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Measuring The Effects Of Monetary Policy: A Factor .

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Measuring the Effects of Monetary Policy: A Factor-Augmented VectorAutoregressive (FAVAR) Approach*Ben S. Bernanke, Federal Reserve BoardJean Boivin, Columbia University and NBERPiotr Eliasz, Princeton UniversityFirst draft: April 2002Revised: December 2003AbstractStructural vector autoregressions (VARs) are widely used to trace out the effect ofmonetary policy innovations on the economy. However, the sparse information setstypically used in these empirical models lead to at least two potential problems with theresults. First, to the extent that central banks and the private sector have information notreflected 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 includedvariables, which generally constitute only a small subset of the variables that theresearcher and policymaker care about. In this paper we investigate one potentialsolution to this limited information problem, which combines the standard structuralVAR analysis with recent developments in factor analysis for large data sets. We findthat the information that our factor-augmented VAR (FAVAR) methodology exploits isindeed important to properly identify the monetary transmission mechanism. Overall, ourresults provide a comprehensive and coherent picture of the effect of monetary policy onthe economy.*Thanks to Christopher Sims, Mark Watson, Tao Zha and participants at the 2003 NBER Summer Institutefor useful comments. Boivin would like to thank National Science Foundation for financial support (SES0214104).

1. IntroductionSince Bernanke and Blinder (1992) and Sims (1992), a considerable literature hasdeveloped that employs vector autoregression (VAR) methods to attempt to identify andmeasure the effects of monetary policy innovations on macroeconomic variables (seeChristiano, Eichenbaum, and Evans, 2000, for a survey). The key insight of thisapproach is that identification of the effects of monetary policy shocks requires only aplausible identification of those shocks (for example, as the unforecasted innovation ofthe federal funds rate in Bernanke and Blinder, 1992) and does not require identificationof the remainder of the macroeconomic model. These methods generally deliverempirically plausible assessments of the dynamic responses of key macroeconomicvariables to monetary policy innovations, and they have been widely used both inassessing 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 todeliver a great deal of useful structural information, especially for such a simple method.Naturally, the approach does not lack for criticism. For example, researchers havedisagreed about the appropriate strategy for identifying policy shocks (Christiano,Eichenbaum, and Evans, 2000, survey some of the alternatives; see also Bernanke andMihov, 1998). Alternative identifications of monetary policy innovations can, of course,lead to different inferences about the shape and timing of the responses of economicvariables. Another issue is that the standard VAR approach addresses only the effects ofunanticipated changes in monetary policy, not the arguably more important effects of the1

systematic portion of monetary policy or the choice of monetary policy rule (Sims andZha, 1998; Cochrane, 1996; Bernanke, Gertler, and Watson, 1997).Several criticisms of the VAR approach to monetary policy identification centeraround the relatively small amount of information used by low-dimensional VARs.Toconserve degrees of freedom, standard VARs rarely employ more than six to eightvariables.1 This small number of variables is unlikely to span the information sets usedby actual central banks, who are known to follow literally hundreds of data series, or bythe financial market participants and other observers. The sparse information sets used intypical analyses lead to at least two potential sets of problems with the results. First, tothe extent that central banks and the private sector have information not reflected in theVAR analysis, the measurement of policy innovations is likely to be contaminated. Astandard 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 VARliterature that a contractionary monetary policy shock is followed by a slight increase inthe price level, rather than a decrease as standard economic theory would predict. Sims’sexplanation for the price puzzle is that it is the result of imperfectly controlling forinformation that the central bank may have about future inflation. If the Fedsystematically tightens policy in anticipation of future inflation, and if these signals offuture inflation are not adequately captured by the data series in the VAR, then whatappears to the VAR to be a policy shock may in fact be a response of the central bank tonew information about inflation. Since the policy response is likely only to partiallyoffset the inflationary pressure, the finding that a policy tightening is followed by rising1Leeper, Sims, and Zha (1996) increase the number of variables included by applying Bayesian priors, buttheir VAR systems still typically contain less than 20 variables.2

prices is explained. Of course, if Sims’ explanation of the price puzzle is correct, then allthe estimated responses of economic variables to the monetary policy innovation areincorrect, not just the price response.A second problem arising from the use of sparse information sets in VARanalyses of monetary policy is that impulse responses can be observed only for theincluded variables, which generally constitute only a small subset of the variables that theresearcher and policymakers care about. For example, both for policy analysis and modelvalidation purposes, we may be interested in the effects of monetary policy shocks onvariables such as total factor productivity, real wages, profits, investment, and manyothers. Another reason to be interested in the responses of many variables is that nosingle time series may correspond precisely to a particular theoretical construct. Theconcept of “economic activity”, for example, may not be perfectly represented byindustrial production or real GDP. To assess the effects of a policy change on “economicactivity”, therefore, one might wish to observe the responses of multiple indicatorsincluding, say, employment and sales, to the policy change.2 Unfortunately, as we havealready noted, inclusion of additional variables in standard VARs is severely limited bydegrees-of-freedom problems.Is it possible to condition VAR analyses of monetary policy on richer informationsets, without giving up the statistical advantages of restricting the analysis to a smallnumber of series? In this paper we consider one approach to this problem, whichcombines the standard VAR analysis with factor analysis.3 Recent research in dynamic2An alternative is to treat “economic activity” as an unobserved factor with multiple observable indicators.That is essentially the approach we take in this paper.3Lippi and Reichlin (1998) consider a related latent factor approach that also exploits the information froma large data set. Their approach differs in that they identify the common factors as the structural shocks,3

factor models suggests that the information from a large number of time series can beusefully summarized by a relatively small number of estimated indexes, or factors. Forexample, Stock and Watson (2002) develop an approximate dynamic factor model tosummarize the information in large data sets for forecasting purposes.4 They show thatforecasts based on these factors outperform univariate autoregressions, small vectorautoregressions, and leading indicator models in simulated forecasting exercises.Bernanke and Boivin (2003) show that the use of estimated factors can improve theestimation of the Fed’s policy reaction function.If a small number of estimated factors effectively summarize large amounts ofinformation about the economy, then a natural solution to the degrees-of-freedomproblem in VAR analyses is to augment standard VARs with estimated factors. In thispaper we consider the estimation and properties of factor-augmented vectorautoregressive models (FAVARs), then apply these models to the monetary policy issuesraised above.The rest of the paper is organized as follows. Section 2 lays out the theory andestimation of FAVARs. We consider both a two-step estimation method, in which thefactors are estimated by principal components prior to the estimation of the factoraugmented VAR; and a one-step method, which makes use of Bayesian likelihoodmethods and Gibbs sampling to estimate the factors and the FAVAR simultaneously.Section 3 applies the FAVAR methodology and revisits the evidence on the effect ofusing long-run restrictions. In our approach, the latent factors correspond instead to concepts such aseconomic activity. While complementary to theirs, our approach allows 1) a direct mapping with existingVAR results, 2) measurement of the marginal contribution of the latent factors and 3) a structuralinterpretation to some equations, such as the policy reaction function.4In 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. Sargent4

monetary policy on wide range of key macroeconomic indicators. In brief, we find thatthe information that the FAVAR methodology extracts is indeed important and leads tobroadly plausible estimates for the responses of a wide variety of macroeconomicvariables to monetary policy shocks. We also find that the advantages of using thecomputationally more burdensome Gibbs sampling procedure instead of the two-stepmethod appear to be modest in this application. Section 4 concludes. An appendixprovides more detail concerning the application of the Gibbs sampling procedure toFAVAR estimation.2. Econometric framework and estimationLet Yt be an M 1 vector of observable economic variables assumed to havepervasive effects throughout the economy. For now, we do not need to specify whetherour ultimate interest is in forecasting the Yt or in uncovering structural relationshipsamong these variables. Following the standard approach, we might proceed byestimating a VAR, a structural VAR (SVAR), or other multivariate time series modelusing data for the Yt alone. However, in many applications, additional economicinformation, not fully captured by the Yt , may be relevant to modeling the dynamics ofthese series. Let us suppose that this additional information can be summarized by anK 1 vector of unobserved factors, Ft , where K is “small”. We might think of theunobserved 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.5

that cannot easily be represented by one or two series but rather are reflected in a widerange of economic variables. Assume that the joint dynamics of ( Ft , Yt ) are given by:(2.1) Ft Ft 1 Y Φ ( L) Y ν t t t 1 where Φ(L) is a conformable lag polynomial of finite order d , which may contain apriori restrictions as in the structural VAR literature. The error term ν t is mean zero withcovariance matrix Q .Equation (2.1) is a VAR in ( Ft , Yt ) . This system reduces to a standard VAR inYt if the terms of Φ (L) that relate Yt to Ft 1 are all zero; otherwise, we will refer toequation (2.1) as a factor-augmented vector autoregression, or FAVAR. There is thus adirect mapping into the existing VAR results, and (2.1) provides a way of assessing themarginal contribution of the additional information contained in Ft . Besides, if the truesystem is a FAVAR, note that estimation of (2.1) as a standard VAR system in Yt —thatis, with the factors omitted—will in general lead to biased estimates of the VARcoefficients and related quantities of interest, such as impulse response coefficients.Equation (2.1) cannot be estimated directly because the factors Ft areunobservable. However, if we interpret the factors as representing forces that potentiallyaffect many economic variables, we may hope to infer something about the factors fromobservations on a variety of economic time series. For concreteness, suppose that wehave available a number of background, or “informational” time series, collectively6

denoted by the N 1 vector X t . The number of informational time series N is “large”(in particular, N may be greater than T , the number of time periods) and will beassumed to be much greater than the number of factors ( K M N ). We assume thatthe informational time series X t are related to the unobservable factors Ft and theobservable factors Yt by:(2.2)X t ' Λ f Ft ' Λ yYt ' et 'where Λ f is an N K matrix of factor loadings, Λ y is N M , and the N 1 vector oferror terms et are mean zero and will be assumed either weakly correlated oruncorrelated, depending on whether estimation is by principal components or likelihoodmethods (see below). Equation (2.2) captures the idea that both Yt and Ft , which ingeneral can be correlated, represent pervasive forces that drive the common dynamicsof X t . Conditional on the Yt , the X t are thus noisy measures of the underlyingunobserved factors Ft . The implication of equation (2.2) that X t depends only on thecurrent and not lagged values of the factors is not restrictive in practice, as Ft can beinterpreted 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 isa two-step principal components approach, which provides a non-parametric way ofuncovering the space spanned by the common components, Ct ( Ft ', Yt ') ' , in (2.2). The7

second is a single-step Bayesian likelihood approach. These approaches differ in variousdimensions 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 ofStock and Watson. In the first step, the common components, Ct , are estimated using thefirst K M principal components of X t .5 Notice that the estimation of the first step doesnot exploit the fact that Yt is observed. However, as shown in Stock and Watson (2002),when N is large and the number of principal components used is at least as large as thetrue number of factors, the principal components consistently recover the space spannedby both Ft and Yt . F̂t is obtained as the part of the space covered by Ĉt that is notcovered by Yt .6 In the second step, the FAVAR, equation (2.1), is estimated by standardmethods, with Ft replaced by F̂t . This procedure has the advantages of beingcomputationally simple and easy to implement. As discussed by Stock and Watson, italso imposes few distributional assumptions and allows for some degree of crosscorrelation in the idiosyncratic error term et . However, the two-step approach implies thepresence of “generated regressors” in the second step. To obtain accurate confidenceintervals on the impulse response functions reported below, we implement a bootstrapprocedure, based on Kilian (1998), that accounts for the uncertainty in the factorestimation.75A useful feature of this framework, as implemented by an EM algorithm, is that it permits one to dealsystematically with data irregularities. In their application, Bernanke and Boivin (2003) estimate factors incases in which X includes both monthly and quarterly series, series that are introduced mid-sample or arediscontinued, and series with missing values.6How this is accomplished depends on the specific identifying assumption used in the second step. Wedescribe below our procedure for the recursive assumption used in the empirical application.7Note that in theory, when N is large relative to T, the uncertainty in the uncertainty in the factor estimatescan be ignored; see Bai (2002).8

In principle, an alternative is to estimate (2.1) and (2.2) jointly by maximumlikelihood. However, for very large dimensional models of the sort considered here, theirregular nature of the likelihood function makes MLE estimation infeasible in practice.In this paper we thus consider the joint estimation by likelihood-based Gibbs samplingtechniques, developed by Geman and Geman (1984), Gelman and Rubin (1992), Carterand Kohn (1994) and surveyed in Kim and Nelson (1999). Their application to largedynamic factor models is discussed in Eliasz (2002). Kose, Otrok and Whiteman (2000,2003) use similar methodology to study international business cycles. The Gibbssampling approach provides empirical approximation of the marginal posterior densitiesof the factors and parameters via an iterative sampling procedure. As discussed inAppendix A, we implement a multi-move version of the Gibbs sampler in which factorsare sampled conditional on the most recent draws of the model parameters, and then theparameters are sampled conditional on the most recent draws of the factors. As thestatistical literature has shown, this Bayesian approach, by approximating marginallikelihoods by empirical densities, helps to circumvent the high-dimensionality problemof the model. Moreover, the Gibbs-sampling algorithm is guaranteed to trace the shape ofthe joint likelihood, even if the likelihood is irregular and complicated.IdentificationBefore proceeding, we need to discuss identification of the model (2.1) – (2.2),specifically the restrictions necessary to identify uniquely the factors and the associatedloadings. In two-step estimation by principal components, the factors are obtainedentirely from the observation equation (2.2), and identification of the factors is standard.9

In this case we can choose either to restrict loadings by Λ f ' Λ f / N I or restrict thefactors by F ′F / T I . Either approach delivers the same common component F Λ f 'and the same factor space. Here we impose the factor restriction, obtaining Fˆ T Zˆ ,where the Ẑ are the eigenvectors corresponding to the K largest eigenvalues of XX ′ ,sorted in descending order. This approach identifies the factors against any rotations.In the “one-step” (joint estimation) likelihood method, implemented by Gibbssampling, the factors are effectively identified by both the observation equation (2.2) andthe transition equation (2.1). In this case, ensuring identification also requires that weidentify the factors Ft against rotations of the form Ft * AFt BYt , where A is K Kand nonsingular, and B is K M . We prefer not to restrict the VAR dynamicsdescribed by equation (2.1), and so we need to impose restrictions in the observationequation, (2.2). Substituting for Ft in (2.2) we obtain(2.3)X t Λ f A 1 Ft* (Λy Λ f A 1 B)Yt etHence unique identification of the factors and their loadings requires Λ f A 1 Λ f andΛy Λ f A 1 B Λ y . Sufficient conditions are to set the upper K K block of Λ f to anidentity matrix and the upper K M block of Λ y to zero. The key to identification hereis to make an assumption that restricts the channels by which the Y ’s contemporaneouslyaffect the X ’s. In principle, since factors are only estimated up to a rotation, the choiceof the block to set equal to an identity matrix should not affect the space spanned by theestimated factors. The specific choice made restricts, however, the contemporaneous10

impact of Yt on those K variables and therefore such variables should be chosen for t

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