift towards a more responsive monetary policy derived fromreductions in inflation volatility as a function of δ, the forward looking parameter in the AS equation.The bottom graph performs an analogous exercise for the gains derived from reductions in outputvolatility. The rest of the parameters values are held fixed at their corresponding 1st period estimatesunder CPI inflation, as shown in Table 4.

49

Panel A: Sensitivity analysis for β

−0.20

0.4

0.8

Inflation to AS

−0.2

−0.1

0

0.1Output gap to AS

0 10 20 30 40 50−0.2

0

0.2

0.4

0.6 Interest Rate to AS

−0.05

0

0.1

0.2

−0.2

0

0.4

0.8

0 10 20 30 40 500

0.2

0.4

0.6Interest Rate to IS

−0.1

−0.05

0 Inflation to MP

−0.2

−0.15

−0.1

−0.05

0 Output gap to MP

0 10 20 30 40 50−0.2

0

0.4

0.8 Interest Rate to MP

β=0.5β=1β=3

Inflation to IS

Output gap to IS

Panel B: Sensitivity analysis for δ

−0.5

0

0.5

1

1.5

2Inflation to AS

−0.8

−0.6

−0.4

−0.2

0Output Gap to AS

0 10 20 30 40 50−0.5

0

0.5

1

1.5

2Interest Rate to AS

−0.05

0

0.05

0.1

0.15

0.2Inflation to IS

−0.2

0

0.2

0.4

0.6

0.8Output Gap to IS

0 10 20 30 40 500

0.1

0.2

0.3

0.4

0.5Interest Rate to IS

−0.06

−0.04

−0.02

0Inflation to MP

−0.15

−0.1

−0.05

0

0.05Output Gap to MP

0 10 20 30 40 50−0.2

0

0.2

0.4

0.6

0.8Interest Rate to MP

δ=0.5δ=0.65δ=1

Figure 7: Impulse Response Functions under different values of β and δ

This Figure presents the impulse response functions which arise under different values of the Fed’sresponse to expected inflation, β, and the private sector’s parameter on expected inflation in the ASequation, δ. For each analysis, the remaining parameter values are held fixed at their corresponding 1stperiod estimates under CPI inflation, as shown in Table 4.

50

Panel A: Numerator slope

0.4 0.8 1.2 1.6 2 2.4 2.8 3.2 3.6 4

−0.2

−0.15

−0.1

−0.05

β

f(β,⋅)

λ=0.0132

λ=0.0072

λ=0.0022

Panel B: Denominator slope

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

−0.18

−0.16

−0.14

−0.12

−0.1

−0.08

−0.06

−0.04

−0.02λ=0.0012

λ=0.0072

λ=0.0132

δ

f(δ,⋅)

Figure 8: f(θ) function used to compute the Implicit Function Derivative

Panel A graphs the function in (49) depending on δ whereas Panel B graphs it as a function of β.The slope of these two functions determine the signs of the numerator and denominator of the ImplicitFunction derivative in equation (51).

51