Measuring the Effects of Monetary Policy: A Factor-Augmented Vector Autoregressive (FAVAR) Approach

Measuring the Effects of Monetary Policy: A Factor-Augmented Vector
Autoregressive (FAVAR) Approach*


Ben S. Bernanke, Federal Reserve Board

Jean Boivin, Columbia University and NBER

Piotr Eliasz, Princeton University




First draft: April 2002

Revised: December 2003





Abstract

Structural vector autoregressions (VARs) are widely used to trace out the effect of
monetary policy innovations on the economy. However, the sparse information sets
typically used in these empirical models lead to at least two potential problems with the
results. First, to the extent that central banks and the private sector have information not
reflected in the VAR, the measurement of policy innovations is likely to be contaminated.
A second problem is that impulse responses can be observed only for the included
variables, which generally constitute only a small subset of the variables that the
researcher and policymaker care about. In this paper we investigate one potential
solution to this limited information problem, which combines the standard structural
VAR analysis with recent developments in factor analysis for large data sets. We find
that the information that our factor-augmented VAR (FAVAR) methodology exploits is
indeed important to properly identify the monetary transmission mechanism. Overall, our
results provide a comprehensive and coherent picture of the effect of monetary policy on
the economy.






* Thanks to Christopher Sims, Mark Watson, Tao Zha and participants at the 2003 NBER Summer Institute
for useful comments. Boivin would like to thank National Science Foundation for financial support (SES-
0214104).

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1. Introduction


Since Bernanke and Blinder (1992) and Sims (1992), a considerable literature has
developed that employs vector autoregression (VAR) methods to attempt to identify and
measure the effects of monetary policy innovations on macroeconomic variables (see
Christiano, Eichenbaum, and Evans, 2000, for a survey). The key insight of this
approach is that identification of the effects of monetary policy shocks requires only a
plausible identification of those shocks (for example, as the unforecasted innovation of
the federal funds rate in Bernanke and Blinder, 1992) and does not require identification
of the remainder of the macroeconomic model. These methods generally deliver
empirically plausible assessments of the dynamic responses of key macroeconomic
variables to monetary policy innovations, and they have been widely used both in
assessing the empirical fit of structural models (see, for example, Boivin and Giannoni,
2003; Christiano, Eichenbaum, and Evans, 2001) and in policy applications.
The VAR approach to measuring the effects of monetary policy shocks appears to
deliver a great deal of useful structural information, especially for such a simple method.
Naturally, the approach does not lack for criticism. For example, researchers have
disagreed about the appropriate strategy for identifying policy shocks (Christiano,
Eichenbaum, and Evans, 2000, survey some of the alternatives; see also Bernanke and
Mihov, 1998). Alternative identifications of monetary policy innovations can, of course,
lead to different inferences about the shape and timing of the responses of economic
variables. Another issue is that the standard VAR approach addresses only the effects of
unanticipated changes in monetary policy, not the arguably more important effects of the

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systematic portion of monetary policy or the choice of monetary policy rule (Sims and
Zha, 1998; Cochrane, 1996; Bernanke, Gertler, and Watson, 1997).
Several criticisms of the VAR approach to monetary policy identification center
around the relatively small amount of information used by low-dimensional VARs. To
conserve degrees of freedom, standard VARs rarely employ more than six to eight
variables.1 This small number of variables is unlikely to span the information sets used
by actual central banks, who are known to follow literally hundreds of data series, or by
the financial market participants and other observers. The sparse information sets used in
typical analyses lead to at least two potential sets of problems with the results. First, to
the extent that central banks and the private sector have information not reflected in the
VAR analysis, the measurement of policy innovations is likely to be contaminated. A
standard illustration of this potential problem, which we explore in this paper, is the Sims
(1992) interpretation of the so-called “price puzzle”, the conventional finding in the VAR
literature that a contractionary monetary policy shock is followed by a slight increase in
the price level, rather than a decrease as standard economic theory would predict. Sims’s
explanation for the price puzzle is that it is the result of imperfectly controlling for
information that the central bank may have about future inflation. If the Fed
systematically tightens policy in anticipation of future inflation, and if these signals of
future inflation are not adequately captured by the data series in the VAR, then what
appears to the VAR to be a policy shock may in fact be a response of the central bank to
new information about inflation. Since the policy response is likely only to partially
offset the inflationary pressure, the finding that a policy tightening is followed by rising

1 Leeper, Sims, and Zha (1996) increase the number of variables included by applying Bayesian priors, but
their VAR systems still typically contain less than 20 variables.

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prices is explained. Of course, if Sims’ explanation of the price puzzle is correct, then all
the estimated responses of economic variables to the monetary policy innovation are
incorrect, not just the price response.

A second problem arising from the use of sparse information sets in VAR
analyses of monetary policy is that impulse responses can be observed only for the
included variables, which generally constitute only a small subset of the variables that the
researcher and policymakers care about. For example, both for policy analysis and model
validation purposes, we may be interested in the effects of monetary policy shocks on
variables such as total factor productivity, real wages, profits, investment, and many
others. Another reason to be interested in the responses of many variables is that no
single time series may correspond precisely to a particular theoretical construct. The
concept of “economic activity”, for example, may not be perfectly represented by
industrial production or real GDP. To assess the effects of a policy change on “economic
activity”, therefore, one might wish to observe the responses of multiple indicators
including, say, employment and sales, to the policy change.2 Unfortunately, as we have
already noted, inclusion of additional variables in standard VARs is severely limited by
degrees-of-freedom problems.

Is it possible to condition VAR analyses of monetary policy on richer information
sets, without giving up the statistical advantages of restricting the analysis to a small
number of series? In this paper we consider one approach to this problem, which
combines the standard VAR analysis with factor analysis.3 Recent research in dynamic

2 An alternative is to treat “economic activity” as an unobserved factor with multiple observable indicators.
That is essentially the approach we take in this paper.
3 Lippi and Reichlin (1998) consider a related latent factor approach that also exploits the information from
a large data set. Their approach differs in that they identify the common factors as the structural shocks,

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factor models suggests that the information from a large number of time series can be
usefully summarized by a relatively small number of estimated indexes, or factors. For
example, Stock and Watson (2002) develop an approximate dynamic factor model to
summarize the information in large data sets for forecasting purposes.4 They show that
forecasts based on these factors outperform univariate autoregressions, small vector
autoregressions, and leading indicator models in simulated forecasting exercises.
Bernanke and Boivin (2003) show that the use of estimated factors can improve the
estimation of the Fed’s policy reaction function.

If a small number of estimated factors effectively summarize large amounts of
information about the economy, then a natural solution to the degrees-of-freedom
problem in VAR analyses is to augment standard VARs with estimated factors. In this
paper we consider the estimation and properties of factor-augmented vector
autoregressive models (FAVARs), then apply these models to the monetary policy issues
raised above.

The rest of the paper is organized as follows. Section 2 lays out the theory and
estimation of FAVARs. We consider both a two-step estimation method, in which the
factors are estimated by principal components prior to the estimation of the factor-
augmented VAR; and a one-step method, which makes use of Bayesian likelihood
methods and Gibbs sampling to estimate the factors and the FAVAR simultaneously.
Section 3 applies the FAVAR methodology and revisits the evidence on the effect of

using long-run restrictions. In our approach, the latent factors correspond instead to concepts such as
economic activity. While complementary to theirs, our approach allows 1) a direct mapping with existing
VAR results, 2) measurement of the marginal contribution of the latent factors and 3) a structural
interpretation to some equations, such as the policy reaction function.
4 In this paper we follow the Stock and Watson approach to the estimation of factors (which they call
“diffusion indexes”). We also employ a likelihood-based approach not used by Stock and Watson. Sargent

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monetary policy on wide range of key macroeconomic indicators. In brief, we find that
the information that the FAVAR methodology extracts is indeed important and leads to
broadly plausible estimates for the responses of a wide variety of macroeconomic
variables to monetary policy shocks. We also find that the advantages of using the
computationally more burdensome Gibbs sampling procedure instead of the two-step
method appear to be modest in this application. Section 4 concludes. An appendix
provides more detail concerning the application of the Gibbs sampling procedure to
FAVAR estimation.

2. Econometric framework and estimation

Let Y be an M ×1 vector of observable economic variables assumed to have
t
pervasive effects throughout the economy. For now, we do not need to specify whether
our ultimate interest is in forecasting the Y or in uncovering structural relationships
t
among these variables. Following the standard approach, we might proceed by
estimating a VAR, a structural VAR (SVAR), or other multivariate time series model
using data for the Y alone. However, in many applications, additional economic
t
information, not fully captured by the Y , may be relevant to modeling the dynamics of
t
these series. Let us suppose that this additional information can be summarized by an
K ×1 vector of unobserved factors, F , where K is “small”. We might think of the
t
unobserved factors as diffuse concepts such as “economic activity” or “credit conditions”

and Sims (1977) first provided a dynamic generalization of classical factor analysis. Forni and Reichlin
(1996, 1998) and Forni, Hallin, Lippi, and Reichlin (2000) develop a related approach.

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that cannot easily be represented by one or two series but rather are reflected in a wide
range of economic variables. Assume that the joint dynamics of ( F , Y ) are given by:
t
t

?F ?
?F ?
(2.1)
t
t 1
= ?(L)
?
+?
? ?
?
?

t
Y
Y
? t ?
? t 1? ?

where ?(L) is a conformable lag polynomial of finite order d , which may contain a
priori restrictions as in the structural VAR literature. The error term ? is mean zero with
t
covariance matrix Q .
Equation (2.1) is a VAR in ( F , Y ) . This system reduces to a standard VAR in
t
t
Y if the terms of ?(L) that relate Y to F are all zero; otherwise, we will refer to
t
t
t 1
?
equation (2.1) as a factor-augmented vector autoregression, or FAVAR. There is thus a
direct mapping into the existing VAR results, and (2.1) provides a way of assessing the
marginal contribution of the additional information contained in F . Besides, if the true
t
system is a FAVAR, note that estimation of (2.1) as a standard VAR system in Y —that
t
is, with the factors omitted—will in general lead to biased estimates of the VAR
coefficients and related quantities of interest, such as impulse response coefficients.

Equation (2.1) cannot be estimated directly because the factors F are
t
unobservable. However, if we interpret the factors as representing forces that potentially
affect many economic variables, we may hope to infer something about the factors from
observations on a variety of economic time series. For concreteness, suppose that we
have available a number of background, or “informational” time series, collectively

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denoted by the N ×1 vector X . The number of informational time series N is “large”
t
(in particular, N may be greater than T , the number of time periods) and will be
assumed to be much greater than the number of factors ( K + M << N ). We assume that
the informational time series X are related to the unobservable factors F and the
t
t
observable factors Y by:
t

(2.2) X '
f
= ? F '
y
+ ? Y '+ e '
t
t
t
t

where f
? is an N × K matrix of factor loadings, y
? is N × M , and the N ×1 vector of
error terms e are mean zero and will be assumed either weakly correlated or
t
uncorrelated, depending on whether estimation is by principal components or likelihood
methods (see below). Equation (2.2) captures the idea that both Y and F , which in
t
t
general can be correlated, represent pervasive forces that drive the common dynamics
of X . Conditional on the Y , the X are thus noisy measures of the underlying
t
t
t
unobserved factors F . The implication of equation (2.2) that X depends only on the
t
t
current and not lagged values of the factors is not restrictive in practice, as F can be
t
interpreted as including arbitrary lags of the fundamental factors; thus, Stock and Watson
(1998) refer to equation (2.2) – without observable factors – as a dynamic factor model.
In this paper we consider two approaches to estimating (2.1)-(2.2). The first one is
a two-step principal components approach, which provides a non-parametric way of
uncovering the space spanned by the common components, C = (F ',Y ') ' , in (2.2). The
t
t
t

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second is a single-step Bayesian likelihood approach. These approaches differ in various
dimensions and it is not clear a priori that one should be favored over the other.

The two-step procedure is analogous to that used in the forecasting exercises of
Stock and Watson. In the first step, the common components, C , are estimated using the
t
first K+M principal components of X .5 Notice that the estimation of the first step does
t
not exploit the fact that Y is observed. However, as shown in Stock and Watson (2002),
t
when N is large and the number of principal components used is at least as large as the
true number of factors, the principal components consistently recover the space spanned
by both F and Y . Fˆ is obtained as the part of the space covered by Cˆ that is not
t
t
t
t
covered by Y .6 In the second step, the FAVAR, equation (2.1), is estimated by standard
t
methods, with F replaced by Fˆ . This procedure has the advantages of being
t
t
computationally simple and easy to implement. As discussed by Stock and Watson, it
also imposes few distributional assumptions and allows for some degree of cross-
correlation in the idiosyncratic error term e . However, the two-step approach implies the
t
presence of “generated regressors” in the second step. To obtain accurate confidence
intervals on the impulse response functions reported below, we implement a bootstrap
procedure, based on Kilian (1998), that accounts for the uncertainty in the factor
estimation.7

5 A useful feature of this framework, as implemented by an EM algorithm, is that it permits one to deal
systematically with data irregularities. In their application, Bernanke and Boivin (2003) estimate factors in
cases in which X includes both monthly and quarterly series, series that are introduced mid-sample or are
discontinued, and series with missing values.
6 How this is accomplished depends on the specific identifying assumption used in the second step. We
describe below our procedure for the recursive assumption used in the empirical application.
7 Note that in theory, when N is large relative to T, the uncertainty in the uncertainty in the factor estimates
can be ignored; see Bai (2002).

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In principle, an alternative is to estimate (2.1) and (2.2) jointly by maximum
likelihood. However, for very large dimensional models of the sort considered here, the
irregular nature of the likelihood function makes MLE estimation infeasible in practice.
In this paper we thus consider the joint estimation by likelihood-based Gibbs sampling
techniques, developed by Geman and Geman (1984), Gelman and Rubin (1992), Carter
and Kohn (1994) and surveyed in Kim and Nelson (1999). Their application to large
dynamic factor models is discussed in Eliasz (2002). Kose, Otrok and Whiteman (2000,
2003) use similar methodology to study international business cycles. The Gibbs
sampling approach provides empirical approximation of the marginal posterior densities
of the factors and parameters via an iterative sampling procedure. As discussed in
Appendix A, we implement a multi-move version of the Gibbs sampler in which factors
are sampled conditional on the most recent draws of the model parameters, and then the
parameters are sampled conditional on the most recent draws of the factors. As the
statistical literature has shown, this Bayesian approach, by approximating marginal
likelihoods by empirical densities, helps to circumvent the high-dimensionality problem
of the model. Moreover, the Gibbs-sampling algorithm is guaranteed to trace the shape of
the joint likelihood, even if the likelihood is irregular and complicated.

Identification

Before proceeding, we need to discuss identification of the model (2.1) – (2.2),
specifically the restrictions necessary to identify uniquely the factors and the associated
loadings. In two-step estimation by principal components, the factors are obtained
entirely from the observation equation (2.2), and identification of the factors is standard.

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