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6/01 c:/ClRC/DatoDue.p65-p. 15

THREE PAPERS IN BAYESIAN EMPIRICAL MACROECONOMICS
By

Paul Richard Corrigan

A DISSERTATION

Submitted to
Michigan State University
In partial fulﬁllment of the requirements
For the degree of

DOCTOR OF PHILOSOPHY
Department of Economics

2004

ABSTRACT
THREE PAPERS IN BAYESIAN EMPIRICAL MACROECONOMICS
By
Paul Richard Corrigan

These three related papers use a variety of Bayesian methods to examine the
speciﬁcation of “New Keynesian” dynamic stochastic general equilibrium (DSGE)
models and their uses in forecasting and business cycle analysis.

In the ﬁrst paper, “Forecasting output and inﬂation with a Bayesian VAR using a
New Keynesian prior,” I evaluate the forecasting performance of a simple New
Keynesian DSGE model. I do this by using the model to calculate a prior mean and
covariance matrix for the coefﬁcients of a VAR forecasting model for output, inﬂation
and interest rates, using Ingram and Whiteman’s (1994) technique for using DSGE
models to calculate priors for Bayesian vector autoregression (BVAR) forecasting
models. The resulting BVAR generates forecasts of inﬂation competitive with those from
a BVAR with a atheoretical prior. However, the New Keynesian BVAR results in very
poor output forecasts, particularly in the short run, suggesting some source of
misspeciﬁcation in the New Keynesian DSGE model.

In the second paper, “Loss—based evaluation of a New Keynesian DSGE model,” I
evaluate and compare the in-sample speciﬁcation of a simple New Keynesian DSGE
model to a cash-credit model with ﬂexible prices and to an identiﬁed Bayesian VAR, in a
manner similar to Schorfheide’s (2000) study of the portfolio adjustment cost model. I
calculate Bayes factors for each model so as to calculate posterior probabilities for all

three models, and construct a benchmark distribution for correlations and impulse

response functions to which I can compare the correlations and impulse responses of the
New Keynesian model, according to a variety of loss functions. I ﬁnd that with a realistic
monetary rule, both the ﬂexible and New Keynesian DSGEs are competitive on a
posterior odds basis with each other, as well as with the identiﬁed VAR. Very different
levels of price stickiness, with very different policy implications, are compatible with the
same data, making use of outside prior information crucial in assessing the role of the
Phillips curve and of “supply-side” and “demand-side” shocks in business cycles.

This point is further underlined in the third paper, “Technology shocks versus
monetary shocks: Identifying sources of business cycle ﬂuctuations with a New
Keynesian DSGE Model.” Following recent papers such as that of Smets and Wouters
(2002), I estimate by Bayesian methods similar to that in the previous paper a three-
variable New Keynesian DSGE model of output, inﬂation and interest rates, including
habit formation in preferences to improve the model’s dynamics as well as stochastic
price rigidity. I calculate Bayes factors for the model along with several Bayesian VAR
models, and ﬁnd it competitive with a four-lag BVAR. I use the DSGE model to estimate
series for monetary shocks, technology/ supply shocks, and autonomous demand shocks
for the United States since 1965. I ﬁnd that monetary policy shocks have probably been
of only secondary importance, as compared to that of supply shocks, in movements in US
inﬂation, as well as output, business cycles since 1965. My results conﬂict with those of
Smets and Wouters for Europe, who found a larger role for monetary shocks; I argue that
our differing prior assumptions regarding price stickiness are more likely to account for

our different results than institutional differences between the US and Europe.

To the memory of Kathleen Flood
1930-2002
for whom I will always be thankful
and look forward to one day seeing again
RIP

iv

ACKNOWLEDGEMENTS

Perhaps you have seen a certain children’s book called [Am a Pencil, which is
designed to teach readers about the marvels of the free market, describing the many
people, tasks, and materials that go into making a lead pencil, the wonder of it all being
the fact that the pencils get made but, says the pencil, “Nobody knows how to make me.”
I doubt anyone knows how to make an economist either; economists are more complex
than pencils. Certainly I don’t know how to make one, least of all single-handedly, and I
couldn’t have done this alone. However, an exhaustive list of all the people who helped
me ﬁnish this dissertation and all their contributions would be as long as the dissertation.
So I’ll save that for my memoirs and just mention a few of the more important ones.

On top of the list surely must come my committee chair, Dr. Rowena Pecchenino.
Many acknowledgements praise committee chairs ﬁrst; aﬁer all, it is the committee chair
that one has to talk into signing the dissertation. However, I truly am convinced I’d never
have completed this dissertation without Dr. Pecchenino; she not only forced me to think
hard about macroeconomics and my research, but kept faith in me even when I had begun
to believe that I would never ﬁnish and would die penniless and alone working in an East
Lansing burger joint (I got into these states of mind frequently in graduate school—if you
don’t, or didn’t, you’re made of pretty stern stuff). I am also grateful to the rest of my
committee, Dr. Ana Maria Herrera, Dr. Jeffrey Wooldridge and Dr. Christine Amsler, for
further indulgences, as a student (in the case of Drs. Herrera and Wooldridge) and as a
teaching assistant (in the case of Dr. Amsler), as well as an advisee.

For input and comments on various portions of this dissertation in its draﬂ stages

as well as other sources of encouragement and support, I am grateful to my fellow

students in the Department of Economics at MSU, including (among others) Scott
Adams, Linda Bailey, Vinit Iagdish, Mark Jahn, Iongbyung Jun, Alina Luca, Iva Petrova,
Artem Prokhorov, Monika Tothova, and David Wetzell. Outside the department, Dr.
Gerhard Glomm of Indiana University encouraged and stimulated my interest in
macroeconomic research, as well as helping Dr. Pecchenino help me keep the faith. I am
grateful to Dr. Glomm and to Dr. Eric Leeper of Indiana University for comments on my
research, as well as to seminar participants at MSU, at the Bank of Canada (especially
Eva Ortega and Greg Tkacz), and at the 2003 and 2004 meetings of the Midwest
Economic Association. One non—economist academic I’m also grateful to for comments
(and attending my defense as Dean’s Representative) is Dr. Dennis Gilliland of the MSU
Department of Statistics.

Going back a bit farther, I am also grateful to Drs. Michael Dowd and David C.
Black of the University of Toledo who ﬁrst encouraged my study of economics, and to
Dr. James Lesage of UT, my ﬁrst teacher in (especially Bayesian) econometrics. Without
them I’d have never started graduate school, never mind ﬁnished; my research presented
here is in a real sense a continued exploration of the topics I became interested in as an
undergraduate at UT.

Food (ideally sushi; Taco Bell is a highly imperfect substitute), drink (Beaner’s
coffee, not beer) and shelter (in Spartan Village) are also vital inputs for an economist. In
a free-market monetary economy of the sort I study on most days, this requires cash.
Hence, I will remain forever grateful to MSU’s University Distinguished Fellowship

program for ﬁnancial support during most of my graduate school career.

vi

The non-academic types who contributed physical, emotional and spiritual
support are numerous enough that they’ll have to be content with slightly briefer
mentions. They fall for the most part into three categories:

My family: For everything. Especially: my mother; who had faith I could do it; my
father, who forced me to prove it; my older brother Robert, who gave me something to
aspire to (including but not limited to a real job); and my younger brother and sister,
Oliver and Amelia, who let me return the favor. (Amelia was eight when I started
college; she turned eighteen in May 2004. So this dissertation has been more than half her
life in the making, and in all that time I still haven’t managed to explain to her what I do
all day. Now you know, Amelia. You still sure you want to go to grad. school?)

Animation fans (don’t laugh): For giving me something to do that wasn’t research
and hence keeping me from going crazy. Most of these are members of the MSU
animation club that I have had the privilege of advising for most of my graduate school
career (on my own time, naturally); these include (among others) Mike and Stephanie
Brandl, Jae-Hwang and Michelle Park, Jacinta Raphael, Al Roderick, Lisa and Melanie
Schoen, Jen Stepanski, Denise Wyatt, and Douglass Weeks. Some non-members to
whom I owe as great a debt include Jonathan Clements and Randall and Rachel Larson.

William G. Wilson and his friends: I would elaborate, but this acknowledgments
section is already too long, and in any case friends of Bill prefer to remain anonymous.

Thank you and God bless you all.

Paul Corrigan
East Lansing, Michigan

Canada Day, July 1, 2004

vii

TABLE OF CONTENTS

LIST OF TABLES
LIST OF FIGURES
INTRODUCTION

CHAPTER I

FORECASTING OUTPUT AND INFLATION WITH A BAYESIAN VAR
USING A NEW KEYNESIAN PRIOR

Introduction

Model

Implementation

Results

Conclusions

CHAPTER 2

LOSS-BASED EVALUATION OF A NEW KEYNESIAN DSGE MODEL
Introduction

Loss-Function Evaluation of DSGE Models

Flexible Prices Versus Sticky Prices: Two Monetary DSGE Models
Empirical Analysis

Conclusions

CHAPTER 3

TECHNOLOGY SHOCKS VERSUS MONETARY SHOCKS:

IDENTIFYING SOURCES OF BUSINESS CYCLE FLUCTUATIONS WITH A
NEW KEYNESIAN DSGE MODEL '

Introduction

A Small Scale New Keynesian Macro Model

How To Estimate The Model

Estimation Results

Conclusions

BIBLIOGRAPHY

viii

ix

xi

18
26
32

40
40
44
48
54
71

79
83
90
98
1 13

124

LIST OF TABLES

Table 1.1: Prior means and standard deviations for structural parameters of New
Keynesian CGE model

Table 1.2a: Theil U and Diebold Mariano-statistics for forecasting models, 1984:1-
1986:IV through l999:lll-2002:II

Table 1.2b: Theil U and Diebold Mariano-statistics for forecasting models, 1984:1-
1986:IV through l991:lV-1994:III

Table l.2c: Theil U and Diebold Mariano-statistics for forecasting models, 1992:]-
1994:IV through I999:III-2002:II

Table 1.3: Multivariate forecasting performance statistics

Table 2.13: Prior distributions for structural parameters of DSGE models
Table 2.lb: Posterior distributions for structural parameters of DSGE models
Table 2.2: Posterior odds for competing models

Table 2.33: Loss function values for output-inﬂation correlations of competing
models, taken individually

Table 2.3b: Loss function values for output-inﬂation correlations of competing
models, taken jointly

Table 2.4: Loss functions for impulse response functions, taken jointly
Table 3. l a: Prior distributions for structural parameters of DSGE model
Table 3.1b: Posterior distributions for structural parameters of DSGE model
Table 3.2: Posterior odds for competing models

Table 3.3a: Cross-correlations of GDP growth and inﬂation

Table 3.3b: Cross-correlations of GDP growth and federal funds rate

Table 3.3c: Cross-correlations of inﬂation and the federal funds rate

Table 3.4: Schwarz criteria and posterior odds for estimated logit models

ix

35

36

37

38

39

74

74

74

75

75

76

115

116

116

117

117

117

117

Table 3.5a: Cross—correlations from estimated DSGE of supply shocks with l 18
endogenous variables

Table 3.5b: Cross-correlations from estimated DSGE of demand shocks with l 18
endogenous variables

Table 3.50: Cross-correlations from estimated DSGE of money shocks with l 18
endogenous variables

LIST OF FIGURES

Figure 2.1: Posterior means of impulse response functions from estimated cash-
credit DSGE model (dotted lines), New Keynesian DSGE model (dashed lines) and
BVAR(4) model (dotted-dashed lines)

Figure 2.2: 90% conﬁdence bands of impulse response functions from New
Keynesian DSGE model (dashed lines) and BVAR(4) model (dotted-dashed lines)

Figure 3.1: 95% conﬁdence bands for impulse response functions from estimated
DSGE; thick band in middle indicates 50th percentile

Figure 3.2a: 50th percentile of posterior distribution for technology/supply shocks
from estimated DSGE model, 1965:] to 2003:I; vertical lines mark NBER
recessions

Figure 3.2b: 50th percentile of posterior distribution for demand shocks from
estimated DSGE model, 1965:I to 2003:I; vertical lines mark NBER recessions

Figure 3.2c: 50th percentile of posterior distribution for monetary shocks from
estimated DSGE model, 1965:] to 2003:I; vertical lines mark NBER recessions

Figure 3.3: Estimated probabilities of NBER recessions from logit models using
series of shocks from estimated DSGE model

xi

77

78

119

120

121

122

123

INTRODUCTION

In an introduction to a book, especially one distilled from a dissertation for
academic readers, one generally gives a roadmap to a book and explains what each
chapter is about, when in fact that’s what you 'would think an abstract and a table of
contents listing the chapter titles were for. At any rate, the papers have each their own
introductions explaining what they’re actually about, so I suggest you read those instead.
I must say I’m ﬂattered you’ve managed to even read this far, mind you, so as a reward,
and/or on the off chance I win the Nobel Prize and any historians of thought happen to go
back and read this (vain hope on either count) I’ll give you a sneak preview of my
forthcoming memoirs and give a bit of background as to how the papers that make up this
dissertation came to be written.

My research program thus far has mostly been devoted to satisfying myself that
there is a marginal value to economic theory in prediction of macroeconomic activity and
that I hadn’t been wasting my time taking bunches of courses in economics when a
simple autoregression could tell me all I needed to know about what the economy was
going to do next week. Obviously mathematical macroeconomic models are grossly
oversimpliﬁed pictures of macroeconomic reality, but surely not all were not so
oversimpliﬁed that they could not have more than pedagogical value. When as a child (at
the University of Toledo) I learned at the knee of James Lesage the rudiments of
economic forecasting using Bayesian vector autoregressions, I quickly grew dissatisﬁed
with the atheoretical priors that were so often used in these models; surely a simple RBC
model, even if it didn’t ﬁt the data, could be used to add information to a VAR that could

help in forecasting economic ﬂuctuations.

I could swear that it was Gerhard Glomm who referred me to the Ingram and
Whiteman (1994) paper, describing a technique for using an RBC model to derive a prior
for a BVAR, though Dr. Glomm himself doesn’t remember doing so. At any rate, I ran
with it, and my third-year paper at MSU, which eventually became the ﬁrst chapter of
this dissertation, was an attempt to apply Ingram and Whiteman’s method, to see if New
Keynesian models could help in forecasting output and inﬂation.

It was an early experiment, and (as will become clear from reading it) not totally
successful. Eventually a more successful exercise, and a somewhat more elegant one,
came to my attention in the form of del Negro and Schorﬂreide (2003), just as I was
ﬁnishing the ﬁrst paper. However, it was thinking about how and why the forecasting
experiments with the New Keynesian prior failed (was it a prior information problem? A
speciﬁcation problem?) led me to try to take the New Keynesian model more seriously
and see what would be required to get the New Keynesian model to ﬁt the data properly,
rather than just write it off as useful only as a guide for a less theoretical model.

What resulted was the second chapter of this dissertation. It was started before the
third, obviously, but ﬁnished not until a little bit after. In part this was because I didn’t
fully trust what my results were telling me, so, being the perfectionist 1 am, I wound up
tweaking and analyzing the sensitivity of my results until the very last moment (much to
the chagrin of Rowena Pecchenino, who’d surely have preferred 1 just put it to bed). It
answered the question of how the model in the ﬁrst paper had failed—poor dynamics that
were easily corrected with addition of another friction—and the result was a DSGE
model that actually ﬁt the data better and gave more precise results than a vector

autoregression regarding the effects of monetary shocks on output (namely, quite small

overall). What 1 did ﬁnd puzzling was that the output and inﬂation data itself didn’t seem
to have much information at all regarding the amount of price stickiness in the
economy—which was rather worrying, given how many studies had essentially mined
just the output and inﬂation data for evidence on price stickiness in the US economy.
More information was necessary, which meant a bigger model. It also meant
disentangling inﬂation from other demand shocks and comparing their effects and
importance on the business cycle.

The second chapter was basically an application of Schorfheide’s (2000) method
of assessing the goodness of ﬁt of DSGE models and evaluating the deﬁciencies thereof
as compared to other less theoretical models like VARs (though ironically now it was the
VARs that were deﬁcient, in part because they weren’t parsimonious or precise enough).
The third chapter was more analogous to Dejong et al. (2000b), which assessed the role
of various types of non-monetary shocks in economic ﬂuctuations, in investment as well
as output. The third chapter is a similar exercise, as far as that goes, but I was
uncomfortable with not checking the goodness of ﬁt of my DSGE model before trying to
construct shocks with my model (if the model doesn’t ﬁt the data, it’s not clear how
meaningful the “shocks” it spits out really are). So I was sure to do that before
constructing series of shocks, and ﬁnding that the experiments of the second and third
chapters of my dissertation were consistent in this: that monetary shocks were relatively
unimportant in output ﬂuctuations, and that technology shocks explain business cycles
best for reasonable levels of price stickiness. A DSGE model with New Keynesian
features (surprisingly for many people, including myself) is not necessarily incompatible

with a supply-shock driven, essentially “old-style RBC” explanation of business cycles.

What is also perhaps surprising, but more comforting, is that these DSGE models
are well-speciﬁed enough that they can be taken seriously when they try to tell us all this.
Praise be. My economics courses were worthwhile after all.

With Bayesian DSGE models proven to be competitive in forecasting and
goodness of ﬁt with VAR macroeconomic models, and offer greater precision in
analyzing the nature of business cycles than the VARs can, there is much interest in them
currently (in 2004) at such research institutions as the Federal Reserve Bank of Atlanta
and the Bank of Canada. I probably shouldn’t speculate on whether such models are the
wave of the future in empirical macro—forecasting of any sort is a formidable task.
Research in this area does look promising enough, though, that I do plan to make more
contributions in this area that (please God) will be of use to macroeconomic researchers,
perhaps (vain hope) even policymakers. For now, though, this opus will have to do.

On a related note (thanks to Dennis Gilliland for reminding me): Bayesian DSGE
research is a young enough literature (in 2004) that routines for estimating such models
aren’t available “canned” in statistical packages. Therefore being partially out of my
mind I decided to cook entirely from scratch my Mathematica 4.1 code for estimating
Bayesian VARs with New Keynesian priors, as well as Bayesian DSGE models, instead
of stealing Ingram and Whiteman’s or Schorﬂieide’s like Ana Maria Herrera wanted me
to. So if you have Mathematica 4.1 or later and want to amuse yourself on a rainy day by
trying to replicate my results, or just want a sample routine for estimating Bayesian
DSGE models that you can adapt to suit your own research needs, please contact me care
of the MSU Department of Economics and I’ll be happy to send you copies of the code.

Eh bien, continuons.

CHAPTER 1

FORECASTING OUTPUT AND INFLATION WITH A BAYESIAN VAR USING
A NEW KEYNESIAN PRIOR

Introduction

So-called New Keynesian pricing models have become the principal models used
by economists to model inﬂation and its effects on real variables such as output and
employment. New Keynesian pricing models incorporate “old Keynesian” features into
modern general equilibrium macroeconomic models by incorporating price setting by
imperfectly competitive ﬁrms that do not have perfect ﬂexibility in their price-setting,
either because they face menu costs of changing their price in any given period
(Rotemberg 1982, 1987), simply do not have a chance to adjust their price every period
(Calvo 1983), or both. Both infrequent opportunities for price setting, pace Calvo, or
menu costs, pace Rotemberg, lead to processes for inﬂation of the structure

it, = ,BE,7r, +1 + lime, , where It, is inﬂation and me, is real marginal cost. Inﬂation in

Calvo-Rotemberg type models is then the present value of real marginal cost, assumed, in
models with imperfect competition, to be high when output is high.

Part of the popularity of New Keynesian pricing models is that such models can
be used to rationalize expectations-augmented Phillips curves, which are widely
depended on as useful models of the output-inﬂation process in applied macroeconomic
work (e. g. Stock and Watson 1999). However, attempts to estimate the implied New
Keynesian Phillips curve have had at best mixed results, in part because of the lack of a
good measure of deviations of marginal cost from trend. Early studies like that of Roberts
(1995) used a measure of the output gap (deviation of real GDP from a calculated trend)

as a proxy for marginal cost. Roberts found average price ﬁxity for the US economy of

about ﬁve quarters. Fuhrer and Moore (1995) criticized Roberts’ study in part because it
attempted to estimate the New Keynesian Phillips’ curve in isolation. Fuhrer and Moore’s
own ﬁndings, using full-information techniques including the Phillips curve as pat of a
system also including an equation describing behavior of the output gap, found that a
standard New Keynesian Phillips curve did not ﬁt the output-inﬂation data very well;
they suggested that the forward-looking New Keynesian Phillips curve be augmented
with elements of backward-looking price setting.

A more sophisticated limited-information study by Gali and Gertler (1999) used
labor income share, rather than deviation of output from trend, as their measure of
marginal cost. They, like Roberts, found average price ﬁxity in the rage of four to six
quarters. However, like Roberts, Gali and Gertler estimated the New Keynesian Philips
curve in isolation. Attempts to estimate a more complete model, examining in particular
the implications for output dynamics of an inﬂation process that ﬁts Calvo-Rotemberg
speciﬁcations, are not nearly as favorable to the model. The ﬁndings of Kurrnann (2001)
are typical. Kurmann, using full-information econometric techniques, found that the
rational expectations restrictions implied by a Calvo-Rotemberg model that generated
reasonable inﬂation dynamics implied an effect of inﬂation on future real marginal cost
(measured by labor income share) that was very much at odds with that implied by an
unrestricted estimation of the forecasting process for real marginal cost would have
implied. Speciﬁcally, the unrestricted model of the process for marginal cost (and by
implication output and employment) implied a small and insigniﬁcant effect of inﬂation.
With the rational expectations restrictions imposed, on the other hand, the inﬂation effect

on marginal cost became implausibly large and the inﬂation terms summed to near zero,

implying that changes in inﬂation might be better predictors of marginal cost/output, as
Fuhrer and Moore (1995) argue.

Kurrnann cites two main suspects that might be to blame for the failure of the
Calvo-Rotemberg model under full-information. Either the Calvo-Rotemberg process for
inﬂation is misspeciﬁed (because, in particular, it ignores the possibility of backward-
looking inﬂation setting pace Fuhrer and Moore), or the labor income share, like the
output gap, is a poor proxy for real marginal cost, or both. Almost certainly the answer is
“both.” If we assume some imperfect competition in the economy, average labor
productivity will not equal the marginal product of labor, and real marginal cost will not
be proportional to labor income share. Part of the problem, though, might come from
taking the Calvo-Rotemberg model of inﬂation too literally, taking it to be a “true” model
rather than as a ﬁrst approximation to a more complex, unknown inﬂation process, and
imposing it exactly on the data.

It might be helpful to impose the restrictions on output and inﬂation dynamics
more loosely on the data, to see if the Calvo-Rotemberg model can give us any guidance
on their probable true form, even if it does not ﬁt exactly. Also, a professional forecaster
less interested in determining the “true” model of inﬂation, which is unknown (and likely
to remain so) than in making accurate forecasts might be interested in determining the
marginal value of New Keynesian theories of the output-inﬂation relationship in making
predictions about the future behavior of the real and nominal sides of the economy.

In this paper, I attempt to evaluate the forecasting performance of a simple
computable competitive general equilibrium model (DSGE or “RBC” model) that allows

for inﬂation to affect output by including price stickiness of the Calvo-Rotemberg type. I

do this by using such a model, with structural parameters corresponding to the values in
US. data, to calculate a prior mean and covariance matrix for the coefﬁcients of a VAR
forecasting model for output, inﬂation and interest rates, using a technique described by
Ingram and Whiteman (1994) for using DSGE models to calculate priors for Bayesian
VAR forecasting models. The Calvo-Rotemberg inﬂation process, in particular, is
calibrated to the speciﬁcations implied by Gali and Gertler (1999). As no good proxies
for real marginal cost exist, I do not attempt to specify a forecasting process for marginal
cost, but rather spin from the New Keynesian model prior information regarding the
dynamics of the output gap and the nominal interest rate, as well as output.

Allowing for a sufﬁciently wide range for the structural parameters of the model
permits me to add this prior information to the data without imposing exact restrictions.
In particular, the Calvo—Rotemberg inﬂation process itself is only regarded as a ﬁrst
approximation to a more general inﬂation process, which could include longer lags of
output than just the ﬁrst (as in the Calvo-Rotemberg model), allowing for backward
looking inﬂation setting of the Fuhrer-Moore type. I do ﬁnd that that prior information
does allow me to generate forecasts of inﬂation that are at least competitive, and oﬁen
signiﬁcantly better, than those from a leading, atheoretical VAR forecasting model.
However, I also ﬁnd that even the loose imposition of the output dynamics from the New
Keynesian model results in very poor output forecasts, particularly in the short run,
suggesting that as a model of the output-inﬂation relationship such models are not
especially promising even as a ﬁrst approximation.

The structure of the rest of the paper is as follows. Section 2 describes the DSGE

model with New Keynesian features, incorporating Calvo-Rotemberg type price

stickiness to allow inﬂation to affect output, that I use to generate a New Keynesian prior
for forecasting. Section 3 describes the technique for generating the New Keynesian prior
and imposing it on a Bayesian VAR (BVAR) forecasting model, a technique closely
following Ingram and Whiteman (1994). Section 4 will describe the performance of the
New Keynesian BVAR relative to competing forecasting models, in particular to a
BVAR using a modiﬁed version of the popular “Minnesota” prior developed by
Litterman (1984). Section 5 concludes with a discussion of the implications of the results
for theory and for practical forecasting applications.
Model

The economy I use to derive the prior is similar in structure to those used in King
and Wolman (1996), Yun (1996), and others in the New Keynesian literature. The
economy is inhabited by a large number of consumers that supply labor and capital to
monopolistically competitive ﬁrms, using the proceeds from wages and rental income to
buy consumer goods and acquire real money balances. I assume consumers demand real

money balances to save them time in making transactions and increase their leisure time.
The typical consumer i has preferences on his own consumption c5 of a composite of

goods supplied by the monopolistically competitive ﬁrms, labor hi, and holdings of real

money balances M g / P, where M 5 is his nominal money holdings and P, is the price

level, a composite of the prices of all goods supplied by the ﬁrms. (The means by which
goods and prices are transformed into the composite consumer good and price level will

be deﬁned more clearly below.) Those preferences are

1-1

°° ' P ‘1 z
Eozﬂt lncg+aln I— g—'1—I—£t 't—cg‘
(:0 _1 Mi:

where B is a discount factor assumed to be less than one. The taste shock 8, follows an

autoregressive process such that In a, = (1— g") In 6‘0 + g In £,_1 + ”at , where Q is less than
or equal to one. Allowing for variations in e, is a crude way of modeling changes in

ﬁnancial technology (a fall in 6, reduces the amount of time one must devote to

shopping, given a certain ratio of money balances to required transactions, allowing one

to reduce one’s money balances).
In each period the consumer starts with M 3:1 / P, = (M 5.1 / P,_1)x (PH / P,)
worth of real money balances, k 54 units of capital and

/P, = (3‘1 /P,_1) x (P,_1 /P,) worth of bonds, carried over from the previous

d
B ir—l

it—l
period. The consumer rents out his k 54 units of capital and hf: units of labor to ﬁrms in

competitive labor and capital markets, getting in return w,h,§ in wages and a gross return

on his ca ital of (1— 6 + r )k.s_ . He also receives a “helicopter drop” of fresh money
p t it 1

balances from the monetary authority of value x, / P, , along with revenue from
maturation of bonds of BS4 / P, . The consumer can now use his income and his
holdings of real money balances to purchase cg units of consumer goods, M {,1 /P, worth

of real money balances and kis, units of capital, along with Bf; /P, worth of bonds at

10

l/(l + R, ) dollars per dollar worth of bonds. The consumer’s budget constraint is

therefore

Pr P: Pr-—l P:

d d
B- P B-

1 —” ———"‘ —"“ —r,k;_1-w,h,=jso
1+R, P, P, P,_1

allowing for free disposal.

d d
M. P_ M._ x

 

Deﬁne xii, as the marginal utility of consumption. The ﬁrst order conditions of the

consumer’s problem are

(Cl: cons. dmd.)

 

 

/ _1\
141 d _1_:l d L’i
——-— M- 1 P . z
.31 ’11—‘05th I [711‘] 1‘ zi_11_51[_t%t‘] =41?
Ci, 1 _ I Mi!
K /
:2: "I
r 0' PrC‘l 1
(C2: labor supply) (1,6 1— 115 ——-————6, ——§—— = xii,w,
1_ 0' Mir

(C3: money demand)

1 1—1

1-1 d “‘ d
-—-— M- I P - I P
aﬂt£,cg I n 1‘th " 0' 5t ,c,, Zlir _Et t ’h’t+1
P 1—0' P

t Mg 1+]

 

(C45 capital SUPPIY) Air = Et ((1 — 5 + rt+l)4ir+1)

P
(C5: dmd. for bonds) -——1—— /l,, = E, —’—- 1,, +1
1 + R! PH]

11

Some rearrangement and combining (Cl) with (C2), and (C3) with (C2) and (C5), yields

the more manageable consumption and money demand functions

 

51% —1—1
(Cl’) ,Bt(l/cg)= 1+g,c;{ z (Mg/1),) I w, 1,,
d I
M- 1— 1+R
(C3:) —P" =44? ’44 R ’l
l l

The composite consumer good demanded by individuals is a composite, as noted
above, of the individual goods produced by a large number of monopolistically

competitive ﬁrms, which I shall index i 6 [0,1] . The outputs Y,, of each ﬁrm i are

w/(w—l)
] , where

l _
aggregated into a single output Y, in the fashion Y, = U0 Yum 1)/ wdi
co>1 is the elasticity of substitution between the two goods. Minimizing the cost

I
of P,Y, a! P,,Y,-,di , given Y, , results in a demand ﬁrnction for ﬁrm i of
0

Y,-, = (P,, /P, )_w Y, . Substituting into the cost function yields the composite price index

. 1 1—(0 l/(l—w)

The typical ﬁrm’s production function implies increasing returns to scale; there is

a ﬁxed cost to production of ho units of labor (which can be thought of as the

entrepreneur’s labor input), but the function, I’ll assume, is otherwise Cobb-Douglas in
capital input kg and “net” labor input hf,’ — kg. The ﬁrm purchases capital at r, a unit

and labor (including the entrepreneur running the ﬁrm) at w,. The ﬁrm’s cost

12

minimization problem, given output of Y,, , is tc, (1') = grind r,lr,-‘,i + w,h,-‘,i such

hit ’kit

that Y,, 2 [(51—19 (2, (kg — ho ))6 , where z, is a measure of technology following the
autoregressive process 1n 2, = yo + (l — p)7t + pln z,_, + UZ, for p < l and

1n 2, = 70 + 1n z,_1 + 122, for p = 1. Solving the problem results in a cost function of
tc,, = w,h0 + mc,Y,-, , where marginal cost me, = z,” 61,170 w? 190—1 (l — 60—0. The
resulting factor demands, along with the production function, are

(F1: labor demand) h,-‘,’ = ho + 13,21" 4“" two-166 (1 — are = ho + mc, 13-, /w,

(F2: capital demand) k,-‘,’ = Y,,z,"9r,'0w,6610_l (1 - a)” = mc, Y,, /r,

- - til-0 d 0
(F3: production function) Y,, = k ,, (2, (h,, — ho ))
The markup at time t, ,u,, is l/mc,. Given that r,k,‘,i = (l — 6)mc,Y,, , it follows that, given

free entry, proﬁts are zero in the long run, the labor share of output is

¢s wh/Y =(1—(1—t9)//1), and so 19 =1—,u(1—¢), whereyis the long run level ofthe

markup (to be derived below) and h is the average amount of labor effort. As in the long

run ch= Y/,u = w(h/h0)+rk, who =(l—l/,u)Y, and so the ratio of ho toh is given

by ho/h=(#-1)/#¢.

The ﬁrm, with its output in hand, can now sell it for the amount
(P,, /P, )Y,-, = (P,, /P, )1—‘0 Y, ; proﬁts in the current period are hence given by

((Pil /P,)—mc,)Y,-, ‘ thO = ((Pit Hill-w ‘mcrui'r /Pr)_w)Yr “ th0°

13

If prices were perfectly ﬂexible, proﬁt maximization would be as simple for the ﬁrm as
maximizing this ﬁJnction over (P,, /P, ) and getting (P,, /P,) = w/(a) —1)mc, ;
furthermore, as all ﬁrms face the same problem, it would be the case that P,, = P, and so
mc, = (a) —1)/a) for all t. (This gives us the long-run value of the markup,

,u = l/mc = w/(a) —1).) However, the whole point of the model is that prices are not
perfectly ﬂexible. Speciﬁcally, following Calvo (1983), in each period ﬁrms face a
probability or<1 that they will not be allowed to adjust their price freely in that period, but
will instead be forced to raise their price by a factor of It, the long run inﬂation rate.
When a ﬁrm does get a chance to adjust its price, of which there is a probability l-or (so

that on average a ﬁrm gets to adjust its price every 1/(1-or) periods, it must face the

probability aj of that the price it sets will remain ﬁxed (except for the rigid adjustments
by a factor of it) for j periods, and adjust the price accordingly so as to maximize the

discounted value of lifetime proﬁts, given Pi, . The ﬁrm solves the expected proﬁt

maximization problem

w . o
maXEtZUYIBVMHj/41)(((7T1Pir /Pt+j)1—w _mct (”Jpn /Pr+j)_w)Yt+j “ Wt+jh0)
Pit 0

If we deﬁne P," as the price chosen by the proportion l-or of ﬁrms allowed to adjust their
prices freely in each period, we get

(I) . .
Er 20W (4+1 Man/(+1- W1 imc.+ja‘fjrr”“’
‘0 0

 

(F4: Price-setting equation) P: =

w

E! Z(aﬂ)j (11+j //I,, )(Yl-i-j /YI)P[$}-'17r—j(w-l)
0

l4

Meanwhile, the form for the price index suggests the function for the evolution of the

1 1/(1—(0)
price P, = [[0 P,-,l‘wdi) is

l to
(M1: Price index process) P27“) 2 (1 — a)P,* + a(7rP,_l )170 .
Finally, deﬁning q, as the economy-wide aggregate for allocation q, the market clearing

conditions for this economy is

(M2: Goods market-clearing condition) cf + k,5 — (1 — 6)k,s_1 = Y,.
(M3: Factor market clearing conditions) h,d = hf , k,d = k [S_,

(M4: Money market-clearing condition) M ,d = M ,5 = M, = M ,_1 + x, .
To close the system completely, I need to deﬁne the rule by which the monetary authority

selects the cash payments x, /P, ; the rule, however, is stated in terms of a log-linearized

form of the model I have just described, and so I will put off describing the rule until

later. An equilibrium for this economy is a collection of allocations
{Y, ,C, ,K, ,h, ,B, ,(M /P), }and prices iw, ,r, ,P, ,P,*,R, } such that the consumers’ and

ﬁrms’ problems are solved and all markets clear.

To get the prior for the Bayesian VAR, Ingram and Whiteman (1994) suggest log-
linearizing the ﬁrst—order conditions for the consumers’ and ﬁrms’ problems and the
market-clearing conditions and using those to solve for the approximate decision rules
and state-variable transition matrix. Then, the VAR representation of the variables of
interest can be derived and a prior mean and covariance matrix calculated.

Log-linearizing the consumer’s ﬁrst-order conditions is straightforward, as is the

goods market clearing condition. Letting a circumﬂex indicate deviation from trend,

15

(C1: cons. dmd.)

 

. R M/PY . — , - , .
c, = O( ) (8,+L—le,-l——Z-(M/P),+W,)+/i,
(C/Y)(1+R0)+(M/PY) I )5

 

(C2: labor supply) xi, + ii), = —7 h, — l

 

h 1‘h_1[ét+‘”‘(c,—(M/P),)]

(C3: money demand) (M 7P), = 15, +(1— me, + pi), +Ri(1 fr R,)
0

r

l—§+r

 

(C4: capital supply) 1:, = E,/i, +1 + E,f,+1

(C5: dmd. for bonds) 2,, — (1 at R, ) = E, i,“ — E, 721+]

Above Rois the long-run average nominal interest rate, r =(y/B)-1+6 is the long-
run average gross rate of return on capital, (M/PY) is the inverse of velocity, (C/Y) is the
average ratio of consumption to output, and it, a P, / P,-, is the inﬂation rate. I, the

average portion of each day not devoted to market activities (including shopping), and h

is the average portion of each day spent working; hence l-h-l will be the average amount

 

 

 

1:1. 1:1
of time spent shopping, viz. £56ng I = I 6( C/Y ] I .The aggregate
1— 1 M 1— x M / PY
1/2’
money demand ﬁrnction implies 6‘ = [M / FY) E— R , and so
C / Y ¢Y 1+ R
l—l

 

 

R
l=l—hl+ I l 0 ﬁ ,orequivalently,h=
1—1¢1+R0PY

 

1, z 1 R0 41 ‘
1— z (15 1+ R0 PY
Meanwhile, the log-linearized market-clearing condition for the goods market is

(M2: Goods market-clearing condition) gc, + —Y— k, — — — k,_, = 1;, .

16

As r =(y/B)-1+8, K/Y will equal (1-¢)/r=B(l-¢)/(y-B(1-5)). It follows that
C/Y=l-I/Y=I-(y-1+5)K/Y.

The log-linearized factor demands are

 

(F 1: labor demand) iv, = mc, + Y, —

(F2: capital demand) r, = mc, + Y, —k,_1

(F3: production function) 1’, = (1 — ,u(1— (15))2, + ,u(1 — ¢)li,_, + ,u¢h,

Log-linearizing (F4) gives us

.33 CD - A A A A
Pt =0 —aﬂ)[z(aﬂ)j(E,mC,+j +Etlr+j +Eth+j +wEtPt+j)]
0

w c A A A
-(l-aﬂ)[2(aﬂ)l (Er’lwwj +E141+j +ErYt+j +(w—1)E,P,+j)]
0

= (1 —aﬂ)[2(aﬂ)j(E,rhc,+j +E113t+jl]’

0
implying P: — aﬂE,P,:, = (1— aﬂ)(rhc, + P, ) .
Log-linearizing the price index process (Ml) gives us P, = (1 — (1)1): + aP,_, , given that
in the long run P: = P, . We can use the fact that 7?, = P, — PM to get

7?, =(1— a)(P,* — P, ). Substituting into (F4) (and a little rearrangement) ﬁnally gets us

our Calvo-Rotemberg type Phillips curve:

 

(F4’: Phillips curve) 72, = 513,72,“ + (1 ‘0‘)“ ‘ aﬂ) ac,
a

17

It remains to deﬁne the monetary policy rule for this economy. Economic agents
assume (correctly) that the central bank sets the nominal interest rate according to the
rule:

(P1: Policy rule)
Er (1 4L R)1+1 = 4114 R), +11 — 25111915, (Y,.; — 9,1,) + R, (Er/‘21“ + E, Yin — Edi ))
This is a combination of a Taylor rule, relating the nominal interest rate target to

deviation of output from “potential” (here assumed to be Ff , the level of output that

would prevail in the absence of price-stickiness) and inﬂation, and of a nominal income
target rule. A rule resembling this one was estimated by McCallum and Nelson (1999) in
the course of estimating a New Keynesian structural macro model for use in policy
analysis. The l; parameter is a measure of the degree to which the monetary authority

smoothes interest rates, adjusting the nominal interest rate only gradually towards the
target R y(1;, — 17/ ) + R,, (7?, + 17', — 17,4 ). To be sure, as McCallum and Nelson freely

admit, actual Fed behavior is best described as discretionary rather than as following any
necessarily simple rule like this one. The “rule” used here is probably best thought of as a
crude attempt to summarize a Fed reaction function of a form unknown even to the
policymakers themselves.
Implementation

Given values for the structural parameters

, ,1,§,¢,y,y,p,c§,R .,R ,(M /PY),R , I can derive an a roximate solution for
y 7r 0 pp

the decision rules for variables of interest, as well as the processes for the state variables,

capital k, , technology 2, , the taste shock for money demand 6‘, , and the nominal interest

18

rate (1 + R,) . In the interest of making the model parsimonious, and because 1 am not
interested in estimating money demand separately from output, I set Q=0 and 8, equal to

a constant for all I. That leaves three state variables, capital, technology, and interest
rates, allowing separate estimation of three endogenous variables. I estimate a three-

variable VAR in output Y , inﬂation 7r, , and the nominal interest rate (1+ R, ) , which

will serve as my measure of monetary policy. Detrended capital, technology and the

interest rate are expected to follow the process

A A

k: kr-l (Oi-k (sz (PM
(1) E,£,+1 = S :2, whereS = 0 ,0 0
E10 5r R)1+1 (1 if R): COR/c £0122 (PRR

Meanwhile, the mapping from the detrended state variables to the detrended endogenous

variables is

Y! kt wyk ¢yz (pyR
(2) 7%! = ? it Where ? = (pair (p/zz ¢7rR
(1 5r R), (1 3r R), 0 0 1

Let y, = [Y, 72, (1410,] and deﬁne 15,9,“ = 1:9,. Using (1) and (2), it is
straightforward to show that F=I'ISIT'. The entries in F are completely determined by the
structural parameters of the underlying log-linearized model.

A sixth-order vector autoregression of 5),, , the ith component of y, ,can be

6

written as 53,, = Eryn, , t =1,..., T ,i = 1,2,3 , where T is the number of observations.
0

My prior mean for the elements F11 , the vector corresponding to the ﬁrst lag coefﬁcients

of equation i, is given by the mean of the associated elements in the ith row of F; my

19

prior mean for the elements of Fy- for J>1 are all zero. The prior covariance matrix for the

ﬁrst lag coefﬁcients is given by the covariance matrix for the corresponding row of F; the
covariance matrix for jth lag coefﬁcients, where j>1 , are equal to that for the ﬁrst lag
coefﬁcients divided by a factor of j2 . I combine the prior with the data to obtain a

posterior estimate for each equation by simple Theil-Goldberger mixed estimation (Theil

and Goldberger 1961; Theil 1971) Let X, the data set used in estimation, be the T x l 8

matrix consisting of observations on y,_, ,. .. y 1—6 , and let )3,- be the Txl vector of
observation on 52,-, . Then we can write the VAR(6) model as 52,- : XB, + e,. Letting B,- be

the prior mean for B,- and Q,- be its prior covariance matrix, the mixed estimate of B ,- is
E _ xlx/ 2 “I 1 I" 2 —I— 1
,‘— 0-,: +0, Xyi/O', +0, Bl

where 0,2 is the estimated variance of the residuals from the corresponding equation

estimated in unrestricted VAR.
It remains to specify a prior mean and covariance matrix for F. I do that by

specifying a prior mean value for the vector of structural parameters

m = ia,,6,1,6,¢,y,y,p,cf,Ry,R,, ,(M/PY),RO}; my prior mean for F will be the value

implied by the prior means of the structural parameters. To calculate the prior covariance,
I assume m has a diagonal covariance matrix with non-zero entries

2 2 2 2 2 2 2 2 2 2 2 2 2
{0a,aﬂ,al,05,a¢,07,0,,,0p,0'§,oRy,0'R,,,a,M,PY),0RO}.lthencantakea

ﬁrst-order Taylor expansion of F about the prior mean. F is then approximately normally

distributed with mean F(ﬁi) and variance VF (m) * var(ﬁ) * VF(E)T.

20

The baseline values for the means and standard deviations of the New Keynesian
model’s structural parameters are given in Table 1.1. The mean values for the discount

factor ,6 , the depreciation rate 6 , the labor share ¢ , and the persistence parameter for
the technology shock p , as well as their standard deviations, are taken from a previous

study by Dejong et al. (2000a), in which a Bayesian technique was used to calibrate a
somewhat different DSGE model. These are well within the range for these variables
used in the RBC literature, and the standard deviations allow for ranges for the variables
the include most of the values used in that literature. The mean value for the gross rate of
per capita economic growth, 7 , is also standard; its standard error was calculated as that
of the trend rate of growth of real GDP estimated over the period 1959:I to 1983le, from
an OLS regression (corrected for serial correlation) of logged real GDP on a constant and
a time trend. For the markup ,u , estimates found in the literature vary widely, from 1.2 to
around 2.0; see Rotemberg and Woodford (1995) for a survey. For the prior mean, I
picked 1.4, near the average for the values found in the literature (and the value used by
Rotemberg and Woodford in their discussion of imperfect competition in DSGE models),
along with a standard error allowing for a fairly wide range (ﬁom 1.2 to 1.6) for this
parameter.

Moving on to the structural parameters for the “demand side” of the economy, the

means and standard deviations for the mean nominal interest rate R0 and the inverse of
mean velocity, (M /PY)0 , were taken to be the historical means and standard errors over

the 1959:I to 1983le period for the (quarterly) Federal Funds rate and the inverse of the
(quarterly) velocity of the Board of Governors adjusted monetary base. The mean value

for the interest elasticity of money demand, 2’ , is that found for the US economy by

21

Goldfeld (1973); the standard error allows for the wide range of values for x found in the
literature. (See Goldfeld and Sichel (1990) for a survey.) 1 should add that for the three
variable VAR discussed here, omitting money, the results simply are not sensitive to
differences in these parameters.

The parameters of the policy rule, 4‘ , R y and R” , are within the range estimated

for the US economy for the post-1979 period by McCallum and Nelson (1999); however,
the standard errors are much larger than theirs, to reﬂect the greater uncertainty 1 have
regarding how well this (or any) assumed policy rule accurately describes the actual
policy behavior of the Federal Reserve.

It remains to set the inﬂation persistence parameter a . Any given value for a
suggests that opportunities for price adjustment are presented to the typical ﬁrm every
77 = 1/(1 — a) quarters. Estimates for the length of time the typical ﬁrm takes to adjust
their prices varies widely in the literature. On the industry level, most overall estimates
are from two to six quarters, but estimates vary widely among industries (see Taylor 1999
for a survey). For the economy as a whole, Roberts (1995) derived an estimate for

(1— a)(l -- aﬂ)/ a of about 0.08, corresponding to a value of a of about 0.75. Gali and

Gertler’s (1999) later study derives a number of estimates of a (their 19) that average
about 0.8; as this is of the same order of magnitude as Roberts’ estimate, this is the
estimate I picked for the prior mean, implying average price ﬁxity of about 5 quarters.
However, Gali and Gertler’s results for various subsamples suggest a higher or (around
0.85) for the peiod since 1980 than for before 1980 (for which their estimates of a are
nearer Roberts’ estimate of 0.75, derived with a slightly earlier data set). The prior

standard deviation for a I picked permits a range for the parameter from 0.75 to 0.85.

22

The baseline values for the structural parameters imply a prior mean for F of

1.028 — 2.582 0.874
0.00904 0.524 0.1 17
0.00757 - 0.403 1.058

The second row, corresponding to the inﬂation equation, suggests that expected
inﬂation is as a ﬁrst approximation best modeled as a ﬁrnction of the output gap and
current inﬂation, pace Stock and Watson (1999), and that inﬂation itself should be
modeled as stationary. The third row suggests that the real interest rate

(1 + R, ) — It, should be modeled as highly persistent but stationary, while the nominal
interest rate (1+ R,) is closely tied to the inﬂation rate (as it should be in any reasonable

model).

Note that the prior implies that the behavior of output, inﬂation and the interest
rate is well described by a VAR(l) process, with the values of longer lags assumed a
priori to be zero. For inﬂation in particular, the Calvo-Rotemberg Phillips curve (F4’)
implies expected inﬂation at time t+1 to be a function only of inﬂation and the output
gap/marginal cost at time t. However, with the prior standard deviations for the structural
parameters speciﬁed here, the imposition of zero values for longer lags is loose enough to
permit a role for longer lags of inﬂation to enter the process, allowing for a backward-
looking element to that process.

I wish to compare the performance of the VARs using the prior derived from the
New Keynesian DSGE model (hereafter the “New Keynesian BVAR”) to a Bayesian
VAR using a standard atheoretical prior. One popular such prior is the so-called
“Minnesota” prior described by Doan, Litterman and Sims (1984). This time, let X be the

T x19 matrix consisting of observations on a T x] vector of ones and vectors

23

y ,_1 ,. . . y 1—6 , where y, is the vector of observations on the levels (not the deviations
from trend) of the three variables, and let y,- be the T x] vector of observation on the level

of variable i. For the ith equation in the VAR, the Minnesota prior imposes a prior mean
of one on the ﬁrst autoregressive lag, and a prior mean of zero on all further
autoregressive and all non—autoregressive lags; the implication is that each variable in the
VAR is best described as a ﬁrst approximation as a random walk. The prior covariance

matrix is a diagonal matrix, with the prior variance for the kth coefﬁcient of variable j in
equation i being 0,12,, =t12/k2 for i=j and 01.12., = (tlztz2 /k2 ) x (0'12- /0',-2 ) for iasi, where t1
is an overall tightness parameter, 12 < 1 is a parameter determining how strictly imposed
is the prior restriction of zero values for non-autoregressive terms, and s,2 is the estimated

variance of the residuals from a sixth-order AR model of variable i, sf /s,2 serving as a

scaling factor accounting for the differences in magnitude between variables i and j. I
used values of 0.2 for both t1 and t2.

Doan, Litterman and Sims (1984) suggest forecasts can be further improved by
including prior information on the sums of coefﬁcients. Such information is usually
included by adding dummy observations to the data set, before mixed estimation. For
example, in a three variable VAR with four lags of output, employment and interest rates,

I add to X the dummy observations

0 yo 0 0 yo 0 0 yo 0 0 yo 0 0
l3 0 O ITO 0 0 ”'0 O 0 IZ'O 0 0 71'0 O
0 0 0 R0 0 0 R0 0 0 R0 0 0 R0

24

When estimating the output equation, the corresponding dummy observations on current

output are t3 [yo 0 0],; similarly, the corresponding dummy observations on inﬂation
for the inﬂation equation are t3 [0 7:0 0] , and those for the interest rate equation are

t3 [0 0 R0] . These dummy observations imply that the autoregressive terms in each

equation sum to one (i.e. that each has a unit root), and that the terms for lags of all other
variables sum to zero (i.e. the changes in each variable, assumed to be permanent, have
no permanent effect on any other others).

Even with the sums of coefﬁcients restrictions imposed, the disadvantage of the
Minnesota prior is that it does not take into account cointegrating relationships between
variables, as in this case one would expect between (at least) inﬂation and the nominal
interest rate. To compensate for this somewhat, Sims (1992) suggests adding to X the

dummy observation (again assuming three variables and four lags)
1411 yo ”0 R0 Yo ”0 R0 yo ”0 R0 yo 7‘0 R0],
with corresponding dummy observation on output t4 DO I , on inﬂation (4 [no] , and on

the interest rate (4 [ROI- This dummy observation implies a cointegrating relationship
between the output gap, the inﬂation rate and the nominal interest rate. t3 and t4 are

parameters setting the tightness by which the prior infomation given by the dummy
observations is imposed. yo , 72’0 , and R0, the implied trend values of output, inﬂation
and interest rates, are measured by the means of the k presample observations, where k is
the number of lags.

Robertson and Tallman (1999), in a recent paper on forecasting with VARs, argue

that a Bayesian VAR estimated imposing the Minnesota/Litterman prior with mixed

25

estimation, and including the Doan-Litterman-Sims (1984) and Sims (1992) sums-of
coefﬁcients restrictions, do about as well in forecasting than Bayesian VARs estimated
using more complex methods of imposing prior information (e. g. Sims and Zha 1998).
Hence, it is this modiﬁed Litterman BVAR (as I shall call it from now on) that 1 used as
the benchmark for the New Keynesian BVAR. For the modiﬁed Litterman BVAR’s
parameter values, I used the values used by Robertson and Tallman in their experiments,

1.6. t] = 0.2,!2 = 0.2,13 = 5.0,2‘4 = 5.0

Results

My data set consisted of quarterly US data on output (seasonally adjusted real
GDP), inﬂation (quarterly rate of change in GDP chain-type price deﬂator), and a
nominal interest rate, the Federal Funds rate, for the period 1959:] to 2002:11. The Federal
Funds rate was used for the nominal interest rate, as it is this interest rate that the Federal
Reserve uses in practice as its policy instrument; hence the interest rate does double duty
as a measure of current monetary policy. (I did experiment with other short-term interest
rates, such as the 3-month Treasury Bill rate, but it made little difference; this is not
surprising, as “market” short-term interest rates tend to track the “administered” federal
funds rate closely.)

The Bayesian VARs using the modiﬁed Litterman prior contained six lags of the
log of each variable and a constant. The VARs using the New Keynesian prior derived
from the New Keynesian DSGE model (hereafter the “New Keynesian BVAR”) were
estimated without a constant, using six lags of detrended output, inﬂation and interest
rates. Detrended output was measured as the deviation of log output from a linear time

trend estimated by OLS over the period of the data set used to estimate the VAR, while

26

detrended inﬂation and interest rates were measured as the deviation of inﬂation and
interest rates from their sample mean over the period of the data set. Such a VAR can be
used to generate out-of-sample forecasts of the deviations of output, inﬂation and interest
rates from trend. To get forecasts of the level inﬂation and interest rates, I simply added
the sample mean to the forecast detrended value of each variable; to get forecasts of level
output, I added the forecast deviation of output from trend to the forecast level of trend
output implied by the estimated trend.

Tables 1.2a through 1.20 presents performance statistics, the Theil-U, for the two
competing models (modiﬁed Litterman and New Keynesian). The statistics are calculated
as follows. Each model is estimated over the ﬁrst 100 periods (1959:] to 1983:1V), and
forecasts are calculated for one quarter, two quarters, four quarters (one year), eight
quarters (two years) and twelve quarters (three years) ahead of 1983:1V. The actual
observation for 1984:] is then added to the data set, forecasts are calculated for one, two,
four, eight and twelve quarters ahead of 1984:], and so forth; in all 63 sets of one, two,
four, eight and twelve step ahead forecasts were generated (the last VAR being estimated
over the 1959:] to 1999:]1). The Theil-U is given by the ratio of the forecast mean
squared error from the model in question to that from a naive forecasting model (random
walk with driﬁ); a value above one indicates that the model forecasts are worse than
those from a naive random-walk model. Lower Theil-U’s indicate better forecasting
performance.

Along with the Theil-U’s, I also calculated the test statistics for equality of loss-
differentials for each model, to both the naive model and each other (corrected for serial

correlation in the loss differentials, i.e. differences in squared forecast errors between

27

each model), as suggested by Diebold and Mariano (1995) as a measure of improvement
or worsening of forecast performance. Under the null that forecast performance is equal,
this statistic should follow a t-distribution; hence large Diebold-Mariano (DM) statistics
suggest rejection of the hypothesis that forecast performance is equal.

Table 1.23 gives the forecast performance statistics for the entire period (1984:]-
1986:IV through 1999:111-2002211). For the nominal side of the economy, the New
Keynesian prior does surprisingly well. For inﬂation and interest rates, the New
Keynesian BVAR easily beats the random walk model, and for inﬂation, the New
Keynesian BVAR consistently beats the modiﬁed Litterman BVAR, with the largest DM
statistics being those for one year to two years ahead. For the nominal interest rate, the
New Keynesian prior is also competitive with the modiﬁed Litterman BVAR, beating it
at horizons of a year or more. While none of the DM statistics are signiﬁcant, this
suggests that the New Keynesian BVAR is at least competitive with the modiﬁed
Litterman BVAR for forecasting nominal variables.

There, however, is where the good news ends. The forecasts generated by the
New Keynesian BVAR for output are consistently worse than those from a naive model,
let alone those from the modiﬁed Litterman BVAR. (The difference is statistically
signiﬁcant at near horizons, and substantial at all.) Some comparison of the Litterman
and New Keynesian prior is in order here. The Litterman prior constrains the inﬂation
and interest terms in the output equation to zero fairly tightly. The sums of coefﬁcients
dummy observations, moreover, constrain the long-run effect of level changes in inﬂation
to zero; this can be interpreted as implying that if anything, it is changes in inﬂation, pace

Fuhrer and Moore, that affect output, not necessarily the level of inﬂation as such. The

28

New Keynesian prior, on the other hand, allows the inﬂation and interest terms to have
coefﬁcients that have values that are individually, and sum to, far from zero.

The prior mean values for the ﬁrst lags (and, hence, the details of how the New
Keynesian pricing model predicts inﬂation will affect output) are probably less important
here for the results than the prior covariance matrix from the New Keynesian BVAR,
which puts much looser restrictions on the values of the inﬂation and interest terms. In
effect, the New Keynesian prior assumes much less information on just what those values
should be. As a result, these terms are much more determined by the data, when
estimating the BVAR, than would be the case if a Litterman prior were used. Given that
the Litterman prior (which holds that “(level) inﬂation doesn’t matter” for output, and
restricts its effect to zero) does better than the New Keynesian prior (which holds that
“inﬂation matters”, and allows its estimated effect to be large) in predicting output, a
reasonable conclusion to draw is that “(level) inﬂation doesn’t matter,” or at any rate its
effect is exaggerated by the Calvo-Rotemberg model.

To check the stability of the results, I also calculated Theil U’s and DM statistics
for two sub-periods, 1984:]-1986:1V through 1991:IV-1994:II] (32 sets of forecasts) and
1992:1-1994:IV through 1999:111—2002:II (31 sets of forecasts). For the earlier period, the
advantage of the New Keynesian BVAR over the modiﬁed Litterman BVAR for inﬂation
is even greater; the Theil-U’s for the New Keynesian BVAR are signiﬁcantly below one
at any reasonable conﬁdence level, and forecasts of inﬂation a year or more ahead are
signiﬁcantly better than those from the modiﬁed Litterman BVAR. However, the New
Keynesian BVAR is still signiﬁcantly worse at forecasting inﬂation than a naive model at

shorter horizons (and certainly than the Litterman model).

29

For the later period, the picture is somewhat different. The Internet boom and
inﬂation rates far below (postwar) historical levels would be bound to hamper forecasting
using models estimated by necessity with data from earlier periods. For inﬂation, the
199221-1994le through 1999:111-2002zll period is not a banner period for either model,
with Theil-U’s well above one for periods of a year or more ahead. However, the New
Keynesian BVAR fares relatively worse, actually being beaten by the Litterman model
for forecasts of inﬂation a year or more ahead. This is not surprising, as the New
Keynesian BVAR (as well as the detrending method used to estimate it) assume inﬂation
is stationary, while the Litterman prior assumes a unit root in inﬂation. Given that, the
Litterman prior can be expected to fare better in the face of an apparent trend shift in
inﬂation. For output, again, neither model fares well, with Theil U’s well above one;
however, the New Keynesian model’s relative disadvantage remains (though the DM
statistics no longer indicate signiﬁcance).

A few general conclusions can be drawn. The good news (for those favoring New
Keynesian models of inﬂation) is that the New Keynesian BVAR does surprisingly well
at forecasting inﬂation. Generally speaking, it can be trusted to be at least competitive
with a leading atheoretical forecasting model (barring trend shifts in inﬂation). The bad
news is that the forecasts for output generated by the New Keynesian BVAR cannot even
beat a random walk model, let alone more sophisticated atheoretical models. This result
is robust to all the speciﬁcation changes I have tried, including, but not limited to, large
changes in the prior means for the “supply side” structural parameters ([3, 8, (l), p, and p)
and increases in their prior standard deviations (particularly for p), to loosen the prior for

the output equation. Starkly put, a New Keynesian prior using reasonable values for the

30

prior means of the structural parameters and standard deviations, and generating
reasonable forecasts for nominal variables, cannot generate adequate forecasts for output.
The implication is that a New Keynesian prior speciﬁed so as to generate good inﬂation
forecasts implies a relationship between output and inﬂation that is inconsistent with the
data.

To get some measure of the “overall ﬁt” of the New Keynesian BVAR, I followed
Ingram and Whiteman (themselves following Doan, Litterman and Sims (1984)) by
calculating the log determinants of the forecast covariance matrices for the naive model,
modiﬁed Litterman BVAR and “baseline” New Keynesian BVAR at each horizon. These
are reported in Table 1.3, along with the log determinants of the forecast covariance
matrices for New Keynesian BVARs with lesser (or=0.75) and greater (or=0.85) price
stickiness.

In general, the New Keynesian BVARs, while certainly beating naive models, are
inferior to the modiﬁed Litterman BVARs in overall forecasting performance, except at
very long forecast horizons (over which the advantage of the Litterman over the New
Keynesian BVAR, or any model, tends to vanish). Among the New Keynesian BVARs,
those with higher or’s tend to do better at shorter horizons (over which prices are more
likely to be sticky), while those with lower or’s do better at farther horizons (where price
stickiness is likely to be less important).

More generally, the New Keynesian BVAR tends to catch up with the modiﬁed
Litterman BVAR as the forecast horizon increases, a tendency robust to sample period.
(Looking back at the DM statistics conﬁrms that for each variable the advantage of the

New Keynesian BVAR over the modiﬁed Litterman BVAR increases with forecast

31

horizon, for inﬂation and interest rates, and its relative disadvantage decreases.) My own
conjecture is that the New Keynesian BVAR does a better job of capturing long-run
tendencies in the data (say, between inﬂation and interest rates) than does a Litterman
BVAR, even supplemented with Sims-type dummy observations to allow for
cointegration. However, this is clearly insufﬁcient to allow the New Keynesian BVAR’s
overall forecasting performance to be up to the standard of the competition.
Conclusions

The results of the forecasting experiments here suggest, at least at ﬁrst sight, that
a “reasonable” New Keynesian model for predicting inﬂation is incompatible with a
reasonable model for predicting output. This ﬁnding parallels that of recent research on
New Keynesian models of the output-inﬂation relationship such Kurrnann (2001), who
found that a New Keynesian model of the Calvo—Rotemberg type that generated
reasonable inﬂation dynamics was incompatible with reasonable dynamics for real
marginal cost, and, by extension, output and employment.

The Bayesian (or quasi-Bayesian) techniques used here to impose loosely the
restrictions on output and inﬂation dynamics implied by the model allow much more
room for misspeciﬁcation than did Kurrnann’s (classical) full-information techniques.
The Ingram-Whiteman technique imposes only inexactly the restriction on output
dynamics implied by a DSGE model with Calvo-Rotemberg type New Keynesian
features. However, even these loose restrictions clearly degrade forecasts of output, to the
point where even a naive random walk does better; this result is robust over a variety of
forecast periods. More generally, any theoretical model that suggests a large amount of

information content in inﬂation (and nominal variables in general) for forecasting output

32

seems to do poorly compared to models that do not (like the modiﬁed Litterman prior,
which assumes a priori no such role). Given that, the relevance of Calvo-Rotemberg type
New Keynesian models for modeling the output-inﬂation relationship is put into question
by these results.

How disastrous one ﬁnds all this, of course, will depend on one’s loss function.
For those more interested in accurate forecasts than proving one theoretical model or
other to be “true” in some sense, the New Keynesian BVAR does generate forecasts
competitive, and, provided inﬂation can be reasonably assumed to be stationary, superior
to those from competing atheoretical models. (Altering the BVAR to allow for trend
shifts in inﬂation, as apparently occurred in the 1990’s, should be straightforward;
alternatively, one could simply correct for secular bias in the forecasts generated by the
model in formulating actual forecasts.) If one is interested in forecasting output as well, a
BVAR for forecasting could very easily be estimated using a New Keynesian prior for
the inﬂation equation, and an atheoretical prior along the lines of the modiﬁed Litterman
BVAR for the output equation. An obvious objection on the grounds of logical
consistency could be defended along these lines. The New Keynesian prior, in a VAR
model with more than one lag, does allow for a non—zero term for long inﬂation lags,
permitting a backward-looking element to the inﬂation process along the lines of Fuhrer
and Moore (1995). The sums-of-coefﬁcients restrictions on the inﬂation terms in the
Doan-Litterman-Sims dummy observations, as well, do allow for changes in inﬂation,
though not levels, to affect output. The hybrid model could then be defended as an
attempt to convey the idea that what effects inﬂation has on output occur along the lines

of a Fuhrer-Moore sticky-inﬂation model.

33

In a sense, what this amounts to is accepting the New Keynesian model as the best
model of inﬂation economics has to offer, and accepting that one should try to forecast
outpout without pretending to have too much theory. What is admittedly not at all
obvious from this exercise, however, is that the failure of the New Keynesian DSGE
model necessarily has to do with the New Keynesian elements, as opposed to other
elements of the DSGE model that make for rather poor dynamics and could easily be
corrected. For instance, adjustment costs in capital or investment could improve the
dynamics of the model. It is also possible that my prior information is bad; Gali and
Gertler’s estimates of price stickiness are on the high end of those estimated in the
literature, and possibly the lower estimates suggested by survey data (e. g. Blinder 1994)
would be appropriate. Examining which modiﬁcations of the model would get the New
Keynesian DSGE prior to improve performance on the real side (to see, as it were,
whether it’s a prior problem or a speciﬁcation problem) would be an interesting exercise,
but if one is going to go to that trouble, one might as well take the plunge and try to
estimate a Bayesian New Keynesian DSGE model, and see what chances it has of being
as true a model as the BVAR. It is that approach, checking what modiﬁcations the DSGE

model needs to “ﬁt the data,” that I will be taking in the next chapter.

34

Table 1.1: Prior means and standard deviations for structural parameters of New

 

 

Keynesian DSGE model
Parameter Mean SD.
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CHAPTER 2
LOSS-BASED EVALUATION OF A NEW KEYNESIAN DSGE MODEL
Introduction

Part of the reason that I only loosely imposed the restrictions on the data implied
by my DSGE model, rather than just estimating a ﬁilly-specified New Keynesian DSGE
model, was an a priori belief that “DSGE models (unaided) just don’t ﬁt the data.”
Certainly the restrictions on economic data imposed DSGE models, especially the older,
simpler ones, are frequently rejected by classical hypothesis tests. If the null is rejected,
however, a simple hypothesis test does not provide a superior alternative model to which
the DSGE model can be compared, so it cannot tell us where to go next or what to
correct.

Many recent studies (like Watson (1993) and Diebold et a1. (1998)) of DSGE
models assume the DSGE is certainly misspeciﬁed (i.e. the probability that the DSGE
model is the “true” model is assumed to be zero), and in essence propose measures of just
how far off the mark the DSGE models are from an atheoretical reference model (like an
identiﬁed VAR). Watson measures misspeciﬁcation by how much uncertainty must be
introduced to the model to make it match sample moments (such as correlations and
impulse responses); Diebold et al. measure misspeciﬁcation using loss functions
penalizing deviations of sample from model moments. However, such an approach
assumes, in effect, that the model can itself teach us nothing that the sample can’t about
how (say) the impulse response of output and inﬂation to various shocks should look.
Given the improvements in the dynamics of DSGE models over the earlier, simpler ones

that failed classical hypothesis tests, it is no longer possible to make that assumption with

40

conﬁdence, unless one really believes that a given DSGE model cannot possibly be the
“true” model of the economy, or at any rate a good description of a subset of time series.

A study by Schorﬂieide (2000) of a cash-in-advance model with partial
adjustment costs uses just such a method. Schorfheide takes a middle-of-the-road
approach, allowing for a non-zero probability that the DSGE model ﬁts the data better
than a given competitor, a probability that can be updated aﬁer looking at the data.
Speciﬁcally, he compares a standard cash-in-advance model and a cash-in-advance
model with portfolio adjustment costs, both including (permanent) technology shocks and
(temporary) monetary shocks, to an identiﬁed VAR with disturbances disaggregated into
permanent and transitory shocks pace Blanchard and Quah (1989). The speciﬁcations of
the cash-in—advance model and the model with portfolio adjustment costs provide
likelihood functions for the data which can be evaluated using a Kalman ﬁlter algorithm;
combining the likelihood function with prior distributions for the structural parameters
gives a posterior distribution for the parameters. Schorﬂieide calculates posterior
probabilities that each model describes the data and overall posterior estimates of the
population characteristics, using so-called Bayes factors (e. g. Kass and Raﬁery 1995).
Unlike classical model selection criteria, which are generally functions of the maximum
likelihood of a given model, the Bayes factor is a measure of the average likelihood of
the model, given the prior density on the model parameters.

Given posterior probabilities, Schorfheide can construct a probabilistic
representation of the data, a weighted mixture of the DSGE models and the identiﬁed
VAR, as a benchmark to which he can compare the DSGE models. He can then calculate

the posterior distributions of various moments of interest, such as correlations and

41

impulse response functions. He goes on to assess the ability of the DSGE models to track
US output and inﬂation time series, generate plausible correlation patterns between
output and inﬂation, and describe the response of output growth to a monetary expansion;
this ability is measured by the values of loss functions penalizing the deviations of the
DSGE model’s moments from those of the benchmark.

The use of a mixture of the DSGE models and the identiﬁed VAR in constructing
the benchmark for comparison allows us in effect to learn from the models regarding how
the data should look. In Schorfheide’s actual application, however, the posterior
probabilities that the cash-in-advance model and the model with portfolio adjustment
costs were true turned out to be vanishingly small. This indicated that we could learn very
little from these models about inﬂation and output dynamics in the actual US economy.
Given that the New Keynesian model is compatible with a negative correlation of output
grth and inﬂation of a magnitude more in line with the data, as King and Watson
(1996) argue, and with a potentially much larger positive response of output to a
monetary shock, a natural question to ask is whether adding New Keynesian features
would not correct the discrepancy between model and posterior.

My purpose in this paper, then, is to evaluate and compare a simple New
Keynesian monetary DSGE model, assessing the discrepancy between its predictions and
a posterior distribution of population characteristics such as cross-correlations and
impulse-response functions, similar to that done by Schorfheide (2000) for the portfolio
adjustment cost model. A necessary ﬁrst step in this process is to estimate a monetary
DSGE model with a New Keynesian Phillips curve using US data for output and

inﬂation, using Bayesian methods to impose prior information regarding the values of the

42

structural parameters. In particular, I use a prior assumption for the measure of price
stickiness in the economy consistent with that found in survey data such as Blinder
(1994). In so doing I obtain an estimate of the degree of price-stickiness in the US
economy, conditional on the model’s being correctly speciﬁed. This is an interesting
exercise in itself, as far as it goes, corresponding to the exercise conducted by Dejong et
al. (2000a, 2000b) for the Greenwood et al. (1988) variable capital utilization model.
However, given that there is a non-zero chance that the New Keynesian DSGE
model is misspeciﬁed, it is only a ﬁrst step towards a full evaluation of the model; my
real purpose is to evaluate just how well speciﬁed such a model, estimated with a prior
using conventional values for the parameters, really is (and so how “plausible” such
values are) and whether it can succeed where the portfolio adjustment cost model failed
in teaching us something about the output-inﬂation relationship. In particular, I want to
see if assuming prices to be somewhat sticky (and as a ﬁrst approximation as sticky as
limited information studies suggest) actually correct some of the problems that cash-in-
advance models with ﬂexible prices have in matching inﬂation dynamics and the effect
of monetary shocks on output. To make my results comparable to Schorﬂieide’s, I
compare the New Keynesian DSGE model to a cash-in-advance model without price
stickiness and to an identiﬁed Bayesian VAR. I estimate all three models to allow me to
calculate posterior probabilities for each model, allowing me to construct a benchmark
distribution for correlations and impulse response functions to which I can compare the
correlations and impulse responses of the New Keynesian model, according to a variety

of loss functions.

43

Among the things I ﬁnd in what follows are that a New Keynesian DSGE model
with sufﬁciently rich dynamics, contrary to Schorﬂieide’s results, is competitive and in
fact superior in ﬁt to a Bayesian VAR in output and inﬂation, with the DSGE being the
model with a much greater posterior probability than the VAR. I ﬁnd, however, that the
goodness of ﬁt derives from modiﬁcations of the basic model—a more realistic monetary
policy in terms of inﬂation rather than money growth as Schorﬂreide used, and well as
costs of adjusting investment—that have little to do with the New Keynesian features of
the model as such. The output and inﬂation data for the US since the mid 1960’s, while
certainly compatible with an amount of price stickiness found in survey data, turn out to
not be especially informative themselves regarding the amount of price stickiness. This is
at least partly because inﬂation shocks can account for very little of US output
ﬂuctuations, and so there is not likely to be much information about the impact of an
inﬂation or money shock on output in the data. What the DSGE model does allow us to
do is to add a good deal of precision regarding out estimation of the output-inﬂation
relationship and the impact of a monetary/inﬂation shock on output, given our best prior
information on price stickiness.

Loss-Function Evaluation of DSGE Models

Before I begin, I should describe the Schorfheide method of loss-function

evaluation of DSGE models in more detail. (My discussion will be somewhat informal;

for the ﬁner details, see Schorfheide (2000).)

Before I do so, some shorthand is in order. Let M1 denote model 1 (say, the
ﬂexible-price cash-credit model) and M 2 denote model 2 (say, the New Keynesian

model). Let 6(1') be the parameter vector for model M 2 . Also, let y, be the n x 1 vector

44

of the values of the n observables to be used in estimation at time t, and let

YT = LVI ,..., yTI be the full n x T data set. The speciﬁcation of each model M,- provides a
likelihood function for Y 7‘, denoted hereafter as p(YT l 90') ,Mi); the product of this
likelihood and the prior distribution for 60-) , p(6(,-) | M ,- ) , is proportional to the posterior
density for 190-), p(6(,-) |YT,M,-) 0c p(YT |6(,-),M,~)p(6(,-) |M,-). As there is a good
chance that both DSGE models are misspeciﬁed, I also consider an identiﬁed VAR as a

reference model, which I shall call model 0 or M 0 ; hence the parameter vector for the

identiﬁed VAR will be labeled 6(0) , its likelihood function p(YT |0(0),Mo), and its
prior density p(9(0) | M 0) .

There are three steps to the model evaluation:
Step I

With the posterior densities p(0(,-) | YT ,M,) in hand for each model, we can
compute posterior distributions for each model’s parameters 6(1') . For model 0, the

identiﬁed VAR, this can be done analytically; for the DSGE models, however, the
posterior densities will be analytically intractable, and so the posterior distributions for

0(1) and 0(2) will have to be derived numerically. The means by which the DSGE model’s

parameter distributions are computed will be discussed in Section 4.

Let 7:0,0 , ”1,0 , and ”2,0 be the prior probabilities we place on models 0, 1, and 2

being the “correct” model, before estimating the models. By Bayes’ rule, the posterior

model probabilities 7:0] , ”LT , and £2] are

45

”i,0P(YT lMi)

2 ’
ZizoﬂiﬁpO/T IMi)

 

”LT 5 [WW lYT) =

where p(YT |Mi) = I p(YT |6(,-),M,-)p(6(,-) | M i)d6,- is the marginal data density, or
Bayes factor, given model Mi. Intuitively p(YT I M i) , Which is the average likelihood of

model M ,- given the prior distribution p09“) | Mi ) , can be thought of as the probability

that an economy that was well described by model M ,- could have produced time series
YT; a relatively large/small p(YT | M i) will result in a relatively large/small posterior

probability 711718 the correct model.
Step 2

The population characteristics (correlations, impulse response functions and so
forth) that we obtain from each model will, of course, be functions of their parameters

60-) . Let a) be an m x 1 vector of population characteristics of interest. With the
posterior distributions of 6( i) in hand, we can obtain posterior distributions of a)

conditional on model M ,- , for short p(a2 | YT ) , for each model. The overall posterior

distribution of (a, given the data Y7- , is a weighted average of the densities p((a | YT,M,- ) ,

2
namely p(<0| Y7) = Zia-revel Yr,Mi).
i=0
Step 3:

With the posterior distribution of a) in hand, we can now calculate the minimum

expected loss L x ((0,5)) associated with each model in terms of the deviation of the “best

46

estimate” of value of a) (which 1 will label a3) given by each model from the “actual”
value a).

(a) For each model M ,- , ﬁnd the value a3“- for the population characteristic vector (0 that

minimizes the expected value of loss function L x (60,60) , given that M ,- is the true model.
That is,
(0,,- = arg minwzQ jL,(w,a3)p(w | YT,M,-)da)
(b) Then, the expected loss or risk Rx (0‘)“- | YT ) associated with the “optimal” estimate
‘bm' for model M, is the expected value of L x (0503,, i) , given the overall posterior
distribution p((a | YT ) derived in step two. In other words,
Mb“ I Y2“) = JL.(w,wx,.-)p(wl Yndw

Following Schorﬂreide (2000), I use three loss functions. Of course, other loss functions

are possible. The virtue shared of the three used here is that the a")Jr ’5 they imply are

reasonably easy to compute. (See Judge et al. (1985, chap. 4), for further discussion of

loss functions commonly used in Bayesian applications.) The ﬁrst, L p (for “L-p-value”),
is

Lp(w,a‘2) = 1{p(a‘)l YT) > pub: m}
where I {x} is an indicator function that equals one when condition x is true and zero

otherwise. The expected L p -loss provides a measure of how far (I) lies in the tail of the

posterior distribution for a).

47

The second loss function, L 2 ( for “L-chi-square”) is deﬁned as follows. Let
I

V a, be the posterior covariance matrix of a) given density p(a)| YT). Deﬁne the function

C12 (gal YT) as a weighted deviation of a) from its posterior mean E (a) | YT) , namely

C12 (wl YT)=(a’—E(a)lYT))'V(;]((0_E(CUIYT))

The loss functionL 2 itself is
I

L a), (I) = I C a) Y > C a) Y
12( ) {12( IT) 12( lr)}
Expected L12 -loss amounts to a measure of the weighted discrepancy of (p from

the posterior mean E (a) | YT) . The third loss function Lq (“L-quadratic”), is
Lq (me) = (<0 - WWW - (13)
where W is a subjective weighting matrix. One virtue of Lq is that it has a

straightforward solution for the expected risk, viz.

Rq(a‘2q,.- IYT)=(a‘2.,,. — E(wl Yr»'W(a3q,.- — E<w| YT»-
It can be shown that the estimate (2)”. that minimizes expected L p -loss given

model M ,- is the posterior mode of p(a)| Y T,M ,-) , while expected L12 - and Lq -loss are

minimized by the posterior mean estimate (13% ,- = E ((p | YT ,Mi).

Flexible Prices Versus Sticky Prices: Two Monetary DSGE Models
To estimate and compare the monetary DSGE models, it is necessary to describe

them in more detail.

48

Once upon a time there were two economies. Both were inhabited by a large
number of individuals (normalized to one) who supplied labor and rented out capital (in
perfectly competitive resource markets) to a large number (also normalized to one) of
imperfectly competitive ﬁrms. Each ﬁrm used labor and capital to produce a particular
differentiated good. All of these goods were then aggregated to produce a single good
which individuals could use either as a consumer good or as a capital good to be added to
their existing stocks of capital. The amount of the composite good y that could be

produced, given amounts of the individual goods yi produced by each ﬁrm i, was

1 (54)“ 8/(8-1)

where a > 1. It is a standard result (Dixit and Stiglitz 1977) that minimizing the cost of

producing y units of the composite good results in a quantity demanded y,- of the good of
ﬁrm i of y,-

yi:(Pi/P)—gy’

where P is the aggregate price level, given by

l I/(l-S)
P = 00 Pil-gdi) .

Consumer ’5 problem
In both economies, the preferences of each individual in terms of consumption

and labor could be described in terms of the lifetime utility function

Zﬂ’wnciz +(1-l)ln021 _§ht)r

1:0

49

where h is labor supply and c is consumption (in terms of the composite good). As in all
good model economies, each consumer attempted to maximize the expected value of
their lifetime utility function in each period. To ensure that there was a demand for
money (following Cooley and Hansen 1989), the custom prevailed in both countries that
a subset of the consumer goods (labeled c1) could only be bought with cash, while the
others (c2) could be bought either with cash or on credit. The weight of cash goods in the
preference function is given by 1.

At the beginning of each period t each individual carried forward bonds with a
nominal value of b,_1 dollars, money balances with a nominal value of m,_1 dollars, and
k,_1 units of the (composite) capital good from the previous period t— 1. He (for so I
shall call the individual for short) would sell h, units of labor in the competitive labor
market at the real wage rate w, per hour, and would rent out his capital stock to ﬁrms in
a competitive capital market, getting back r,k,_1 units, r, being the market rental rate.

Finally, the individual received a helicopter drop of x, dollars on top of the receipts from

maturing bonds of b,_1 dollars, leaving him with total cash balances of b,_1 + m,_1 + x,

dollars. He could then purchase consumer goods (either cash or credit), investment goods

i , , bonds and money balances subject to the budget constraint

it] 1 bt_bt—l+ﬂt_ mt-l x,

cl,+c2,+i,+<b 7— + —
’t—l ”RIP: P: P: B P:

where R, is the prevailing nominal interest rate. Following Smets and Wouters (2002), I

add a cost of adjustment of investment, with the cost function (I) equaling zero in the

50

steady state and having a constant second derivative, so that (13(7) = mill—3:? = (a >
DSGE models of all kinds are notorious for their poor dynamics; adding these adjustment
costs improves the dynamics considerably in the DSGE models used here, and hence, as
will become evident below, their ﬁt to the data (and posterior probabilities).

We have related that cash goods c1 could only be bought with real money
balances. The custom also prevailed bonds could only be bought with cash; so,

individuals were also subject to the cash-in-advance constraint

 

in both economies, ensuring a demand for money.
Firm ’5 problem
Each ﬁrm i purchased labor and capital in perfectly competitive labor markets to

produces good y,-. The amount of good y,- that ﬁrm i could produce in period twith labor
input hi, and capital input ki, was given by the production function
]_
Yit = kit (p (Z: (hit _ ’10))"; ,

ho being a ﬁxed cost of production in terms of labor (which serves as a barrier to entry)

and 2, was the level of technology. 2, evolved according to the random walk process
lnz,—lnz,_1=1n7 +821,
where the technology shock 62, is white noise, so that technology grew on average at a

rate of? per period. The cost minimization problem of ﬁrm i was

min tc,, = w,h,-, + r,k,-, such that y, = k},“” (z, 01,, — 120))‘0 .

51

The solution of that problem can be expressed (Yun 1996) in the form

tc,, = w,h0 + mc, yi, , where me is the real marginal cost of producing an additional unit

of y. The instantaneous real proﬁt of firm i at time t was then
“it = (Pit /Pt)yil _tcit = (Pi! /Pt)yit - WIhO —mctyit
=(Pn/P,)1‘£y, "WthO‘mcz(Pit/Pt)_£)’t .
The difference between the two economies lay in the ﬂexibility with which ﬁrms
could set their prices. In one economy, the “ﬂexible-price” economy, ﬁrms could adjust

their prices as often as they pleased, and did so as to maximize proﬁt. As all ﬁrms were

identical, it always turned out that Pi, = P, and me = (a —1)/a in each period. Hence, in

the ﬂexible-price economy, real marginal cost was always a constant.

However, in the other economy, the “New Keynesian” economy, prices were not
perfectly ﬂexible, and the real marginal cost of output could vary over the business cycle.
In the “New Keynesian” economy, following a queer custom ﬁrst described in Calvo
(1983), a fraction (1 > 0 of ﬁrms, chosen by lot, were not allowed to freely adjust their

price, but only to raise it by the average rate of inﬂation Ito. This is a standard

assumption in the New Keynesian literature, meant to convey the idea that ﬁrms might be
locked into long-lasting price contracts. Its chief drawback is not allowing the degree of
price ﬂexibility to change over time, particularly in response to long-lasting changes in
the average rate of inﬂation (ruled out by assumption, as the long-run rate of inﬂation is
assumed given). The remaining fraction 1— a of ﬁrms were allowed to adjust their prices,

and did so so as to maximize the expected present value of their proﬁts, allowing for the

chance that the chance was or" that the ﬁrm might not be able to change that price for k

periods in the future, no matter what the price level was doing.

52

Now as the lucky 1— a ﬁrms in the group that could adjust their prices were
identical to each other, as were the a unlucky ﬁrms in the group that could not, the price
charged by each ﬁrm in each group was the same, so that the price level at time t was

given by
—8
P,” = (1 — a)(P,*)' + aPtljlg,

where P: was the price charged by the ﬁrms allowed to adjust their prices in period t.

Monetary policy

We do not know exactly what was the method by which the helicopters
determined x,, but several methods come to mind. The choice of methods calls for
wisdom. Schorﬂieide (2000) assumed a rule of money growth without feedback, so that

the rate of money growth followed an AR(1) process such that

lay, = (1 —pm)ln,u0 +pmln,u,_1+am,,am, ~ N(0,0'3,),0 3 pm <1.
While such a policy is very simple, a money growth model without feedback does not at
all resemble US monetary policy during the post-1959 period (with the possible
exception of the “monetarist” experiment from 1979 to 1982). Among the side effects of
assuming such a policy is that, with no feedback to account for supply shocks (in either
direction), is an effect on the price level from a permanent supply shock that is far too
large, which Schoriheide (2000) observed in his experiments with monetary DSGE
models, as well as a spike in the price level from a monetary shock. These are generally
not observed in VARs, however monetary and supply shocks are measured. Also, in

practice, monetary policy accommodates supply shocks to at least some degree.

53

In many papers in the New Keynesian literature, monetary policy would be
speciﬁed in terms of a Taylor rule, with the nominal interest rate being written in terms of
the output gap and inﬂation. However, given that I will be estimating my models with
only output and inﬂation data, I cannot expect the data to be all that informative
regarding the Taylor rule. For that reason, and the fact that all the real effects of money in
New Keynesian DSGE models come from its effects on inﬂation, I will specify monetary
policy in terms of the resulting behavior of inﬂation. Inspection of a VAR process for
output and inﬂation suggests that logged inﬂation, In 72}, is well described by an AR(1)
process, so that

Inn, 2 (l —p)7r0 + prr,_1+a,,,

where ln IIO = ln ,uo —ln7 and 8,, is an inﬂation term with variance 0,7, that is
correlated with the technology shock 82. Speciﬁcally, 8,, = 7:26, + am , where 7:, < Ois
the contemporaneous response of inﬂation to a technology shock and am is the portion of

inﬂation shocks uncorrelated with technology, which I assume to be the monetary shock.

With the correlation of technology and inﬂation shocks corr(z,7r) in hand it is

. a
straightforward to calculate II, = corr(£z , 8,, )—”—.
Z

Empirical Analysis

Given that there are only two shocks in my DSGE models (technology/”aggregate
supply” and money/inﬂation/”aggregate demand”), to ensure non-degenerate
distributions for the data I can only estimate my models for two time series at a time. As
my interest is primarily in the relationship between output and inﬂation, I estimated each

of the competing models with data on output (real GDP deﬂated by population) and the

54

price level (chain-type GDP price deﬂator), logged and differenced to obtain series on
(per capita) economic growth and inﬂation. The data is quarterly data for the United
States from l964:l to 2003:].

As a reference model, I will be using a VAR in output growth and inﬂation:

B(L)y, = u, = Ae,

where y, = [d 1n GDP,,7I,]' , B(L) is a lag polynomial and e, = [awe ] is a 2x1 vector of

a technology shock and a monetary shock, related to the prediction errors u, for output
and inﬂation by the 2x2 matrix A. To identify the four elements of A requires one
restriction above and beyond those given by the correlations of the prediction errors from
the VAR. I do so here by imposing the restriction that the long-run effect on output of the
temporary monetary shocks is zero, as implied by both the cash-in-advance and New
Keynesian models. So, following Blanchard and Quah (1989), I disaggregate the
prediction error into a permanent element, with a long-run effect on output, and a
temporary element, which does not, and assume that the permanent shock measures
technology and the temporary shock money. (I do this a bit reluctantly, as the temporary
shock could easily be measuring non-monetary types of demand shocks, such as
government spending shocks.

Key to my analysis is calculating posterior model probabilities, which requires
that I estimate the VAR with a proper Bayesian prior. There are several ways of doing
this; here, I supplemented my VAR models of output and inﬂation with a version of the
Normal-Wishart variant on the Minnesota prior suggested by Kadiyala and Karllson
(1997). Kadiyala and Karllson (1997) assumed a priori that the prior covariance matrix of

the disturbances from the VAR equations followed an inverse Wishart distribution with

55

low degrees of freedom, while the coefﬁcients of the VAR conditional on the covariance
matrix of the disturbances followed a multivariate normal distribution centered on the
random-walk process assumed by Litterman (i.e. ﬁrst own lag coefﬁcient equals one, all
other lag coefﬁcients equal zero). They set the parameters of their prior so that prior

variances of the lag coefﬁcients in the BVAR satisﬁed:

7r, / k for the kth own lag;

It's,2 Hts} for the kth lag of variablej; and
7:33,? for the constant term in equation 1'.

Following Kadiyala and Karllson It, was set to 0.012, suggesting a range of the own lags
from 0.8 to 1.2. For It, , a value of 0.3 was used. The inverse Wishart distribution used

four degrees of freedom. The prior means of the BVAR coefﬁcients differed slightly
from those of Kadiyala and Karllson’s original Kadiyala-Karllson prior. The prior mean
of the constant in the output equation was set at 0.005 (not zero). The prior for the ﬁrst
lagged term in the BVAR output equation was set to zero, not one, to imply a random
walk in output (not output growth). The last modiﬁcation was to discard, when drawing
from the BVAR, any draw that implied non-stationarity in the system.

The prior distributions for the DSGE model parameters were chosen to ﬁt the
domain of the structural parameters. As calculated posterior probabilities can be
somewhat sensitive to prior distributions, I imposed highly informative priors only on
parameters for which I was fairly sure there was little if any information in the data; if I
judged the data were highly informative, 1 used as loose a prior as I could while keeping
the prior proper (which is necessary for calculating posterior probabilities). A couple of

parameters were ﬁxed a priori. When the weight of cash goods ,1; in the preference

56

function was allowed to ﬂoat between zero and one, the data strongly favored values near
zero; I set it a priori to 0, thus shutting down inﬂation tax effects which the data suggest
are not very important (Very similar results would have obtained for small but positive
values for Z .) The data also strongly favor very low values for the markup ,u, tending
towards one (perfect competition); as there is much evidence in recent years that markups
are fairly low (cite someone here), but I need to keep the markup above one to ensure
monopolistic competition, I set ,u to 1.01.

The prior distributions for the rest of the DSGE model parameters are given in
table 2. la. The mean and standard deviation for the normal priors of the growth rate ln ?
and inﬂation rate ln p center the distributions near the sample means for the period, while
allowing a wide range for the parameters, so as not to make these priors too informative.
With a view to a minimally informative prior on the inﬂation process, I also use a ﬂat
(uniform) prior on the autoregressive term for the inﬂation process p (between 0 and 1),

the correlation between technology shock and inﬂation disturbance corr(ez , 8,, ) ,

(between -1 and 0, given that this correlation is almost certainly negative). The variance

terms for technology 0'? and inﬂation 0,2, are given inverse gamma prior distributions,

with 2 degrees of freedom (resulting in inﬁnite prior variances for these terms). For the

investment adjustment cost parameter (p , I wanted a reasonably diffuse but still proper
prior. To ensure that, instead of estimating 90 itself, I estimated (p/(l — (p) , which is
constrained to fall between zero and one, assuming a prior distribution for (p /(1 — go)

uniform between zero and one.

57

The data have very little information for the depreciation rate d and the labor
share of output a}, I chose priors that allow most values for these parameters used in the
RBC literature to fall comfortably within two standard deviations of the mean. The
discount rate ,6 is also not identiﬁed. The best analogy to the nominal interest rate in my

model is a short-term interest rate such as federal funds, the real mean value over most of

the sample period would suggest a ,6 = 1),er
+

 

of just about one; I set the prior mean in the

end at 0.999. This results in a capital-output ratio that is probably too high; however, the
results presented here are not sensitive to reasonable prior distributions for 6', ¢ or ,B.

As for price stickiness, I have stated the prior in terms of the average period of
price stickiness in quarters, 1/(1 — a). Most micro surveys of ﬁrms’ price behavior
suggest a range for this parameter of two to four quarters; an estimate of 2.5 to 3.5 is
obtained by a macro study by Sbordone (2002) using a measure of marginal cost instead
of output, so I picked a prior constraining l/(l - a).to be above one and assuming the
most probable values (within two standard errors) to be two to four. Gali and Gerlter
(1999) found rather higher estimates in the 5 to 6 range, but they conjecture that these are
probably biased upward, and are sensitive to sample period; at any rate, as labor contracts
and the “menus” of various ﬁrms are usually revised at most annually, there is no reason
to expect l/(l — a) to be much above four.

Step I

As the posterior distributions of the Bayesian VARs are Normal-Wishart like the

priors, it is straightforward to take draws from them, and calculating moments of interest

(as well as posterior probabilities). However, the posterior distribution of the DSGE

58

models, being the product of the likelihood (evaluated using a Kalman ﬁlter algorithm)
and prior distributions of various shapes, clearly isn’t analytically tractable, so I cannot
generate draws from them directly. So, I used a Metropolis-Hastings algorithm to
generate parameter draws from the distribution. The results reported here are based on
80,000 draws from that algorithm. The posterior means and standard errors for the
parameter draws from the Metropolis algorithm, for both the ﬂexible-price and New
Keynesian models, are given in Table 2.1b. Studying these gives some preliminary clues
regarding the performance (good or bad) of the DSGE models.

Starting on a positive note, for both models the posterior distributions for the
parameters suggest not a great deal of misspeciﬁcation. For the structural parameters,
neither model updates the prior distributions very much, suggesting that for both models
the data is quite compatible with reasonable values of 6, ¢ or ,6. For both models the
measure of investment adjustment costs (a /(l —— (p) is substantial, though the estimated
range is quite wide. The inﬂation process in both models is fairly well behaved as well;

the persistence term pin both models is around 0.9, with a fairly small error. corr(z,rr)

is constrained to be negative; this constraint seems to be binding, but not unduly so, for
the ﬂexible price model (zero falling within two standard errors of the mean), while for
the New Keynesian model it seems not to be too binding at all.

In the New Keynesian model, the posterior mean average price ﬁxity l/(l - a) is

almost spot on at three quarters. Hence the data on output and inﬂation and their behavior
are compatible with survey data on the stickiness of prices. However, given no obvious

signs of misspeciﬁcation in the ﬂexible-price model, it is fair to ask whether the data

59

really suggest an average price ﬁxity of three quarters, or whether the data are simply
uninforrnative about average price ﬁxity and so are compatible with a range of priors.

This is conﬁrmed by the calculated posterior odds for the various models, given in
Table 2.2. l assigned prior probabilities of one-third to both DSGE models (ﬂexible price
versus sticky price/New Keynesian), leaving a probability of one-third for the reference
model (the VAR). Choice of lag length is always a matter of uncertainty with VAR
models, so I split the one-third probability four ways, with one-twelfth probability each
going to VARs with one, two, three and four lags.

The posterior model probabilities are proportional to the Bayes factors, for which,
again, the formula is p(YT |M,) = Jp(YT 10(i),M,-)p(0(,-) |M,-)dl9,~ . For each model I

calculated a modiﬁed harmonic mean estimator (Geweke 1999) of the Bayes factor of the

form

1 "Sim f(9((lis)) )

”Sim 5:1 P(YT [6((5),M,-)p(9((3) lMi)

 

 

PHM(YTlMi)=

_ . —l/ — _. — —
where f (6) = (27:) ‘1”leng 2 exp[—- 0.5(0 — 6,)1/9: (t9 — 9,)], (9“) being the calculated

posterior mean of 60) and V9,,- being the posterior covariance matrix of 60-) generated

1/ 2
by the Metropolis algorithm. The IVg,,-l term can be interpreted as penalizing high

dimensionality, as do more common model selection criteria as the Schwarz criterion
(which can be interpreted as a rough approximation to the Bayes factor; see the next
chapter below and Kass and Raftery 1995 for more details).

One clear result is that the DSGE models dominate the Bayesian VARs; the

combined probabilities of the two DSGE models compared to the VARs suggest a

60

DSGE/BVAR posterior odds of 104:1. Changes in the BVAR prior distribution might
alter those odds somewhat, but what is clear is that the DSGE models are not only
competitive in ﬁt but also possibly superior. Among the advantages of the DSGE models
are greater parsimony (11 free parameters for the New Keynesian model versus 21 for the
best of the BVARs, the BVAR(4)) allowing for greater precision in estimation of
moments.

Here, however, the good news for New Keynesians ends. As expected, the
posterior probability is greater for the New Keynesian than the ﬂexible-price model.
However, the posterior odds of the sticky-price model are only about three to one; as it
were, if we allow price ﬂexibility and price stickiness to be equally likely a priori (as my
prior probabilities imply), we wind up not being able to “reject the null” of perfect price
ﬂexibility at any reasonable level (above 90%, that is, a nine to one ratio). To be sure, to
put a one-half prior probability on price ﬂexibility implies a prior distribution assigning a
probability of one-half on 1/(l — a) equaling one, when my actually prior beliefs are the
gamma distribution on 1/(1 — a) centered at three and giving a prior probability of

l/(l — a) equaling one of zero. More consistent with such a prior belief would have been

to set the prior probability of the ﬂexible price model equal to zero. Relaxing that
assumption, however, brings into sharper relief just how little information the inﬂation
and output data really possess; this calls into question the usefulness of such macro
studies as those of Gali and Gertler (1999) or Sbordone (2002), who essentially mine the

output and inﬂation data in search of the most plausible value of 1/(1— a).

To underline the point that the evidence from the output and inﬂation data alone

that there is substantial price stickiness and/or New Keynesian elements in the business

61

cycle, I re-estimated the New Keynesian model, this time with a ﬂat prior on a (i.e. a
uniform prior between 0 and l). The posterior mode of a was about 0.77, suggesting
average price ﬁxity of slightly over a year, and consistent with Gali and Gertler’s results
using a classical generalized method of moments estimator. However, the posterior
distribution was very ﬂat, with values for a from as low as 0.2 to as high as 0.85 not
possible to rule out. The modiﬁed harmonic mean estimator of the Bayes factor for the
“ﬂat prior” New Keynesian model was about 1188.22, below that for the ﬂexible price
model that restricted or to zero; Occam’s razor would then suggest taking the ﬂexible-
price model as a benchmark, and assuming no price-stickiness at all. It is clear that prior
information on the amount price-stickiness has to be imposed from without for us to be
able to learn much about inﬂation’s impact on output, given that the data are compatible
with such a wide range of degree of price stickiness.

The difference between the results here and Schorﬂreide’s (2000) results that
found monetary DSGE models come far less from the imposition of New Keynesian
effects as with other changes to the model. By stating monetary policy in terms of
inﬂation instead of monetary growth, the “spike” in inﬂation generated by a persistent
change in monetary growth, which is inconsistent with the data, is removed here. The
addition of investment adjustment costs also improves output dynamics to the degree that

the overall ﬁt of the model is much improved. If (p is restricted to zero, the Bayes factor

of the ﬂexible price model falls to 1180.74 and that of the New Keynesian model to

1182.93, leaving posterior odds of about 7:1 in favor of the BVAR(4).

62

Step 2

The good in-sample ﬁt of (correctly-speciﬁed) DSGE models is encouraging, but
probably of more interest is the behavior these models imply for the output-inﬂation
relationship. It would also be helpful to know along what lines the “bad” models (the
VARs) fail. To do this, I calculated loss functions based on predicted population
characteristics. Speciﬁcally, following Schorheide, I considered: (a) correlations between

output growth and inﬂation corr(A ln GDP, , A In P, + h ), h = —2,...,2 , and (b) the responses

of output growth dA ln GDP/ d8, and inﬂation dA In P/ d8, to a permanent technology
shock that increases technology by one standard deviation, and the responses of output

growth dA 1n GDP/ d8m and dA In P/ d8m to a one standard deviation monetary shock.

To calculate the posterior distribution of the correlations and the lRFs, I obtain the
means and covariance matrices of the vectors of correlations and IRFs from each model.
Then I assume the posterior density of the correlations and IRFs to be approximately
normal with mean

2
Elwl YT]: gﬂiJElwl YTaMi]
,2

and covariance matrix

2
leYT = [Zﬂi,T(Vw|YT,M,- + Elw'wl YnMill] — Elw'wl YT]-
i=0
I have argued above why my prior beliefs suggest I should set the prior (and so
posterior) probability of the ﬂexible price model equal to zero, and so ignore it in

calculating posterior distributions of moments. Since the posterior probabilities of the

Bayesian VAR models are also very low, I can also ignore those, and take the posterior

63

distributions from the New Keynesian model as my posterior distribution. In what
follows, then, the loss functions calculations taking the distributions from the New
Keynesian model to be the “true” one can be regarded as the “true” values of the loss
functions. However, I also calculate for most moments the loss functions assuming the
BVAR(4) to be the “true” model, for two reasons. First, it is of some interest to examine
just how the DSGE and BVAR models differ. It will turn out that, in general, the
distribution of the moments is tighter under the DSGE models than under the BVAR,
resulting the DSGE moments ﬁtting nicely within the BVAR moment distributions, but
the BVAR moments having minimal probability under the DSGE models. Also, in most
of the literature classical and Bayesian VARs are the models actually used to study the
effects of monetary shocks, and in Schorﬂreide (2000) the VAR was taken as the “true”
or benchmark model, so that the loss functions for the DSGEs, calculated taking the
BVAR(4) to be the “true” or reference model, are most closely analogous to those for the
DSGE models examined in his paper.

Step 3

Correlations

The posterior distributions for the cross correlations, and the L p -losses for the

correlations implied by the DSGE models taken individually, are given in table 2.3a. At
each lead and lag the posterior (that is, the New Keynesian model) suggests a correlation
between output growth and inﬂation between zero and -0.2 at all leads and lags. By
contrast, the ﬂexible price model given a range between zero and —0.1 for correlation of
current output with leads of inﬂation, but a tight constraint around zero for correlations of

current output with lagged inﬂation. The lower absolute values for the ﬂexible price

64

model are in part due to the lower estimated correlation between technology and inﬂation
disturbances; the zero correlations with lagged inﬂation are due to the effect of a
monetary/inﬂation shock being constrained to zero.

In contrast to the narrower range for correlations of the ﬂexible price model (a
special case of the New Keynesian model), the BVAR(4) suggests a much wider range of
the correlations, between zero and —0.4 for all leads and lags. The distribution of the
correlations from the BVAR(4) nest those for the New Keynesian and the ﬂexible-price
models comfortably. The ﬂexible-price correlations are on the high end of the range

suggested by the BVAR(4), but at no lead or lag are the L p -risks higher than 0.95; in the

case of the New Keynesian model, all are well below 0.9. However, the mean of the
correlations for the BVAR (approximating the mode) is on the low end of the plausible
levels from the New Keynesian models, and well beyond those from the ﬂexible price

model; the L p -risks of the BVAR correlations assuming ﬂexible prices are all essentially

one, and assuming the New Keynesian model all above 0.99 (calling for the BVAR
model’s “rejection at the 99% level”). The BVAR(4) correlation distribution can be
interpreted as the range of correlations between output and inﬂation compatible with the
data; the contribution of the restrictions on dynamics given by the DSGE model is to
improve the precision of the correlation estimates. If we assign a substantial posterior
probability to the hypothesis that the DSGE models are the “true” models, as the Bayes
factors suggest we should, then the lower precision of the BVAR estimates makes for a
much greater expected loss from using the results of the BVAR over the DSGE.

By the same token, if we assign a high prior probability (and so a high posterior

probability) to the idea that prices are perfectly ﬂexible, the New Keynesian model turns

65

out to be substantially riskier than the ﬂexible price model, given the wider range of
correlations it allows without a substantially higher posterior probability (unless the prior
odds in favor of price stickiness are extremely higher). The New Keynesian models’
distribution of correlations nests the ﬂexible price correlations fairly comfortably; in no

case, given the distribution from the New Keynesian model, are the L ,, -risks of the

ﬂexible price mean correlations higher than 0.95, and most are below 0.9. However,
given the distribution from the ﬂexible price model, which essentially restricts the
correlations of output and lagged inﬂation to zero (which the New Keynesian model does

not), the L p -risks of the New Keynesian model’s mean correlations of output and lagged

inﬂation are essentially one, though the risks for the correlations with current and leads of
inﬂation are well below 0.9.

The cash-in-advance model suggests a slightly too negative correlation between
current output and future inﬂation; the New Keynesian model compensates for that

somewhat, as indicated by L ,, -losses closer to 0.5 which would indicate a “perfect ﬁt.”

(The New Keynesian model, as it were, includes the Phillips curve relation allowing high
output to spur an eventual rise in inﬂation.) Under the cash in advance model, there is
essentially no correlation between current output and past inﬂation; the New Keynesian
model allows for a more realistic negative correlation between output and inﬂation
lagged one period, but for essentially none before then.

The ﬁt for the correlations individually is considerably better than that reported in
Schorﬂreide (2000), especially for the relation between output and current inﬂation,
which he found to be too sharply negative, and the relation between output and future

inﬂation (which he found not negative enough). The reason for the better ﬁt, both by the

66

“eyeball metric” and by L p -loss measure, has, however, as much to do with the more

general preferences in my model as opposed to Schorfheide’s. In particular, the
(implausibly) low risk aversion parameter makes consumption (and hence real money
balances) more closely related to the capital stock and less sensitive to technology

shocks, reducing ﬂuctuations in consumption (and so inﬂation) over the cycle.

To get a sense of how risky overall is each model, I evaluated the L ,, -losses of all

ﬁve correlations jointly. The probability p(¢p, | Y,) of the posterior modes of the

correlations from the DSGE models is approximately $791145qu | YT,MJ-), the

j=0
posterior mean of the probabilities of the posterior means of the DSGE correlations. In
Table 2.3b are reported the probabilities of the posterior mean of each model, given each
model’s distribution, where the distributions are assumed to be multivariate normal
distributions with the mean and covariance matrices being the posterior mean and

covariance matrices of the correlation draws from each model taken from the Metropolis

algorithm. If the New Keynesian model is taken to be the true model (so that

n2] = 1 and no, T ,rrl, T = 0 , the probability of the BVAR(4) mean correlations (model

0) are essentially zero, while those from the ﬂexible price model (model 1) are only about
8% as likely as the mean from the New Keynesian model (model 2). By that formulation,
clearly not assuming price stickiness is riskier than assuming it. However, this result is

sensitive to the assumption that the chance of price stickiness is negligible; if we put take

the posterior odds for model 1 versus model 2 given in Table 2.2, the implied probability

2 30.6 times that

 

27.11 1990
of the ﬂexible price mean correlations is (0.224)e 392 4+ (0.760)e 22 47
_ (0.224)e‘ ' + (0.760)e '

67

of the New Keynesian mean correlations. In effect, unless one assumes the probability of
perfect price ﬂexibility is very low a priori, the least risky hypothesis a posteriori is that
of perfect price ﬂexibility—or in other words, that inﬂation-related demand shocks have
no impact on output ﬂuctuations.

A fairly similar conclusion can be drawn from the calculated L 2 -losses for the
I

competing models. The L 2 -losses of a given model i given another model j can be
I

approximated by the p-value of a test of the null hypothesis

H 0 : C12 (gaw- | YT,Mj) ~ 12 with ﬁve degrees of freedom. Taking the BVAR(4) ﬁrst
as the benchmark, the weighted distance C12 (gqu- | YT , M 0) between the mean
correlations from the two DSGE models and the means from the BVAR(4) give the edge

to the New Keynesian model. However, for both DSGE models the L12 -risks are low

enough to make “rejection” impossible at any reasonable conﬁdence level. If we take

instead the ﬂexible-price model as the benchmark, the values for C12 ((bq’, | YT ,M 1)
reported here imply L12 -losses of near one for both the BVAR(4) and the New

Keynesian model (and rejection of both models, as it were). Taking ﬁnally the New

Keynesian model as the benchmark, the value for C12 ((qu | YT ,Mz) reported for the
ﬂexible price model has an implied L12 -loss that cannot allow rejection at any
reasonable conﬁdence level; however the value C12 (gbqﬂ | YT , M 2) reported for the

BVAR(4) implies an L12 -loss of about one (and clear rejection). Clearly, unless a priori

68

perfect price ﬂexibility is ruled out, inference using a model assuming perfect price
ﬂexibility is a much less risky proposition, in part because inﬂation effects on output are
small enough, given the size and persistence of inﬂation disturbances that theycan be for
many purposes be ignored in analyzing output ﬂuctuations.

Impulse-response functions

Looking (in Figure 2.1) at the posterior means of the impulse responses of output
grth and inﬂation to one-standard-deviation shocks in technology and in money (i.e. in
inﬂation disturbances unrelated to technology shocks) will help make this clear. Looking
ﬁrst at the impact of a technology shock on output, the impulse-response function for the
ﬂexible price and for the New Keynesian model are visually indistinguishable. On the
other hand, that from the BVAR(4) using Blanchard-Quah ordering suggests a much
lower impact of technology on output—and by implication a much larger role for
monetary/aggregate demand shocks in output ﬂuctuations. The posterior mean response
from the BVAR(4) of a monetary shock on output growth is more than twice that of the
New Keynesian model. (By assumption, the impact of a monetary shock on output in the
ﬂexible price model is zero.)

The mean impact of a technology shock on inﬂation implied by the BVAR(4) is
also more than twice that of the New Keynesian model, while that for the ﬂexible price
model is only about a third that of the New Keynesian model. These results match those
found with the correlations, with the highest correlations between output and inﬂation
coming from the BVAR and the lowest from the ﬂexible price model. The BVAR
suggests a larger role for technology/supply shocks in inﬂation determination than do the

DSGEs, and a correspondingly smaller role for monetary/demand shocks; the difference

69

between the mean responses of inﬂation to a monetary shock for the two DSGE models is
small, and both are considerably higher than that implied by the BVAR(4).

Simply looking at the posterior means is a little misleading, however, for in fact,
again, the DSGE provides a good deal more precision in disaggregating output and
inﬂation into technology and monetary shocks than the BVAR can. Figure 2.2 reports
90% conﬁdence intervals for the IRFs from the New Keynesian DSGE model along with
those from the BVAR(4). Except for the money-on-inﬂation IRF, the BVAR(4)
conﬁdence interval for each IRF comfortably nests the New Keynesian IRF conﬁdence
interval. The BVAR(4) conﬁdence intervals for technology and money shocks on outpupt
are particularly wide; for both shocks the 90% conﬁdence interval allows for an impact of
essentially zero to up to 0.8 percent, near the average size of an output disturbance in
quarterly US GDP growth data. In other words the data is compatible with either all
output disturbances being attributed to permanent (technology) shocks, all being
attributed to temporary (monetary/demand) shocks, or any case in between. The
restrictions imposed by the Blanchard-Quah identiﬁcation scheme alone are not strict
enough to shed light on which shock, temporary or permanent, plays a larger role in
output ﬂuctuations. The restrictions imposed by the DSGE model go far enough beyond
those of the Blanchard-Quah scheme (though inasmuch as technology shocks are
permanent and inﬂation shocks temporary, the Blanchard-Quah restrictions are satisﬁed
as well) to greatly increase the precision of the IRFs and reveal, at last, that permanent
technology shocks are by far the more important, the impact of a technology shock being

near the maximum the Blanchard-Quah identiﬁcation scheme would allow as plausible.

70

Table 2.4 reports the loss function calculations for the IRFs (taken at lags 0
through 11) of the various competing models, given as benchmarks either the New

Keynesian model or the BVAR(4). (As the distribution of dA ln GDP/ d8," is degenerate

for the ﬂexible price model, being zero at all lags, L12 -risk for any other model
implying non-zero responses; hence, L12 -risks taking ﬂexible-price as the benchmark

are not reported.) For the most part, they conﬁrm what the “eyeball metric” suggests for
the precision, and the compatibility of the various models. Taking the New Keynesian

model as benchmark, the calculated C12 (an | YT ,M 2 ) values for all four BVAR(4)
mean IRFs are enormous and L12 -risk essentially one. However, the calculated

C12 (gqu | YT , M2 ) values for the four ﬂexible-price IRFs are extremely low. For all but
dAln GDP/ d8”, , L12 -risk is close to zero, and for dAln GDP/ (18," itself L12 -risk is

only about .31, suggesting that a hypothesis of zero (or rather negligible) impact of
inﬂation on output cannot be rejected at any reasonable conﬁdence level. Meanwhile, if

we take the BVAR(4) as a benchmark, the L12 -risks are very low for the mean IRFs of

both DSGE models, indicating they are well within the range of possibilities suggested by
the VAR. For all but the response of inﬂation to a temporary (monetary) shock, the

L12 risks are near zero; only for dAln P/ (18", are they even above .1, and even then

none are less than .4.

I also report the Lq ~losses, given either the New Keynesian model or BVAR(4)

as benchmark, for the competing models, using as a weighting matrix a 12 x 12 identitiy

7l

matrix times 1/12 (giving the average square deviation of the benchmark IRF from the
competing model). Using the New Keynesian model as benchmark, the BVAR(4) has

consistently larger Lq -losses than the ﬂexible price model; for dA In GDP/ d8z , in
particular, the L,, -losses are over 25 times greater, and for dAln P/ d8m nearly ten times

greater. Using the BVAR(4) on the other hand, the margins between the two DSGE
models are not nearly as large; however, the New Keynesian model’s losses are
consistently lower. If one takes, as many would, the BVAR to be the true model, the New
Keynesian DSGE would appear to be the least risky. However, when a substantial or
even overwhelming probability exists that the DSGE is the true one, the BVAR(4)
becomes much riskier by several orders of magnitude.
Conclusions

The loss-ﬁmction based evaluation of a New Keynesian DSGE model estimated
using Bayesian methods leads to several conclusions. Contrary to Schorﬂreide’s (2000)
ﬁndings for a cash-in-advance Bayesian DSGE model, it is possible to construct and
estimate Bayesian DSGE models with New Keynesian features that are competitive, if
not superior, in ﬁt to a Bayesian vector autoregression in output and inﬂation. The
advantage of the Bayesian DSGE model is its ability to greatly improve the precision of
estimates of such moments as correlations between output and inﬂation as well as
impulse response functions, without an undue cost in terms of ﬁt.

Another conclusion may be more surprising. The improved performance of the
New Keynesian DSGE model used here comes from elements such as investment
adjustment costs and a monetary rule in terms of inﬂation (not, as in Schorﬂieide, a

monetary growth rule without feedback) that have little to do with the New Keynesian

72

features of the model. Disappointingly, perhaps, for a researcher devoted to a New
Keynesian explanation of business cycles, the output and inﬂation data bear surprisingly
little information about the average price stickiness in the US economy. The problem is
that inﬂation shocks have contributed such a small amount to US business cycles (enough
that they can easily be regarded as negligible) that the amount of information one could
reasonably expect the US GDP series alone to bear regarding the impact of inﬂation on
output is quite small. One conclusion that could easily be drawn is that studies that mine
output data, or marginal cost data derived from output data, for information on the slope
of the New Keynesian Phillips curve may well be best abandoned in favor of micro
surveys of price stickiness, and plausible levels of price stickiness imposed on the data a
priori for the purpose of analyzing the impact of monetary shocks.

It should be added that there are, naturally, other sources of information besides
improved priors, that is, other sources of data such as nominal interest rates. If anything,
the monetary policy model in this paper is unrealistically simple, not least because central
banks set nominal interest rates, not inﬂation as such. Also, it is rather doubtﬁrl all the
variation in output attributed to temporary shocks in the Blanchard-Quah identiﬁcation
scheme comes from monetary/inﬂation shocks; much could easily come from non-
monetary sources of demand shocks, such as government spending and preference
shocks. In the next chapter, we will take a closer look at how to disaggregate data using a
DSGE model into technology, money and non-monetary demand shocks, and explore

each one’s roles in business cycle ﬂuctuations.

73

Table 2.1a: Prior distributions for structural parameters of DSGE models

 

 

Parameter Range Density Mean Std. deviation
ln 7 9% Gaussian 0.005 0.005
In 77 9? Gaussian 0.010 0.005
)6 [0,1] Beta 0.999 0.0005
6 [0,1] Beta 0.015 0.0025
¢ [0, 1] Beta 0.700 0.025
go /(1 — (a) [0,1] Uniform 0.5 0.289
P [0, 1] Uniform 0.5 0.289
corr(8Z , 6),) [-1,0] Uniform -0.5 0.289
a /(1 — a) 93+ Gamma 2 0.5
Scale d.f.
0', ‘R 7“ Inverse gamma 0.008
0',r s}; + Inverse gamma 0.002

 

Table 2.lb: Posterior distributions for structural parameters of DSGE models

 

 

 

 

Model
Flexible price New Keynesian
Parameter Mean Std. deviation Mean Std. deviation
In y 0.0053 0.0009 0.00512 0.0009
In 7? 0.0102 0.0021 0.0103 0.0018
,3 0.999 0.0005 0.999 0.0005
6 0.0150 0.0025 0.0149 0.0024
¢ 0.700 0.0251 0.701 0.0237
(a/(l - (0) 0.361 0.161 0.301 0.154
,0 0.891 0.0378 0.878 0.0327
corr(82,8,,) -0.0932 0.0577 -0.312 0.130
1 /(1 — a) 3.099 0.467
0', 0.0116 0.0012 0.0118 0.0012
0",, 0.0028 0.0002 0.0028 0.0002

 

Table 2.2: Posterior odds for competing models

 

 

Flex. price N—K

Model DSGE DSGE BVAR(l) BVAR(Z) BVAR(3) BVAR(4)
Priorprob. 1/3 1/3 1/12 1/12 1/12 1/12

”Egg“ 1188.36 1189.58 1184.70 1184.94 1185.95 1186.32
Post. prob 0.224 0.760 0.001 0.002 0.005 0.007

Post. odds,

, , 1 3.39 0.006 0.008 0.022 0.032

”1,7 ”1,7

 

74

Table 2.3a: Distributions and loss ﬁrnction values for output-inﬂation correlations of
competing models, taken individually

 

Correlations, corr(A ln GDP, , A In P, + h )

 

Model

Flex. price

Drstnbutrons N-K

BVAR(4)
Benchmark

Flex-price

-risk

L
1’ N-K

BVAR(4)

Mean
Std. err.
Mean
Std. err.
Mean
Std. err.
Model
N-K
BVAR(4)
Flex-price
BVAR(4)
Flex-price
N-K

h = —2
-0.009
0.007
-0.090
0.045
~0.258
0.128
h = -2
~l
~l
0.930
~l
0.948
0.810

h = -1
-0.014
0.010
-0.083
0.043
-0.253
0.128
h = —l
~1
~l
0.891
~l
0.939
0.817

h = 0
-0.054
0.037
-0.092
0.059
-0.247
0.129
h = 0
0.702
~l
0.478
0.991
0.866
0.771

h =1
-0.047
0.031
-0.080
0.051
-0.219
0.130
h = 1
0.698
~1

0.477
0.994
0.809
0.71]

h = 2

-0.042
0.027
-0.070
0.043

~0.198
0.132
h = 2

0.690

~1

0.475
0.997
0.763
0.669

Table 2.3b: Loss function values for output-inﬂation correlations of competing models,

taken jointly

 

1111519,,- 1M ,11
J' = 0 j =1 J' = 2
LP‘IOSS i = 0 (BVAR(4)) 12.16 -5876 453.45
1' = 1 (ﬂex-price) 8.92 27.11 19.90
i = 2 (N-K) 10.97 -3924 22.47
C12 (wqj IYTan)
J' = 0 j =1 j = 2
i = 0 (BVAR(4)) 11807 951.79
1' =1 (ﬂex-price) 6.48 0 5.10
L 2 -,isk 1'=2(N-K) 2.38 839.09 0
l L 2 -I‘ISI(
I
j = 0 j = j = 2
i = 0 (BVAR(4)) ~1 ~1
i=1 (ﬂex-price) 0.737 0 0.596
1' = 2 (N-K) 0.205 ~1 0

 

75

Table 2.4: Loss function values for impulse response functions of competing models,

 

 

 

 

 

taken jointly
Impulse respgnse functions
dA ln GDP dA In P dA ln GDP dA In P
d8Z d8, d8m d8m
Benchmark
1 = 0 0.0086 0.0033 0.0086 0.0033
i = 1 0.00023 0.0012 0.0046 0.00034
New C 2 (wq l. l YT) = 0 1.12E12 8.09E11 3.421312 3.78E12
Keynesian I ’ z 1 1.67 2.78 9.14 0.92
1 = 0 ~1 ~1 ~1 ~1
i — 1 0.00023 0.0031 0.309 9.01E-6
i 1 0.0116 0.0084 0.0260 0.0050
i - 2 0.0086 0.0033 0.0086 0.0033
. i —-1 5.38 6.05 4.86 10.03
BVAR(4) C12 ((0‘“ | YT ) = 2 4.36 3.76 1.87 9.77
z 1 0.056 0.087 0.038 0.386
1' - 2 0.024 0.012 0.00042 0.363

 

76

Technology on outputgrowth Technologl on inﬂation
0.8 _ 00,5

0.6 - 605 ’/¢D

- 0.075 ,0” w,....----"

0.4 - 0.1 ,.—-
'\ - 0.125 ,-
O.2 \ - 015 /'

4M ' 01.75 ’.

 

 

Moneyon output gowth

 

 

 

Figure 2.1: Posterior means of impulse response functions from estimated cash-credit
DSGE model (dotted lines), New Keynesian DSGE model (dashed lines) and BVAR(4)
model (dotted-dashed lines)

77

0.8
0.6
0.4

Technology on output gowth

 

 

- 0.05
- 0.1
- 0.15
- 0.2
- 0.25

0.25
0.2
0. 15
O. l
0.05

Technology on inﬂation

 

 

Moneyon inﬂation

\\
\‘:\~\

\

‘\~°"\~.\.

 

Figure 2.2: 90% conﬁdence bands of impulse response functions from New Keynesian
DSGE model (dashed lines) and BVAR(4) model (dotted-dashed lines).

78

CHAPTER 3

TECHNOLOGY SHOCKS VERSUS MONETARY SHOCKS:
IDENTIFYING SOURCES OF BUSINESS CYCLE FLUCTUATIONS WITH A
NEW KEYNESIAN DSGE MODEL
Introduction

To answer the questions of “What does monetary policy do?” and “Does
monetary policy generate recessions?” we need to be sure, keeping the critique of
Rudebusch (1998) in mind, that the measure of monetary policy shocks used by our
models make sense. Most of the methods used to identify monetary shocks in the vector
autoregression (VAR) literature, which still comprises the bulk of empirical literature on
the effects of monetary policy, are not wholly satisfactory in that regard. The most widely
used method for identifying monetary shocks are zero restrictions on contemporaneous
effects (6. g. Bemanke and Mihov 1998; Christiano, Eichenbaum and Evans 1996, 1998;
Leeper et al. 1996), but the institutional rigidities used to defend these restrictions are at
best debatable.

Other restrictions that have sometimes been used are long-run restrictions on the
effects of shocks (e. g. Gali 1992) and sign restrictions on their effects (e. g. Uhlig 1999);
Identiﬁcation schemes based on the heteroskedasticity of structural shocks have also been
proposed, for example by Rigobon (2003). Identiﬁcation schemes restricting signs and
long-run effects of temporary monetary shocks can be defended with a number of
models, among them so-called “New Keynesian” models, which have become quite
popular in recent work on monetary policy and its effects (e. g. Clarida, Gali and Gertler

2000; Leeper and Zha 2001; McCallum and Nelson 1999). New Keynesian models are

dynamic stochastic general equilibrium models (DSGE models) of various levels of

79

elaboration, descended from the “real business cycle” (RBC) models of Kydland and
Prescott (1982). New Keynesian models add such features as imperfect competition and
price and wage stickiness to a basic RBC model to permit monetary and other demand
disturbances to play a larger role in economic ﬂuctuations. The long-run effect
restrictions imposed by Gali (1992) were inspired by restrictions that would be imposed
by an IS-LM model, or a modern New Keynesian DSGE equivalent.

Arguably more appealing from a theoretical (or an aesthetic) standpoint would be
to have done with ad-hoc identiﬁcation schemes and use an estimated DSGE model to
identify sets of monetary shocks. In practice, however, little attention has been paid to the
possibility of identifying monetary shocks with DSGE models. The purpose of this paper
is to ﬁll that gap.

The earliest papers in the DSGE literature devoted to estimating series of shocks
(e. g. Hansen and Prescott 1993) did so using calibrated models. The implicit assumption
was that DSGE models were too highly stylized to be worth estimating. However, more
recent, richer DSGE models have done a better job of ﬁtting US macro time series than
their predecessors. A large literature now exists using classical maximum likelihood
methods and/or generalized method of moments (e. g. Christiano and Eichenbaum 1992;
McGrattan 1994; McGrattan et al. 1997) to ﬁt RBC models containing several shocks to
the data. More recent work has used Bayesian methods to estimate and assess non-
monetary DSGE models. Dejong et al. (2000a, 2000b) studied a non-monetary DSGE
model containing total factor productivity and marginal efficiency of investment shocks.

Among the advantages of a Bayesian approach is the ability to supplement the data with

80

prior information on structural parameters, which is helpful when parameter estimates
conﬂict markedly with other evidence.

Several recent papers answer the common criticism that DSGE models, monetary
or non-monetary, are not competitive with classical VAR models in terms of ﬁt. Kim
(2000) ﬁnds a monetary DSGE model with a realistic monetary rule ﬁts the output,
inﬂation, money supply and interest rate data about as well as a classical VAR. del Negro
and Schorfheide (2003) found that a stylized monetary DSGE model could be used to
construct a prior for a Bayesian VAR that performed as well, or better, than the random-
walk “Minnesota” prior (Litterman 1986) in forecasting US output and inﬂation. Smets
and Wouters (2002) estimated a DSGE model of the euro area using seven data series and
ten types of shocks, including such real rigidities as habit formation as well as nominal
wage and price rigidity. They assessed the ﬁt of their model by calculating J effreys-
Bayes posterior odds (Zellner 1971), and found it comparable to that of a Bayesian VAR
using a Minnesota prior for each time series.

The present paper tries to use a monetary DSGE model to answer the question:
Do monetary policy shocks generate recessions? For that matter, have they been as
important as supply shocks in US business cycles? The approach used here to answer the
question is to estimate a series of technology/supply shocks, monetary shocks and non-
monetary demand shocks using a well-ﬁtted DSGE model, and examine what
information content these shocks—in particular monetary shocks—can provide in post-
war US business cycles.

The technique of the paper is similar to Dejong et al. (2000a, 2000b). The DSGE

model used is estimated with output, inﬂation and interest rates only, similar to the del

81

Negro and Schorﬂieide (2002) study. Following Dejong et al., the likelihood ﬁanction
implied by the monetary DSGE model is combined with a prior distribution for the model
parameters to derive a posterior distribution for model parameters, impulse responses and
series of shocks.

Dejong et al. found that their model forecast output and investment about as well
as a Bayesian VAR using a “Minnesota” prior, but did not directly compare in-sample ﬁt.
To compare the ﬁt of this paper’s DSGE model to that of competing models, and assess
how trustworthy the model is for business cycle analysis, this paper follows Schorﬂieide
(2000) and Smets and Wouters (2002) in calculating Bayesian posterior odds for the
DSGE model versus a competing model, speciﬁcally a Bayesian VAR model of output,
inﬂation and the interest rate. The DSGE model is competitive with the Bayesian VAR,
suggesting that it describes the dynamics of output, inﬂation and interest rates about as
well as the BVAR. Among the advantages of using the DSGE model (as opposed to a
BVAR of roughly equal ﬁt) lies in its identifying the shocks with direct appeal to the
theoretical model, giving the shock series ﬁrmer basis in theory than the ad hoc methods
used to construct monetary shocks from a VAR. Another is that the restrictions ﬁom the
DSGE allow a good deal more precision in describing the relationships of the various
time series used to estimate the model.

With the various shock series in hand, it is possible to study the behavior and
interactions of the shocks around NBER recessions. As a rule, monetary shocks lead US
recessions, while supply shocks coincide with recessions. However, supply shocks are a

more reliable indicator of NBER recessions. Logit models of the probability of

82

recessions, estimated using current and lagged shock series as explanatory variables,
yield accurate predictions of the turning points.

This paper’s results differ ﬁom those of Smets and Wouters in their implications
for the relative importance of shocks. Smets and Wouters found technology shocks to be
of secondary importance in European business cycles compared to preference and labor
supply shocks, as well as monetary shocks. Most of the explanatory power from the logit
model for NBER recessions estimated here comes from supply shocks. Models omitting
supply shocks are inadequate, and by standard model selection criteria suggest the
evidence that monetary shocks have had much predictive content for recessions,
particular post-1982 recessions, is quite weak.

The main source of the difference between the results below versus those of
Smets and Wouters, this paper will argue, is the different prior assumptions about the
levels of nominal price rigidity and intertemporal substitution of consumption. The prior
distributions for nominal price rigidity used here restricts it to levels that micro data
suggest are most plausible for the US economy. The prior used for intertemporal
substitution of consumption reﬂects the very low estimates found in the consumption
literature (e. g. Hall 1988). Given the fairly steep IS curves and Phillips curves that result,
demand-side shocks generally, and monetary shocks in particular, are given less role in
business cycles as compared to supply or real technology shocks than Smets and Wouters
implied for Europe. The results below come closer to the conclusions of the VAR
literature that has found for the most part “responses of real variables to monetary policy
shifts are estimated as modest or nil, depending on speciﬁcation” (Sims 1998, p. 933).

A Small Scale New Keynesian Macro Model

83

The model used in what follows is similar to models used in del Negro and
Schorﬂreide (2002) and Smets and Wouters (2002), the biggest difference being a
modiﬁcation to make preferences compatible with balanced growth. Consumers'
preferences also allow for habit formation in consumption and leisure, following recent
research such as Fuhrer (2000). Nominal rigidity is added by following Calvo (1983) and
allowing imperfectly competitive ﬁrms only a positive probability (less than one) of
being able to adjust their prices in a given period. The systematic part of monetary policy
is taken to be a modiﬁed Taylor rule (Taylor 1993), which speciﬁes the interest rate
target as a function of output and inﬂation.

Preferences
A large number of consumers have preferences that can be expressed by the

lifetime utility function

 

0° 1 1— 1— )
E,Z,Bt(8,l ((xt‘¢Xt—1) 0+(mt/Pt) a ,
_ — 0'
t—O
m, /P, is individual holdings of real money balances at time t, with m, being holdings of
nominal balances and P, being the economy-wide price level. x, is a composite of
consumption and leisure so:
x = 81—771”
t 1 r ,

where c, is individual consumption and 1, =1 — h, , where h, is the fraction of period t

engaged in labor. (In this section small letters will denote individual decision variables,
and capitals will denote economy-wide aggregates.)

Economy wide average consumption and leisure (denoted by X ,_1 ), which

individuals take as given, enters individual utility functions to generate habit formation in

84

consumption and leisure; the parameter (15 determines the amount of habit formation.

Lifetime expected utility has a ﬁnite value if and only if ,3 = lg},(1-r7)(1-0) < 1 .

The term 8, measures shifts in preferences that follow the AR(1) process

It will become clear that the effect of these shocks is to raise output, employment
and inﬂation, suggesting an aggregate demand shock. With that in mind, preference
shocks will be referred to as "autonomous demand shocks" or just “demand shocks” ﬁom
now on.

While labor/leisure is homogeneous in this economy, consumer goods are not. c,

is a composite of a large number of imperfectly substitutable consumer goods c,-, , each

produced by an individual ﬁrm i. Speciﬁcally, aggregate consumption as a function of the
individual goods is

)5/(5—1)

1 5—1/5
C, :(J-Ocl‘t ) (11

where 5 > 1. Cost minimization yields individual demand functions for each good c,-, of
the form
—6
Cit =(Pi: /Pt) Ct,

where p,-, is the price of good i. 6 then is the elasticity of demand for each good i. P,

can be expressed in terms of the individual p,-, ’s as

l 15 ”(l—5)

85

At the beginning of each period t each individual carries forward bonds with a

nominal value of b,_1 dollars, and money balances with a nominal value of m,_1 dollars.
Individuals sell h, units of labor in the competitive labor market at the real wage rate w,
per hour. They also receive dividends from ownership of monopolistically competitive
ﬁrms in the amount of r, . Finally, individuals receive a helicopter drop of x, dollars each
on top of the receipts from maturing bonds of b,_1 dollars, leaving them with total cash
balances of b,_1 + m,_1 + x, dollars. Individuals then purchase consumer goods,

privately issued bonds and money balances subject to the budget constraint

b b
c,+ 1 —’——Ll—+ﬂ—-’—nt—_l-—x—t—w,h,—r,50,
1+R.P. P. P. P. P.

 

R, being the prevailing nominal interest rate.

The ﬁrst order conditions of the consumer’s problem are

(c1: consumption demand) (1 — 7])c,_'7 (1 — h)? (x, — ¢X,_1)_0 = .1,,

 

(C2: labor supply) 77c,1 _'7 (1 — h)?1 (x, - ¢X,_1)_0 = w, .1, ,
I P.
(C3: demand for bonds) ,1, = E, —/1, +1 ,
1+Rt Pt+l
(C4' money demand) (m /P )‘a = xi. —E ii = ill .
- t t t 1 3+1 7+1 1+Rr t

(C1) and (C2) easily yield

(C5)

 

while (C1) and (C4) yield

86

Rt
1+R,

 

(C4’) (1 — n)c,"’(1 — h)? (x. — «5X.-. 1‘“ = (m. /P. )‘0.

the money demand function for this economy.
The assumed preferences for consumption and leisure ﬁx the marginal rate of
substitution between consumption and leisure at one. This ensures balanced growth, with

individual consumption and the wage rate following the same trend, and labor effort

being a constant h at the steady state.
Production

Consumers own, and supply labor to, monopolistically competitive ﬁrms that
produce according to a technology expressible as
(P1: Technology) y. = (2. (It. — 120)).
where y,-, is the output of ﬁrm i, h;, is the amount of labor used by that ﬁrm and z, is
economy-wide level of technology, the evolution process for which follows a random

walk with drift process
_ .. _ 2 _ 2
A 1n 2, — 1117 + V2, ,Vztlld, E,_1Vz, — 0, E,_1Vz, — OZ ,
the technology shock v2, being white noise and y being the long-run rate of economic

growth. ho is a ﬁxed cost in terms of labor that ensures a nontrivial barrier to entry.

Technology can be broadly interpreted here as an aggregate of non-labor inputs to
production, including institutional factors (for instance, regulatory regimes) as well as
technology. Supplies of natural resources are also a “supply-side” constraint on
productive capacity; the measure of technology shocks reported below will certainly pick

up the disruptions in oil supplies that so commonly caused economic disruptions in recent

87

US history. Given this, “supply shocks” and “technology shocks” will be used
interchangeably in what follows.

The cost minimization problem of ﬁrm i is
mintc,, = w,h,-, + r, such that y,-, = (2, (h,, —h0 )).
The solution of that problem can be expressed in the form tc,, = w,h0 + mc, y,, , where
mc, is the real marginal cost of producing an additional unit of y,- , given as
(F2: Labor demand) me, = w, / z, .
The instantaneous real proﬁt of ﬁrm i at time t is then

“it = (Pit /Pt)yit _tcit :(pit /Pt)yit _Wth0 ‘mczyiz
1—6 -6
=(Pir/Pt) yt_wth0—mct(pit/Pt) Yr

where Y , is economy-wide aggregate supply. Under perfect price ﬂexibility, maximizing

 

proﬁts would imply (p,, /P, ) = mc, for all ﬁrms; as it would be the case, all ﬁrms

6 -1
. . . . 6 —1
be1ng1dent1cal, that p,-, = P, , 1t would follow that me, = —6_ for all t. The average

Y . .
economy wide markup would be ,u E t = —1— = A. While this rs indeed the long-

 

run trend of the markup, it will not hold in general for all ﬁrms if prices are not perfectly
ﬂexible.

The precise mechanics of price-stickiness, as in the previous sections, follow that
in Calvo (1983). Each ﬁrm faces a probability a > 0 of getting a “stop signal” and being
unable to adjust their prices as it wishes, being only able to raise it by the long-run

average rate of inﬂation I? . The remaining fraction 1- a of ﬁrms receives a “go signal” to

88

adjust their prices freely. If a ﬁrm is allowed adjust its price, its choice for its product’s
price is the solution to the maximization problem
oo
13?." E1Eol(a,3)j(/1.+j 71, )(((7rjpt. /P..,- )'”" — mc. (.1 p. /,P..,- rim..- — w..,-ho).
which has the solution
(F 3: Price-setting equation)

00 . '
512101314401 //l,)(Y,+j /Yt)ma+thij7r_J§

t: 5 0
Pt:

 

oo . .
EtZ(aﬂ)J(/lt+j /1,)(Y,+j/n)ﬁi}lﬂ—j(§—1)
0

Meanwhile, ﬁrms whose prices are “stopped” choose the price p ,-, = 77p,,_1 . As all ﬁrms,

given their status as “stop” or “go” ﬁrms, are otherwise identical, these ﬁrst order

1/(1—6)
) imply a

l _
conditions and the price level aggregation function P, = U0 p}, 6di

process for the price level obeying

1
(F4: Price-level evolution process) P,1_5 = (1 — a)p: + a(7rP,_1)1_6 .

Monetary policy

To close the system completely requires specifying a monetary rule for the central
bank. Here it is assumed the central bank follows a monetary rule that can be expressed
in the form

(MP1: Monetary policy)

(1+ R, ) = v,,R((1+ R,_l ), Y, /(z, (77 — 110)),27, , Y, / Y,_1,7r, /7r,_1).

89

That is, the Fed sets the interest rate with regard to the output gap (the deviation of

aggregate output Y, from its long-run trend 2, (h — ho) where h is the long-run average
for h,, the current inﬂation rate 7r,, the economic growth rate Y, / Y,_] , and the rate of
change of the inﬂation rate, It, /n,_1. The central bank also smoothes changes in the

interest rate, so that the lagged interest rate (1 + R,_1 ) enters the reaction function. Also

inﬂuencing the target interest rate is a multiplicative, non-systematic and iid monetary

shock v" which has the properties E ,_1 1n 17,, = 0, E ,_1 (ln v,, )2 = 0,2 .

Market clearing
The aggregate demand and supply for consumer goods are equal, so that
(MKl: Goods market clearing) Ct = Yr;
there being no inventories, this is a result of all ﬁrms selling their production at the going

price for their good, so that Cit = yit for all 1'. Labor demand and supply also equalize at

the competitive wage rate w, , so that

l
(MK2: Labor market clearing) h, = H, = [h,,di.
0

Aggregate demand and supply for money and bonds also equalize. In the case of bonds,

as all individuals are identical, net demand for bonds is zero at the market interest rate.

We have
(MK3: Money market clearing) m, = M r .
(MK4: Bond market clearing) b, = 0,

How To Estimate The Model

90

The model above must be log-linearized and put in state-space form to make estimation

feasible. Log-linearizing (C 1) through (C4), imposing the goods market clearing
condition (MKl) (which log-linearizes as C, = 13,) and a little manipulation, gives me at

last

CS’ 1?:w— -67,
( ) . ’1-1.’

where the deviation of a variable from its long-run trend is denoted by marking it with a

circumﬂex. Log-linearizing the production function (F1) yields
(F1 ’1 i. = ,.11, ;
The fact that the output gap Y, is proportional to employment hi, can be used to write

(C3) in the form of an expectations-augmented Phillips curve. (C5) and the fact that in

the long run Y, = ,uw, h, can be used to write the leisure preference parameter in terms of

the average markup and the average labor effort as 77 = —1_h— . With those facts in

1— h + ,uh
hand, and the fact that the demand shock process log-linearizes
as 8, = ,0, 8H + 133, implying E, 51+] = p58, , the log-linearized form of (C3) can be
written in the form of an expectations-augmented IS curve, so:

(142R): —E,7?,+1
(IS) =0"[1- (l—h)#+h I ] EY MY,+—¢—Y,_1—%I,-vz,]

 

 

#(l—h+‘u]7) 1_¢ tt+1‘1_¢ 1_¢
1—77 17 .
“Lllu—Iiihlw’m‘ “YIN—p.16.-

Log-linearizing (F3) yields

91

At 00 - .. .. A ..
P1 =(1—aﬂ)[z(aﬂ)J(E,mc,+j +E141+j +EtY1+j +£1Pt+j)]
O

— (1 — aﬂ)[2(aﬂ)j (8,150,“- + 5,1,, ,- + E, 15,, + (6 —1)E,f>,,,)]
0

oo
= (1 — aﬂ)[2(aﬂ)l (5,111on + E,P,+j )J ,
0
. . , :1: .. a: .. ..
Implymg p. - aﬂEt p1+1 = (1 - aﬂXmC. + P.) .
Log-linearizing the price index process (F4) gives us P, = (l — a)P,* + aP,_1 , given that

in the long runP* = P . The fact that 7? =P —P_ implies ii =(l-a)(P* —P).
1 1 1 1 1 1 1 1 1

Substituting into (F3) (and a little rearrangement ﬁnally gets us our Calvo-type Phillips

 

curve:

9 A A 1— a 1— a A
(F3) ”1:.BE1”1+1+( )2 '8) mc,.
(F2) implies
(F2’) the, = 13), ,

so that combining (F2’), (F 3’) and (CS’) gives a Phillips curve of the form:

1.}? 171— 1- ~ ~ ~
Illa—)5: ( a); aﬂ)Yt=7rt—16Et”t+l-

 

(Phillips curve)

Finally, taking the elasticity of the target interest rate with regard to each of the
endogenous variables to be constant, log-linearizing the monetary policy reaction

function gives a rule of the form

_ 01R.)=p.(14R._.)+0-p.xer“. +8.4.)
(Monetary polrcy) .. . . . .
+rAy(Yt ’Y1—1 +Vz1)+”Azt(”1 ‘”1—1)+Vr1.

92

The three equations of the model can be written as

(IS curve)

(1 71 R)1-E1ﬁt+1

1-77 h 1 . 1 ~ ~

 

 

141—bit?)
(1—11')11+I? ~ _ ~ _ .
+(#(l—h+#l;)l(Eth+l Ytl+0 pg)?”

, (141%.)=p.04R._.)+(1—p.)(ryi. +0.4.)
(Monetary polrcy) A A . . .
+rAy(Yt -Yt—l +vzt)+rA7t(7rt -7rt—l)+vrt’

where p, is the interest rate smoothing parameter, and

p(1—17)_+77(1—a)(1—a/3)Y.
40—h) a ’

 

(Phillips curve) = 7?, — ,6E,7i, +1.

The log-linearized system can be interpreted as an expectations-augmented IS-
LM model. Positive demand shocks inﬂuence aggregate demand by shifting the IS curve,
while monetary shocks do so by shifting the “LM curve” formed by targeting of the
interest rate. However, the ﬁll] impact of the shocks on aggregate demand is not felt in

the period of the shock because of habit formation. The Phillips curve describes the
deviation of aggregate supply from trend Y, as proportional to the current inﬂation rate

minus expected future inﬂation.
The log-linearized version of the model can be written (following Hamilton 1994)
in the state space form:

93

The vector of observable variables is y, = {A In GDP, , A In P, , ln(1 + R), }' . The vector of

deterministic variables is x, = {In 7, In 77, In E} for all t, mapped into the constant terms

1 0 0
for the dependent variable equations by A = 0 1 0 . The state variables
1 1 —1

are ?, = {132, , 8, , 13,, , (1 5r R,_] ), Y,_1, 721—1 } , and the mapping matrix H and the transition
matrix F are

1+ $yz $ye $yv $yr $yy —1 $y7r
H’ = (I)... 19.... 10.... to... (0., (0.... and
$77: $re $rv $rr $132 $17!

( 0 O 0 0 O 0
0 p g 0 0 0 O
0 0 O O O 0

$rz $re $rv $rr $ry $r7r

$yz $ ye $yv $ yr $yy $y7r

($712 $718 $7zv $727 $71)» $7M l

 

 

where the go’s are complicated functions of the model parameters. The mapping matrix

uses the fact that the observables can be approximated by

AlnGDP, =9,,+1?, —1?,_,
AlnP, =In7?+/i,
ln(l+R), =1ny+1n11—1n3+(14R),.

The shock vector v, = {122, ,v,, ,0, v,, ,0,0} has the covariance matrix

94

 

 

(03000001
0630000
Q=0000200.
0000,00
000000
(000000)

The Kalman ﬁlter can easily be used to calculate the likelihood L(Y | B), of the system,
where Y = {y , } ,T: 1 are the 3xT matrix observations of y, over the full period from time 1
to time T, andB = {y,3,y,h,o,n,a,pg,p,,ry,r,,,rAy,rA,,,0'2,0'£,0',} is the vector of
the model’s structural parameters. Combining the likelihood with a suitable prior
distribution f pm, (B) will imply a posterior distribution with a PDF satisfying

f... (B) cc L(Y 1 B)fp.-.-o. (B).

By taking draws from the posterior, it is straightforward to calculate posterior

distributions for the moments of interest. These include the structural parameters, impulse

response functions, and the values of the structural shocks (122,, v8, , vr, } at time t. In a

Kalman ﬁlter framework (see Hamilton 1994 for details) it is straightforward to calculate
a historical series of the values of the state variables, given the information we have at

time T, ?1|T , with the values for the expectation of the states at each time t given
information up to M , ?tll-1°With those in hand, the historical series of shocks, given the
data up to time Tand a certain value of B , arejust V1|T,B = ?1|T,B — F?1—11T,B-

0n the data and the priors
The data used to calculate the likelihood were quarterly data from 1965:I to

200321 on economic growth (logged growth rate of per capita real GDP), inﬂation (logged

95

growth rate of the chain type GDP deﬂator) and the federal funds rate. These were
obtained from the Federal Reserve Bank of St. Louis FRED database. Data from 1959:]
to 1964:IV were used to initialize the Kalman ﬁlter, but not to actually estimate the
likelihood.

The prior used to estimate the Bayesian VARs were Normal-Wishart priors of the
Kadiyala-Karllson type, similar to those used in chapter 2; the hyperparameter values
used for the experiments in chapter 2 were also used here, and so it is pointless discussing
the prior for the VARs in detail here. Of more interest is the prior distributions for the 17
model parameters for the DSGE model are given in table 3.1. All the prior distributions
must be proper (i.e. integrate to one) for estimated Bayes factors to be meaningful (Kass
and Raftery 1995). However, the results were for the most part not sensitive to the priors
(with a few exceptions), so most were picked to be in line with values common in the
literature, where the data were likely to be uninforrnative, or as “gently uniform” as
possible when the data were likely to be informative. The prior distributions for the
economic growth rate rand the long-run inﬂation rate ii are centered on their sample
means for the 1965-2003 period. Large prior standard deviations were assumed to
prevent either prior from being too informative.

The standard deviations of the technology shock 02 , the demand shock 0,. and

the monetary shock a, were assumed to have inverse gamma distributions with 2 degrees

of freedom, implying an inﬁnite prior variance. The scale parameters 3 were picked after
some experimentation, guided by results from estimation with a truly diffuse prior.

Uniform distributions were used on the persistence parameter for demand shocks p g and

on the monetary policy smoothing parameter p,. The prior distributions on the output

96

gap term ry and the inﬂation term r,, were picked so as to center them at values

consistent with Taylor's original rule, giving values of 0.5 for the output gap and 1.5 for

inﬂation. There was less guidance regarding the output growth term rm and the inﬂation
growth term rA,,; some experimentation suggested prior means of 0.125 for rAy and 0.5

for rm, , with wide standard deviations. All the terms in the monetary rule were assigned

gamma distributions, to constrain them above zero. The data were also fairly informative
regarding the discount factor ,6, suggesting a value in the ballpark of 0.99875 for
quarterly data (essentially one).

The prior mean for the average markup a was 1.2, in line with the fairly low
estimates for the parameter found in recent studies (e. g. Basu and F emald 1995). The

distribution was a gamma distribution in terms of ,u—l , to ensure a value for ,u above one.

The prior mean of average amount of free time devoted to labor h , was 0.3, a common
value in the RBC literature. The prior distribution of habit formation ¢, was assumed
uniform between 0 and 1.

More critical are the priors for the price ﬁxity parameter, a, and the curvature
parameter for the utility function, 0'. Many older studies (Roberts 1995; Gali and Gertler
1999) suggested price ﬁxity in the range of ﬁve to six quarters; however, this is at odds
with survey data (e. g. Blinder 1994) that suggests price ﬁxity for a typical ﬁrm is closer
to three quarters. Some more recent macro studies (e. g. Sbordone 2002) concur. The

results below assume a prior gamma distribution on a /(1 —- a) = 1/(1 — a) —1 centered at

three with a standard deviation of one; this implies values for 1/(1— a) from two to six, 3

97

range most researchers would ﬁnd plausible, and centers average price ﬁxity a priori at
one year.

The value of the reciprocal of the curvature parameter for the utility function, 1/ a,
is related to the intertemporal elasticity of substitution of consumption. The value of this
parameter remains somewhat controversial; what little has been determined regarding the
intertemporal elasticity of substitution of consumption in the US economy suggests that it
is probably not large. A large literature starting with Hall (1988) has been unable to show
that intertemporal elasticity of consumption is much above zero with aggregate
consumption data. By contrast, research with DSGE models generally assumes values for
the reciprocal of the intertemporal elasticity of consumption from one to ﬁve; in the
Bayesian New Keynesian DSGE literature, del Negro and Schorﬂreide use a prior
implying a range for the reciprocal from one to three, while Smets and Wouters center
their prior for the reciprocal at one.

The results reported below assume a gamma prior distribution for 1/ 0', with a
prior mean of 0.01 and one degree of freedom, so that the prior variance of 013 inﬁnite
and the prior mode of 1/ oitself is zero. The implication of the prior for intertemporal

elasticity can be best understood by recalling the IS equation

(171R): —Et7?t+l
(IScurve) =o{l— (1 h)#+h I 1 E}; —+—¢Y1+i-1—l‘£§vzt]

 

 

141—1014?) 1— —¢ ”“ —Y’¢ 1—¢
(l—h),u+h _ __
+[ Ill—(I-h+,uh)J(El Yt+l Yt)+(l p£)£[’

98

Ignoring habit formation, the response of the output gap to a unit change in the

 

 

interest rate is not 1/0'but rather 1/£0[1— (1—h),u +£ )+( (l-hl’LH-é j]; the prior
1110-11 +7111) 110—11 +141)

mean for this term has a value of about 0.1, Hall’s (1988) point estimate for intertemporal
elasticity of substitution of consumption.
Estimation results
Posterior distributions of structural parameters

Following Schorﬂreide (2000) and Smets and Wouters (2002), the well-known
Metropolis-Hastings algorithm was used to get the DSGE model’s posterior distribution
and the posterior odds. (See, for example, Geweke (1999) for a discussion of the
Metropolis-Hastings algorithm.) Table 2a gives the posterior means and standard
deviations of the structural parameter values drawn from the Metropolis-Hastings
algorithm. Most values are not far from their prior means, and in most cases standard
deviations are smaller, suggesting that the data is informative and that the priors are
consistent with the data. The demand shock persistence parameter is high, approximately
in the 0.9 range, as is the interest rate smoothing parameter. The amount of habit
formation is substantial, with a posterior estimate of (15 of around 0.3.

The posterior mean of 1/(1 — a) is slightly higher than the prior mean, suggesting

price ﬁxity of about four quarters (a bit longer than a year). 1/ ois also not too far from its
prior mean; its value along with the value of habit formation suggests intertemporal
substitution of around 0.07. Both have posterior variances not far from the prior,
suggesting little updating. The lack of updating suggests though that the data are actually

fairly uninformative about 1/(1— a) and 1/0', and so of the IS and Phillips curves.

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Compare the results in Table 2a to those of Smets and Wouters (2002), who,
using a rather different prior, get different results for Europe. As noted above, they use a
prior suggesting a prior mean of l/ o of around one, along with a loose prior on a
restricting it to be between 0.5 and 1. Their model also ﬁts about as well as a Bayesian
VAR. However, their posterior estimate for a is about 0.9, suggesting average price ﬁxity
l /(1 — or) of about ten quarters. Other evidence suggests price stickiness is much less than
this, in Europe as well as in the US. Gali, Gertler and Lopez-Salido (2001) ﬁnd a more
realistic estimate of l/(l — a) for the euro area would be closer to three or four. Like Gali
and Gertler (1999), Smets and Wouters do not trust their high point estimate for
a, attributing it to misspeciﬁcation of the marginal cost process.

The ﬁnding of counterfactually high 6;, however, is not robust to the prior
distribution, particularly a more informative one on a constraining it away from such
values. Part of Smets and Wouters’ problem derives from their relatively uninformative
prior for a, which places a non-trivial weight on values well above plausible levels. The
real problem, however, is subtler than this. Study of the posterior correlation matrix (not
reported in full here) suggests that draws of a and 1/ 0' are positively correlated; the
correlation coefﬁcient is about 0.3. The data, then, are somewhat informative for a
combination of the two parameters, but not for each individually. The data are compatible
with both high price ﬁxity and high intertemporal elasticity of consumption/output (and a
large role for monetary and other demand shocks), or for both low price ﬁxity and low

intertemporal elasticity of consumption (and a small role for monetary and demand

shocks).

100

In essence we have an identiﬁcation problem, in that the data alone do not give
the model much guidance in disaggregating disturbances into supply, demand or
monetary shocks. To be able to say anything useful about which shocks are important in
the business cycle, then, we need the best prior information we have available regarding
parameters for price ﬁxity and intertemporal elasticity of consumption. The marginal
value of Bayesian estimation, where outside information on these parameters can be
taken into account, is clearly positive here.

What about regime shifts in monetary policy?

The time period of the data set used here is similar to that of data sets commonly
used in VAR studies, but it spans a period usually considered to contain several policy
regimes. Rudebusch (1998) strongly objected to use of such data sets to estimate VARs.
Rudebusch claimed that one reason monetary policy shocks from VAR models did not
make sense was their counterfactual assumption of a time-invariant reaction function.

It turns out that splitting the sample to allow for the policy to vary over different
time periods (especially the 1979-1982 period) did not alter the qualitative nature of the
results. Among the more noticeable effects was a reduction in the estimated long-run rate
of inﬂation—allowing for a higher volatility of shocks in the 1979:111-1982:II period (the
period of the Volcker “monetarist” experiment) down-weights these observations
somewhat, and this period was marked by high inﬂation. The policy rules for the pre-
Volcker experiment period (1965:1-1979zll) and post-Volcker experiment period
(1982:111-2003zl) were also more accommodating; during the Volcker experiment,
monetary shocks were less persistent than during the Martin-Burns-Miller and late

Volcker-Greenspan periods. However, the differences were not large enough to affect

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overall conclusions. The models with multiple regimes did soundly beat the model
without regime shifts, on a posterior odds basis, but most of the improvement came from
allowing for the variance of monetary shocks (and to a lesser extent, supply shocks) to
vary over periods.

Posterior odds-based evaluation of the model

Following the method described in chapter 2, to compare the DSGE model to the
competition, posterior odds were calculated for the DSGE model as well as a number of
Bayesian VARs, estimated with up to four lags, by calculating Bayes factors for each
model. As in Chapter 2, the approximate Bayes factor reported for each model is the
modiﬁed harmonic mean estimator suggested in Geweke (1999). Table 3 gives the
estimated Bayes factors and posterior odds for the DSGE model along with those
estimated for BVARs with the Normal-Wishart prior. For the DSGE model, the results
are based on 50,000 draws from the Metropolis-Hastings algorithm; for the BVARs, they
are based on 10,000 draws from the Normal-Wishart posterior.

Of the BVARs, the model with four lags is best from a posterior odds viewpoint.
However, the posterior odds of the BVAR (4) are lower than those for the DSGE model.
Using the prior distribution in Table 1, the posterior odds are more than 5 to 1 in favor of
the DSGE model versus the BVAR (4). For the DSGE versus all the BVARs, it is slightly
less than 4 to 1. The margin is nowhere large enough to suggest abandoning VAR models
entirely in favor of DSGE models. However, it does suggest that the DSGE model is
competitive in ﬁt compared to a four lag BVAR, and so is well speciﬁed enough to be
useful in business cycle analysis. The Bayes factor results parallel the results of the

forecasting experiments of Dejong et al. (20003). Geweke (1999) argues that Bayes

102

factors can be interpreted as a measure of forecasting performance over the sample period
and so the results here suggest that the forecasting performance of the DSGE model is
roughly that of the BVAR.

The impulse responses: What do monetary policy shocks do?

The 95% conﬁdence bands for estimated impulse responses of output, inﬂation
and interest rates to a one-standard deviation shock of each type are illustrated in Figure
1. The distributions are taken from 10,000 draws from the posterior distribution of the
DSGE model.

All three sets of impulse responses are of reasonable shape. A technology shock
raises output, and also lowers inﬂation slightly. The response of interest rates to a
technology shock is ambiguous; the median is slightly above zero, but the conﬁdence
band nests zero easily. Less ambiguous is the response of economic variables to a
demand shock. A demand shock raises inﬂation and output, and also raises interest rates,
partly from higher expected future inﬂation, partly from anti-inﬂationary monetary
policy. A monetary policy shock raises the interest rate and lowers output and inﬂation.
However, notice that the scale of output responses to “average” demand and monetary
shocks is considerably smaller in magnitude than the responses of output to the
technology shock. Visual inspection of the impulse responses alone suggests that the
contribution of typical monetary shocks to the movement of output will be minor
compared to that of technology.

Because of habit formation, the full effects of shocks on output only come with a
slight lag. The lag, however, is not nearly as long for monetary shocks as VAR studies

often imply; the output effect peaks here at two quarters, whereas in many VAR studies

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output does not “hit bottom” in response to a monetary shock until up to three years after
the shock, which seems too long for a transitory shock. Also encouraging is the
unambiguously negative response of inﬂation to the monetary shock; the “price puzzle”
that arises in small-scale VARs, which postulate a positive response of inﬂation to a
monetary shock, is absent here.

Moving from real to nominal variables, study of the impulse responses of
inﬂation and interest rates to the various shocks suggests that the demand shock is most
important for determining inﬂation, with monetary shocks of at most secondary
importance. Finally, the systematic portion of monetary policy seems to be mostly made
up of responses to demand shocks, not so much technology shocks. This is as it should
be, if we presume countercyclical monetary policy is designed to moderate deviations of
output from the long-run trend determined by technology, not to counteract expansions
driven by genuine growth in the economy’s productive capacity. This fact adds to the
plausibility of the model’s measure of monetary shocks (which are deviations from the
demand shock-driven systematic policy). Thanks to interest rate smoothing, the ﬁrll
adjustment of the target only comes with a lag of up to two quarters.

What might not be obvious from the impulse responses is that a positive
technology shock results in output rising by less than the full proportion of the
technology shock. Price stickiness results in ﬁrms being unable to adjust their prices
quickly enough to be able to sell a proportionately large number of goods. The response
of ﬁrms whose prices are "stuck" to a technology shock that lowers marginal cost is to
lay off workers, rather than increase output. Hence labor demand, real wages and

employment fall in response to a technology shock.

104

This is a common phenomenon in New Keynesian models, interesting here mostly
for its implications for the importance of demand versus supply shocks. The data favor
(in both VAR and DSGE models) a mildly countercyclical inﬂation rate. If prices are
assumed to be stickier than they actually are, the response of output to a technology
shock will be measured as much smaller than is the case. As a result inﬂation will be too
countercyclical; to patch up the problem demand side shocks that imply a procyclical
output-inﬂation relationship will be assigned a excessive role in recessions by the model.
Cross-correlations of time series: How do the BVARs and the DSGE diﬂer?

An "identiﬁcation-free" method of comparing the implications of the competing
models for the data is to compare the correlations at various leads and lags of the various
series implied by the DSGE model versus a BVAR. These are reported in Tables 3.3a-c.
Only the cross-correlations for the best—ﬁtting BVAR (the BVAR (4)) are reported. The
95% conﬁdence intervals of the cross-correlations for both the BVAR and DSGE models
are calculated from 10,000 draws from the posterior distributions of each.

The DSGE and BVAR agree on a negative correlation between output and
inﬂation, though the DSGE model's correlations are smaller in magnitude. They also
agree on a positive correlation between inﬂation and nominal interest rates. Where they
differ qualitatively is the correlations between output and nominal interest rates; while the
correlation between present output and future (nominal) interest rates is negative in the
BVAR, it is positive in the DSGE. This is consistent with the positive relation of
economic growth and monetary policy assumed in the monetary rule. The correlation of

lagged nominal interest rates and present output is positive in both, but those for the

105

DSGE are much smaller in magnitude, the median being about 0.03, essentially zero,
versus the median from the BVAR of about 0.3.

Another qualitative difference in the DSGE versus the BVAR correlations is the smaller
standard errors of the DSGE correlations. To quantify the greater precision of the DSGE

correlations versus the BVAR correlations, I calculated the minimal L 2 -losses from
I

using each model’s set of correlations, given that the other was the true model (for

details, see chapter 2). The L 2 —loss statistic penalizes imprecision, and it is obvious that
Z

the BVAR’s estimates of the correlations are much less precise across the board; the

BVAR’s L12 -losses are anywhere from 30 to 200 times larger than those of the DSGE.
Also reported is the CDF of the L12 -loss given a chi-square distribution with 7

degrees of freedom; taking as a ﬁrst approximation that the distributions of each set of
correlations in each model are approximately multinormal, this will provide a test of the
hypothesis that a given model could have produced the posterior mean of the correlations
ﬁom the other model. The p-values of the posterior means of the BVAR correlations
given the DSGE are all about one; clearly, the BVAR’s estimates of the correlations are
inconsistent with the restrictions of the DSGE. Meanwhile, the cross-correlations of
output and inﬂation, and inﬂation and interest rates, reported by the DSGE cannot be
rejected by the BVAR, as it were, at any reasonable signiﬁcance level. However, the
BVAR does reject the DSGE’s estimates of the output-interest rate relationship at the
95% level.

The greater restrictions placed on the data by the DSGE result in much greater

precision being put on the ranges in which the cross-correlations may fall; as a result, the

106

BVAR correlations have a much lower probability of being produced if the DSGE is the
"true" model than vice versa, suggesting a much greater risk from choosing the BVAR
estimates if the choice is wrong. Among the beneﬁts of using the DSGE model is that it
permits greater precision regarding the relationships of economic time series by adding
more information about economic structure, without large costs in terms of ﬁt.

The shocks themselves: Do monetary policy shocks generate recessions?

It is safe to conclude at this point that the DSGE model is well speciﬁed; its ﬁt is
competitive with a VAR, and it generates well-behaved impulse responses. It also allows
more precision regarding the relationship of economic variables. Given this, some
conﬁdence is possible in assessing the effects of individual shocks as measured by the
DSGE model in particular US business cycles.

The lowest-cost way of doing this is to visually inspect the shock series and their
behavior around individual recessions. In ﬁgures 3.2a-c are reported, in standard
deviation terms, the medians of the distributions of the technology (3.2a), demand (3.2b)
and monetary (3.2c) shocks. By far the best indicator of NBER recessions (marked in the
graphs by vertical lines) is the technology shock series; above-average and persistent
technology shocks coincide with all the recessions of the post-1965 period (1970, 1974-5,
1980, 1981-2, 1990-1, 2001). All but the 1970 and 2001 recessions were associated with
disruptions in oil supplies. More likely culprits for the 1970 and 2001 recessions, both
fairly mild, were a decline in investment demand after the prolonged booms both
followed, coupled with changes in tax policy just before the 1970 and the 1990-1

recessions.

107

The demand shock series that by rights should best capture shocks to investment
demand, unrelated to long-run technology, has a much harder time matching the
historical record of cycles. The 1974-5 recession is associated with a positive demand
shock; while there was a negative shock in the 1970 recession, it was no larger than
average for the pre-l982 period. There is no obvious demand shock associated with the
1990-1 recession; there is for the 2001 recession, however. A possible interpretation is
that both the observed demand and technology shocks capture some elements of the
decline in investment after the end of the 1990’s boom.

Looking at monetary shocks, perhaps the best general conclusion isthat a
recession is generally associated with a lagged monetary shock; this is apparent for the
1974-1975, 1980 and 1981-2 recessions, as well as the 2001 recession, before which the
Greenspan Fed was often taken to task for excessively tight monetary policy in response
to the boom. However, shocks of the scale seen before the 2001 recession occurred
without incident in the 1980s and 1990s, and no such shock is obvious before 1970 or
1990-1.

As a test of a DSGE model’s ability to account for business cycles, Dejong et al.
(2000b) suggested estimating a logit model of the probability of NBER recessions
(explained variable equals one if a quarter was within an NBER recession, 0 otherwise),
using as explanatory variables the current values and values lagged up to four quarters of
the standardized levels of each shock series. The estimated probabilities of NBER
recessions from the various logit models are reported in Figure 3.3. For each model,
results are reported from the model with a lag length that maximized the Schwarz

criterion; the motivation for using this criterion will become clearer.below. The best

108

model using all the shock series does a good job of picking out recessions; all the genuine
recessions are predicted, and if we take a predicted probability of 0.5 as predicting a
recession, only one "false recession" is predicted, in the mid 1990's.

More informative regarding the sources of recessions is examining how well each
shock series does individually in predicting recessions. The performance of logit models
using only demand shocks and/or monetary shocks is not promising. The pure demand
shock model picks out the 1974-5, 1980 and 1981—2 recessions, all three of which were
associated with disinﬂation; however, it cannot pick out the non-inﬂationary recessions
of the 1990's. The pure monetary shock model is not much better; it picks out the 1979-
82 recessions, associated with the large variations in interest rates during the Volcker
period, as well as the 1970 and 2001 recessions, but misses the much larger 1974-5
recession, and is prone to predicting false recessions, most obviously in 1995. By
contrast, a model using only technology shocks does much better; it picks all the post-
1965 recessions (with only one possible "false alarm" in the late 1970's), and the false
1995 recession disappears.

Contrasting that "supply-side" model to a model with a demand side model
(including only demand and monetary shocks) is instructive. During the 1970's, when
volatile monetary policy and high and volatile inﬂation prevailed, a "demand-side" model
of recessions does about as well as a supply-side one with technology shocks. However,
in the 1980's and 1990's, the "demand-side" model's performance deteriorates, while the
"supply-side" model does not. To the extent that aggregate demand shocks can generate
business cycles, they clearly have taken a back seat to supply shocks in the 1980's and

1990's. This is not entirely surprising, of course; much of it is due to stricter adherence to

109

systematic policy (and hence smaller variance of deviations from policy) during the
tenure of Chairman Alan Greenspan than during the tenures of Chairmen Martin, Burns
and Miller. Even for the 1970's, however, the effects of monetary shocks are hard to
disentangle from those of supply shocks (i.e. oil shocks).

The last two logit models reported, one with supply and monetary shocks and one
with supply and demand shocks, are hard to distinguish visually from the pure supply
shock model. To ﬁgure out if accounting for monetary shocks adds anything to the pure
supply shock model, it would help to calculate posterior odds for the various logit
models. Complicating calculation of Bayes factors is a lack of prior information on the
values of the logit model parameters. However, Kass and Raftery (1995) argue that the
Schwarz criterion can be used as a rough "prior-free" approximation of the average
likelihood. So, the Schwarz criteria are reported in Table 3.4 as approximations of the
Bayes factors for each logit model.

The probabilities of models omitting supply shocks are near zero, conﬁrming the
inadequacy of purely "demand-side" theories of the cycle. As a general rule, models
including non-monetary demand shocks are inferior to models without them; monetary
shocks appear to be the best candidate for the source of the demand-side shocks that have
buffeted the US economy. The “demand shock only” model is particularly bad, unable to
beat a naive model assuming recessions to be equally likely each period.

However, a pure supply shock model dominates all the models including
monetary and other demand shocks; the next best model is a model including monetary
shocks, but the posterior odds are about 7 to 1 against that model. While this is not a

decisive rejection of the importance of monetary shocks in recessions, it is fair to

110

conclude that they have at best been of secondary importance to supply or technology
shocks. Such a result is not surprising in the context of much of the previous identiﬁed
VAR literature on the effects of monetary shocks; Sims (1998) summarizes the
literature’s ﬁndings by stating that the effects of monetary shocks in business cycles is
“modest to nil.” In the context of the DSGE model, however, a structural interpretation of
why this is so is much easier to propose; given the fairly steep IS curve the model
assumes, only monetary shocks on the scale observed during the 1979-1982 period are
likely to disrupt the real economy appreciably.
Correlations between shocks and time series

An alternative perspective on the role of the various shocks in the cycle is to look
at the distributions of the cross-correlations of the shock series with output, inﬂation and
interest rates. The results are reported in detail in Tables 3.5a-c. The signs of
relationships between the shock series and the time series can, however, be summed up

succinctly in the following table:

 

 

Technology Demand Monetary
Lags Current Leads Lags Current Leads Lags Current Leads
GDP growth + + 0 - + + - - +
Inﬂation - - - + + + 0 - 0
Fed. funds - - - + + + + + +

 

For output growth these correlations largely conﬁrm what eyeballing the shock
series and the estimated logit models did. The contemporaneous correlation of the supply
shock with output growth is very high (it is no lower than 0.8 and may be as high as
0.95). For the other two shocks it is much weaker. Correlations of output growth with

demand shocks range between 0.17 and 0.36 for the demand shocks. The correlations for

111

monetary shocks are if anything even weaker; they are no lower than -—0.3 and may well
be approximately zero. It is clear supply shocks are the best indicator for ﬂuctuations in
economic growth.

For nominal variables such as inﬂation and interest rates, the situation is more
ambiguous, but some general tendencies can be noted. The highest median correlation for
inﬂation is with demand shocks, between 0.3 and 0.5. Next highest in absolute value are
supply shocks, fairly precisely estimated at around -0.22 to -0.27. Weakest, again, were
monetary shocks, with correlations ranging from about zero to at most -0.25. The
evidence is quite weak for a role in monetary shocks (as opposed to systematic policy)
having had a great impact on inﬂation or output in the post-1965 period. Even for
inﬂation the case for supply shocks (i.e. oil and other supply shocks) being more
important for inﬂation ﬂuctuations overall is more persuasive. Inﬂation was not
obviously “always and everywhere a monetary phenomenon” even in the 19708;
monetary shocks were very large, but so were oil shocks.

The correlation of the rate with demand shocks and monetary shocks is
unambiguously positive. However, the correlation with supply shocks is negative,
suggesting that nominal interest rates fall in response to a supply shock. This result
conﬂicts with the impulse response results reported above, which suggest interest rates
should rise. One interpretation is that the positive effects on the target rate from a rise in
output from the shock are swamped by the negative effect on inﬂation, with which the
federal funds rate is strongly correlated.

Looking now at leads and lags of the shock series, lagged supply is positively

correlated with output growth, but leads of supply are essentially uncorrelated with

112

output. However, inﬂation and nominal interest rates are negatively correlated with
supply at all leads and lags. Since supply shocks are assumed to be white noise, the
correlations of the leads of the shocks with endogenous variables should all be zero. So,
to the extent that the leads deviate from zero, they suggest potential sources of
misspeciﬁcation. The supply shock series is highly enough correlated with output growth
to give the negative correlation with inﬂation and interest rates that we saw in the BVAR.

The correlations of lagged demand shocks are better behaved: lagged demand is
negatively correlated with output growth, as output peaks in the period of the shock then
falls back towards trend; correlations of lagged demand with inﬂation and interest rates
are positive. However, while leads of demand shocks are uncorrelated with the federal
funds rate, with conﬁdence intervals all nesting zero, leads of demand shocks are
positively correlated with current output growth, and, more weakly, with inﬂation.
Lagged monetary shocks are negatively correlated with output growth, and essentially
uncorrelated with inﬂation, and, unsurprisingly, positively correlated with interest rates.
Leads of monetary shocks are essentially uncorrelated with inﬂation and interest rates;
more problematic is the unambiguously positive correlation of leads of monetary shocks
with output growth, that is, of current monetary shocks with lagged output growth. This
suggests that the monetary shock series is still picking up policy responses to output
growth not captured in the monetary rule. Steps should be taken in future work to
minimize this problem and improve the accuracy of monetary policy shock estimates.
Conclusions

To sum up, then, there is little evidence, ﬁ'om the results reported above, that the

DSGE model's estimated monetary policy shocks have had much information content for

113

the real business cycle, at least since the 1980's. The estimated logit models for NBER
recessions do not indicate that adding monetary shocks adds much to a model estimated
using just supply shocks in explaining recessions. Also, the correlation between the
model’s estimate of monetary shocks with inﬂation as well as output growth is fairly
weak. There is also little evidence that other, non-monetary demand-side shocks have
much impact on the real business cycle either. Given the best evidence on price stickiness
and intertemporal substitution of consumption, the chief culprits for US business cycles,
contrary to Smets and Wouters' results for Europe, appear to be supply and technology
shocks. What is apparent for now is that the degree of culpability for business cycles that
a Bayesian DSGE model assigns to supply shocks versus monetary shocks is sensitive to
its priors on the structural parameters, as these determine how our models disaggregate
disturbances into various shocks. Our ability to learn the causes of recessions from
Bayesian DSGE models depends on how well we specify our prior knowledge about the
structure of the economy, and, by extension, how well that prior knowledge can be
reﬁned. Let that stand as the moral to this whole tangled tale.

The results are naturally conditional on the accuracy of the shock series, and their
reﬁnement is certainly possible. Information from other measures of output, inﬂation and
interest rates (differentiated by measurement error) could be mined for information on the
timing of monetary, supply, and other shocks. DSGE models could give a structural
interpretation to the dynamic factors estimated from big data sets that have proven useful
for forecasting business cycles (Stock and Watson 2002a, 2002b). Unfortunately, such a
study must be left to future research, that is, to another day.

Sldn agat agus beannacht Dé leat.

114

Table 3.1a: Prior distributions for structural parameters of DSGE model

 

Parameter Distribution Range Mean Standard deviation
Long-run trend parameters
ln 7 Normal (-oo,oo) 0.005 0.005
In 7: Normal (-oo,oo) 0.010 0.0075
Preference parameters
73 Beta [0,1] 0.99875 0.000625
¢ Uniform [0,1] 0.500 0.289
1/0 Gamma [0,oo) 0.010 0.010
h Beta [0,1] 0.300 0.050
p 5 Uniform [0,1] 0.500 0.289
Nominal/real rigidity parameters
,u-l Gamma [0,oo) 0.200 0.100
a/( 1 - a) Gamma [0,oo) 3.000 1.000
Monetary policy parameters
Pr Uniform [0,1] 0.500 0.289
r y Gamma [0,oo) 0.125 0.050
r,, Gamma [0,oo) 1 .500 0.250
rAy Gamma [0,oo) 0.125 0.050
rA ,, Gamma [0,oo) 0.500 0.250
Shock variance parameters
5 v
0' z Inverse gamma [0,oo) 1.00 2
0' 1: Inverse gamma [0,oo) 3.00 2
0’, Inverse gamma [0,oo) 0.25 2

 

115

Table 3.1b: Posterior distributions for structural parameters of DSGE model

 

 

 

 

 

Parameter Quantile
0.025 0.50 0.975
In 7 0.00324 0.00522 0.00716
In it 0.00656 0.0101 0.00140
,5 0.99752 0.99886 0.999649
45 0.199 0.333 0.474
1/0 0.00333 0.0105 0.0227
h 0.195 0.290 0.387
P8 0.864 0.919 0.966
p 1.051 1.163 1.420
1/(1 -a) 3.423 4.532 6.058
Pr 0.832 0.887 0.927
ry 0.051 1 0.126 0.271
r,, 1.194 1.535 2.073
"Ay 0.0843 0.134 0.201
’15:: 0.269 0.476 0.720
0. 1.025 1.237 1.536
0'5 3.089 4,903 8.853
0r 0.255 0,293 0.343
Table 3.2: Posterior odds for competing models
Model DSGE BVAR(I) BVAR(Z) BVAR(3) BVAR(4)
Prior probability 1/5 1/5 1/5 1/5 1/5
Approx. Bayes factor
(Modiﬁed harmonic mean) 1873.01 1869.47 1868.65 1870.20 1871.26
Posterior probability 0.78 0.02 0.01 0.05 0.14

 

116

Table 3.3a: Cross-correlations of GDP growth and inﬂation
Model Quantile '=-3 i=-2 '=-1 i=0 i=1 i=2 '=3
0.025 -0.165 -0. 185 -0.197 -0.213 -0. 136 -0.104 -0.086
DSGE 0.50 -0.095 -0.1 10 -0.122 -0.124 -0.058 -0.034 -0.024
0.975 -0.047 -0.055 -0.065 -0.050 -0.002 0.015 0.019
L12 |BVAR = 5.60, p = 0.412

 

0.025 -0.583 -0.584 -0.577 -0.571 -0.560 -0.548 -0.525
BVAR(4) 0.50 -0.210 -0.226 -0.218 -0.210 -0. 180 -0.156 -0.1 17
0.975 0.015 0.005 0.01 1 0.019 0.045 0.069 0.106

1.)!2 |DSGE = lO77,p 01

 

Table 3.3b: Cross-correlations of GDP growth and federal funds rate
Model Quantile i=-3 i=-2 i=-1 i=0 i=1 i=2 i=3
0.025 -0.066 -0.066 -0.048 -0.010 0.041 0.047 0.044
DSGE 0.50 -0.029 -0.027 -0.014 0.064 0.098 0.104 0.098
0.975 -0.007 -0.005 0.013 0.155 0.169 0.177 0.168
L12 | BVAR = 16.9, p = 0.982

0.025 -0.606 -0.621 -0.623 -0.587 -0.560 -0.545 -0.533
BVAR(4) 0.50 -0.293 -0.329 -0.338 -0.255 -0.179 -0.139 -0.1 15
0.975 -0.099 -0. l 33 -0.142 -0.060 0.020 0.059 0.086

L12 IDSGE =1394,p 01

 

 

Table 3.3c: Cross-correlations of inﬂation and the federal funds rate
Model Quantile '=-3 i=-2 '=-1 i=0 '=1 i=2 i=3
0.025 0.123 0.158 0.197 0.238 0.237 0.232 0.222
DSGE 0.50 0.374 0.416 0.462 0.510 0.519 0.518 0.510
0.975 0.686 0.713 0.744 0.776 0.785 0.788 0.785
L12 | BVAR = 2.63, p = 0.083

0.025 0.084 0.135 0.178 0.216 0.220 0.227 0.217
BVAR(4) 0.50 0.505 0.552 0.598 0.639 0.653 0.660 0.658
0.975 0.893 0.903 0.913 0.924 0.929 0.931 0.931

L12 |DSGE=71.1,sz

 

 

 

Table 3.4: Schwarz criteria and posterior odds for estimated logit models.
Model Naive T D M TD TM TDM DM
Lag length NA 2 3 4 2 2 2 3
Prior prob. 1/8 1/8 1/8 1/8 1/8 1/8 1/8 1/8
Schwarz crit. -73.44 -36.63 -73.52 -68.82 -40.93 -38.45 -41.88 -69.57
Posterior prob. 9E-17 0.846 8E-17 9E-15 0.01 1 0.138 0.004 4E-15

Code in “Model” row denotes series included: T=technology, D=demand, M=monetary.

 

 

ll7

Table 3.5a: Cross-correlations from estimated DSGE of supply shocks with endogenous

 

 

variables

Model Quant. i=-3 i=-2 '=-1 i=0 i=1 i=2 i=3
0.025 0.085 0.196 0.185 0.797 -0.081 0.072 -0.020
Output growth 0.50 0.095 0.123 0.230 0.903 0.031 0.120 -0.073
0.975 0.102 0.239 0.262 0.959 0.144 0.162 0.023
Inﬂation 0.025 -0.1 14 -0.178 -0.21 1 -0.278 -0.130 -0. 173 -0. l 32
0.50 -0.095 -0.162 -0.189 -0.249 -0.077 -0.132 -0.097
0.975 -0.080 -0.l45 -0.166 -0.220 -0.008 -0.077 -0.056
Federal funds 0.025 -0.072 -0.094 -0.l34 -0.240 -0.302 -0.260 -0.165
0.50 -0.048 -0.078 -0.1 16 -0.215 -0.255 -0.203 -0. 109
0.975 -0.031 -0.064 -0.093 -0.178 -0.192 -0.132 -0.044

 

Table 3.5b: Cross-correlations from estimated DSGE of demand shocks with endogenous

 

 

 

 

 

variables

Model Quantile '=-3 i=-2 z—-1 '=0 i=1 i=2 i=3
0.025 -0.139 -0.264 -0.085 0.172 -0.069 0.050 0.112

Output growth 0.50 -0.108 -0.219 -0.032 0.263 0.021 0.089 0.148
0.975 -0.079 -0.l69 0.016 0.361 0.1 17 0.124 0.176

0.025 0.183 0.178 0.243 0.302 -0.087 -0.028 0.021

Inﬂation 0.50 0.272 0.270 0.337 0.398 0.047 0.088 0.126
0.975 0.358 0.364 0.437 0.509 0.190 0.213 0.243
0.025 0.212 0.169 0.143 0.1 10 -0.099 -0.116 -0.108
Federal funds 0.50 0.318 0.284 0.271 0.240 0.004 -0.016 -0.012
0.975 0.429 0.408 0.41 1 0.3 86 0.132 0.104 0.105

Table 3.5c: Cross-correlations from estimated DSGE of money shocks with endogenous

variables

Model Quantile i=-3 '=-2 i=-l i=0 i=1 i=2 i=3
Output growth 0.025 -0.102 -0.290 -0.209 -0.319 0.104 0.050 -0.049
0.50 -0.075 -0.239 -0.255 -0.151 0.182 0.104 0.001

0.975 -0.047 -0.184 -0.158 0.007 0.245 0.151 0.049
0.025 -0.133 -0.106 -0.156 -0.254 -0.073 -0.075 -0. 148
Inﬂation 0.50 -0.023 0.006 -0.039 -0.126 0.057 0.037 -0.047
0.975 0.055 0.085 0.047 -0.027 0.155 0.1 18 0.025
Federal funds 0.025 0.061 0.095 0.189 0.249 -0.007 -0.016 -0.006
0.50 0.137 0.171 0.267 0.331 0.081 0.067 0.067

0.975 0.214 0.249 0.345 0.417 0.173 0.152 0.140

 

118

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All shocks

0.8 |
0.6
0.4
0.2

197075 80 85 9O 952(11)

 

197075 80 85 90 952(XX)

Tech shocks only

0.8
0.6
0.4
0.2

197075 80 85 90 952000
Tech. and noney shocks

0.8 I
0.6
0.4
0.2

197075 80 85 90 952(XX)

oney shocks only

 

197075 80 85 90 952000
Dermnd and rmney shocks

1
0.8 H
0.6

0.4 1 I

197075 80 85 90 952000
Tech.anddermnd shocks

0.8

0.4
0.2

197075 80 85 90 952(0)

Figure 3.3: Estimated probabilities of NBER recessions from logit models using series of
shocks from estimated DSGE model

123

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