Data-Rich DSGE and Dynamic Factor Models

## Contributor Notes

Dynamic factor models and dynamic stochastic general equilibrium (DSGE) models are widely used for empirical research in macroeconomics. The empirical factor literature argues that the co-movement of large panels of macroeconomic and financial data can be captured by relatively few common unobserved factors. Similarly, the dynamics in DSGE models are often governed by a handful of state variables and exogenous processes such as preference and/or technology shocks. Boivin and Giannoni(2006) combine a DSGE and a factor model into a data-rich DSGE model, in which DSGE states are factors and factor dynamics are subject to DSGE model implied restrictions. We compare a data-richDSGE model with a standard New Keynesian core to an empirical dynamic factor model by estimating both on a rich panel of U.S. macroeconomic and financial data compiled by Stock and Watson (2008).We find that the spaces spanned by the empirical factors and by the data-rich DSGE model states are very close. This proximity allows us to propagate monetary policy and technology innovations in an otherwise non-structural dynamic factor model to obtain predictions for many more series than just a handful of traditional macro variables, including measures of real activity, price indices, labor market indicators, interest rate spreads, money and credit stocks, and exchange rates.

## Abstract

Dynamic factor models and dynamic stochastic general equilibrium (DSGE) models are widely used for empirical research in macroeconomics. The empirical factor literature argues that the co-movement of large panels of macroeconomic and financial data can be captured by relatively few common unobserved factors. Similarly, the dynamics in DSGE models are often governed by a handful of state variables and exogenous processes such as preference and/or technology shocks. Boivin and Giannoni(2006) combine a DSGE and a factor model into a data-rich DSGE model, in which DSGE states are factors and factor dynamics are subject to DSGE model implied restrictions. We compare a data-richDSGE model with a standard New Keynesian core to an empirical dynamic factor model by estimating both on a rich panel of U.S. macroeconomic and financial data compiled by Stock and Watson (2008).We find that the spaces spanned by the empirical factors and by the data-rich DSGE model states are very close. This proximity allows us to propagate monetary policy and technology innovations in an otherwise non-structural dynamic factor model to obtain predictions for many more series than just a handful of traditional macro variables, including measures of real activity, price indices, labor market indicators, interest rate spreads, money and credit stocks, and exchange rates.

## I. Introduction

Dynamic factor models (DFM) and dynamic stochastic general equilibrium (DSGE) models are widely used for empirical research in macroeconomics. The traditional areas of DFM application are the construction of coincident and leading indicators (e.g., Stock and Watson 1989, Altissimo et al. 2001, Matheson 2011) and the forecasting of macro time series (Stock and Watson 1999, 2002a, b; Forni, Hallin, Lippi and Reichlin 2003; Boivin and Ng 2005). DFMs are also used for real-time monitoring (Giannone, Reichlin, Small 2008; Aruoba, Diebold, and Scotti 2009; Aruoba, Diebold 2010), in monetary policy applications (e.g., the Factor Augmented VAR approach of Bernanke, Boivin, and Eliasz 2005, Stock and Watson 2005) and in the study of international business cycles (Kose, Otrok, Whiteman 2003, 2008; Del Negro and Otrok 2008; Aruoba, Diebold, Kose, Terrones 2011). The micro-founded optimization-based DSGE models primarily focus on understanding the sources of business cycle fluctuations and on assessing the importance of nominal rigidities and various types of frictions in the economy. Recently, they appear to have been able to replicate well many salient features of the data (e.g., Christiano, Eichenbaum, and Evans 2005; Smets and Wouters 2003, 2007). As a result, the versions of DSGE models extended to open economy and multisector contexts are increasingly used as tools for projections and policy analysis at major central banks (Adolfson et al. 2007, 2008; Edge, Kiley and Laforte 2009; Coenen, McAdam and Straub 2008).

The empirical factor literature argues that the co-movement of large panels of macroeconomic and financial data can be captured by relatively few common unobserved factors. Early work by Sargent and Sims (1977) found that the dynamic index model with two indices fits well the real variables in their panel. Giannone, Reichlin and Sala (2004) claim that the number of common shocks, or, in their terminology, the stochastic dimension of the U.S. economy, is two. Based on recent theoretical work developing more formal number-of-factors criteria, several authors (e.g., Bai and Ng 2007; Hallin and Liška 2007; Stock and Watson 2005) have argued for a higher number of dynamic factors that drive large U.S. macroeconomic panels – ranging from four to seven.

The dynamics in DSGE models are also often governed by a handful of state variables and exogenous processes such as preference and/or technology shocks. Boivin and Giannoni (2006) combine a DSGE and a factor model into a data-rich DSGE model, in which DSGE states are factors and factor dynamics are subject to DSGE model implied restrictions. They argue that the richer information coming from large macroeconomic and financial panels can provide better estimates of the DSGE states and of the structural shocks driving the economy. In addition, Boivin and Giannoni (2006) showed – and we confirm their conclusions in a related work in Kryshko (2011) – that the data-rich DSGE model delivers different estimates of deep structural parameters of the model compared to standard non-data-rich estimation.

In this paper, we take both a data-rich DSGE model and an empirical dynamic factor model to the same rich data set, and ask: How similar or different would be the latent empirical factors extracted by a factor model versus the estimated data-rich DSGE model states? Do they span a common factor space? Or – in other words – can we predict the true estimated DFM latent factors from the DSGE model states with a fair amount of accuracy? We ask this question for three reasons. First, the factor spaces comparison may serve as a useful tool for evaluating a DSGE model. Recent research has shown that misspecification remains a concern for valid inference in DSGE models (Del Negro, Schorfheide, Smets and Wouters 2007 – DSSW hereafter). If a DSGE model is taken to a particular small set of observables, misspecification often manifests itself through the inferior fit. Dynamic factor models usually fit well and perform well in forecasting. So if it turns out that the spaces spanned by two models are close, that is good news for a DSGE model. This means that a DSGE model overall captures the sources of co-movement in the large panel of data as a sort of a core, and that the differences in fit between a data-rich DSGE model and a DFM are potentially due to restricted factor loadings in the former. Second, a well known weakness of dynamic factor models is that the latent common components extracted by DFMs from the large panels of data do not mean much in general. If factor spaces in two models are closely aligned, this facilitates the economic interpretation of a dynamic factor model, since the empirical factors become isomorphic to the DSGE model state variables that have clear economic meaning. Third, if factor spaces are close, we are able to propagate the structural shocks in an otherwise completely non-structural dynamic factor model to obtain predictions for a broad range of macro series of interest.2 This way of doing policy analysis is more reliable, because, in addition to the impulse responses derived in the data-rich DSGE model, which might be misspecified, we are able to generate a second set of responses to the same shocks in the context of a factor model that is primarily data-driven and fits better.

We compare a data-rich DSGE model with a standard New Keynesian core to an empirical dynamic factor model by estimating both on a rich panel of U.S. macroeconomic and financial data compiled by Stock and Watson (2008). The specific version of the data-rich DSGE model is taken from Kryshko (2011). The estimation involves Bayesian Markov Chain Monte Carlo (MCMC) methods.

We find that the spaces spanned by the empirical factors and by the data-rich DSGE model states are very close meaning that, using a collection of linear regressions, we are able to predict the true estimated factors from the DSGE states fairly accurately. Given the accuracy, we can use this predictive link to map in every period the impact of any structural DSGE shock on the data-rich DSGE states into the empirical factors. We then multiply the responses of empirical factors by the DFM factor loadings to generate the impulse responses of data indicators to structural shocks. Applying this procedure, we propagate monetary policy and technology innovations in an otherwise non-structural dynamic factor model to obtain predictions for many more series than just a handful of traditional macro variables, including measures of real activity, price indices, labor market indicators, interest rate spreads, money and credit stocks, and exchange rates. For instance, contractionary monetary policy realistically leads to a decline in housing starts and in residential investment, to a hump-shaped positive response of the unemployment rate peaking in the 5th quarter after the shock before returning to normal, to the negative rates of commodity price inflation, to a widening of interest rate spreads, to a contraction of consumer credit and to an appreciation of the dollar – despite the fact that our DSGE model does not model these features explicitly.

The paper is organized as follows. In Section II we present the variant of a dynamic factor model and a quick snapshot of the data-rich DSGE model to be used in the empirical analysis. Our econometric methodology to estimate two models is discussed in Section III. Section IV describes our data set and transformations. In Section V we proceed by conducting the empirical analysis. We begin by discussing the choice of the prior distributions of dynamic factor model’s parameters. Second, we analyze the estimated empirical factors and the posterior estimates of the DSGE model state variables and explore how well they are able to capture the co-movements in the data. Third, we compare the spaces spanned by the latent empirical factors and by the data-rich DSGE model state variables. Finally, we use the proximity of the factor spaces to propagate the monetary policy and technology innovations in an otherwise non-structural dynamic factor model to obtain the predictions for the macro series of interest. Section VI concludes.

## II. Two Models

In this section, we begin by describing the variant of a dynamic factor model. Then, we present a quick snapshot of the data-rich DSGE model with a New Keynesian core to be estimated on the same large panel of macro and financial series.

### A. Dynamic Factor Model

We choose to work with the version of the dynamic factor model as originally developed by Geweke (1977) and Sargent and Sims (1977) and recently used by Stock and Watson (2005). If the forecasting performance is a correct guide to choose the appropriate factor model specification, the literature remains rather inconclusive in that respect. For example, Forni, Hallin, Lippi and Reichlin (2003) found supportive results for the generalized dynamic factor specification over the static factor specification, while Boivin and Ng (2005) documented little differences for the competing factor specifications.

Let Ft denote the N×1 vector of common unobserved factors that are related to a J ×1 large3 (JN) panel of macroeconomic and financial data Xt according to the following factor model:

$\begin{array}{ccc}{X}_{t}=\mathbf{\Lambda }{F}_{t}+{e}_{t}& \phantom{\rule{7.0em}{0ex}}& \left(1\right)\end{array}$

where Λ is the J × N matrix of factor loadings, et is the idiosyncratic errors allowed to be serially correlated, G is the N×N matrix that governs common factor dynamics and ηt is the vector of stochastic innovations. The factors and idiosyncratic errors are assumed to be uncorrelated at all leads and lags: E(Ftei, s) = 0, all i, t and s. As in Stock and Watson (2005), we assume that matrices Q, R and Ψ are diagonal, which implies we have an exact dynamic factor model: E(ei, tej, s) = 0, ij, all t and s. This is in contrast to the approximate DFM of all Chamberlain and Rothschild (1983) that relaxes this assumption and allows for some correlation across idiosyncratic errors ei, t and ej, t, ij. As written, the model is already in static form, since data series Xt load only on contemporaneous factors and not on their lags.4

### B. Data-Rich DSGE Model

The specific version of the data-rich DSGE model that we work with in this paper is taken from Kryshko (2011), Section II.

Its New Keynesian business cycle core features capital as the factor of production, nominal rigidities in price setting, and investment adjustment costs. The real money stock enters households’ utility in additively separable fashion. The economy is populated by households, final and intermediate goods-producing firms and a central bank (monetary authority). A representative household works, consumes, saves, holds money balances and accumulates capital. It consumes the final output manufactured by perfectly competitive final good firms. The final good producers produce by combining a continuum of differentiated intermediate goods supplied by monopolistically competitive intermediate goods firms. To manufacture their output, intermediate goods producers hire labor and capital services from households. Also, when optimizing their prices, intermediate goods firms face the nominal price rigidity a la Calvo (1983), and those firms that are unable to re-optimize may index their price to lagged inflation. Monetary policy is conducted by the central bank setting the one-period nominal interest rate on public debt via a Taylor-type interest rate feedback rule. Given the interest rate, the central bank supplies enough nominal money balances to meet equilibrium demand from households.

In Kryshko (2011), Section II we have shown that if θ is the vector of deep structural parameters characterizing preferences and technology in our DSGE model and εt is the vector of exogenous shocks, then the equilibrium dynamics of the data-rich DSGE model can be summarized by the transition equation of the non-redundant DSGE model state variables St:

and the collection of measurement equations connecting the core macro series ${X}_{t}^{F}$ and the non-core informational macro series ${X}_{t}^{S}$ to the DSGE model states:

$\begin{array}{ccc}\underset{{X}_{t}}{\underbrace{\left[\frac{{X}_{t}^{F}}{{X}_{t}^{S}}\right]}}=\underset{\mathbf{\Lambda }\left(\mathbf{\theta }\right)}{\underbrace{\left[\frac{{\mathbf{\Lambda }}_{\mathbf{F}}\left(\mathbf{\theta }\right)}{{\mathbf{\Lambda }}_{\mathbf{S}}}\right]}}{S}_{t}+\underset{{e}_{t}}{\underbrace{\left[\frac{{e}_{t}^{F}}{{e}_{t}^{F}}\right]}},& \phantom{\rule{7.0em}{0ex}}& \left(5\right)\end{array}$

where the measurement errors et may be serially correlated, but uncorrelated across different data indicators (Ψ, R are diagonal):

Notice that the state-space representation of the data-rich DSGE model (4)-(6) is very much like the dynamic factor model (1)-(3) in which transition of the unobserved factors is governed by a DSGE model solution and where some factor loadings are restricted by the economic meaning of the DSGE model concepts.

## III. Econometric Methodology

This section discusses the estimation techniques for the two models considered in this paper. First, we refer the reader to Kryshko (2011) on the details about a Markov Chain Monte Carlo algorithm to estimate the data-rich DSGE model, including the choice of the prior for factor loadings. Second, we describe the Gibbs sampler to estimate a dynamic factor model.

### A. Estimation of the Data-Rich DSGE Model

We refer the reader to Kryshko (2011), Section III.A and that paper’s appendices regarding the implementation details of the MCMC algorithm to estimate our data-rich DSGE model.

### B. Estimation of the Dynamic Factor Model

Consider the original dynamic factor model described in Section II.A:

$\begin{array}{ccc}{X}_{t}=\mathbf{\Lambda }{F}_{t}+{e}_{t}& \phantom{\rule{7.0em}{0ex}}& \left(7\right)\end{array}$

Let us collect the state-space matrices into Γ={Λ, Ψ, R, G} and the latent empirical factors into FT={F1, F2, …, FT}. Similar to the data-rich DSGE model (4)-(6), (7)-(9) is a linear Gaussian state-space model, and we are interested in joint inference about model parameters Γ and latent factors FT. Unlike in the data-rich DSGE model, though, we no longer have deep structural parameters determining the behavior of matrices in transition equation (8).

We sidestep the problem of a proper dimension of factor space by assuming that dim (Ft) =N =6, the number of non-redundant model states in the data-rich DSGE model. In contrast, the dynamic factor literature has devoted considerable attention to developing the objective criteria that would determine the proper number of static factors by trading the fit against complexity (Bai and Ng, 2002) and of dynamic factors (e.g., Bai and Ng 2007, Hallin and Liska 2007, Amengual and Watson 2007, Stock and Watson 2005) in DFMs similar to the one above. However, our choice is indirectly supported by the work of Stock and Watson (2005) and Jungbacker and Koopman (2008), who, roughly based on these criteria, find seven dynamic and seven static factors driving a similar panel of macro and financial data.

A principal components analysis of the data set XT reveals that our choice for the number of factors is not an unreasonable one. As Table C1 demonstrates, the first 6 principal components account for about 75 percent of the variation in the data. The scree plot in Figure C1 shows a very flat slope of the ordered eigenvalues curve when going from the 6th to 7th eigenvalue. Putting in the 7th principal component would add 4.4 percent to the total variance of the data explained, a fairly marginal improvement over the already high cumulative proportion of 75 percent.

Another problem associated with the dynamic factor model (7)-(9) is that the scales and signs of factors Ft and of factor loadings Λ are not separately identified. Regarding scales, take any invertible N×N matrix P and notice that the transformed model is observationally equivalent to the original one:

$\begin{array}{ccc}{X}_{t}=\underset{\stackrel{˜}{\mathbf{\Lambda }}}{\underbrace{{\mathbf{\Lambda }\mathbf{P}}^{-1}}}\underset{{\stackrel{˜}{F}}_{t}}{\underbrace{\mathbf{P}{F}_{t}}}+{e}_{t}& \phantom{\rule{7.0em}{0ex}}& \left(10\right)\end{array}$

Regarding signs, for the moment think of (7)-(9) as a model with only one factor. Then multiply by -1 the transition equation (8), as well as the factor loading and the factor itself in measurement equation (7). We obtain the new model, yet it is observationally equivalent to the original.

We follow the factor literature (e.g. Geweke and Zhu 1996; Jungbacker and Koopman 2008) and make the following normalization assumptions to tell factors apart from factor loadings: (i) set Q = IN to fix the scale of factors; (ii) require one loading in Λ to be positive for each factor (sign restrictions); and (iii) normalize some factor loadings in Λ to pin down specific factor rotation.

Denote by Λ1 the upper N×N block of Λ so that $\mathbf{\Lambda }=\left[{\mathbf{\Lambda }}_{\mathbf{1}}^{\prime };{\mathbf{\Lambda }}_{\mathbf{2}}^{\prime }{\right]}^{\prime }$. One way to implement (ii) and (iii) would be to assume that Λ1 is lower triangular (i.e., λij = 0 for j >i, i = 1, 2, …, N−1) with strictly positive diagonal ${\lambda }_{ii}>0,i=\overline{1,N}$ (see Harvey 1989, p.451). However, our data set in estimation, to be described later in the Section IV, will consist of core and non-core macro and financial series. Furthermore, within the core series we will have four blocks of variables: real output, inflation, the nominal interest rate and the inverse velocity of money, respectively; each block contains several measures of the same concept. For example, the output block comprises real GDP, total industrial production and industrial production in the manufacturing sector; the inflation block includes GDP deflator inflation, CPI inflation and personal consumption expenditures inflation. For this reason, we choose another alternative to implement normalizations (ii) and (iii) – the block-diagonal scheme that to some degree exploits the group structure of the core series in data Xt :

where 1s stand for non-zero elements in Λ.

We acknowledge that our block-diagonal scheme imposes some overidentifying restrictions on factor loadings beyond those minimally necessary. However, scheme (12) can also be interpreted as a special case of the appealing dynamic hierarchical factor model of Moench, Ng, and Potter (2008), which – on top of aggregate common factors – introduces intermediate block factors and makes use of the block structure of the data.

Now, to estimate the model (7)-(9) under normalizing assumptions (i)-(iii), we again apply the Bayesian MCMC methods as in the estimation of the data-rich DSGE model (Kryshko 2011, Section III.A). We construct a Gibbs sampler that iterates on a complete set of known conditional posterior densities to generate draws from the joint posterior distribution p(Γ, FT|XT) of model parameters Γ={Λ, Ψ, R, G} and latent factors FT:

$\begin{array}{ccc}p\left({F}^{T}/{\Gamma };{X}^{T}\right)\propto p\left({F}^{T}/{\Gamma }\right)p\left({X}^{T}/{\Gamma },{F}^{T}\right)& \phantom{\rule{7.0em}{0ex}}& \left(13\right)\end{array}$
$\begin{array}{ccc}p\left({\Gamma }/{F}^{T};{X}^{T}\right)\propto p\left({\Gamma }\right)p\left({F}^{T}/{\Gamma }\right)p\left({X}^{T}/{\Gamma },{F}^{T}\right)& \phantom{\rule{7.0em}{0ex}}& \left(14\right)\end{array}$

The main steps of the Gibbs sampler are:

1. Specify initial values Γ(0) and FT, (0).

2. Repeat for g=1, 2, …, nsim

• 2.1.Generate latent factors FT,(g) from p(FT(g−1);XT) using the Carter-Kohn (1994) forward-backward algorithm;

• 2.2.Generate state-space parameters Γ(g) from p(Γ|FT(g); XT) by drawing from a complete set of known conditional densities.

3. Return ${\left\{{{\Gamma }}^{\left(g\right)},{F}^{T,\left(g\right)}\right\}}_{g=1}^{{n}_{\mathit{\text{sim}}}}$

Compared to the MCMC algorithm for the data-rich DSGE model, this Gibbs sampler is easier and it differs in two key respects: (i) we no longer have the complicated Metropolis step, since there are no deep structural parameters θ coming from the economic model; and (ii) inside Γ, we have to draw matrix G from the transition equation of factors (in the data-rich DSGE model it was pinned down by numerical solution of a DSGE model given structural parameters θ).

To draw the latent factors FT from p(FT|Γ; XT), we use the familiar Carter-Kohn (1994) machinery. First, we apply the Kalman filter to the linear Gaussian state-space system (7)-(9) to generate filtered latent factors ${\stackrel{^}{F}}_{t/t},t=\overline{1,T}$. Then, starting from ${\stackrel{^}{F}}_{T/T}$, we roll back in time along the Kalman smoother recursions and generate FT = {F1, F2, …, FT} by recursively sampling from a sequence of conditional Gaussian distributions.

To sample from the conditional posterior p(Γ|FT; XT), we notice the following: with diagonality of matrices Ψ and R and conditional on factors FT, (7) and (9) are a set of standard multivariate linear regressions with AR(1) errors and Gaussian innovations $\left(k=\overline{1,j\right)}$ :

Hence, under the conjugate prior p(Λ, Ψ, R) we can apply the insight of Chib and Greenberg (1994) to derive the conditional posteriors [R|(Λ, Ψ); G, FT, XT], [Λ|(R, Ψ); G, FT, XT], [Ψ|(Λ, R); G, FT, XT] and to sample accordingly.

What remains to be drawn is the transition matrix G. Given factors FT, the conditional posterior p(G|(Λ, R, Ψ); FT, XT) can be derived from a VAR(1) in (8):

We assume the so-called Minnesota prior (Doan, Litterman and Sims, 1984; the specific version comes from Lubik and Schorfheide, 2005) on transition matrix G and truncate it to the region consistent with the stationarity of (16). We implement our prior by a set of dummy observations that tilt the VAR to a collection of univariate random walks (details are in Appendix A).

To estimate the empirical DFM, in the actual implementation of the Gibbs sampler we have applied the Jungbacker-Koopman (2008) computational speed-up presented in Kryshko (2011), Section III.B (and already utilized to improve the speed of computations in the data-rich DSGE model’s estimation). We find that the “improved” estimation of the empirical DFM runs 10.5 times faster than the no-speedup estimation, a magnitude consistent with the CPU gains reported by Jungbacker and Koopman (2008) for a DFM of a similar size in their study.

## IV. Data

To estimate the dynamic factor model and the data-rich DSGE model, we employ the large panel of U.S. quarterly macroeconomic and financial time series compiled by Stock and Watson (2008). The panel covers 1959:Q1 – 2006:Q4, however, our sample in this paper is restricted only to 1984:Q1 – 2005:Q4 so as to avoid dealing with the issue of the Great Moderation5 and to concentrate on a period with a relatively stable monetary policy regime.

Our data set is identical to the one employed in Kryshko (2011) and consists of 12 core series that either measure specific DSGE model concepts or are used in the DFM normalization scheme (12), and 77 non-core informational series that load on all DSGE states (DFM factors) and may contain useful information about the aggregate state of the economy. The core series include three measures of real output (real GDP, the index of total industrial production and the index of industrial production: manufacturing), three measures of price inflation (GDP deflator inflation, personal consumption expenditure (PCE) deflator inflation, and CPI inflation), three indicators of the nominal interest rates (the federal funds rate, the 3-month T-bill rate and the yield on AAA-rated corporate bonds), and three series measuring the inverse velocity of money (IVM based on the M1 aggregate and the M2 aggregate and IVM based on the adjusted monetary base). The 77 non-core series include the measures of real activity, labor market variables, housing indicators, prices and wages, financial variables (interest rate spreads, exchange rate depreciations, credit stocks, stock returns) and, together with appropriate transformations to eliminate trends, are described in Appendix B. To save space, we refer the reader to Kryshko (2011), Section IV that describes in detail the construction of all data indicators included in our data set.

Because measurement equations (5) and (7) are modeled without intercepts, we estimate a dynamic factor model and a data-rich DSGE model on a demeaned data set. Also, in line with standard practice in the factor literature, we standardize each time series so that its sample variance is equal to unity (however, we do not scale the core series when estimating the data-rich DSGE model).

## V. Empirical Analysis

The next step in our analysis is to take a dynamic factor model and a data-rich DSGE model to the data using the MCMC algorithms described above and to present the empirical results. We begin by discussing the choice of the prior distributions of dynamic factor model’s parameters. Second, we analyze the estimated empirical factors and the estimates of the DSGE model state variables and explore how well they are able to capture the co-movements in the data. Third, we compare the spaces spanned by the latent empirical factors and by the data-rich DSGE model state variables. Finally, we use the proximity of the factor spaces to propagate the monetary policy and technology innovations in an otherwise non-structural dynamic factor model and obtain the predictions from both models for the core and non-core macro and financial series of interest.

### A. Priors and Posteriors

Since we estimate the DFM (7)-(9) and the data-rich DSGE model (4)-(6) using Bayesian techniques, we have to provide prior distributions for both models’ parameters.

Let us first turn to a dynamic factor model. Let Λk and Rkk be the factor loadings and a variance of the measurement error innovation for the kth measurement equation, k =1..J. Similarly to Boivin and Giannoni (2006) and Kose, Otrok and Whiteman (2008), we assume a joint Normal-InverseGamma prior distribution for (Λk, Rkk) so that Rkk ~ IG2(s0, v0) with location parameter s0 =0.001 and degrees of freedom v0 = 3, and the prior mean of factor loadings is centered around the vector of zeros ${\mathbf{\Lambda }}_{k}|{R}_{kk}~N\left({\mathbf{\Lambda }}_{k,0},{R}_{kk}{\mathbf{M}}_{0}^{-1}\right)$ with Λk, 0 = 0 and M0 = IN. The prior for the kth measurement equation’s autocorrelation Ψkk, all k, is N(0, 1). We are making it perfectly tight, however, because there could be data series with stochastic trends we seek to capture with potentially highly persistent dynamic factors and not with highly persistent measurement errors. This implies that all measurement errors are iid mean-zero normal random variables. Finally, as explained in Section III.B, for the factor transition matrix G, we implement a version of a Minnesota prior (Lubik and Schorfheide, 2005) and tilt the transition equation (8) to a collection of univariate random walks.6

In our data-rich DSGE model, we have two groups of parameters: state-space model parameters comprising matrices Λ, Ψ and R, and deep structural parameters θ of an underlying DSGE model. The prior for the state-space matrices is elicited differently for the core and the non-core data indicators contained in Xt. Regarding the non-core measurement equations, the prior for (Λk, Rkk) and for Ψkk is identical to the one assumed in DFM above. The prior distribution for the factor loadings in the core measurement equations follows the same scheme as elaborated in Kryshko (2011), Section V.A. Our choice of prior distribution for the deep structural parameters of a DSGE model is exactly identical to the one presented in Section V.A of Kryshko (2011).

We use the Gibbs sampler presented above in Section III.B and the Gibbs sampler with Metropolis step outlined in Kryshko (2011), Section III.A to estimate our empirical dynamic factor model and the data-rich DSGE model, respectively. The only parameters of direct interest are the deep structural parameters θ of an underlying DSGE model, and we have already discussed them extensively in Kryshko (2011). We do not discuss the posterior estimates of DFM parameters here either, since we are more interested in comparing factor spaces spanned by the estimated latent factors and by the DSGE model states. However, all the parameter estimates are collected in the technical appendix to this paper, which is available upon request.

### B. Empirical Factors and Estimated DSGE Model States

Our empirical analysis proceeds by plotting the estimated empirical factors extracted by a dynamic factor model and the estimated DSGE state variables from our data-rich DSGE model.

Figure C2 depicts the posterior means and 90 percent credible intervals of the estimated data-rich DSGE model states. These include three endogenous variables (model inflation ${\stackrel{^}{\pi }}_{t}$, the nominal interest rate ${\stackrel{^}{R}}_{t}$ and real household consumption ${\stackrel{^}{X}}_{t}$) and three structural AR(1) shocks (government spending gt, money demand χt and neutral technology Zt). In Kryshko (2011) we have noted four observations. First, all three structural disturbances exhibit large swings and prolonged deviations from zero capturing the persistent low frequency movements in the data. Second, the estimated data-rich DSGE model states are much smoother than their counterparts in the regular DSGE model, because in the data-rich context, the model states are the common components of a large panel of data, and they have to capture well not only a few core macro series (as is the case in the regular DSGE model), but also very many non-core informational series. The third observation is that the money demand shock χt appeared to be very different in the data-rich versus the regular DSGE model estimation, owing primarily to the fact that in the data-rich DSGE model it helped explain housing variables, consumer credit and non-GDP measures of output at the cost of the poorer fit for the IVM_M2S. The fourth observation was a counterfactual behavior of government spending shock and real consumption during recessions: the former tended to fall and the latter to rise when times are bad.

We proceed by discussing the latent empirical factors extracted by our DFM from the same rich data set. Figure C3 plots the posterior means and 90 percent credible intervals of the estimated factors. First, note that unlike the DSGE model states, these factors have in general no economic interpretation. This is less true of factors F3-F6, because of the assumed normalization scheme (12). Second, while factors 3 and 5 indeed look much like the data on real output and nominal interest rate, factors 4 and 6 – despite the normalization – do not. This shows that the exclusion normalizations favoring a certain ex-ante meaning of a particular factor are not a sufficient condition to guarantee this meaning ex-post after estimation. The third observation is that the credible intervals for F1 and F2 – the latent factors common to all macro and financial series in the panel – are not uniformly wide or narrow, as is more or less the case for factors F3-F6. During several years prior to 1990-91 recession, the 90 percent credible bands for factor F1 expand, and then quickly shrink after recession is over. The same pattern is observed for factor F2 for several years preceding the 2001 recession. One interpretation of this finding could be that the volatility of these two factors is not constant over time and follows a regime-switching dynamics over the business cycle. Clearly, to have a stronger case, one might like to estimate a DFM on the full postwar sample of available U.S. data.

### C. How Well Factors Trace Data

Let us now turn to the question of how well the factors and the DSGE states are able to trace the actual data. A priori we should expect that the unrestricted dynamic factor model will do a better job on that dimension than the data-rich DSGE model whose cross-equation restrictions might be misspecified and the factor loadings in which might be unduly restricted. And that’s indeed what we find and what can be concluded from inspecting Table C2 and Table C3 which present the (posterior mean of) fraction of the unconditional variance of the data series captured by the empirical factors and by the DSGE model states. On average, the data-rich DSGE model states “explain” about 75 percent of variance for the core macro series and 72 percent of variance for the non-core. The latent empirical factors extracted by a DFM are able to account for 95 and 94 percent of the variance for the core and non-core series, respectively. So overall, the empirical factors capture more than the DSGE states.

More specifically, within the core series it is the measures of inflation and of inverse money velocities that are traced relatively more poorly than the real output and nominal interest rates in both models. The same picture is observed in the non-core block of series: price and wage inflation measures and the financial variables in both models tend to have a higher fraction of unconditional variance due to measurement errors. In the data-rich DSGE model, the state variables capture about 15 to 25 percent of the variance in exchange rate depreciations and stock returns, but about 65 to 85 percent of the variance of interest rate spreads and credit stocks. This is not surprising given that our theoretical model does not have New Open-Economy Macroeconomics mechanisms (e.g., Lubik and Schorfheide, 2005 or Adolfson, Laseén, Linde, Villani, 2005, 2008) and does not feature financial intermediation (e.g., Bernanke, Gertler, Gilchrist, 1999). In the dynamic factor model, these percentages are much higher: the latent factors explain about 97-98 percent of the variance of the interest spreads and credit stocks, about 65-82 percent of the variability in exchange rate depreciations and 80-82 percent of stock returns (Table C4). This suggests that our DSGE model is potentially misspecified along this “financial” dimension.

### D. Comparing Factor Spaces

Up to this point, we have done two things: (i) we have estimated the empirical latent factors in a dynamic factor model and the DSGE states in a data-rich DSGE model; and (ii) we have established that both factors and DSGE states are able to explain a significant portion of the co-movement in the rich panel of U.S. macro and financial series. From Figure C2 and Figure C3 we have learned that the states and the factors look quite different; therefore now we come to our central question: can the empirical factors and the estimated DSGE model state variables span the same factor space? Or, in other words, can we predict the true estimated DFM latent factors from the DSGE model states with a fair amount of accuracy?

Let denote the posterior means of the empirical factors and of the data-rich DSGE model state variables. For each latent factor ${F}_{i,t}^{\left(\mathit{\text{pm}}\right)}$, we estimate, by Ordinary Least Squares, the following simple linear regression:

$\begin{array}{ccc}{F}_{i,t}^{\left(\mathit{\text{pm}}\right)}={\beta }_{0,i}+{\mathbf{\beta }}_{l,i}^{\prime }{S}_{t}^{\left(\mathit{\text{pm}}\right)}+{u}_{i,t}& \phantom{\rule{7.0em}{0ex}}& \left(17\right)\end{array}$

with mean zero and homoscedastic error term ui,t. We report the R2 s for the collection of linear predictive regressions (17) in Table C7. Denoting the OLS estimates by ${\stackrel{^}{\mathbf{\beta }}}_{0}=\left[{\beta }_{0,1},\dots ,{\beta }_{0,N}{\right]}^{\prime }$ and by ${\stackrel{^}{\mathbf{\beta }}}_{1}=\left[{\mathbf{\beta }}_{1,1},\dots ,{\mathbf{\beta }}_{1,N}{\right]}^{\prime }$, we then construct the predicted empirical factors ${\stackrel{^}{F}}_{t}^{\left(\mathit{\text{pm}}\right)}$ :

$\begin{array}{ccc}{\stackrel{^}{F}}_{t}^{\left(\mathit{\text{pm}}\right)}={\mathbf{\beta }}_{0}+{\stackrel{^}{\mathbf{\beta }}}_{1}{S}_{t}^{\left(\mathit{\text{pm}}\right)}& \phantom{\rule{7.0em}{0ex}}& \left(18\right)\end{array}$

The Figure C4 overlays true estimated DFM factors ${F}_{t}^{\left(\mathit{\text{pm}}\right)}$ versus those predicted by the DSGE states ${\stackrel{^}{F}}_{t}^{\left(\mathit{\text{pm}}\right)}$.

From both Table C7 and Figure C4 we can clearly conclude that the DSGE states predict empirical factors really well and therefore the factor spaces spanned by the DSGE model state variables and by the DFM latent factors are very closely aligned. What are the implications of this important finding? First, this implies that a DSGE model indeed captures the essential sources of co-movement in the large panel of data as a sort of a core and that the differences in fit between a data-rich DSGE model and a DFM are potentially due to restricted factor loadings in the former. Second, this also implies a greater degree of comfort about propagation of structural shocks to a wide array of macro and financial series – which is the essence of many policy experiments. Third, the proximity of factor spaces facilitates economic interpretation of a dynamic factor model, as the empirical factors are now isomorphic – through the link (18) – to the DSGE model state variables with clear economic meaning.

### E. Propagation of Monetary Policy and Technology Innovations

The final – and the most appealing – implication of the factor spaces proximity in the two models is that it allows us to map the DSGE model state variables into DFM empirical factors every period and therefore propagate any structural shocks from the DSGE model in an otherwise completely non-structural dynamic factor model to obtain predictions for a broad range of macro series of interest. Suppose Λdfmdsge and Λdfm denote the posterior means of factor loadings in the data-rich DSGE model (4)-(6) and in the empirical DFM (7)(9), respectively. Then, for any structural shock εi, t we can generate two sets of impulse responses of a large panel of data Xt:

$\begin{array}{ccc}{\left(\frac{\partial {X}_{t+h}}{\partial {\epsilon }_{i,t}}\right)}_{\mathit{\text{dfm}}-\mathit{\text{dsge}}}={\mathbf{\Lambda }}^{\mathit{\text{dfm}}-\mathit{\text{dsge}}}×\frac{\partial {S}_{t+h}}{\partial {\epsilon }_{i,t}}& \phantom{\rule{7.0em}{0ex}}& \left(19\right)\end{array}$
$\begin{array}{ccc}{\left(\frac{\partial {X}_{t+h}}{\partial {\epsilon }_{i,t}}\right)}_{\mathit{\text{dfm}}}={\mathbf{\Lambda }}^{\mathit{\text{dfm}}}×\frac{\partial {F}_{t+h}}{\partial {\epsilon }_{i,t}}={\mathbf{\Lambda }}^{\mathit{\text{dfm}}}\left[{\stackrel{^}{\mathbf{\beta }}}_{1}\frac{\partial {S}_{t+h}}{\partial {\epsilon }_{i,t}}\right],& \phantom{\rule{7.0em}{0ex}}& \left(20\right)\end{array}$

where ∂St+h/∂εi, t is computed from the transition equation of the data-rich DSGE model for every horizon h =0, 1, 2, … and where we have used the link between St and Ft determined by (18).

In what follows we focus on propagating monetary policy (εR, t) and technology (εZ, t) innovations in both the data-rich DSGE and the dynamic factor model to generate predictions for the core and non-core macro series. The corresponding impulse response functions (IRFs) are presented in Figure C5, Figure C6, Figure C7 and Figure C8. It is natural to compare our results to findings in two strands of the literature: Factor Augmented Vector Autoregression (FAVAR) literature (e.g. Bernanke, Boivin, Eliasz, 2005; Stock and Watson, 2005) and the regular DSGE literature (e.g. Christiano, Eichenbaum, Evans, 2005; Smets and Wouters, 2003, 2007; DSSW 2007; Aruoba and Schorfheide, 2009; Adolfson, Laseén, Linde, and Villani, 2008). In FAVAR studies, we are able to obtain predictions for a rich panel of U.S. data similar to ours, but only of the monetary policy innovations. In the regular DSGE literature, one can propagate any structural shocks including monetary policy and technology innovations, but to a limited number of core macro variables (e.g., real GDP, consumption, investment, inflation, the interest rate, the wage rate and hours worked in Smets and Wouters, 2007). The framework that we propose in this paper delivers on both fronts: we are able to compute the responses of the core and non-core variables to both monetary policy and technology shocks. Moreover, we will have two sets of responses: from the data-rich DSGE model, which might be misspecified, and from the dynamic factor model that is primarily data-driven and fits better.

At least from the perspective of monetary policy innovations, we tend to favor the predictions obtained from the empirical dynamic factor model (20). It turns out (we provide evidence below) that the two models’ predictions for the non-core variables are fairly close. The responses of the core series, though, seem more plausible in the empirical DFM case, since, for example, channeling the shock through the DFM helps eliminate the puzzling behavior of price inflation observed in the data-rich DSGE model context that we have documented in Kryshko (2011), Section V.E.

One general observation from comparing IRFs should be emphasized from the very beginning. The responses of core variables like real GDP, real consumption and investment, and inflation in regular DGSE studies are often hump-shaped, matching well the empirical findings from identified VARs. Our IRFs do not have many humps, because the underlying theoretical DSGE model, as presented in Kryshko (2011), Section II.B, abstracts from, say, habit in consumption or variable capital utilization – mechanisms that help get the humps in those often more elaborate models. This, however, can be fixed by replacing the present DSGE model with a more elaborate one.

Let us turn first to the effects of monetary policy innovation, which are summarized in Figure C5 and Figure C6. A contractionary monetary policy shock corresponds to 0.75 percent (or 75 basis points) increase in the federal funds rate. As the nominal policy rate rises and the opportunity costs of holding money for households increase, we observe a strong liquidity effect associated with falling real money balances. Also, high interest rates make the saving motive and buying more bonds temporarily a more attractive option. This raises households’ marginal utility of consumption and discourages current spending in favor of the future consumption. Because the household faces investment adjustment costs and cannot adjust investment quickly, and government spending in the model is exogenous, the lower consumption leads to a fall in aggregate demand. The firms respond to lower demand in part by contracting real output and in part by reducing the optimal price. Hence, the aggregate price level falls, but not as much given nominal rigidities in the intermediate goods-producing sector.

Why do the monopolistically competitive firms respond to falling demand in part by charging a lower price? The short answer is that because they are able to cut their marginal costs. On the one hand, higher interest rates inhibit investment and the return on capital is falling. On the other hand, firms may now economize on real wages. The market for labor is perfectly competitive, since we assume no wage rigidities. This implies that the real wage is equal to the marginal product of labor, but also that it is equal to the household’s marginal rate of substitution between consumption and leisure, as in Kryshko (2011), Equation (78). Since the disutility of labor in our model is fixed, and the marginal utility of consumption is higher, the household accepts lower real wage and the firms are able to pass on their losses in revenues to households by reducing their own wage bills.

Now given lower marginal costs, the New Keynesian Phillips curve suggests we should observe falling aggregate prices and negative rates of inflation (in terms of a deviation from the steady-state inflation). That’s what we see in the second column of Figure C5. Notice that channeling the monetary policy shock through the pure dynamic factor model helps correct the so-called “price puzzle7 for the data-rich-DSGE-model-implied responses of PCE deflator inflation and CPI inflation. Interestingly, a positive response of CPI inflation to a monetary policy contraction is also documented in Stock and Watson (2005), despite the fact that they use a data-rich Factor Augmented VAR. It has been argued (e.g., Bernanke, Boivin and Eliasz, 2005) that the rich information set helps eliminate this sort of anomaly.

As can be seen from the first column of Figure C5, the response of industrial production (IP) to the monetary policy tightening seems counterfactual compared to FAVAR findings (we have documented this finding in Kryshko, 2011 too). First, this may have something to do with the inherent inertia of IP in responding to monetary policy. It continues to be driven by excessive optimism from the previous phase of the business cycle and it takes time to adjust to new conditions. But once IP falls below the trend, it remains subdued for a long time. Second, this may have something to do with the way the monetary policy shock is identified in the FAVAR literature. By construction, in a FAVAR the industrial production is contained in the list of “slow moving” variables, and the identification of the monetary policy shock is achieved by postulating that it does not affect slow variables contemporaneously. Regarding the responses of real GDP, we document that the data-rich DSGE and DFM models disagree about the magnitude of the contraction. The DFM-implied response is almost negligible implying that the costs of disinflation are very small (which is hard to believe), whereas the data-rich-DSGE-model-implied response is about minus 0.5 percent – hump shape aside, a value in the ballpark of findings in the regular DSGE literature.

If we look at the effects of the monetary policy tightening on non-core macro and financial variables (Figure C6), they complete the picture for the core series with details. Real activity measures, such as real consumption of durables, real residential investment and housing starts, broadly decline. Prices go down as well; in particular, we observe negative rates of commodity price inflation and investment deflator inflation. The measures of employment fall (e.g., employment in the services sector) indicating tensions in the labor market, while unemployment gains momentum with a lag before eventually returning to normal. The interest rate spreads (for instance, the 6-month over the 3-month Treasury bill rate) widen considerably, reflecting tighter money market conditions and increased liquidity risks and credit risks. Consumer credit contracts, in part due to lower demand from borrowers facing higher interest rates and in part owing to the reduced availability of funds. The dollar appreciates, reflecting intensified capital inflows lured by higher returns in the domestic financial market. As a result, both export and import price indices fall, thereby translating – according to the magnitudes in Figure C6 – into a deterioration of the U.S. terms of trade.

Broadly speaking, the reported results are qualitatively very similar to the FAVAR findings of Bernanke, Boivin and Eliasz (2005) and Stock and Watson (2005). Except for the humps, they also accord well with the monetary policy effects on the core variables documented in the regular DSGE literature. On top of that, the responses of the non-core variables seem to provide a reasonable and consistent picture of monetary tightening as well.

We plot the effects of a positive technology innovation in Figure C7 (core series) and Figure C8 (non-core series). Following the positive TFP shock, real output broadly increases (although there is a disagreement between the DFM and the data-rich DSGE model as to the response of real GDP), as our economy becomes more productive and the firms find it optimal to produce more. New demand comes primarily from higher capital investment, reflecting much better future return on capital, and also from additional household consumption fueled by greater income. The higher output on the supply side plus improved efficiency implies a downward pressure on prices. Through the lenses of the New Keynesian Phillips curve, the current period inflation is positively related to expected future inflation and to current marginal costs. A positive technology shock has raised production efficiency and reduced the current marginal costs (the elevated real wage resulting from increased labor demand was not enough to prevent that). However, because technology innovation is very persistent, the firms expect future marginal costs and thus future inflation to be lower as well. This anticipation effect, coupled with currently low marginal costs, leads to prices falling now, as is evident from column 2 of the Figure C7.

The increase in real output above steady state and the fall of inflation below target level, under the estimated Taylor (1993) rule, requires the Fed to move the policy rate in opposite directions. The fact that the Fed actually lowers the policy rate means that the falling prices effect dominates, with other interest rates following the course of the federal funds rate (column 3, Figure C7). Declining interest rates boost real output even more, which in turn raises further the return on capital. As the positive impact of technological innovation dissipates, this higher return, through the future marginal costs channel, fuels inflationary expectations that ultimately translate into contemporaneous upward price pressures. The Fed reacts by increasing the policy rate, which explains the observed hump in the interest rate IRF. Given temporarily lower interest rates, households choose to hold, with some lag, relatively higher real money balances (from column 4, Figure C7, this applies more to M1S and the monetary base, and less to the M2S aggregate that comprises a hefty portion of interest-bearing time deposits). A part of the growing money demand comes endogenously from the elevated level of economic activity.

These results – both in terms of the magnitudes and shapes of responses – align fairly closely with findings in the regular DSGE literature (e.g., Smets and Wouters, 2007; Aruoba, Schorfheide, 2009; and DSSW 2007).

The responses of the non-core macroeconomic series (Figure C8) appear to enrich the story for core variables with additional insights. Following a positive technology innovation, the subcomponents of real GDP (real consumption of durables, real residential investment) or the components of industrial production (e.g., production of business equipment) generally expand (although there is weaker agreement between the predictions of the DFM and the data-rich DSGE model). Measures of employment (e.g., employment in the services sector) increase. However, this stands in contrast to the results in Smets and Wouters (2003) and Adolfson, Laseén, Linde, Villani (2005), who find in European data that employment actually falls after a positive stationary TFP shock. As marginal costs fall, commodity price inflation (P_COM) and investment deflator inflation (PInv_GDP) follow the overall downward price pressures trend. The interest rate spreads (SFYGM6) shrink, in part reflecting the lower level of perceived risks, while credit conditions ease, leading to growth in business loans. Despite the interest rates being below average for a prolonged period of time, the dollar appreciates, but by less than after the monetary tightening. Finally, the real wage (RComp_Hour) increases, while average hours worked (Hours_AVG) decline. The rise in the real wage and the initial fall in hours worked are in line with evidence documented by Smets and Wouters (2007). However, the subsequent dynamics of hours are quite different: in Smets and Wouters the hours turn significantly positive after about two years. Here they stay below steady state for much longer. This may have something to do with a greater amount of persistence in the technology process in our model.

## VI. Conclusions

In this paper, we have compared a data-rich DSGE model with a standard New Keynesian core to an empirical dynamic factor model by estimating both on a rich panel of U.S. macroeconomic and financial indicators compiled by Stock and Watson (2008). We have established that the spaces spanned by the empirical factors and by the data-rich DSGE model states are very closely aligned.

This key finding has several important implications. First, it implies that a DSGE model indeed captures the essential sources of co-movement in the data and that the differences in fit between a data-rich DSGE model and a DFM are potentially due to restricted factor loadings in the former. Second, it also implies a greater degree of comfort about the propagation of structural shocks to a wide array of macro and financial series. Third, the proximity of factor spaces facilitated economic interpretation of a dynamic factor model, since the empirical factors have become isomorphic to the DSGE model state variables with clear economic meaning.

Most important, the proximity of factor spaces in the two models has allowed us to propagate the monetary policy and technology innovations in an otherwise completely non-structural dynamic factor model to obtain predictions for many more series than just a handful of traditional macro variables, including measures of real activity, price indices, labor market indicators, interest rate spreads, money and credit stocks, and exchange rates. The responses of these non-core variables therefore provide a more complete and comprehensive picture of the effects of monetary policy and technology shocks and may serve as a check on the empirical plausibility of a DSGE model.

Data-Rich DSGE and Dynamic Factor Models
Author: Mr. Maxym Kryshko
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