feds · December 3, 2020

Latent Variables Analysis in Structural Models: A New Decomposition of the Kalman Smoother

Abstract

This paper advocates chaining the decomposition of shocks into contributions from forecast errors to the shock decomposition of the latent vector to better understand model inference about latent variables. Such a double decomposition allows us to gauge the inuence of data on latent variables, like the data decomposition. However, by taking into account the transmission mechanisms of each type of shock, we can highlight the economic structure underlying the relationship between the data and the latent variables. We demonstrate the usefulness of this approach by detailing the role of observable variables in estimating the output gap in two models. Accessible materials (.zip)

Finance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C. Latent Variables Analysis in Structural Models: A New Decomposition of the Kalman Smoother Hess Chung, Cristina Fuentes-Albero, Matthias Paustian, and Damjan Pfajfar 2020-100 Please cite this paper as: Chung, Hess, Cristina Fuentes-Albero, Matthias Paustian, and Damjan Pfajfar (2020). “Latent Variables Analysis in Structural Models: A New Decomposition of the Kalman Smoother,” Finance and Economics Discussion Series 2020-100. Washington: Board of Governors of the Federal Reserve System, https://doi.org/10.17016/FEDS.2020.100. NOTE: Staff working papers in the Finance and Economics Discussion Series (FEDS) are preliminary materials circulated to stimulate discussion and critical comment. The analysis and conclusions set forth are those of the authors and do not indicate concurrence by other members of the research staff or the Board of Governors. References in publications to the Finance and Economics Discussion Series (other than acknowledgement) should be cleared with the author(s) to protect the tentative character of these papers.

Latent Variables Analysis in Structural Models: A New ∗ Decomposition of the Kalman Smoother Hess Chung†,‖ Cristina Fuentes-Albero†,‡ Federal Reserve Board Federal Reserve Board Matthias Paustian†,§ Damjan Pfajfar†,¶ Federal Reserve Board Federal Reserve Board December 2, 2020 Abstract This paper advocates chaining the decomposition of shocks into contributions from forecast errors to the shock decomposition of the latent vector to better understand model inference about latent variables. Such a double decomposition allows us to gauge the influence of data on latent variables,likethedatadecomposition. However,bytakingintoaccountthetransmissionmechanismsofeachtypeofshock, wecanhighlighttheeconomicstructureunderlyingtherelationship between the data and the latent variables. We demonstrate the usefulness of this approach by detailing the role of observable variables in estimating the output gap in two models. JEL Classification: C18, C32, C52 Keywords: Kalmansmoother,latentvariables,shockdecomposition,datadecomposition,double decomposition ∗We thank Mariano Kulish and seminar participants at the ECB, the Banque de France, the 4th Workshop on Empirical Macroeconomics of the Macroeconomics, Policy and Econometrics Research group of Ghent University, the 8th Conference on Growth and Business Cycles in Theory and Practice at the University of Manchester, RCEF ConferenceinRimini,the12thDynareConference,theXIXAnnualInflationTargetingSeminaroftheBancoCentral do Brasil, and 44th Simposio of the Spanish Economic Association. The views expressed in this paper are solely the responsibility of the authors and should not be interpreted as reflecting the views of the Board of Governors of the Federal Reserve System. ∗Address: Board of Governors of the Federal Reserve System, 20th and Constitution Ave NW, Washington, DC 20551, U.S.A. †E-mail: Hess.T.Chung@frb.gov. ‡E-mail: Cristina.Fuentes-Albero@frb.gov. §E-mail: Matthias.O.Paustian@frb.gov. ¶E-mail: Damjan.Pfajfar@frb.gov. 1

1. Introduction Researchers estimating models using Bayesian methods or maximum likelihood have employed several different ways of decomposing the latent state vector path to better understand and communicate model behavior. Shock decompositions, for example, in which researchers decompose the estimated latent vector into contributions of the estimated structural shocks, have long been standardintheliterature. IncaseswherethemodelislinearandtheshocksareGaussian, moreover, the latent variables are linear functions of the data. Thus, following Sander (2013) and Andrle (2013), researchers can decompose the estimated latent vector into contributions of the observed variables using the so-called data decomposition. In this paper, we propose linking the data decomposition of the vector of structural shocks and the shock decomposition of the latent vector in what we label as the double decomposition. This way of analyzing the latent variable path can be particularly illuminating when the focus is on inference regarding highly theoretical constructs, such as the natural rate of interest or the“flex-price”output gap, where the relation between the observable variables and the latent variable is not intuitive and depends heavily on the model’s theoretical structure. We first show how the double decomposition can be used to study the behavior of a latent variable—the output gap—in a simple model by Ireland (2011). After illustrating the mechanics of the double decomposition, we focus on the output gap in a more complex model—a version of the model presented in Del Negro, Giannoni and Schorfheide (2015). Using this model, where inference is more complex, we illustrate the value of the double decomposition for practitioners. A data decomposition suggests that the estimated path of the output gap in this model shows the influence of a large number of observables. At the same time, a shock decomposition shows that the outputgapisprimarilydrivenbypermanenttechnologyshocksandshockstoidiosyncraticfirmrisk (effectively, shocks to the spread between the return on capital and the return on risk-free assets). Thus, it is illuminating to start with the relationship between the observables that carry a large signal about the output gap, according to the data decomposition, and the model’s estimates of the shocks which largely drive the output gap, according to the shock decomposition. This process is formalized in the double decomposition. We also illustrate the value of the double decomposition for interpreting the model’s reaction to incoming news, examining in detail the effects of news flow between the end of 2013 and the beginning of 2014. The incoming news over this period featured large forecast errors in several highly influential observables and correspondingly large movements in important latent variables, such as the output gap. In such circumstances, the double decomposition can provide a clear economic interpretation of the effects of news on these variables. Formally, let us consider a model with a linear-Gaussian structure. Let Y be the vector of t observablevariables,X bethelatentvector,andη bethevectorofstructuralshocks. Theestimated t t vectors of latent variables and structural shocks depend on current, past, and future forecast errors. We decompose inference about endogenous latent variables in two steps. First, the linear-Gaussian structure of the model implies that structural shocks at some arbitrary time t, η , and the matrix t 2

of forecast errors for the observable vector over the sample, ν, are jointly normally distributed.1 As a result, we can write E(η |ν) = K(t)ν for some matrix K(t), which is a function of time, but, t importantly, independent of the matrix of one-step ahead forecast errors. The data decomposition exploits this fact to decompose estimated shocks into contributions of each observable at every date in the sample, holding the other observables constant. Second, linearity also allows us to write the estimated value of a latent variable i at time t as E (cid:0) Xi|Y (cid:1) = [ (cid:80)∞ Φ(θ)sK(t−s)]ν, where t s=0 Y = [Y Y ... Y ](cid:48), which facilitates tracing the influence of news on the estimates of the 1 2 T latent variables operating through the transmission mechanism of each type of structural shock separately. Both steps of our decomposition have been known in the literature. The shock decomposition is used to understand model dynamics but does not provide a linkage between data and latent variables. The data decomposition provides such a linkage but without a transparent relation to the underlying causal mechanisms which, as we emphasize, explain why the observables and the latent variables are linked. Because the double decomposition traces the influence of the data on latent variables through estimates of the structural shocks and their subsequent propagation, this causal narrative is highlighted. Moreover, as the double decomposition emphasizes inference about structural shocks, we focus on the role of forecast errors, rather than the level of the data, as is morecommonintheliterature. Thisaccountingisbetterforourpurposesbecausecontemporaneous forecasterrorsaretightlylinkedtomodelestimatesofcontemporaneousstructuralshocks,butmuch less so to past shocks. By contrast, the influence of a quarter of data in levels corresponds to large structural shocks both in that quarter and also offsetting shocks in the surrounding periods, which complicates the relationship between the observables and the latent variables of interest.2 The rest of the paper is organized as follows. Section 2 overviews the analysis of the latent vector in structural models. Section 3 then illustrates how the double decomposition works in a simple 3-equation New Keynesian model. We then turn to a more complex case, a medium-scale DSGE model, in Section 4. Section 5 concludes. 2. Latent Variable Analysis in Structural Models Traditionally, empirical macroeconomic researchers focus on the so-called shock decomposition, which decomposes the latent variables into contributions of structural shocks (see Appendix A.2). Recently, exploiting the linearity of the model, Andrle (2013) and Sander (2013) propose the data decomposition,whichtracestheindependenteffectofeachobservableontheestimatedlatentvector (see Appendix A.3). While the shock decomposition does not provide a link between the data and the latent vector, the data decomposition, which does describe such a relationship, does not provide a causal narrative underlying the relation. We address these shortcomings by proposing a new decomposition that does a data decomposition shock by shock. Given the nature of our 1The matrix of forecast errors is defined as ν = [ν ν ... ν ](cid:48), where ν is the vector of forecast errors at 1 2 T t time t. 2Adatadecompositioninlevelsdoeshavetheadvantagethatthecontributionofagivenquarterofdatatendsto be more closely aligned in time with the corresponding effect on latent variables, while as we discuss below, this is not always the case in a data decomposition in forecast errors. 3

decomposition, we label it as the double decomposition. All these decompositions can be applied to any linear model with Gaussian shocks. Let us consider the state space representation of a linear model X = Φ(θ)X +R(θ)η (1) t t−1 t Y = Z(θ)X +ε (2) t t t where X is an m × 1 vector of state variables, η is a p × 1, random vector with innovations t t or structural shocks such that η ∼ N (0,Q), Y is an n × 1 vector of observable variables and t t ε ∼ N (0,H) is an n×1 vector of measurement errors. We assume that the structural shocks and t measurement errors are uncorrelated. In this framework, estimates of the latent vector are only updated if there is a forecast error. Hence, we proceed to derive the historical decompositions in terms of forecast errors instead of in terms of the data matrix.3 Let ν be the one-step ahead forecast error t (cid:16) (cid:17) ν t = Y t −Y(cid:98)t|t−1 = Z X t −X(cid:98)t|t−1 +ε t (3) We first derive the historical data decomposition of the estimated vector of structural shocks. As shown in Appendix A.1, the smoothed estimate of the vector of structural shocks can be written as a weighted sum of current and future one-step ahead forecast errors, ν , t+h (cid:104) η = QR(cid:48) Z(cid:48)F−1 ν +L(cid:48) Z(cid:48)F−1 ν (cid:98)t|T t|t−1 t t|t−1 t+1|t t+1 (cid:105) +...+L(cid:48) L(cid:48) ...L(cid:48) Z(cid:48)F−1 ν t|t−1 t+1|t T|T−1 T|T−1 T (cid:104) (cid:105) with L = Φ I −P Z(cid:48)F−1 Z , F = ZP Z(cid:48) +H, and P = ΦP Φ(cid:48) +RQR(cid:48). t|t−1 t|t−1 t|t−1 t|t−1 t|t−1 t|t−1 t−1|t−1 Let us define the observation weights for the forward recursions as (cid:40) I τ = t Ω(cid:63) = t,τ L(cid:48) L(cid:48) ...L(cid:48) τ > t t|t−1 t+1|t τ−1|τ−2 Then, the data decomposition of the smoothed vector of structural shocks is given by T (cid:88) η = QR(cid:48) Ω(cid:63) Z(cid:48)F−1 ν (4) (cid:98)t|T t|τ τ|τ−1 τ τ=t n T (cid:88)(cid:88) = QR(cid:48) Ω(cid:63) Z(cid:48)F−1 νj (5) t|τ τ|τ−1 τ j=1 τ=t where νj is an n− dimensional vector with all rows equal to zero but the jth row, which is set to τ the actual forecast error. 3Appendix A.4 provides the double decomposition in terms of the data matrix. 4

The second step in the inference about the latent vector is its shock decomposition. Under the assumption of stationary dynamics, the autoregressive model for the state vector in equation (1) has an infinite moving average representation t−1 (cid:88) X(cid:98)t|T = ΦtX(cid:98)0|T + ΦτRη (cid:98)t−τ|T (6) τ=0 where X(cid:98)0|T is the estimated vector of initial conditions, given data through time T. Let us define ηi as a vector containing the estimated path for the ith structural shock conditional on the (cid:98)t−τ|T remaining(p−1)structuralshocksbeingzeroforallτ. Therefore, theestimatedvectorofstructural shocks is given by η = (cid:80)p ηi and the estimated realization of the state vector by (cid:98)t−τ|T i=1(cid:98)t−τ|T p t−1 (cid:88)(cid:88) X(cid:98)t|T = ΦtX(cid:98)0|T + ΦτRη (cid:98)t i −τ|T (7) i=1τ=0 where (cid:80)t−1 ΦτRηi states the contribution of the ith structural shock to the estimated vector τ=0 (cid:98)t−τ|T of latent variables at time t. Thedoubledecompositionchainsthedatadecomposition, whichrelatesobservablestoestimates of shocks, to the shock decomposition, which relates shocks to latent variables. Using equation (4), we define the contribution of news about the jth observable to the latent vector through the ith shock as follows t−1 (cid:40) T (cid:41) X(cid:98) i,j = ΦtX(cid:98) j + (cid:88) ΦτR QR(cid:48) (cid:88) Ω(cid:63) Z(cid:48)F−1 νj (8) t|T 0|T t|τ(cid:48) τ(cid:48)|τ(cid:48)−1 τ(cid:48) τ=0 τ(cid:48)=t i where X(cid:98) j is the contribution of news about the jth observable to the vector of initial conditions 0|T for the latent variables. Hence, the double decomposition allows us to gauge the influence of data on the path of latent variables, like the data decomposition. However, by taking into account the transmission mechanisms of each type of shock, we can highlight the economic structure underlying the relationship between the data and the latent variables. The overall contribution of news about the jth observable on the latent vector is given by p p t−1 (cid:40) T (cid:41) X(cid:98) j = (cid:88) X(cid:98) i,j = ΦtX(cid:98) j + (cid:88)(cid:88) ΦτR QR(cid:48) (cid:88) Ω(cid:63) Z(cid:48)F−1 νj (9) t|T t|T 0|T t|τ(cid:48) τ(cid:48)|τ(cid:48)−1 τ(cid:48) i=1 i=1τ=0 τ(cid:48)=t i and the estimated vector of latent variables is p n n p t−1 (cid:40) T (cid:41) X(cid:98)t|T = (cid:88)(cid:88) X(cid:98) t i | , T j = (cid:88) ΦtX(cid:98) 0 j |T + (cid:88)(cid:88) ΦτR QR(cid:48) (cid:88) Ω(cid:63) t|τ(cid:48) Z(cid:48)F τ − (cid:48)| 1 τ(cid:48)−1 ν τ j (cid:48) (10) i=1 j=1 j=1 i=1τ=0 τ(cid:48)=t i 3. A Simple Model Demonstration In this section, we provide a practitioner’s view of the logic behind the double decomposition using a small-scale model. By using the double decomposition, we are able to better understand the 5

economicmechanismsinthismodelbywhichsomevariables,suchasinflation,arehighlyinformative about the output gap, while others, such as GDP growth, are less so. We illustrate how the double decomposition works using an estimated version of the canonical New Keynesian model presented in Ireland (2011). Formally, the behavioral core of the model consists of four equations governing the shadow value of wealth (λ ), the log-level of detrended t output (Y ), inflation (π ) and the federal funds rate (R ): t t t (z¯−βγ)(z¯−γ)λ = γz¯Y −(z¯2+βγ2)Y +βγz¯E Y +(z¯−βγρ )(z¯−γ)a −γz¯z (11) t t−1 t t t+1 a t t λ = R +E (λ −π ) (12) t t t t+1 t+1 (1+βα)π = απ +βE π −ψλ +ψa +e (13) t t−1 t t+1 t t t R−R = ρ π +ρ (Y −Y +z )+(cid:15)R (14) t−1 π t g t t−1 t t The model features four exogenous processes driven by structural shocks: the growth rate of total factor productivity (z ), a shock to household preferences (a ), a price mark-up shock (e ), and a t t t monetary policy shock ((cid:15)R). Ireland (2011) estimates the parameters of the model by maximum t likelihood, using as observable variables GDP growth, GDP price inflation, and the 3-month T-bill rate from 1983:Q1 to 2009:Q4. In our analysis, we use the parameter estimates reported in Ireland (2011). To illustrate the double decomposition, let us suppose we are interested in understanding how the model infers movements in the output gap from the path of observable variables.4 In line with our previous discussion, our analysis proceeds sequentially in an ”onion”structure. First, we show how the main features of the data decomposition for structural shocks follow from some properties of the impulse response functions for observables. The impulse response functions for the output gap then allow us to understand how the model translates news about observable variables into estimates of the output gap. Finally, we describe the double decomposition of the output gap over the estimation sample. Figure 1 reports impulse responses following one-standard-deviation innovations to the exogenous processes. Certain familiar qualitative differences in the response of the observable variables are visible. First, there are two shocks that drive inflation and output growth in the same direction on impact –the preference and monetary policy shocks–, which are, in turn, distinguishable from each other by the reaction of the T-bill rate. Second, the remaining two shocks – the mark-up and technology shocks– move inflation in opposite directions and can be again distinguished from each other by the comovement of GDP growth with the T-bill rate: while following a mark-up shock, GDP growth falls and the T-bill rate rises, both growth and the T-bill rate rise following a technology shock. With these qualitative differences, inference about structural shocks given forecast errors in the observable variables is easy to explain intuitively. First, the fact that the systematic component of monetary policy depends only on observable variables means that the monetary policy shock is 4Inthisexample,theoutputgapisdefinedrelativetothelevelofoutputthatwouldhaveprevailedintheabsence of nominal rigidities. 6

Figure 1. Impulse Response Functions Following Structural Shocks 0.2 0.1 0 0 10 pref shocks htworG PDG 0.02 0 -0.02 -0.04 -0.06 -0.08 0 10 markup shocks htworG PDG 0.4 0.2 0 10 tech shocks htworG PDG 0 -0.05 -0.1 -0.15 0 10 mp shocks htworG PDG 0.1 0.05 0 0 10 pref shocks noitalfnI 0.1 0.05 0 0 10 markup shocks noitalfnI -0.02 -0.04 -0.06 -0.08 0 10 tech shocks noitalfnI -0.02 -0.04 -0.06 -0.08 -0.1 0 10 mp shocks noitalfnI 0.14 0.12 0.1 0 10 pref shocks llibT 0.03 0.02 0.01 0 10 markup shocks llibT 15 10 5 0 10 tech shocks llibT 10-3 0.06 0.04 0.02 0 10 mp shocks llibT 0.2 0.1 0 0 10 pref shocks paG tuptuO -0.02 -0.04 -0.06 -0.08 -0.1 0 10 markup shocks paG tuptuO -0.05 -0.1 -0.15 0 10 tech shocks paG tuptuO -0.05 -0.1 -0.15 -0.2 0 10 mp shocks paG tuptuO Note: Impulseresponsesarecalculatedfollowingone-standard-deviationshocksandarereportedinpercentagepoints. measurableandinferenceaboutitspathistrivial. Accordingly, weonlyfocusoninferenceaboutthe other three structural shocks. The model’s inference about these shocks is most easily understood by considering the model’s response to forecast error vectors with the property that the implied monetary policy shock is zero. This restriction implies that the forecast errors in the T-bill do not have any information beyond what is contained in the forecast errors for GDP growth and inflation. Therefore, we only discuss the information content of these variables. In this context, we consider two cases: (i) a case featuring a one-standard-deviation forecast error in GDP growth and a zero error in inflation and (ii) a case featuring a zero forecast error in GDP growth and a one-standarddeviation error in inflation. Given the linear-Gaussian structure of the model, inference for all of the other cases can be represented as a linear combination of cases (i) and (ii). Figure 2 reports the results of these exercises. Let us first suppose that the model is presented with a one-standard-deviation forecast error in GDP growth, a zero forecast error in inflation, and a forecast error in the T-bill consistent with the monetary policy rule. From Figure 1, we know that the dominant drivers of GDP growth are the 7

preference and technology shocks. Therefore, the model accounts for the positive forecast error in GDP growth using one-standard-deviation shocks to preferences and technology, as shown in the left column in Figure 2.5 Moreover, as shown in the second row of Figure 1, the effect on inflation of this combination of shocks is relatively small. Thus, such a combination also mostly accounts for the zero inflation forecast error, with only a slight movement in the markup shock. Let us now consider the alternative case: the model is presented with a one-standard-deviation forecast error in inflation, a zero forecast error in GDP growth, and a forecast error in the T-bill consistent with the monetary policy rule. This case is more complicated because, as shown in Figure 1, all one-standard-deviation structural shocks have similar effects on impact for inflation. The right column of Figure 2 shows that the model can account for this case by sizable movements in the three non-monetary-policy shocks. In particular, the model infers a 0.8 standard deviation increaseinthepreferenceshock,which,byitself,increasesGDPgrowth. Themodeloffsetstheeffect ofthepreferenceshockonGDPgrowthwithcontractionarysupplyshocks: apositivemarkupshock and a negative shock to technology, so that the overall effect on GDP growth is zero. These shocks also help explain the positive forecast error in inflation. Combining the information contained Figure 1 and Figure 2, we obtain Figure 3, which shows how the model’s estimate of the output gap responds to the two vectors of forecast errors described above. This two-way decomposition – effects of news about observable variables on structural shock contributions – is essentially what we formalize in the double decomposition. Let us consider the left column in Figure 3, which reports the case of a one-standard-deviation forecast error in GDP growth. In this case, individually, the associated positive preference and technology shocks have sizable effects on the model estimate of the output gap. However, taken jointly, these two shocks are offsetting, leaving a much smaller imprint on the output gap than on GDP growth. By contrast, in the case of a positive forecast error only in inflation, all the associated shocks (positive preference and markup shocks and a negative technology shock) move the output gap in the same direction. Therefore, we conclude that the estimated path for the output gap is quite sensitive to inflation news, but not GDP growth. Wecannowusethesefindingstounderstandthemodel’sestimatesoftheoutputgapinasample from 1983:Q1 to 2009:Q4. The resulting double decomposition is shown in Figure 4. The upper-left panel displays the data decomposition of the estimated path for the output gap. The remaining panels show the data decomposition of the contribution of each structural shock to the estimated path for the output gap. For example, the middle panel in the first row of Figure 4 shows the estimated path for the output gap if only technology shocks were active and all other shocks were zero. Consistentwithourprevious discussion, the data decomposition of the output gapreportedin the upper-left panel of Figure 4 shows that the estimated path of the output gap is almost entirely driven by inflation news. The shocks associated with news about GDP growth and the T-bill have largely offsetting effects on the output gap. 5These effects are calculated for an observation 54 quarters from both the beginning and end of the sample. This corresponds to an observation in the middle of the 1983:Q1-2009:Q4 (i.e., arbitrary period in the middle between 1996:Q2 and 1996:Q3) sample used to estimate the model. 8

Figure 2. Effect of News at Time 0 on Estimated Paths for Structural Shocks pref shocks pref shocks 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0 -20 -10 0 10 20 -20 -10 0 10 20 GDP Growth News Inflation News markup shocks markup shocks 0.05 0.4 0.2 0 0 -0.05 -0.2 -0.4 -20 -10 0 10 20 -20 -10 0 10 20 GDP Growth News Inflation News tech shocks tech shocks 0 0.6 -0.1 0.4 -0.2 0.2 0 -0.3 -20 -10 0 10 20 -20 -10 0 10 20 GDP Growth News Inflation News Note: Shock responses are reported in standard-deviation units. Responses represent the effects of news at time 0 in the given variable on the estimated path of the shocks (labelled in bold) 20 quarters before and after that time. Given the simplicity of the model, this outcome could have been anticipated by noting that, compared to DSGE models estimated on data after the Great Recession, the Phillips curve is relatively sensitive to movements in real activity. Correspondingly, in the Ireland (2011) model, the role of markup shocks is smaller than in those DSGE models. In this context, inflation is naturally highly informative regarding the output gap. However, even in this simple case, running through the formal logic of the double decomposition helps to narrow the key dynamic features of the model underlying this outcome, i.e., that inflation is relatively responsive to both technology and preference shocks. 9

Figure3. Effect of News at Time 0 on Estimated Contributions of Structural Shocks to the Output Gap pref shocks pref shocks 0.15 0.2 0.1 0.15 0.1 0.05 0.05 0 0 -20 -10 0 10 20 -20 -10 0 10 20 GDP Growth News Inflation News markup shocks markup shocks 10-3 0 0.04 -2 0.02 -4 -6 0 -20 -10 0 10 20 -20 -10 0 10 20 GDP Growth News Inflation News tech shocks tech shocks -0.02 0.06 -0.04 -0.06 0.04 -0.08 -0.1 0.02 -0.12 -20 -10 0 10 20 -20 -10 0 10 20 GDP Growth News Inflation News Note: Responses represent the effects in percentage point of news at time 0 in the given variable to the contribution of a specified shock (labelled in bold) to the path of the output gap 20 quarters before and after that time. 4. A Large-Model Demonstration 4.1. The Model To further demonstrate the utility of the double decomposition, we now turn to a larger model, more like the workhorse models used in academia and central banks, featuring a much larger set of observable variables and more complex transmission dynamics. In particular, we use an estimated version of the model originally developed by Del Negro, Giannoni and Schorfheide (2015) (DGS). Broadly speaking, the DGS model extends the baseline Smets and Wouters (2007) model with financial frictions on the firm side and a time-varying inflation target in the monetary policy rule. Our version of the model differs from the one described by DGS in two respects. First, our Taylortype interest rate feedback rule responds to the change in real GDP, rather than to the flex-price 10

Figure 4. Decomposition of Structural Shock Contributions to the Output Gap in the Ireland 2011 Model Note: Upperrightpanel: datadecompositionoftheoutputgapinpercentagepoints. Otherpanels: datadecomposition of the contribution of the given shock to the output gap in percentage points. Sample: 1983q1-2009q4. outputgap. Thisassumptionimpliesthatthesettingofthefederalfundsrateisnot,byconstruction, a strong signal about the output gap. The monetary policy rule in our model is given by R = ρr(R )+(1−ρr)[φ (π −π∗)+φ (y −y )]+(cid:15)r t t−1 π t t y t t−1 t wheretheparameterρr reflectsthedegreeofinterestratesmoothing,π∗ isthetime-varyinginflation t target, and (cid:15)r represents a monetary policy shock. The time-varying inflation target follows t π∗ = ρ π∗+σ (cid:15) (15) t π∗ t π∗ π∗,t 11

where 0 < ρ < 1 and (cid:15) is an iid shock. The time-varying inflation target accounts for the π∗ π∗,t low frequency movements in longer-horizon inflation expectations in our estimation sample and potentially any effects of unconventional monetary policy, such as forward guidance, on agents’ expectations. The inflation target is informed by matching model-based average expected inflation over the next 10 years with the Survey of Professional Forecasters median 10 year-ahead inflation expectations. Second, following Barsky, Justiniano and Melosi (2014), we use several indicators of inflation and wages, incorporating a factor structure in the corresponding measurement equations. For price inflation, we consider the log difference of the GDP deflator, core CPI price inflation, and core PCE price inflation. For wage inflation, we use average hourly earnings and the employment cost index.6 Themodelincludeselevenshocks: shockstotransientTFP,permanentTFP,investmentspecific technology, the aggregate risk premium, firm-specific risk, marginal bankruptcy costs, net worth, price markups, wage markups, exogeneous spending, the inflation target, and the monetary policy rule. For the estimation, we use data on GDP growth, consumption growth, investment growth, the federal funds rate, the spread between corporate bonds and Treasuries of comparable maturity, net worth, corporate debt, long-run inflation expectations, wage inflation measured by average hourly earnings, wage inflation measured by the employment cost index, GDP deflator price inflation, core CPI price inflation, and core PCE price inflation. The estimation sample is 1987:Q1-2015:Q4.7 4.2. Comparison with Standard Decompositions In this subsection, in Figures 5-6, we present results from the standard shock and data decompositions for the output gap, defined as the log-difference between the level of actual output and the levelofoutputinacounterfactualeconomy(the”flex-priceeconomy”)withflexiblepricesandwages and without price and wage markup shocks. The output gap was mostly positive in the second half of the 1990s and in the 2000s. In 2009, the output gap decreased rapidly to about -9 percent where it remained for a couple of years before gradually recovering. The shock decomposition suggests that financial shocks, notably the firm-specific risk shock, drive most of the business cycle variation togetherwithpermanentproductivityshocks. SincetheGreatRecession, investmentspecificshocks and monetary policy shocks play an important and very persistent role preventing the output gap from declining further. 6To illustrate the procedure, let’s consider the case of wages. The vector of L quarterly wage growth data series w is given by: wp(cid:48) = (cid:2) wp ,...,wp (cid:3)(cid:48) (16) t 1,t Lw,t Themeasurementequationlinkingthemodelconceptofwageinflationwiththejth elementofthisvector,wp ,isas j,t follows wp =(w¯+c )+Λ wp +u (17) j,t j j m,t j,t wherew¯ istheestimatedsteadystatelevelandc capturesthedifferencesinestimatedmeansamongdifferentseries. j The idiosyncratic disturbances, u , which follow an AR(1) structure, and loading factors, Λ , are series specific. j,t j 7Our results may be specific to the sample period we use and the fact that we abstract from the issues that arise due to the ZLB. We leave the estimation that would explicitly take into account the ZLB for future research. 12

Figure 5. Shock Decomposition of the Output Gap Note: Bars represent the contribution of shocks to the output gap in percentage points. Sample: 1987q1-2015q4. Figure 6. Data Decomposition of the Output Gap Note: Bars represent the contribution of forecast errors in the given observable to the estimated path of the output gap in percentage points. Sample: 1987q1-2015q4. Looking at the data decomposition in Figure 6, we see that a number of observables influence the estimate of the output gap. In particular, the federal funds rate, real consumption growth, and the corporate bond spread seem to have the largest influence on the estimate of the output gap. 13

Some features of these results resonate with other analyses of the output gap over our sample, but others are more counter-intuitive. For example, the model estimates that the output gap was far above zero in the second half of the 1990s, peaking at 9.6 percent at the end of 2000. Noting the low level of unemployment, coupled with strong investment growth and subdued inflation over that period, contemporary accounts from these years emphasized the role of productivity shocks and this model agrees with that assessment, finding that almost all of the rise in the output gap in the late 1990s is accounted for by shocks to permanent productivity. However, as shown by the data decomposition of the output gap, news about investment growth, hours and inflation played a very small role in the model’s estimates of the output gap at this time; rather, as in most other times,contributionsfromconsumptiongrowth,thefederalfundsrateandthecorporatebondspread dominate. In the next section, we show that a consideration of the double decomposition helps to explainwhytheseobservablesaresoinfluentialandwhytheysostronglysignalshockstopermanent productivity. 4.3. Decomposing Estimates of the Output Gap Inthissection,weemploythedoubledecompositiontoexaminethemodel’sinferencesaboutthe output gap. To provide an intuitive explanation of the double decomposition, we proceed as we did in Section 3: we first examine the relationship between forecast errors in key observable variables and the model’s estimates of structural shocks (described in Table 1) and then use impulse response functions to relate these effects on shock estimates to the model’s estimate of the path of the output gap (described in Tables 2 and 3). We conclude that the output gap path is chiefly informed by two classes of observable variables: spendingcomponentsandfinancialvariables. Inthefirstclassarespendingcomponents–consumptiongrowthmostimportantly–whichproviderelativelycleansignalsaboutpermanentproductivity and firm-level risk shocks, the main contributors to the variance of the output gap. In the second class,themodel’sfinancialobservablevariablesalsosignallargemovementsinproductivityandfirm risk. These data are influential in the longer term, after the effects of transient offsetting shocks also associated with these observables have faded. To streamline our discussion, we therefore focus on this set of observable variables. Table1displaystheeffectofaone-standard-deviationforecasterrorintheseselectedobservables on the model’s estimate of contemporaneously occurring shocks.8 The first two columns of Table 1 describetheeffectsofone-standard-deviationforecasterrorsinconsumptionandinvestmentgrowth, respectively, holding all other forecast errors at zero. Because GDP, consumption and investment growth are all observed, a forecast error in consumption or investment alone, holding the other two fixed, requires an exactly offsetting movement in exogenous spending, equal to a little more than one standard deviation (-1.16) in the case of a consumption forecast error and -0.74 in the case of an investment forecast error. The crowding-in of consumption and investment occasioned 8For expositional ease, in this first step, we abstract from finite-sample effects and present results for inference at anobservationveryfarfromthestartandendofthesample(200quarters). Asintheprevioussection,inthistable, the magnitudes of the shocks are measured in standard-deviation units. 14

by the decline in exogenous spending helps the model account for some of the higher-than-expected privatespending, butthemodelmustalsorelyonotherstructuralshocks. Forconsumptionforecast errors, the offsetting shock is chiefly a positive shock to permanent productivity, which boosts both consumption and investment growth, and a much smaller increase in firm-level risk, which offsets the impetus to investment that the first two shocks would otherwise deliver. Table 1: Effects of News on Shock Estimates Positive news on Shock Cons Inv Hours FFR Spread Net Worth Prod (temp) -0.11 0.02 -1.45 -0.06 -0.05 -0.02 Prod (perm) 1.01 -0.12 0.02 0.50 0.46 0.21 Firm-level Risk 0.14 -1.11 -0.01 -0.39 -0.35 -0.48 Risk premium -0.15 0.82 0.00 0.28 1.50 0.47 Exog. spending -1.16 -0.74 0.12 -0.01 -0.01 -0.00 Inv. efficiency -0.03 0.52 0.02 -0.21 -0.27 -1.12 Mon Pol -0.02 -0.02 -0.04 0.94 -0.02 -0.01 Price markup 0.06 0.02 -0.11 0.20 0.05 0.06 Wage markup -0.23 -0.03 -0.11 -0.08 -0.14 0.00 Net worth -0.01 -0.32 -0.02 -0.15 -0.33 -0.18 Infl Target 0.11 0.19 -0.23 0.04 0.09 0.25 Mar. bankruptcy 0.08 0.24 0.02 0.15 0.56 0.17 Note: Table entries are the effect of a one-standard deviation forecast error in the given observables on the model’s estimate of the contemporaneous shock. Shocks are measured in standard-deviation units. The negative shock to exogenous demand and the positive shock to firm-risk both imply lower output gaps, while the positive productivity shock would imply a higher output gap. However, the exogenous spending shock has a relatively weak impact on the output gap, while only a small firm-level risk shock is necessary to account for the zero forecast error in investment. Accordingly, we find that positive news on consumption is associated with higher output gaps, mostly because such news signals the arrival of a permanent boost to productivity. This outcome is quantified in the first column of Table 2, which shows that, all else equal, a one-standard-deviation forecast error in consumption raises the estimated contribution of permanent productivity shocks to the output gap by 0.44 percentage points, with small effects on contributions from other shocks. Turning now to the case of a forecast error in investment growth (the second column of Table 1), wefindthat,again,theimplieddeclineinexogenousspendingisnotsufficienttoexplaintheforecast error. In fact, the forecast error is accounted for by a negative shock to firm-level risk, accompanied by a positive shock to the marginal efficiency of investment (further boosting investment and, to a lesserextent,consumption). Byitself,suchaconfigurationofshockswouldentailasizeablenegative forecast error in spreads and positive forecast errors in consumption, which, by assumption, did not occur. Themodelthusinfersthattheaggregateriskpremiummusthaveexperiencedalargepositive shock, while permanent productivity suffered a modest downward shock; both of these shocks boost 15

Table 2: Effect of News on Shock Contributions to the Contemporaneous Output Gap Positive news on Contribution to gap from Cons Inv Hours FFR Spread Net Worth Prod (temp) 0.03 -0.01 0.34 0.02 0.01 0.01 Prod (perm) 0.44 -0.09 -0.02 0.23 0.18 0.08 Firm-level Risk -0.04 0.29 -0.01 0.13 0.08 0.13 Risk premium 0.02 -0.08 0.01 -0.04 -0.22 -0.05 Exog. spending -0.04 -0.02 0.00 0.00 0.01 0.00 Inv. efficiency -0.01 -0.00 0.00 -0.02 -0.04 -0.06 Mon Pol 0.00 0.01 -0.00 -0.25 0.01 -0.00 Price markup -0.00 0.00 -0.00 -0.00 0.00 0.00 Wage markup 0.01 -0.00 0.04 -0.01 0.01 -0.00 Net worth -0.00 0.00 0.00 -0.02 -0.05 -0.01 Infl Target 0.01 0.01 -0.01 -0.00 -0.00 0.02 Mar. bankruptcy 0.00 -0.01 -0.00 -0.00 -0.02 -0.01 Total 0.43 0.11 0.36 0.04 -0.02 0.11 Note: Table entries are the effect of a one-standard deviation forecast error in the given observables on the model’s estimate of the contemporaneous output gap, measured in percentage points. spreads and lower consumption by much more than investment. Putting the pieces together, from the second column of Table 2, we see that investment forecast errors are associated with an increase in the contribution of the firm-level risk shocks to the output gap of 0.29 percentage points, about half of which is offset negative contributions from the aggregate risk premium and permanent technology shocks. Overall, then, news about investment growth emerges as less informative about the contemporaneous output gap than news about consumption, essentially because consumption growth errors are cleanly associated with a permanent productivity shock, which has strong effects on the output gap, while investment growth is associated with a mix of offsetting shocks. A data decomposition would find that the second most influential observable for the model’s estimate of the contemporaneous output gap is aggregate hours: as shown by the third column of Table 2, a one-standard-deviation forecast error in hours would raise the estimate of the output gap atthatdateby0.36percentagepoints,similartotheeffectofaone-standard-deviationforecasterror in consumption growth. In the case of aggregate hours, however, the forecast error is accounted chieflybyanegativeshocktotemporaryproductivity. Becausetheproductivityshockistemporary, its effects on spending growth are largely muted by a rise in labor inputs – a desirable feature when accounting for a positive forecast error in hours, but zero forecast errors in spending observable variables. The association of hours with temporary productivity shocks accounts for nearly all of the effect of hours forecasts errors on estimates of the output gap. Finally, wenowconsidertheeffectofnewsonfinancialobservablevariables, severalofwhichcan be influential on the model’s estimates of the output gap. Overall, these observable variables are associated with sizeable movements in permanent productivity and firm-level risk. However, they 16

are not, taken separately, very informative about the contemporaneous output gap, as the absence of consistent forecast errors in spending implies the presence of strong offsetting shocks. As we discuss below, these offsetting shocks are relatively transient and financial observables emerge as informative about the output gap several quarters in the future. We start with news about the federal funds rate, shown in the third column of Table 1. In this model,systematicmonetarypolicydependsonlyonobservables,theunobservedinflationtargetand monetary policy shocks. In fact, the forecast error in the federal funds rate alone is accounted for mostly by the monetary policy shock. To account for the zero forecast errors in spending, the model offsets the contractionary implications of this monetary policy shock with several expansionary shocks, including positive shocks to permanent productivity and negative shocks to firm-level risk and the aggregate risk premium. From Table 2, we see that the positive contributions to the output gap from the permanent productivity shock and firm-level risk dominate the offsetting negative contributions. On balance, a positive forecast error in the federal funds rate is associated with a modest increase in the model’s estimate of the contemporaneous output gap. Thenextfinancialindicatorthatwewillconsider,thecorporatebondspread,isheavilydrivenby shocks to the aggregate risk premium that appear directly as wedges in the equation characterizing the laws of motion of this variable. As with forecast errors in the federal funds rate, however, the positive shock to the aggregate premium must be offset by other expansionary shocks, in order to be consistent with zero forecast errors in spending observables and, as we have also seen before, the model infers positive shocks to permanent productivity and negative shocks to firm-level risk for this reason. Again consulting Table 2, we see that these contributions are roughly offsetting and the total effect of forecast errors in the spread is quite small. The last observable that we will consider is net worth. A forecast error in net worth is largely explained by a one-standard-deviation decline in the marginal efficiency of investment, which raises the price of installed capital because investment flows are a less effective source of new capital. However, as in the previous cases, this shock alone has counterfactual implications for consumption and investment. Therefore, they must be accompanied by offsetting expansionary shocks, namely positive permanent productivity shocks and negative firm-level risk shocks. These expansionary shocks, as we have previously discussed, have much larger effects on the output gap than the other main contributors in this case and so a one-standard-deviation forecast error in net worth is associated with an 0.11 percentage point increase in the estimated output gap. So far, we have focused on the link between forecast error and estimates of contemporaneous structural shocks and output gaps. However, as we showed earlier, the contribution of the data to latentvariablescanbethoughtofascontributionstoestimatedshocksatvarioushorizons, weighted by the impulse responses of the latent variable of interest at those horizons. In the current model, thepersistenceofoutputgapeffectsdiffersquiteabitacrossdifferentshocks. Inparticular,asshown in the Appendix, permanent shocks to productivity and risk premium shocks induce movements in the output gap that are substantially more persistent than other major sources of variation; indeed, the output gap impulse response functions for those shocks are hump-shaped, peaking more than two years after the shock at roughly double their value at impact. Hence, news about variables that 17

Table 3: Effect of News on Shock Contributions to the Output Gap 8 quarters in the Future Positive news on Contribution to gap from Cons Inv Hours FFR Spread Net Worth Prod (temp) 0.03 -0.01 0.40 0.02 0.01 0.01 Prod (perm) 1.29 -0.21 -0.00 0.65 0.58 0.26 Firm-level Risk -0.18 1.38 0.02 0.55 0.47 0.62 Risk premium 0.03 -0.15 0.00 -0.06 -0.30 -0.09 Exog. spending -0.04 -0.03 0.00 0.00 0.00 0.00 Inv. efficiency -0.01 0.09 0.01 -0.06 -0.09 -0.27 Mon Pol 0.00 0.00 0.01 -0.34 0.00 -0.00 Price markup -0.01 -0.00 0.02 -0.01 -0.00 -0.00 Wage markup 0.06 0.01 0.05 0.01 0.03 -0.00 Net worth -0.00 0.14 -0.00 0.00 0.02 0.04 Infl Target 0.03 0.04 -0.05 0.01 0.01 0.06 Mar. bankruptcy -0.00 -0.01 -0.00 -0.01 -0.02 -0.01 Total 1.18 1.28 0.47 0.77 0.72 0.62 Note: Table entries are the effect of a one-standard deviation forecast error in the given observables on the model’s estimate of the output gap eight quarters later, measured in percentage points. are linked to large movements in permanent productivity or risk premium shocks tend to be highly influential about future output gaps, even this influence is somewhat obscured in the near-term by signals about offsetting shocks. For this reason, we now examine the dynamic contributions of news about observables to estimates of the future path of the output gap. To get a sense of how the persistence of the shocks affects the contribution of the different observables, in Table 3, we examine the effect of the important observables on the contributions of the various shocks to the output gap 8 quarters after impact.9 From this table, it is immediately apparent that, at this horizon, this set of observables is still highly influential regarding the output gap, sometimes even more so than on impact as in the case of the federal funds rate, and that almost all of that influence is mediated through the persistent effects of permanent productivity and firm-level risk shocks. To summarize our findings thus far, then, two classes of observables tend to contribute most strongly to the estimated path of the output gap in this model. The first class of observables are spending components, i.e., consumption and investment growth, whose forecast errors have sparse and large loadings on permanent productivity and firm-level risk shocks, because these shocks account for a large share of those observables’ variance and have much larger effects on them than on the others variables in the information set. For the second class of observables (financial observables) by contrast, productivity and firm-risk shocks are not necessarily the dominant source 9A more comprehensive set of coefficients from the data decomposition for these observables is available in the Appendix. We note that, because of the effect of shocks on the gap is persistent, estimates of the time-T output gap can depend on forecast errors after time T. However, as shown by Figure A.4, the influence of future forecast errors is much smaller than the past and contemporaneous forecast errors, justifying our focus on the latter. 18

ofvariance. Rather,forthesevariables,thedominantsourceofvariance(monetarypolicyshocks,for example, in the case of the funds rate) would, by itself, cause large movements in spending. Thus, a non-zero forecasterrorinsuch avariablealonesignalsoffsetting movements inthedominantdrivers of consumption and investment, muting the impact on the contemporaneous output gap. Figure 7. Dynamic contributions to two-sided estimates for the flex-price output gap over the estimation sample Note: Upperleftpanel: datadecompositionoftheoutputgapinpercentagepoints. Otherpanels: datadecomposition of the contribution of the given shock to the output gap in percentage points. Sample: 1987q1-2015q4. Having laboriously worked our way through this underlying logic, we now confirm that these patterns are, indeed, reflected in the relation between observables and shock contributions in the estimation sample. By doing so, we also show that the double decomposition can shed light on why observables are particularly important for the data decomposition. Our results are presented in Figure 7. Consistent with our theoretical priors, we see that forecast errors in investment growth and net worth make significant contributions to the path of the output gap through their role in signalling the level of firm-specific risk, while consumption growth is also influential, but chiefly as an indicator of permanent productivity. The federal funds rate emerges as a fourth important 19

observable and, in this case, conveys information about the level of both permanent productivity and firm-level risk. Tracing the impact of observables on the output gap through their effects on estimates of the shocksclarifieswhythesevariablesemergeasinfluential,andwhytheobservablesemphasizedbythe contemporaryaccountsarenotsoinfluential. Forexample,inthismodel,forecasterrorsinhoursare mostly driven by temporary productivity shocks, whose effects on the output gap fade rapidly. By contrast, forecast errors in consumption, driven by permanent shocks to productivity, have similar effects to hours errors on estimates of the contemporaneous output gap but build substantially over subsequent years. Regarding investment growth, our earlier analysis shows that investment growth can be highly informative about the output gap, but only at longer horizons, because the shocks underlying investment growth errors have offsetting effects on the contemporaneous output gap. Moreover, earlier forecast errors, especially in the federal funds rate, were associated by the model with permanent technology shocks, explaining a significant fraction of the higher investment over these years, further reducing the contribution of investment growth to the output gap. 4.4. Interpreting the News: An Illustrative Example In this section, we illustrate the use of the decompositions derived above to construct simple, easilyinterpretablenarrativesconnectingnewsabouttheobservablesinthemodeltorevisionstothe forecast. For our example, we focus on the model’s reaction to news between the end of the fourth quarter of 2013 and the middle of the second quarter of 2014. We focus on this period because the configuration of incoming news (e.g., very disappointing readings on labor productivity, but unexpectedly strong consumption growth) is not naturally explained by a single shock, highlighting the value of the decomposition that we presented above. Specifically, in this experiment, we took a snapshot of the real-time data for two points in time: the first one at 9am EST on December 31, 2013 and the second one at 9am EST on May 15, 2014. That guarantees us that the first dataset includes real-time estimates for the third quarter of 2013, including the advance estimate of GDP and its components. The second dataset includes real-time estimates for the first quarter of 2014. We do not attempt to condition on partial information on the forecast quarter from monthly data sources, such as the Employment Situation report. Figure 8 displays the real-time data corresponding to both vintages, as well as the associated forecasts and estimates of the output gap. As is apparent from the Figure, the main feature of the newsbetweenthesetwodatasetsisadrasticpositiveforecasterrorfor2014:Q1consumptiongrowth, while the forecast errors for 2014:Q1 investment growth and output growth are negative. In the data, the most obvious sources of weakness in 2014:Q1 output growth are in inventory investment and net exports. Correspondingly, exogenous spending in this quarter was weak. The model was surprised, to a lesser extent, by inflation, inflation expectations, and federal funds rates. Finally, aggregate hours were only slightly below expectation, implying a large negative error in average labor productivity. Perhaps surprisingly, the model reacts to this constellation of news by keeping its forecast for bothoutputgrowthandhoursataverysimilarlevel,butrevisingitsestimateoftheoutputgapfrom 20

Figure 8. Comparison of forecasts for selected variables, December 2013 vs. May 2014 Note: The gray line reports the real-time data and forecast as of December 2013. The red line reports the real-time and forecast as of May 2014. The shaded area covers the common period of forecast for both dates. about -8.5 percent to -7 percent in 2014:Q1. Furthermore, inflation and inflation expectations have beenrevisedupward, whiletheweaknessininvestmentsisprojectedtopersistforafewperiods. We can use the general properties of the double decomposition to develop a crisp narrative explaining this reaction. Figure9displaysthedoubledecompositionfortherevisiontotheoutputgrowthforecast. Inline with our earlier discussion, we can see that the negative forecast error in output growth signalled to themodelaverytransientdropinexogenousdemand, byvirtueoftheaggregateresourceconstraint (the panel labelled“Exog Demand”). At the same time, the positive error in consumption growth was associated with positive contributions to output growth from shocks to productivity (shown by the positive green bars in the panel labelled “Productivity”), boosting the forecast for output growth. Similarly, monetary policy shocks, almost exclusively informed by surprises in the federal fundsrate,wereexpansionary. However,thesetwoeffectswerebalancedbynegativefinancialshocks 21

(informed by the negative error in investment growth), so that the forecast for output growth was little changed at the end of the day. Figure 9. Decomposition of the revision in the forecast for output growth between December 2013 and May 2014 Note: Upper left panel: data decomposition of the revision to the forecast. Other panels: data decomposition of the contribution of the given shock to the revision. Financial shocks include shocks to firm-specific risk, net worth and bankruptcy costs. Productivity shocks include both permanent and transient shocks. Mark-up shocks include both price and wage mark-ups. Turning now to the revision in the output gap, as shown in Figure 10, the situation is somewhat surprising, but consistent with the analysis of the output growth. As mentioned before, news about consumption growth is mostly informative about permanent productivity in this model. Given that we observed a sizeable positive surprise in consumption growth, the model inferred a shock to permanentproductivity,andnotademandshockasitwouldbethepriorformostpractitioners. The output gap narrowed in the response, as growth in potential output increased. The estimated shock to permanent productivity due to the consumption forecast error is attenuated by the negative surprise in the federal funds rate. The revision to the output gap is further attenuated by the 22

Figure10. Decomposition of the revision in the forecast for the output gap between December 2013 and May 2014 Note: Upper left panel: data decomposition of the revision to the forecast. Other panels: data decomposition of the contribution of the given shock to the revision. Financial shocks include shocks to firm-specific risk, net worth and bankruptcy costs. Productivity shocks include both permanent and transient shocks. Mark-up shocks include both price and wage mark-ups. negative contributions from financial shocks inferred as a result of the surprisingly meagre growth in real investment. Tosummarize,thepositivenewsonconsumptiongrowth,accompaniedbylittlechangeinoutput growth, suggested to the model that aggregate supply conditions in 2014:Q1 were better than expected and the output gap has been revised up. This example demonstrates the utility of the double decomposition for interpreting the reaction of the model to incoming data. 23

5. Conclusion Inthispaper, weadvocatechainingthedecompositionofshocksintocontributionsfromforecast errors to the shock decomposition of the latent vector in order to better understand model inference about latent variables. This double decomposition allows us to gauge the influence of data on the path of latent variables, like the data decomposition. However, by taking into account the transmission mechanisms of each type of shock, we can highlight the economic structure underlying the relationship between the data and the latent variables. We demonstrate the usefulness of this approach by detailing the role of observable variables in estimating the output gap within the DSGE model `a la Del Negro, Giannoni and Schorfheide (2015). Tracing the logic of the double decomposition, we find that real economic factors, such as consumption and investment, and certain financial observables, such as the corporate bond spread, are highly informative about the output gap largely because they provide strong signals about the arrival of permanent productivity and firm-level risk shocks – the main drivers of variation in the output gap. Our double decomposition results also highlight the mechanisms that explain some puzzling features of the data decomposition, such as the weak role of aggregate hours and the strong role of the federal funds rate in informing the model’s estimates of the path of the output gap. As these features are key to the model’s reaction to data, the double decomposition enables practitioners to better understand and communicate model estimates of latent variables. 24

References Andrle,Michal.2013.“UnderstandingDSGEFiltersinForecastingandPolicyAnalysis.”IMFWorking Paper 13/98. Barsky, Robert, Alejandro Justiniano, and Leonardo Melosi. 2014.“The Natural Rate of Interest and Its Usefulness for Monetary Policy.”American Economic Review, 104(5): 37–43. DelNegro,Marco,MarcP.Giannoni,andFrankSchorfheide.2015.“InflationintheGreatRecession and New Keynesian Models.”American Economic Journal: Macroeconomics, 7(1): 168–96. Durbin,J.,andS.J.Koopman.2002.TimeSeriesAnalysisbyStateSpaceModels.OxfordUniversity Press:Oxford Statistical Science Series. Ireland, Peter N. 2011.“A New Keynesian Perspective on the Great Recession.”Journal of Money, Credit and Banking, 43(1): 31–54. Sander,N.2013.“FreshPerspectivesonUnobservableVariables: DataDecompositionoftheKalman Smoother.”Reserve Bank of New Zealand Analytical Notes AN 2013/09. Smets, Frank, and Rafael Wouters. 2007.“Shocks and Frictions in US Business Cycles: A Bayesian DSGE Approach.”American Economic Review, 97(3): 586–606. 25

Appendix A.1. The Kalman Filter and Smoother Let us consider the following state space system: X = ΦX +Rη t t−1 t Y = ZX +ε t t t Let us assume that the structural innovations η ∼ N (0,Q), the measurement errors ε ∼ t iid t iid (cid:16) (cid:17) N (0,H), and the initial state X 0 ∼ N X(cid:98)0|0 ,P 0|0 . In stationary models, usually X(cid:98)0|0 and P 0|0 correspond to the invariant distribution associated with the law of motion for the state vector, X . t The Kalman filter and smoother recursions are based on the following lemma. Lemma A.1.1. Let (x(cid:48),y(cid:48))(cid:48) be jointly normal with (cid:32) (cid:33) (cid:32) (cid:33) µ Σ Σ x xx xy µ = and Σ = µ Σ Σ y yx yy Then the conditional probability density function p(x|y) is a multivariate normal with µ = µ +Σ Σ−1(y−µ ) x|y x xy yy y Σ = Σ −Σ Σ−1Σ xx|y xx xy yy yx The Kalman filter recursions can be summarized as follows: X(cid:98)t|t−1 = ΦX(cid:98)t−1|t−1 P = ΦP Φ(cid:48)+RQR(cid:48) t|t−1 t−1|t−1 ν t = Y t −Y(cid:98)t|t−1 = Y t −ZX(cid:98)t|t−1 X(cid:98)t|t = X(cid:98)t|t−1 +K t|t−1 ν t P = P −M F−1 M(cid:48) t|t t|t−1 t|t−1 t|t−1 t|t−1 M = P Z(cid:48) t|t−1 t|t−1 K = M F−1 t|t−1 t|t−1 t|t−1 F = ZM +H t|t−1 t|t−1 (cid:104) (cid:105) L = Φ I −P Z(cid:48)F−1 Z t|t−1 t|t−1 t|t−1 Note that ΦP = L P t|t t t|t−1 We can rewrite the one-step ahead forecast errors or innovations, ν , as t (cid:16) (cid:17) ν t = Y t −Y(cid:98)t|t−1 = Z X t −X(cid:98)t|t−1 +ε t = Zξ t|t−1 +ε t 26

where   0 τ < t     E(cid:2) ε ν(cid:48)(cid:3) = H τ = t t τ   E(cid:2) ε ξ (cid:3) Z(cid:48) τ > t   t τ|τ−1 with E(cid:2) ε ξ (cid:3) = −HK(cid:48) Φ(cid:48) t t+1|t t|t−1 E(cid:2) ε ξ (cid:3) = −HK(cid:48) Φ(cid:48)L(cid:48) t t+2|t+1 t|t−1 t+1|t ... E(cid:2) ε ξ (cid:3) = −HK(cid:48) Φ(cid:48)L(cid:48) L(cid:48) ...L(cid:48) t T|T−1 t|t−1 t+1|t t+2|t+1 T−1|T−2 Then, the smoothed estimate for the vector of measurement errors is given by (cid:104) ε = H F−1 ν −K(cid:48) Φ(cid:48)Z(cid:48)F−1 ν −K(cid:48) Φ(cid:48)L(cid:48) Z(cid:48)F−1 ν (cid:98)t|T t|t−1 t t|t−1 t+1|t t+1 t|t−1 t+1|t t+2|t+1 t+2 (cid:105) −...−K(cid:48) Φ(cid:48)L(cid:48) ...L(cid:48) Z(cid:48)F−1 ν t|t−1 t+1|T T−1|T−2 T|T−1 T (cid:104) (cid:105) = H F−1 ν −K(cid:48) Φ(cid:48)r t|t−1 t t|t−1 t Similarly, the smoothed estimate for the vector of structural shocks is T (cid:88) η = E[η ν ]F ν (cid:98)t|T t τ τ|τ−1−1 τ τ=1 where (cid:40) (cid:41) 0 τ < t E(cid:2) η ν(cid:48)(cid:3) = t τ E(cid:2) η ξ (cid:3) Z(cid:48) τ ≥ t t τ|τ−1 with E(cid:2) η ξ (cid:3) = QR(cid:48) t t|t−1 E(cid:2) η ξ (cid:3) = QR(cid:48)L(cid:48) t t+1|t t|t−1 ... E(cid:2) η ξ (cid:3) = QR(cid:48)L(cid:48) L(cid:48) ...L(cid:48) t T|T−1 t|t−1 t+1|t T|T−1 28

The smoothed estimate for the vector of structural shocks can also be rewritten as a weighted sum of future one-step ahead forecast errors or innovations, ν ,: t (cid:104) η = QR(cid:48) Z(cid:48)F−1 ν +L(cid:48) Z(cid:48)F−1 ν (cid:98)t|T t|t−1 t t|t−1 t+1|t t+1 (cid:105) +...+L(cid:48) L(cid:48) ...L(cid:48) Z(cid:48)F−1 ν t|t−1 t+1|t T|T−1 T|T−1 T = QR(cid:48)r (A.2) t−1 A.2. Historical Shock Decomposition The historical shock decomposition represents the estimated latent variables in terms of contributions of the estimated structural shocks. Let e be the estimated path for the ith structural (cid:98)t|T shock obtained from the Kalman smoother and let ηi be an n-dimensional vector of zeros with (cid:98)t−τ|T e in the ith row. Under the assumption of independently and identically distributed structural (cid:98)t|T shocks, the autoregressive model for the state vector in equation (1) has an infinite moving average representation so that the contribution of the ith structural shock to the vector of latent variables at time t is given by t−1 (cid:88) X(cid:98) t i |T = ΦtX 0 + ΦτRη (cid:98)t i −τ|T (A.3) τ=0 Therefore, X(cid:98) i is the estimated model-implied path for the state vector given the estimated path t|T for the ith structural shock conditional on the remaining (n−1) structural shocks being zero for all τ. The estimated actual realization of the state vector, X(cid:98)t|T , is equal to the sum across the structural shocks contributions n t−1 X(cid:98)t|T = ΦtX 0 + (cid:88)(cid:88) ΦτRη (cid:98)t j −τ|T (A.4) i=1τ=0 A.3. Historical Data Decomposition Given the linearity of the econometric model, each observable variable has an independent effect on the Kalman smoother’s estimate of a latent variable. The historical data decomposition traces these independent contributions of each observable variable to the estimated smoothed realization of each latent variable. We first compute the data decomposition for the filtered vector of state variables, X(cid:98)t|t , which is given by. t (cid:88) X(cid:98)t|t = Ω t,0 X(cid:98)0|t + Ω t,τ K τ|τ−1 Y τ (A.5) τ=1 29

In order to obtain the sequence of weights {Ω }T , let us consider the updated state vector t,τ τ=1 from the Kalman filter recursion X(cid:98)t|t = X(cid:98)t|t−1 +K t|t−1 ν t (cid:2) (cid:3) = K t|t−1 Y t + I −K t|t−1 ΦX(cid:98)t−1|t−1 Let us define (cid:2) (cid:3) N = I −K Φ t|t−1 t|t−1 Note that N = Φ−1L Φ. Then, t t|t−1 X(cid:98)t|t = K t|t−1 Y t +N t|t−1 X(cid:98)t−1|t−1 = K Y +N K Y +N N K Y +...+ t|t−1 t t|t−1 t−1|t−2 t−1 t|t−1 t−1|t−2 t−2|t−3 t−2 N N ...N K Y +N N ...N N X t|t−1 t−1|t−2 2|1 1|0 1 t|t−1 t−1|t−2 2|1 1|0 0|0 Thus, the observation weights for the filtered state vector are given by (cid:40) I τ = t Ω = t,τ N N ...N τ < t t|t−1 t−1|t−2 τ+1|τ and the weight for the initial conditions is Ω = N N ...N t,0 t|t−1 t−1|t−2 1|0 In order to compute the data decomposition for the smoothed vector of state variables, X(cid:98)t|T , let us write the smoother recursion as (cid:16) (cid:17) X(cid:98)t|T = X(cid:98)t|t +P t|t Φ(cid:48)P t − + 1 1|t X(cid:98)t+1|T −ΦX(cid:98)t|t = X(cid:98)t|t +P t|t−1 Φ(cid:48)r t−1 We have already computed the observation weights for the filtered vector of state variables, X(cid:98)t|t . Thus, we only need to compute the data decomposition of r , which is a weighted sum of future t−1 forecast errors r = Z(cid:48)F−1 ν +L(cid:48) Z(cid:48)F−1 ν +...+L(cid:48) L(cid:48) ...L(cid:48) Z(cid:48)F−1 ν t−1 t|t−1 t t|t−1 t+1|t t+1 t|t−1 t+1|t T−1|T−2 T|T−1 T = Z(cid:48)F−1 Y +L(cid:48) Z(cid:48)F−1 Y +...+L(cid:48) L(cid:48) ...L(cid:48) Z(cid:48)F−1 Y t|t−1 t t|t−1 t+1|t t+1 t|t−1 t+1|t T−1|T−2 T|T−1 T −Z(cid:48)F t − |t 1 −1 ZΦX(cid:98)t−1|t−1 −L(cid:48) t|t−1 Z(cid:48)F t − + 1 1|t ZΦX(cid:98)t|t −...−L(cid:48) t|t−1 ...L(cid:48) T−1|T−2 Z(cid:48)F T − | 1 T−1 ZΦX(cid:98)T−1|T−1 Let the observation weights for the forward recursions in r be t−1 (cid:40) I τ = t Ω(cid:63) = t,τ L(cid:48) L(cid:48) ...L(cid:48) τ > t t|t−1 t+1|t τ−1|τ−2 30

Then, T T−1 (cid:88) (cid:88) r t−1 = Ω(cid:63) t,τ Z(cid:48)F τ − |τ 1 −1 Y τ − Z(cid:48)Ω(cid:63) t−1,τ F τ+1|τ ZΦX(cid:98)τ|τ τ=t τ=t−1 T T−1 τ (cid:88) (cid:88) (cid:88) = Ω(cid:63) Z(cid:48)F−1 Y − Z(cid:48)Ω(cid:63) F ZΦ Ω K Y t,τ τ|τ−1 τ t−1,τ τ+1|τ τ,j j,j−1 j τ=t τ=t−1 j=1 The data decomposition of the smoothed state vector is t (cid:88) X(cid:98)t|T = Ω t,0 X(cid:98)0|T + Ω t,τ K τ|τ−1 Y τ τ=1   T T−1 τ (cid:88) (cid:88) (cid:88) +P t|t−1 Φ(cid:48)  Ω(cid:63) t,τ Z(cid:48)F τ − |τ 1 −1 Y τ − Z(cid:48)Ω(cid:63) t−1,τ F τ+1|τ ZΦ Ω τ,j K j|j−1 Y j (A.6) τ=t τ=t−1 j=1 which states that the time-varying observable weights depend on the relative position of τ with respect to t. The contribution of the jth observable variable to the estimated latent variable vector at time t, X(cid:98) j , is given by t|T t X(cid:98) t j |T = Ω t,0 X(cid:98) 0 j |T + (cid:88) Ω t,τ K τ|τ−1 Y τ j τ=1 (cid:34) T T−1 τ (cid:35) +P (cid:88) Ω(cid:63) Z(cid:48)F−1 Yj − (cid:88) Z(cid:48)Ω(cid:63) F ZΦ (cid:88) Ω K Yj (A.7) t|t−1 t,τ τ|τ−1 τ t−1,τ τ+1|τ τ,j l|l−1 l τ=t τ=t−1 l=1 where Yj is an n−dimensional vector with all rows equal to the unconditional mean of the corre- τ sponding observable variable but the jth row, which is set to the actual observed value. The estimated actual realization of the smoothed state vector, X(cid:98)t|T , is equal to the sum across the observable data series contributions. n n n t X(cid:98)t|T = (cid:88) X(cid:98) t j |T = ω t (cid:63)Ω t,0 (cid:88) X(cid:98) 0 j |T + (cid:88)(cid:88) Ω t,τ K τ|τ−1 Y τ j j=1 j=1 j=1τ=1 n (cid:34) T T−1 τ (cid:35) +P Φ(cid:48) (cid:88) (cid:88) Ω(cid:63) Z(cid:48)F−1 Yj − (cid:88) Z(cid:48)Ω(cid:63) F ZΦ (cid:88) Ω K Yj (A.8) t|t−1 t,τ τ|τ−1 τ t−1,τ τ+1|τ τ,j l|l−1 l j=1 τ=t τ=t−1 l=1 Similarly, the data decomposition of the smoothed vector of structural shocks is given by   T T−1 τ (cid:88) (cid:88) (cid:88) η (cid:98)t|T = QR(cid:48)  Ω(cid:63) t,τ Z(cid:48)F τ − |τ 1 −1 Y τ − Z(cid:48)Ω(cid:63) t−1,τ F τ+1|τ ZΦ Ω τ,j K j|j−1 Y j τ=t τ=t−1 j=1 31

A.4. Historical Double Decomposition The historical double decomposition links the historical data decomposition of the structural shocks and the historical shock decomposition of the latent vector. The data decomposition of the smoothed estimate for the vector of structural shocks is given by (cid:34) T T−1 τ (cid:35) (cid:88) (cid:88) (cid:88) η = QR(cid:48) Ω(cid:63) Z(cid:48)F−1 Y − Z(cid:48)Ω(cid:63) F ZΦ Ω K Y (A.9) (cid:98)t|T t,τ τ|τ−1 τ t−1,τ τ+1|τ τ,l l|l−1 l τ=t τ=t−1 l=1 Substituting this expression for the vector of smoothed structural shocks in the ∞−MA representation of the smoothed vector of latent variables in equation (A.4), we obtain n t−1 (cid:34) T T−1 k (cid:35) X(cid:98)t|T = ΦtX(cid:98)0|T + (cid:88)(cid:88) ΦτRQR(cid:48) (cid:88) Ω(cid:63) τ−1,k Z(cid:48)F k − |k 1 −1 Y k j − (cid:88) Z(cid:48)Ω(cid:63) t−τ−1,k F k+1|k ZΦ (cid:88) Ω k,l K l|l−1 Y l j j=1τ=0 k=τ−1 k=t−τ−1 l=1 (A.10) A.5. Tables and Figures Table A.1: Effects of News on Shock Estimates Positive news on Shock Real GDP Infl core PCE core CPI Exp Infl Debt Wages ECI Prod (temp) 1.17 0.00 0.00 0.00 0.02 -0.01 -0.01 -0.02 Prod (perm) 0.09 -0.01 -0.02 -0.02 -0.16 0.00 0.02 0.07 Firm-level Risk 0.14 0.01 0.01 0.01 0.08 -0.05 -0.00 -0.02 Risk premium -0.10 -0.00 -0.01 -0.01 -0.05 0.04 0.01 0.02 Exog. spending 1.76 0.00 0.00 0.00 -0.00 -0.00 -0.00 -0.01 Inv. efficiency 0.02 0.01 0.01 0.01 0.04 -0.11 -0.00 -0.00 Mon Pol -0.23 -0.06 -0.16 -0.16 0.02 0.02 0.01 0.03 Price markup 0.02 0.14 0.36 0.37 -0.11 -0.03 -0.04 -0.14 Wage markup 0.02 0.02 0.06 0.06 0.28 -0.01 0.09 0.36 Net worth 0.05 0.00 0.01 0.01 0.05 -0.02 0.00 0.00 Infl Target -0.08 -0.02 -0.04 -0.04 1.02 0.03 -0.02 -0.09 Mar. bankruptcy -0.04 -0.00 -0.01 -0.01 -0.04 0.01 -0.00 -0.01 Note: Table entries are the effect of a one-standard deviation forecast error in the given observables on the model’s estimate of the contemporaneous shock. Shocks are measured in standard-deviation units. 32

Table A.2: Effects of News on Estimates of the Contribution of Shocks to the Output Gap Positive news on Contribution to gap from Real GDP Infl core PCE core CPI Exp Infl Debt Wages ECI Prod (temp) -0.28 0.01 0.02 0.02 -0.02 -0.00 0.00 0.00 Prod (perm) 0.07 0.00 0.01 0.01 -0.09 0.05 0.02 0.07 Firm-level Risk -0.05 0.00 0.00 0.00 -0.02 0.13 0.00 0.02 Risk premium -0.00 -0.00 -0.00 -0.00 0.01 -0.07 -0.00 -0.01 Exog. spending 0.05 -0.00 -0.00 -0.00 0.00 -0.00 -0.00 -0.00 Inv. efficiency 0.01 0.00 0.00 0.00 0.00 0.00 -0.00 -0.00 Mon Pol 0.07 0.03 0.07 0.07 -0.02 -0.01 0.00 0.00 Price markup 0.00 -0.01 -0.03 -0.03 0.01 -0.00 0.00 0.01 Wage markup -0.03 -0.01 -0.04 -0.04 0.01 0.01 -0.01 -0.04 Net worth 0.01 -0.00 -0.00 -0.00 0.00 -0.06 0.00 0.00 Infl Target -0.01 -0.01 -0.02 -0.02 0.10 0.01 -0.01 -0.02 Mar. bankruptcy 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 Total -0.15 0.01 0.01 0.01 -0.01 0.03 0.01 0.03 Note: Table entries are the effect of a one-standard deviation forecast error in the given observables on the model’s estimate of the contemporaneous output gap, measured in percentage points. Table A.3: Effect of News on Shock Contributions to the Output Gap 8 quarters in the Future Positive news on Contribution to gap from Real GDP Infl core PCE core CPI Exp Infl Debt Wages ECI Prod (temp) -0.32 0.01 0.01 0.01 -0.01 -0.00 0.00 0.01 Prod (perm) 0.15 -0.00 -0.01 -0.01 -0.23 0.08 0.04 0.14 Firm-level Risk -0.20 -0.00 -0.01 -0.01 -0.10 0.33 0.01 0.04 Risk premium 0.02 -0.00 -0.00 -0.00 0.01 -0.04 -0.00 -0.01 Exog. spending 0.07 -0.00 -0.00 -0.00 0.00 -0.00 -0.00 -0.00 Inv. efficiency 0.02 0.00 0.00 0.01 0.01 -0.02 -0.00 -0.00 Mon Pol 0.08 0.03 0.07 0.07 -0.02 -0.00 -0.00 -0.00 Price markup -0.01 -0.02 -0.04 -0.04 0.02 0.00 0.00 0.02 Wage markup -0.03 -0.02 -0.04 -0.05 -0.03 0.02 -0.03 -0.13 Net worth -0.00 -0.00 -0.01 -0.01 -0.01 -0.23 0.00 0.00 Infl Target -0.03 -0.01 -0.02 -0.02 0.27 0.01 -0.01 -0.04 Mar. bankruptcy 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 Total -0.25 -0.02 -0.05 -0.05 -0.10 0.14 0.01 0.03 Note: Table entries are the effect of a one-standard deviation forecast error in the given observables on the model’s estimate of the output gap eight quarters later, measured in percentage points. 33

Figure A.1. Impulse responses functions for major output gap drivers 0.4 0.2 0 10 20 30 40 Tech (perm) htworG noC 0 -0.05 -0.1 -0.15 10 20 30 40 Firm Risk htworG noC 0 -0.05 -0.1 -0.15 -0.2 10 20 30 40 Risk Premium htworG noC 0 -0.1 -0.2 -0.3 10 20 30 40 Mon Pol htworG noC 0.3 0.2 0.1 0 10 20 30 40 Tech (perm) htworG vnI 0 -0.5 -1 -1.5 -2 10 20 30 40 Firm Risk htworG vnI 0 -0.2 -0.4 10 20 30 40 Risk Premium htworG vnI 0 -0.2 -0.4 -0.6 10 20 30 40 Mon Pol htworG vnI 1 0.8 0.6 0.4 0.2 10 20 30 40 Tech (perm) sruoH 0 -0.5 -1 10 20 30 40 Firm Risk sruoH 0 -0.1 -0.2 -0.3 10 20 30 40 Risk Premium sruoH 0 -0.2 -0.4 -0.6 10 20 30 40 Mon Pol sruoH 15 10 5 0 10 20 30 40 Tech (perm) sdaerpS 10-3 0.15 0.1 0.05 10 20 30 40 Firm Risk sdaerpS 0.15 0.1 0.05 0 10 20 30 40 Risk Premium sdaerpS 5 0 -5 10 20 30 40 Mon Pol sdaerpS 10-3 0.1 0.05 0 10 20 30 40 Tech (perm) htroW teN 0 -0.5 -1 -1.5 10 20 30 40 Firm Risk htroW teN 0 -0.2 -0.4 -0.6 10 20 30 40 Risk Premium htroW teN 0 -0.5 -1 10 20 30 40 Mon Pol htroW teN 1.5 1 0.5 10 20 30 40 Tech (perm) paG tuptuO 0 -0.5 -1 -1.5 10 20 30 40 Firm Risk paG tuptuO 0 -0.1 -0.2 -0.3 10 20 30 40 Risk Premium paG tuptuO 0 -0.2 -0.4 -0.6 -0.8 10 20 30 40 Mon Pol paG tuptuO Note: Impulse responses are calculated following one-standard-deviation shocks and are reported in percentage points. 34

snoitcnuf sesnopser eslupmi rehtO .2.A erugiF 50.0 0 50.0- 04 03 02 01 )pmet( hceT htworG noC 1.0 50.0 0 04 03 02 01 IEM htworG noC 0 20.0- 40.0- 60.0- 80.0- 04 03 02 01 dnepS goxE htworG noC 20.0 0 20.0- 40.0- 60.0- 80.0- 04 03 02 01 krM lfnI htworG noC 40.0 20.0 0 20.0- 04 03 02 01 )pmet( hceT htworG vnI 8.0 6.0 4.0 2.0 0 04 03 02 01 IEM htworG vnI 60.0 40.0 20.0 0 04 03 02 01 dnepS goxE htworG vnI 0 1.0- 2.0- 04 03 02 01 krM lfnI htworG vnI 0 2.0- 4.0- 04 03 02 01 )pmet( hceT sruoH 6.0 4.0 2.0 0 04 03 02 01 IEM sruoH 3.0 52.0 2.0 51.0 1.0 04 03 02 01 dnepS goxE sruoH 0 1.0- 2.0- 04 03 02 01 krM lfnI sruoH 0 50.0- 1.0- 04 03 02 01 krM egaW htworG noC 0 10.0- 20.0- 04 03 02 01 ycprknB htworG noC 5 0 5- 04 03 02 01 htrW teN htworG noC 3-01 51.0 1.0 50.0 0 04 03 02 01 graT lfnI htworG noC 50.0 0 50.0- 1.0- 04 03 02 01 krM egaW htworG vnI 1.0 0 1.0- 2.0- 3.0- 04 03 02 01 ycprknB htworG vnI 1.0 0 1.0- 2.0- 3.0- 04 03 02 01 htrW teN htworG vnI 2.0 51.0 1.0 50.0 0 04 03 02 01 graT lfnI htworG vnI 0 1.0- 2.0- 3.0- 04 03 02 01 krM egaW sruoH 0 1.0- 2.0- 04 03 02 01 ycprknB sruoH 50.0 0 50.0- 1.0- 04 03 02 01 htrW teN sruoH 3.0 2.0 1.0 04 03 02 01 graT lfnI sruoH 3 2 1 04 03 02 01 )pmet( hceT sdaerpS 3-01 40.0 30.0 20.0 04 03 02 01 IEM sdaerpS 1 0 1- 04 03 02 01 dnepS goxE sdaerpS 3-01 0 2- 4- 04 03 02 01 krM lfnI sdaerpS 3-01 0 20.0- 40.0- 04 03 02 01 )pmet( hceT htroW teN 0 1- 2- 04 03 02 01 IEM htroW teN 30.0 20.0 10.0 0 10.0- 04 03 02 01 dnepS goxE htroW teN 0 1.0- 2.0- 04 03 02 01 krM lfnI htroW teN 0 2.0- 4.0- 04 03 02 01 )pmet( hceT paG tuptuO 4.0 2.0 0 2.0- 04 03 02 01 IEM paG tuptuO 1.0 50.0 0 04 03 02 01 dnepS goxE paG tuptuO 0 1.0- 2.0- 04 03 02 01 krM lfnI paG tuptuO 0 2- 4- 6- 04 03 02 01 krM egaW sdaerpS 3-01 20.0 510.0 10.0 500.0 04 03 02 01 ycprknB sdaerpS 1.0 50.0 0 04 03 02 01 htrW teN sdaerpS 6 4 2 0 04 03 02 01 graT lfnI sdaerpS 3-01 20.0 0 20.0- 40.0- 04 03 02 01 krM egaW htroW teN 0 1.0- 2.0- 3.0- 04 03 02 01 ycprknB htroW teN 0 2.0- 4.0- 04 03 02 01 htrW teN htroW teN 2.0 51.0 1.0 50.0 0 04 03 02 01 graT lfnI htroW teN 0 1.0- 2.0- 3.0- 4.0- 04 03 02 01 krM egaW paG tuptuO 0 1.0- 2.0- 3.0- 04 03 02 01 ycprknB paG tuptuO 50.0 0 50.0- 1.0- 04 03 02 01 htrW teN paG tuptuO 4.0 3.0 2.0 1.0 04 03 02 01 graT lfnI paG tuptuO .stniop egatnecrep ni detroper era dna skcohs noitaived-dradnats-eno gniwollof detaluclac era sesnopser eslupmI :etoN 35

Figure A.3. Effects of data news on two-sided estimates of structural shocks. 0 -0.05 -0.1 -5 0 5 Cons Growth PFT tneisnarT 0.01 0 -5 0 5 Inv Growth PFT tneisnarT 0 -0.05 -5 0 5 Fed Funds PFT tneisnarT 0 -0.02 -0.04 -5 0 5 Risk Spread PFT tneisnarT 0 -0.01 -0.02 -5 0 5 Net Worth PFT tneisnarT 1 0.5 0 -5 0 5 Cons Growth PFT tnenamreP 0 -0.05 -0.1 -5 0 5 Inv Growth PFT tnenamreP 0.4 0.2 0 -5 0 5 Fed Funds PFT tnenamreP 0.4 0.2 0 -5 0 5 Risk Spread PFT tnenamreP 0.2 0.1 0 -5 0 5 Net Worth PFT tnenamreP 0 -0.05 -0.1 -0.15 -5 0 5 Cons Growth muimerP ksiR etaerggA 0.8 0.6 0.4 0.2 0 -5 0 5 Inv Growth muimerP ksiR etaerggA 0.2 0.1 0 -5 0 5 Fed Funds muimerP ksiR etaerggA 1 0.5 0 -5 0 5 Risk Spread muimerP ksiR etaerggA 0.4 0.2 0 -5 0 5 Net Worth muimerP ksiR etaerggA 0 -0.5 -1 -5 0 5 Cons Growth gnidnepS suonegoxE 0 -0.2 -0.4 -0.6 -5 0 5 Inv Growth gnidnepS suonegoxE 0 -5 -10 -5 0 5 Fed Funds gnidnepS suonegoxE 10-3 2 0 -2 -4 -6 -8 -5 0 5 Risk Spread gnidnepS suonegoxE 10-3 1 0 -1 -2 -3 -5 0 5 Net Worth gnidnepS suonegoxE 10-3 5 0 -5 -10 -15 -5 0 5 Cons Growth yciloP yratenoM 10-3 0.01 0 -0.01 -5 0 5 Inv Growth yciloP yratenoM 0.8 0.6 0.4 0.2 0 -5 0 5 Fed Funds yciloP yratenoM 0 -10 -20 -5 0 5 Risk Spread yciloP yratenoM 10-3 10 5 0 -5 -5 0 5 Net Worth yciloP yratenoM 10-3 0 -0.01 -0.02 -0.03 -5 0 5 Cons Growth ycneiciffE tnemtsevnI 0.5 0 -5 0 5 Inv Growth ycneiciffE tnemtsevnI 0 -0.1 -0.2 -5 0 5 Fed Funds ycneiciffE tnemtsevnI 0 -0.1 -0.2 -5 0 5 Risk Spread ycneiciffE tnemtsevnI 0 -0.5 -1 -5 0 5 Net Worth ycneiciffE tnemtsevnI 0.1 0.05 0 -5 0 5 Cons Growth ksiR mriF 0 -0.5 -1 -5 0 5 Inv Growth ksiR mriF 0 -0.1 -0.2 -0.3 -5 0 5 Fed Funds ksiR mriF 0 -0.1 -0.2 -0.3 -5 0 5 Risk Spread ksiR mriF 0 -0.2 -0.4 -5 0 5 Net Worth ksiR mriF 0.02 0.01 0 -0.01 -5 0 5 Cons Growth htroW teN 0 -0.2 -0.4 -5 0 5 Inv Growth htroW teN 0.05 0 -0.05 -0.1 -5 0 5 Fed Funds htroW teN 0.1 0 -0.1 -0.2 -0.3 -5 0 5 Risk Spread htroW teN 0 -0.05 -0.1 -0.15 -5 0 5 Net Worth htroW teN 0.08 0.06 0.04 0.02 0 -5 0 5 Cons Growth yctpurknaB 0.2 0.1 0 -5 0 5 Inv Growth yctpurknaB 0.1 0.05 0 -5 0 5 Fed Funds yctpurknaB 0.4 0.2 0 -5 0 5 Risk Spread yctpurknaB 0.15 0.1 0.05 0 -5 0 5 Net Worth yctpurknaB 0.1 0.05 0 -5 0 5 Cons Growth tegraT noitalfnI 0.15 0.1 0.05 0 -5 0 5 Inv Growth tegraT noitalfnI 0.04 0.02 0 -5 0 5 Fed Funds tegraT noitalfnI 0.08 0.06 0.04 0.02 0 -5 0 5 Risk Spread tegraT noitalfnI 0.2 0.1 0 -5 0 5 Net Worth tegraT noitalfnI 0.06 0.04 0.02 0 -0.02 -5 0 5 Cons Growth pu-kraM ecirP 0.02 0 -0.02 -5 0 5 Inv Growth pu-kraM ecirP 0.15 0.1 0.05 0 -0.05 -5 0 5 Fed Funds pu-kraM ecirP 0.04 0.02 0 -0.02 -5 0 5 Risk Spread pu-kraM ecirP 0.06 0.04 0.02 0 -0.02 -5 0 5 Net Worth pu-kraM ecirP 0 -0.1 -0.2 -5 0 5 Cons Growth pu-kraM egaW 0 -0.01 -0.02 -5 0 5 Inv Growth pu-kraM egaW 0 -0.02 -0.04 -0.06 -5 0 5 Fed Funds pu-kraM egaW 0 -0.05 -0.1 -5 0 5 Risk Spread pu-kraM egaW 3 2 1 0 -1 -5 0 5 Net Worth pu-kraM egaW 10-3 Note: Each row shows the effect of a one-standard deviation forecast error in a selected observable at time 0 on the two-sided estimate of various shocks, also expressed in standard deviations. 36

Figure A.4. Effects of selected forecast errors at time 0 on the path of the output gap Cons Inv Hours 1.4 1.4 1.4 1.2 1.2 1.2 1 1 1 0.8 0.8 0.8 0.6 0.6 0.6 0.4 0.4 0.4 0.2 0.2 0.2 0 0 0 -0.2 -0.2 -0.2 -10 0 10 20 30 40 -10 0 10 20 30 40 -10 0 10 20 30 40 FFR Spread Net Worth 1.4 1.4 1.4 1.2 1.2 1.2 1 1 1 0.8 0.8 0.8 0.6 0.6 0.6 0.4 0.4 0.4 0.2 0.2 0.2 0 0 0 -0.2 -0.2 -0.2 -10 0 10 20 30 40 -10 0 10 20 30 40 -10 0 10 20 30 40 Note: Panelsrepresenttheeffectofaone-standard-deviationforecasterrorinthegivenvariableattime0onthe estimated path of the output gap between 10 quarters before that date and 40 quarters after. Effects are calculated assuming that the news arrives 200 quarters after the beginning of the data set. 37

Cite this document

APA

Hess Chung, Cristina Fuentes-Albero, Matthias Paustian, & and Damjan Pfajfar (2020). Latent Variables Analysis in Structural Models: A New Decomposition of the Kalman Smoother (FEDS 2020-100). Board of Governors of the Federal Reserve System, Finance and Economics Discussion Series. https://whenthefedspeaks.com/doc/feds_2020-100

BibTeX

@techreport{wtfs_feds_2020_100,
  author = {Hess Chung and Cristina Fuentes-Albero and Matthias Paustian and and Damjan Pfajfar},
  title = {Latent Variables Analysis in Structural Models: A New Decomposition of the Kalman Smoother},
  type = {Finance and Economics Discussion Series},
  number = {2020-100},
  institution = {Board of Governors of the Federal Reserve System},
  year = {2020},
  url = {https://whenthefedspeaks.com/doc/feds_2020-100},
  abstract = {This paper advocates chaining the decomposition of shocks into contributions from forecast errors to the shock decomposition of the latent vector to better understand model inference about latent variables. Such a double decomposition allows us to gauge the inuence of data on latent variables, like the data decomposition. However, by taking into account the transmission mechanisms of each type of shock, we can highlight the economic structure underlying the relationship between the data and the latent variables. We demonstrate the usefulness of this approach by detailing the role of observable variables in estimating the output gap in two models. Accessible materials (.zip)},
}