A Markov switching factor-augmented VAR model for ...

A Markov switching factor-augmented VAR model for analyzing US business cycles and monetary policy Florian Huber∗

Manfred M. Fischer

Vienna University of Economics and Business

Abstract This paper develops a Markov switching factor-augmented vector autoregression to investigate the transmission mechanisms of monetary policy for distinct stages of the US business cycle. We assume that autoregressive parameters and covariance matrices of the error terms are regime-dependent, driven by an unobserved Markov indicator. Endogenously determined transition probabilities are governed by an underlying probit model that features a large set of possible predictors. The empirical findings provide evidence for differences in the transmission of monetary policy shocks that mainly stem from heterogeneity in the responses of financial market quantities.

Keywords:

Non-linear FAVAR, business cycle, monetary policy, structural model, US economy

JEL Codes:

C30, E52, F41, E32

November 14, 2017

∗

Corresponding author: Florian Huber, Vienna University of Economics and Business. E-mail: [email protected].

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1

Introduction

The present paper introduces a new econometric model – a synthesis of multivariate Markov switching (MS) models1 with time-varying transition probabilities and factoraugmented vector autoregressive (FAVAR) models (see Bernanke et al., 2005). The proposed model allows for discrete shifts in the autoregressive parameters and the error variances over time, with switches being determined by a latent Markov indicator with time-varying transition probabilities. The transition probabilities depend on a large set of possible covariates within a probit formulation. A combination of different shrinkage priors permits reliable estimation of the model and provides additional inferential opportunities. Within this framework we analyze the transmission of monetary policy shocks to the wider macroeconomy for distinct stages of the business cycle. Compared to the existing literature on MS models, our approach explicitly allows for discriminating between expansionary and recessionary regimes while exploiting, in a parsimonious way, large information sets through a relatively small number of latent factors. These factors capture the co-movement of a wide range of macroeconomic and financial variables throughout the evolution of the business cycle, with any nonlinearities stemming from changes in the law of motion of the latent factors. This is in contrast to existing small-scale business cycle models that exploit the co-trending behavior of a handful of real activity indicators (see Filardo, 1994; Kim and Nelson, 1998). Another important novelty of our modeling approach is that the transition distributions of the underlying hidden Markov chain are parameterized by a probit model that features a great variety of predictors, as opposed to the existing literature that either assumes constant transition probabilities (Hamilton, 1989) or introduces timevarying transition probabilities that are governed by only a small number of possible exogenous regressors (Filardo, 1994; Amisano and Fagan, 2013; Kaufmann, 2010; 2015; Billio et al., 2016). Given the fact that our model is highly parameterized, we adopt a Bayesian approach to estimation and inference, and use shrinkage priors to alleviate potential overfitting issues. Since the set of possible regressors that determine the time-varying transition probabilities may be large, we impose a stochastic search variable selection (SSVS) prior (George and McCulloch, 1993) on the underlying regression coefficients of the probit regression to draw inferences about the (relative) importance of the variables for business cycle transitions. For the coefficients of the state equation of the FAVAR model we utilize a hierarchical variant of the Minnesota prior (Doan et al., 1984; Sims and Zha, 1998; Giannone et al., 2015) to push the system towards a multivariate random walk. In the empirical application we first evaluate whether our proposed model is supported by the data, and then provide evidence that transition probabilities change over time, a feature that proves to be important if the goal is to achieve a high level of concordance between the extracted, model-based cycle and the NBER (National Bureau 1

For textbook introductions to MS models, see Krolzig (1997), Kim and Nelson (1999), Fr¨ uhwirthSchnatter (2006).

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of Economic Research) reference cycle. To gain further insights into the driving forces of business cycle transitions, the proposed SSVS prior is used to identify a small subset of important covariates. Turning to the structural analysis, we investigate whether the transmission mechanisms of monetary policy depend on the prevailing state of the business cycle. The monetary policy shock is identified by means of standard sign restrictions in the spirit of Uhlig (2005) that allow for simultaneous relations between monetary policy and financial market quantities. Our findings, corroborating recent empirical evidence provided in Eickmeier et al. (2016), suggest that the impact of monetary policy on financial markets is more pronounced during expansions than recessions. This result, however, does not carry over to variables representing the real side of the economy. To gain a comprehensive picture on how monetary policy actions impact business cycle transitions, our model allows the analysis of changes of transition probabilities with respect to exogenous monetary policy shocks. The paper is organized as follows. Section 2 presents the proposed model with time-varying transition distributions along with a detailed account of the Bayesian estimation strategy. Section 3 describes the data employed and provides model evidence. Section 4 investigates the time-variation in the transition probabilities and uses the nonparametric concordance statistic to check the degree of synchronization of the business cycle extracted with our model and the NBER reference cycle. The regime-dependent dynamic responses, discussed in Section 5, suggest differences in the transmission of exogenous monetary policy shocks and reveal that the likelihood of a recessionary regime is reduced by expansionary monetary policy. The final section concludes. 2 2.1

Econometric framework A non-linear FAVAR model

Let xt be an N × 1 vector of economic variables including measures of the monetary policy stance. These variables are assumed to be observable at time t = 1, . . . , T and to drive the dynamics of the economy. Standard practice is to use data for xt and to estimate a VAR, a structural VAR or another multivariate time series model. In monetary application contexts, however, it is useful to take additional economic information, not fully captured by xt , into account. Assume that this additional information can be summarized by a K × 1 vector of unobserved factors, say f t , where K is small. The unobserved factors are extracted from a large panel of M indicators, y t , providing information on several important sectors of the economy. Note that M may be much greater than the number of factors and observed variables, implying that M  R = K + N . We assume that the factors and variables are related by the following observation equation y t = Λf f t + Λx x t + e t , (2.1) with Λf and Λx representing M × K and M × N matrices of factor loadings, while et is an M × 1 vector of normally distributed zero mean disturbances with a diagonal 3

M × M variance-covariance matrix Σe . Equation (2.1) captures the idea that both f t and xt , which in general may be correlated, represent common forces that drive the dynamics of y t . Non-linearities are introduced by assuming that the joint dynamics of z t = (f 0t , x0t )0 follow a regime-switching VAR model with state-dependent autoregressive coefficients and error variance-covariance matrices,2 z t = aSt + A1St z t−1 + · · · + AQSt z t−Q + εt ,

(2.2)

where Q denotes the number of lags of z t . St is a Markovian regime indicator that takes values zero or one, aSt is an R-dimensional intercept vector, the coefficient matrices AqSt (q = 1, . . . , Q) are of dimension R × R, and εt is a normally distributed zero mean error term with R × R variance-covariance matrix ΣεSt . The subscript St in aSt , AqSt and ΣεSt indicates that all parameters are allowed to change across regimes. Accounting for changes in ΣεSt represents one of the key empirical features needed to capture salient characteristics of macroeconomic time series (Primiceri, 2005; Sims and Zha, 2006; Koop et al., 2009; Clark, 2011; Clark and Ravazzolo, 2015; Casarin et al., 2017). We assume that St is an unobserved binary Markov switching variable indicating whether the economy is in an expansionary (St = 0) or recessionary (St = 1) phase. The assumption of two regimes is consistent with the existing literature that deals with the identification of US business cycles (see, for example, Hamilton, 1989; Filardo, 1994; Kim and Nelson, 1998; Kaufmann, 2010), and with the stylized facts raised by Burns and Mitchell (1946). The transition probabilities of St are given by   p00,t p01,t Pt = . (2.3) p10,t p11,t P Hereby pij,t = P rob(St = j|St−1 = i) with 1j=0 pij,t = 1 for all i and t are the transition probablities from St = j to St−1 = i. This implies that the transition probabilities are allowed to vary over time, in contrast to the traditional literature on Markov switching models (see Hamilton, 1989) that postulates constant transition probabilities over time. Note that the higher pjj,t is, the longer the process is expected to remain in state j. Several authors (see, e.g., Kaufmann, 2010; Billio et al., 2016) emphasize that using constraints on the duration of the business cycle phases might be used to improve the extraction of the cycle by inducing more structure on the transition probabilities.

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This is in contrast to Primiceri (2005), Canova and Gambetti (2009), Koop et al. (2009), and Korobilis (2013) who use time-varying parameter models that imply smoothly evolving autoregressive coefficients and changing error variances.

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We use a probit specification (Filardo, 1994; Amisano and Fagan, 2013) to determine the time-variation in the transition probabilities,3 P rob(St = j|St−1 = i, wt−1 ) = pij,t = φ(γ0i + γ 0 wt−1 ).

(2.4)

γ0i is a regime-specific intercept and wt−1 a G-dimensional vector including variables that may be used to predict regime shifts. γ is a set of coefficients with the gth element γg measuring the sensitivity of probability pij,t with respect to wgt−1 , i.e. the gth element of wt−1 . In accordance with Amisano and Fagan (2013) we assume that the slope coefficients in Eq. (2.4) are symmetric across regimes but the intercept is allowed to change. Equation (2.4) resembles a standard probit model with an underlying latent variable formulation given by rt = γ0i + γ 0 wt−1 + t , (2.5) where rt ∈ R is a continuous latent variable and t denotes the normally distributed error with variance normalized to unity for identification purposes. The system given by Eqs. (2.1)-(2.5) is a Markov switching factor-augmented VAR (MS-FAVAR) with time-varying transition probabilities. This framework extends existing models by introducing a Markov indicator in the state equation of an otherwise standard FAVAR, and by allowing for dependency structures between a large set of potential predictors and the transition probabilities. 2.2

A Bayesian approach to estimation and inference

Densely parameterized models, such as the MS-FAVAR model outlined above, are known to yield a good in-sample fit, but imprecise out-of sample forecasts. To address this issue, we use a full Bayesian approach to estimation and inference and impose a set of informative priors on the parameters of the model. To simplify prior implementation let us rewrite Eq. (2.2) as z t = A0St dt + εt ,

(2.6)

where ASt = (A1St , . . . , AQSt , aSt )0 is a C × R matrix of stacked coefficients (with C = RQ + 1), and dt = (z 0t−1 , . . . , z 0t−Q , 1)0 denotes a C-dimensional data vector. Conditional on St and f t , the model can be represented as a standard regression model (Zellner, 1973). Stacking the rows of z t and dt yields the corresponding TSt × R and TSt × C regime-specific full data matrices, denoted by Z St and D St , where TSt is the number of observations located within a given regime.

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An alternative would be to use a logit specification that provides advantages if the number of regimes is greater than two. See, for example, Kaufmann (2015) and Billio et al. (2016) for recent applications.

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Prior distributions for the state equation We impose a set of conditionally conjugate priors given by vec(ASt )|ΣSt ∼ N (vec(A), ΣεSt ⊗ V A (θSt )),

(2.7)

where A denotes the C × R prior mean matrix, while V A (θSt ) is a C × C prior variancecovariance matrix that depends on a regime-specific hyperparameter θSt which controls the overall tightness of the prior. The prior variance on the coefficients is governed by the Kronecker product ΣεSt ⊗ V A (θSt ), a matrix of dimension RC × RC. On the variance-covariance matrix we impose an inverted Wishart prior, Σst ∼ IW(Ψ, v),

(2.8)

with Ψ being a R × R prior scale matrix, while v are the prior degrees of freedom. Concerning A and V A , we specify ( ai for q = 1 and i = j (2.9) A such that E{[AqSt ]ij } = 0 for q > 1 and i 6= j, V A such that V ar{[AqSt ]ij } =

θS2 t σi q 2 σj

(2.10)

for q = 1, . . . , Q; i, j = 1, . . . , R, with E{·} and V ar{·} denoting the expectation and variance operators, respectively. We let [AqSt ]ij select the (i, j)th element of the matrix concerned. The prior mean associated with the first own lag of variable i is given by ai , whereas for higher lag orders and other lagged variables the prior mean is set equal to zero. σi and σj are empirical standard deviations obtained by estimating a set of univariate autoregressions on z t using OLS.4 They serve to account for the different variability of the data. This prior is a conjugate variant of the Minnesota prior put forward by Doan et al. (1984) and Litterman (1986). The rationale behind the Minnesota prior is that, a priori, a random walk proves to provide a good representation of the data. Thus it might be sensible to center the system on a (multivariate) random walk process that implies setting aij = 1 for i = j for non-stationary data while aij = 0 for i = j in the stationary case. Moreover, we specify the prior on the intercept to be relatively uninformative. Typically, the overall shrinkage parameters are specified by maximizing the marginal likelihood based on a grid of possible values of θSt . However, due to the non-linear and non-conjugate framework at hand, we follow Giannone et al. (2015) to impose a set of Gamma priors on θSt , θSt ∼ G(c0 , c1 ), (2.11) with c0 and c1 being prior hyperparameters typically specified to be weakly informative. This permits us to integrate out θSt in a Bayesian fashion. 4

We obtain the standard deviations by estimating separate autoregressive models of order Q using the principal components estimator for the latent factors.

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Prior distributions for the probit model We also have to specify priors on the latent regression model given by Eq. (2.5). Since the dimensionality of wt−1 is potentially large, and the number of potential models is vast, we control for model uncertainty by imposing a stochastic search variable selection (SSVS) prior (George and McCulloch, 1993) on the elements of γ.5 Specifically, the prior on the gth parameter in Eq. (2.5) is given by γg |δg ∼ N (0, τ02 )δg + N (0, τ12 )(1 − δg ) for g = 1, . . . , G,

(2.12)

where δg is a binary random variable that controls which component of the normally distributed prior to use. The prior variances τ02 and τ12 are set such that τ02  τ12 . Thus, if δg equals one, the prior on the gth coefficient is effectively rendered noninfluential. This captures the notion that no significant prior information for that parameter is available, centering the corresponding posterior distribution around the maximum likelihood estimate. If δg equals zero, we impose a dogmatic prior, shrinking γg towards zero. This case would lead to a posterior which is strongly centered around zero, implying that we can safely regard that coefficient to be equal to zero. We impose a Bernoulli prior on the elements of δ = (δ1 , . . . , δG ), δg ∼ Bernoulli (pg ),

(2.13)

where P rob(δg = 1) = pg denotes the prior inclusion probability. In this specific application context the SSVS prior allows us to investigate the relative importance of different factors for the evolution of the business cycle. Finally, the prior on the regime-specific constant is Gaussian, γi0 ∼ N (0, ϕi ),

(2.14)

with ϕi being the prior variance typically specified to be large and symmetric across regimes, i.e. ϕ0 = ϕ1 . Prior distributions for the observation equation To complete the prior setup we also have to specify a suitable set of prior distributions on the factor loadings in Eq. (2.1). To simplify prior implementation let us collect Λf and Λx in a M × (K + N ) matrix Λ = (Λf , Λx ). Similar to the prior choice discussed above we impose a mixture Gaussian prior on the jth element of λ = vec(Λ), λj |ιj ∼ N (0, %20 )ιj + N (0, %21 )(1 − ιj ) for j = 1, . . . , M (K + N ). 5

(2.15)

Other possible prior specifications that aim to control for model uncertainty include the Bayesian lasso (Park and Casella, 2008; Belmonte et al., 2014) and the elastic net (Zou and Hastie, 2005), which can be cast in the form of a global-local shrinkage prior (see Polson and Scott, 2010; Griffin and Brown, 2010; Bhattacharya et al., 2015). Recent literature also suggests that such priors can be adopted within a VAR model (Huber and Feldkircher, 2016). This, however, would lead to an even conditionally non-conjugate model which loses the computational advantage of the Kronecker structure on the likelihood function.

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Here, ιj is a binary random variable, while %0 and %1 are hyperparameters controlling the tightness of the prior. Similar to the prior on δ, we impose a set of Bernoulli priors on the elements of ι = (ι1 , . . . , ιM (K+N ) ), (2.16) ιj ∼ Bernoulli(ρj ), with P rob(ιj = 1) = ρj denoting the prior inclusion probability of a given variable in the observation equation. Finally, the last ingredient still missing is the prior on the innovation variances of the state equation, where we use inverted Gamma priors on the M diagonal elements of Σe , denoted by ςj (j = 1, . . . , M ), ςj ∼ IG(αj , β j ),

(2.17)

with αj and β j denoting the prior shape and scale parameters, respectively. The Markov chain Monte Carlo algorithm Up to now we have remained silent on how to obtain estimates for f t . The literature suggests two routes. The first route to produce consistent estimates of the latent factors (see, for example, Bernanke et al., 2005; Korobilis, 2013) involves using a two-step estimation approach in which the factors are estimated by principal components, prior to estimation of the FAVAR. That is, one estimates the space spanned by the first K principal components of y t . This yields consistent (in the large T, M case) estimates of the true space spanned by f t and xt . Conditional on the principal components one can proceed as in the standard Markov switching VAR case. This approach has the advantage to be computationally fast and easy to implement. One disadvantage, however, is that estimation based on principal components treats the factors f t to be known, thus neglecting the uncertainty associated with fˆt , the estimate of f t . The second route, which we are going to follow, uses simulation based methods that estimate the latent factors along the other model parameters in one step by using Markov Chain Monte Carlo (MCMC) techniques (see, for example, Kim and Nelson 1999). This can be implemented by means of the well-known algorithms put forth in Carter and Kohn (1994) and Fr¨ uhwirth-Schnatter (1994). However, while still straightforward to implement, this increases the computational burden considerably. Conditional on the factors and the latent states {St }Tt=1 , the parameters of the state equation (2.2) can be simulated using simple Gibbs steps, iteratively sampling from the (conditional) posterior distributions of the parameters in Eq. (2.2). In practice, under the conjugate prior, this procedure is quite fast, implying that even if we increase the number of factors, computation does not become prohibitively slow. Unfortunately, the conditional posterior distributions of θSt , however, take no convenient form, rendering simple Gibbs updating steps infeasible. We thus opt for a random walk MetropolisHastings step.

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Sampling the latent regime indicators is simplified by the fact that we face a numerical integration problem with discrete support. We employ the filter put forward by Kim and Nelson (1999) and Amisano and Fagan (2013). The implementation of this procedure is described in detail in Appendix A. 2.3

Identification of the model

The model described above is econometrically unidentified. To achieve identification, we have to impose three different sets of restrictions on the model. The first involves a minimum set of normalization restrictions on the observation equation needed to identify the latent factors and the corresponding loadings. The second relates to the label switching problem that controls the prevailing phase of the business cycle. Finally, the identification of the structural shocks in the state equation requires further restrictions. Identification problems associated with the latent factors The factors and their loadings in Eqs.(2.1)-(2.2) are not separately identified. We identify the sign and the scale of the factors and the loadings by imposing a standard identification scheme commonly used in the literature on FAVAR models (see Bernanke et al., 2005). That is, we set the upper K × K block of Λf to an identity matrix and the upper K × N block of Λx to zero.6 Label switching problem Since the likelihood function of the model is invariant with respect to permutation of the labels of the Markov states we have an identification problem, known as the label switching problem. For a discussion of this problem and a review of different routes for dealing with this issue, see Fr¨ uhwirth-Schnatter (2001; 2006). It appears natural to interpret the two different regimes as different phases (recession and expansion) of the business cycle, and hence to impose constraints on certain elements of aSt to achieve identification.7 More specifically, we achieve identification by assuming that ajSt =0 > ajSt =1 .

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(2.18)

Note that this choice implies that our findings could be sensitive with respect to the ordering of the variables in y t . However, the robustness of our findings can be assessed quite easily by resorting to the two-step estimation approach mentioned in the previous subsection. This approach is order invariant and thus can be used to check the impact of different orderings of the variables contained in y t . It is worth noting that the two-step procedure leads to qualitatively similar insights for the application that follows. 7 See Kim and Nelson (1998), Kaufmann (2000), Billio et al. (2016) for a similar approach to identification.

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Here we let ajSt select the intercept term of the jth equation that is related to output growth, implying that the conditional mean is lower in a recessionary than in an expansionary regime.8 Structural identification Before describing the sign restrictions imposed on the impulse responses, it is worthwhile to outline the general identification strategy adopted in this paper. Noting that Eq. (2.2) presents the reduced form of the model, the (regime-specific) structural form of the model may be written as ˜0St z t = A

Q X

˜qSt z t−q + ε˜t . A

(2.19)

q=1

˜0St denotes a R × R matrix of impact coefficients, A ˜qSt (q = 1, . . . , Q) are R × R A matrices of lagged structural coefficients, and ε˜t are standard normally distributed ˜−1 structural errors. Multiplying with A 0St from the left yields the reduced form of the ˜−1 model in Eq. (2.2). Note that the reduced form errors are given by εt = A ˜t . We 0St ε impose sign restrictions to recover the structural shocks of our model. More specifically, we can decompose the regime-specific variance-covariance matrix ΣεSt as −1

0

−1

−1

−1

˜0S R ˜ St R ˜ S (A ˜0S )0 = A ˜0S (A ˜0S )0 ΣεSt = A t t t t t

(2.20)

˜ St with R ˜ St R ˜ 0S = I R . After for any R × R-dimensional orthonormal rotation matrix R t specifying a set of sign restrictions, we follow the approach outlined in Rubio-Ramirez et al. (2010) to sample rotation matrices. The exact sign restrictions and the implementation used are discussed later in more detail (see subsection 5.1). 3

Data and model evidence

3.1

Data description and specification issues

To estimate our model we use a panel of 110 monthly macroeconomic time series drawn from the FRED-MD, a macroeconomic database of 134 monthly US indicators (see McCracken and Ng, 2016). Our series range from 1959:01 to 2014:07 and may be classified into eight categories (number of series in parentheses): output and income (16), labor market (28), housing (5), consumption, orders and inventories (7), money and credit (10), interest and exchange rates (19), prices (16), and stock market (2). All data series are seasonally adjusted, if applicable, and transformed to be approximately stationary according to the transformation codes outlined in McCracken and Ng (2016). The list 8

Experimenting with other identification strategies based on the unconditional mean of the process and the average variance differences led to the same regime allocation.

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of the series is given in Appendix B. M = 103 variables are included in the y t vector, while we use the Federal Funds Rate, the credit spread (measured in terms of the spread difference of the yields on Baa-rated bonds minus the Federal Funds Rate), the industrial production index, the consumer price index, personal consumption expenses, average weekly hours (in manufacturing), and average hourly earnings (in construction) as observable variables in xt , implying N = 7. This choice closely resembles the variables typically included in medium-scale dynamic stochastic general equilibrium models (Christiano et al., 2005; Smets and Wouters, 2007). Since a wide variety of commonly identified indicators are already included in y t and xt we set wt = (y 0t , x0t )0 . This specification allows us to link the responses of the variables in y t and xt (with respect to a monetary policy shock) to the transition probabilities, effectively obtaining responses of recession probabilities. Before proceeding to the empirical results a brief word on the specification of the proposed model is necessary. Consistent with the vast majority of papers in the literature, the lag order is set equal to 13 (Q = 13). For the regime-specific shrinkage parameter θSt we specify the Gamma prior to be relatively uninformative with c0 = c1 = 0.01. Based on using the deviance information criterion (Spiegelhalter et al., 2002) as well as Bayesian and classical information criteria, we set the number of factors equal to four. The prior on the intercept vector in the state equation is normally distributed with zero mean and variance set equal to 102 , implying a high prior variance and relatively little shrinkage. Finally, since the data involved is (approximately) stationary we specify ai = 0 for all i. Since we standardize the variables in wt , the hyperparameters of the mixture normal priors are set equal to τ02 = 1 and τ12 = 0.1, while the prior variance on the constant is set equal to ϕ0 = ϕ1 = 102 . The prior on the free elements of Λ is equal to %20 = 10 and %21 = 0.1. Experimenting with different choices of %0 and %1 has led to qualitatively similar results. Finally, we take αj = β j = 0.01 to render this prior effectively noninfluential. Model estimation is based on the MCMC algorithm described in subsection 2.2. More specifically, we simulate a chain consisting of 70,000 draws where we discard the first 35,000 draws as burn-in. Inspection of selected trace plots and traditional convergence criteria indicates that our algorithm mixes well and shows satisfactory convergence properties.9 3.2

Model comparison

The model proposed is densely parameterized and possesses a set of additional features that can be used to conduct inference. In this section, we evaluate whether this additional flexibility is supported by the data in comparison to a set of nested specifications such as the VAR model, the MS-VAR model, the FAVAR model and the MS-FAVAR model with constant transition probabilities. 9

See the online appendix for additional information on the mixing and convergence properties of the algorithm.

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We assess the model performance in terms of the well-known log predictive score (LPS), motivated in Geweke and Amisano (2010). The main reason is that the onestep-ahead LPS can be interpreted as a training sample marginal likelihood, and thus implicitly rewards model fit and punishes model complexity. As training sample we use the period from 1959:01 to 1998:07, while the remaining part of the sample serves as a hold-out sample. After obtaining a set of one-step-ahead predictive densities we expand the initial estimation period until the end of the sample is reached. [Table 1 about here.] The results in the upper part of Table 1 indicate that for one-step-ahead forecasts of the seven variables considered, our MS-FAVAR model with time-varying transition probabilities (termed MS-FAVAR-2 in Table 1) improves the accuracy of density predictions relative to its nested model specifications in terms of marginal log predictive scores.10 Factor-augmentation improves scores in some cases (e.g., the industrial production index and the consumer price index) and lowers them in others (e.g., the credit spread). Incorporating time-varying transition probabilities typically improves density forecast accuracy by amounts that are sometimes sizable. The final row of Table 1 depicts the results obtained by evaluating the joint predictive density of xt obtained after marginalizing out the factors in the case of the FAVAR models. The findings here corroborate the results based on marginal LPS, i.e. allowing for time-variation in the transition distributions yields more precise predictive densities for the seven variables included in xt . Overall these results show that allowing for time-variation in the transition probabilities translates into improved forecast accuracy as measured by log predictive scores, underpinning our model choice. Inspection of the cumulative log predictive scores over time (not shown) suggests that accuracy gains stem from improved density predictions during the recent financial crisis. 4

Replicating US business cycle behavior

4.1

Evidence on the time-variation of transition probabilities

In this subsection we first present evidence on the time-variation in the transition probabilities and then check the degree of synchronization of the cycle extracted with our model and the NBER reference cycle. Figure 1 reports the posterior mean of the transition probabilities P rob(St = 1|St−1 = 0) in the lower part of the figure, and P rob(St = 0|St−1 = 1) in the upper part, i.e. the probability of entering a recession at time t when being in an expansion in t − 1 and the probability of getting out of a recessionary phase, respectively. This figure clearly indicates that transition probabilities change markedly, beginning with the recession in the early 1970s. This recession 10

The log predictive scores have been obtained by integrating out the remaining variables of the joint predictive density.

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was caused by sharp increases in government spending and energy prices, most notably the price of oil, leading to a stagflationary period within the US. The two recessions in the early 1980s were a consequence of the Federal Reserve’s pronounced regime shift, when chairman Paul Volcker started to fight inflation by increasing the policy rate dramatically. In 1990, the US experienced a relatively short period of negative growth caused by high oil prices, high debt levels and a low level of consumer confidence. Note that the probability of entering a downturn during that time period displays only a modest reaction while the probability of leaving a recessionary regime declines considerably. For the recession following the burst of the dot-com bubble and the September 11th attacks, the probabilities of moving into a recession increase only slightly whereas the probabilities of getting out of a recession decrease markedly. The rather weak increase in the probability of entering a recession might be due to the fact that this recession was by far the mildest one, merely resulting in an aggregate GDP loss of 0.3% from peak to trough. Finally, the last recession in our sample is the recent financial crisis, which led to sustained losses in output. Again, the model captures this period rather well, allocating high recession probabilities to the corresponding months. In conclusion, Fig. 1 clearly indicates that transition probabilities do vary over time and depend on the variables included in wt . However, we also see pronounced movements in the transition probabilities for periods which are not characterized as being a recession according to the NBER Business Cycle Dating Committee. [Fig. 1 about here.] To assess whether allowing for time-variation in the transition probabilities yields a model-based business cycle that is well in accordance with the NBER reference cycle, we compute the posterior distribution of the concordance statistics (Billio et al., 2016) for the MS-VAR model, the MS-FAVAR model with fixed transition probabilities and our MS-FAVAR model with time-varying transition probabilities. The concordance statistic (CS) is a non-parametric measure of the proportion of time during which two time series are in the same regime (expansion or recession). Originally proposed by Harding and Pagan (2002), the CS statistic is given by T 1X {(St St∗ ) + (1 − St ) (1 − St∗ )}, CS = T t=1

(4.1)

where T is the sample size. Recall that {St }Tt=1 is the model-implied regime indicator series taking the value zero when the series is in an expansion and unity when it is in a recession. The reference series {St∗ }Tt=1 , constructed by the Business Cycle Dating Committee of the NBER, is defined in the same way. The CS statistic ranges between zero and one, with zero marking perfectly countercyclical switches and one indicating perfect synchronization. 13

[Fig. 2 about here.] Figure 2 shows the posterior distribution of the CS statistic computed by taking draws of St and then using Eq. (4.1) to obtain the full posterior distribution. Comparing the posteriors of the MS-VAR and the MS-FAVAR models reveals that introducing more information leads to more synchronization with the reference cycle. This suggests that exploiting a broader panel of macroeconomic data (i.e., by moving from a 7-dimensional MS-VAR to a 110-dimensional MS-FAVAR) evidently improves the extracted cycle. Moving from a model with fixed transition probabilities to a specification that allows for time-variation in the transition distributions leads to an even stronger degree of synchronization between the model-based cycle and the reference cycle. Overall, the findings of this subsection provide evidence that time-varying transition probabilities also lead to regime-allocations that appear to be more in line with the reference business cycle regimes identified for the US. 4.2

Key determinants of US business cycle transitions

To answer the question which variables drive the transition between recessionary and expansionary regimes we calculate the corresponding posterior inclusion probabilities (PIPs) and the elasticities of transition probabilities at time t = T computed from the MCMC output. Table 2 lists the posterior distribution of the elasticities (with the corresponding 16th and 84th percentiles, henceforth referred to as credible sets) and the PIPs of the top 20 covariates included in Eq. (2.5). [Table 2 about here.] The table shows that the most important determinant for predicting a change in the business cycle at time t is the average weekly hours in manufacturing (AWHMAN) at time t − 1. This variable is included in 92 percent of all models sampled. Taking a look at the elasticities reveals that a one percent increase in the average of weekly hours in production leads to a 4.4 percent lower probability of moving into a recession at time t. This finding suggests that companies tend to act in a forward looking manner, reducing the amount of hours worked if the economic outlook deteriorates. The next important predictor is the lagged ISM Manufacturing: New Orders Index (NAPMNOI) with a PIP of 0.56. The elasticities indicate that if the growth in new orders increases by one percent, the risk of entering a recession in the next period decreases by 1.9 percent. The next two covariates that receive PIPs exceeding 0.5 are the industrial production index (INDPRO), the spread between AAA-rated bonds and the Federal Funds rate (AAAFFM). The table provides evidence that increases in the spread between highest quality corporate bonds and the policy rate have some predictive content for business cycle transitions. Specifically, we observe that an increase in the AAAFFM spread tends to reduce the probability of moving into a recession one period later by 1.76 percent.

14

5 5.1

The dynamic responses of the US economy to monetary policy shocks Specification of sign restrictions

In the spirit of Uhlig (2005) we identify a monetary policy shock by means of sign restrictions. Sign restrictions have been traditionally used within small to mediumscale VAR models (Uhlig, 2005; Dedola and Neri, 2007; Peersman and Straub, 2009). Recently, Mumtaz and Surico (2009) and Ahmadi and Uhlig (2015) implemented sign restrictions within a dynamic factor model framework closely related to our MS-FAVAR model with time-varying transition probabilities. Mumtaz and Surico (2009) introduced a mixture of sign and zero impact restrictions on the responses of the latent factors in Eq. (2.2), while Ahmadi and Uhlig (2015) restrict the responses of the observed variables in Eq. (2.1) only. Due to the large number of macroeconomic quantities that cover different segments of the economy, simple identification strategies based on zero impact restrictions are potentially problematic since they rule out contemporaneous relations between financial markets and the monetary policy authority. Sign restrictions overcome this issue by allowing for instantaneous reactions of key financial market quantities and the central bank. In this paper we impose restrictions on selected quantities in y t and xt along with average restrictions on groups of variables. We recover a monetary policy shock under the assumption that the corresponding impulse response vector is identified by imposing the restrictions described in Table 3. [Table 3 about here.] The first restriction rules out positive responses of the short-term interest rate while the restrictions on industrial production and its components capture the notion that production increases in response to monetary easing. Within our modeling framework, higher levels of industrial production are accompanied by a decline in the unemployment rate. Consistent with looser financial conditions and a higher level of available credit to private households, we constrain housing starts across the US to increase on average. Furthermore, selected monetary quantities are restricted to rise while selected interest rates are assumed to decrease. Finally, we rule out an initial price puzzle by restricting prices to rise to be in consistency with expansionary monetary policy. Technical implementation is achieved by using the algorithm proposed in Arias et al. (2014) which collapses to the method stipulated in Rubio-Ramirez et al. (2010) in the absence of zero restrictions. For each draw from the posterior distribution we draw a rotation matrix and assess whether the sign restrictions are satisfied. If this is the case, we keep the rotation matrix and store the associated structural coefficients whereas if the restrictions are not fulfilled we reject the draw and repeat the procedure. 5.2

Responses in different business cycle regimes

For the purpose of analyzing how monetary policy operates within the different regimes of the business cycle, we simulate a 100 basis points (bps) expansionary monetary 15

policy shock conditional on the regime, and trace its effect on a subset of six important macroeconomic and financial variables included in xt and y t . Figure 3 displays the impulse response functions. All plots include the 16th and 84th percentiles of the posterior distribution in light grey in addition to the median response (in black) for the next 60 months after impact. The first column of the figure displays the responses in expansions and the second the responses in recessions, while the third and final column shows the posterior distribution of the differences between the responses in expansions and recessions. [Fig. 3 about here.] The first two panels, (a) and (b), show rather strong immediate reactions of real activity in both business cycle stages. The increase in output growth (panel (a)) is mirrored by declines in the growth rate of unemployment (panel (b)) during the very first few months under consideration. It is noteworthy that the sign of the responses is consistent with standard New Keynesian dynamic stochastic general equilibrium models (see, for example, Christiano et al., 2005; Smets and Wouters, 2007) and with the empirical literature on the impact of monetary policy within a FAVAR framework (see Bernanke et al., 2005; Boivin et al., 2009; Korobilis, 2013). The posterior distribution of the differences between expansions and recessions (see the final column of Fig. 3(a) and (b)) indicates that for both, output and unemployment, zero is included within the 16th and 84th credible set. It is worth noting that this does not imply that there is (necessarily) no discernible difference between regimes, but only that the possibility is not rejected, or is potentially consistent with the data. Turning to the responses of the credit spread in panel (c) suggests that expansionary monetary policy lowers credit spreads over the first two years in expansions. In recessions, credit spreads also decline but the effect appears to be rather short-lived. The impact magnitudes imply a decline by 2.5 percentage points in both regimes, quickly returning to the baseline value in the recessionary regime while falling more persistently in an expansion. Short-run responses appear to be similar in magnitude whereas the medium-run reactions tend to differ markedly across down- and upturns (see the final column of Fig. 3(c)). This result corroborates established findings which report strong evidence in favor of the credit channel of monetary policy, indicating that decreasing yields on corporate debt lead to an increase in investments that in turn lift economic activity (see Bernanke and Gertler, 1995; Gilchrist et al., 2009; Gilchrist and Zakrajˇsek, 2012). This finding also carries over to the dynamic responses of the term spread (panel (d)). The shapes of the impulse response functions again depend on the stage of the business cycle. In expansions we observe a rather long-lasting response, with regimespecific differences providing evidence for changes in the transmission channel. These differences can be traced back to institutional constraints that force investors to reduce the duration of their fixed income portfolios during recessions, leading to a steepening of the long end of the yield curve (see Eickmeier and Hofmann, 2013; Baumeister and Benati, 2013). 16

It is worth emphasizing that the similarity between panels (c) and (d) stems from the fact that one of the factors in f t is highly correlated with selected interest rates and spreads (IDs 74-88 in Table B in the Appendix), with the associated factor being strongly loaded by financial market quantities. As a consequence, the responses of such quantities are similar and display the same persistent pattern during expansions while being rather short-lived in recessionary episodes. The differences in impulse responses appear to be substantial in the medium term. This result is consistent with recent findings in Eickmeier et al. (2016). We conjecture that these differences stem from a shift in the functioning of the balance sheet transmission channel (Bernanke and Gertler, 1989; Bernanke et al., 1999) alongside a possible change in the risk taking transmission mechanism (Campbell and Cochrane, 1999). By construction, the Federal Funds rate is reduced by 100 bps on impact. As presented in panel (e), this effect dies out after around two months in both regimes. Interestingly, we find some evidence that the short-term interest rate overshoots its baseline value during recessions after around one quarter, with a high degree of estimation uncertainty.11 Since our model exploits a large information set, we expect prices (panel (f)) to increase without producing a considerable price puzzle (Sims, 1992). The impact responses seem to confirm this conjecture. However, as implied by our identification strategy, immediate negative price reactions are effectively ruled out. Finally, panels (g) and (h) depict the dynamic responses of housing starts and the S&P 500. We observe a positive relationship between expansionary monetary policy and housing conditions, with the positive reactions of housing starts being slightly longer lasting during expansionary stages of the business cycle as compared to the reactions observed in recessions. Turning to the responses of equity prices yields only limited evidence.12 So far we have focused on a small subset of the 110 variables included in the model. The specification of wt = (y 0t , x0t )0 allows us to compute the endogenous response of recession probabilities in both regimes, providing an aggregate measure of how monetary policy affects the wider economy, taking into account the dynamic responses of all elements in wt . This is possible by computing the response of the latent variable in Eq. (2.5), and then exploiting basic properties of the probit model to compute the dynamic responses of recession probabilities. The non-linear nature of the probit model implies that the specific responses depend on the current level of wt . For facilitating easier interpretation, we compute the impulse response function at the mean of wt . The (regime-specific) responses of wt allow us 11

The transient response of the short-term interest rates is not due to our identification strategy but possibly caused by shrinking the coefficients of the first lag of the Federal Funds rate in the monetary policy rule to zero. 12 One question that typically arises is whether our results are driven by the identification scheme adopted or whether a looser identification strategy would lead to the same insights. To address this issue we explored the robustness of our findings with respect to a weaker identification scheme imposed exclusively on the responses of xt while leaving the remaining responses unrestricted. The corresponding responses are similar to the ones reported here.

17

to compute a hypothetical scenario that provides information on how the probability of entering a recession (see panel (a) in Fig. 4) and the probability of getting out of a recession (see panel (b) in Fig. 4) changes in response to a 100 bps expansionary monetary policy shock. [Fig. 4 about here.] The shaded areas, again, represent the credible intervals of the posterior distribution of impulse responses. Figure 4 reveals two noteworthy features. First, panel (a) indicates that expansionary monetary policy reduces the probability of entering a recession. Second, panel (b) provides evidence that the impact of monetary policy on the probability of getting out of a recession is rather limited, corroborating the findings provided in Eickmeier et al. (2016) who argue that if uncertainty increases, the balance sheet transmission mechanism becomes less effective since banks decrease leverage ratios, effectively limiting the ability of the central bank to influence funding conditions. 6

Closing remarks

In the present paper we propose a novel MS-FAVAR model with time-varying transition probabilities. The proposed approach combines Markov switching VAR models with factor-augmented VARs, leading to a high-dimensional non-linear model that enables the analysis of large datasets. Overfitting issues are solved by relying on a set of hierarchical shrinkage priors in the Minnesota tradition. A probit model is adopted to parameterize a two regime Markov switching process with endogenous transition probabilities. The transition distributions are driven by a large set of predictors that may influence the regime switching behavior of the model. This is achieved by parameterizing the transition distributions with a probit model and using a Bayesian shrinkage prior to identify the most promising covariates. The results achieved show that the specific combination of large datasets, a nonlinear modeling approach, and time-varying transition structures translates into a better description of the nexus between the US business cycle and monetary policy. Considering an expansionary monetary policy shock reveals pronounced effects in expansions and slightly weaker effects in recessions, mainly driven by a change in the balance sheet transmission channel of monetary policy. The key determinant of the differences between responses is state-dependent duration effects of financial market reactions. Acknowledgments: The authors gratefully acknowledge the valuable advice of two anonymous reviewers, Gregor Kastner, and the Associate Editor, Jonathan Temple.

References Ahmadi PA and Uhlig H (2015) Sign restrictions in Bayesian FAVARs with an application to monetary policy shocks. NBER Working Paper Series, No. 21738 18

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22

23 2,819.04

-54.78 -81.65 636.45 37.36 715.68 723.13 740.33

VAR

2,826.91

-54.28 -87.53 652.64 36.69 714.58 726.00 743.72

FAVAR

2,816.38

-54.92 -81.73 637.17 37.66 715.83 722.80 740.26

MS-VAR

2,826.66

-54.26 -87.28 652.80 36.43 714.30 725.43 744.02

MS-FAVAR-1

2,962.53

43.76 -41.84 661.87 48.35 720.07 727.53 751.90

MS-FAVAR-2

Notes: VAR stands for a vector autoregressive model with a Minnesota prior, FAVAR for a factor-augmented VAR model, MS-VAR for a Markov switching VAR model, MS-FAVAR-1 for a Markov switching FAVAR model with constant transition probabilities, and MS-FAVAR-2 for the Markov switching FAVAR model with time-varying transition probabilities. See Appendix B for definition of the variables.

Sum of LPS

FEDFUNDS BAAFFM INDPRO AWHMAN CES2000000008 DPCERA3M086SBEA CPIAUCSL

Variables

Table 1: Out-of-sample performance in terms of log predictive scores (LPS), 1998:08 to 2014:07

Table 2: Posterior distribution of elasticities along with posterior inclusion probabilities Elasticities

Low0.16

Mean

High0.84

PIP

AWHMAN NAPMNOI INDPRO AAAFFM

-6.66 -4.12 -4.21 -4.33

-4.36 -1.90 -1.63 -1.76

-1.97 -0.04 0.21 0.17

0.92 0.56 0.53 0.52

AAA IPFINAL IPFPNSS IPDCONGD CES0600000007 NAPM T10YFFM NAPMEI IPMAT BAAFFM BAA IPCONGD IPMANSICS T5YFFM HOUST W875RX1

-0.03 -0.33 -0.35 -2.08 -0.96 -0.62 -1.38 -0.24 -1.91 -1.35 -0.09 -1.85 -0.65 -1.00 -0.41 -0.09

1.08 1.18 1.05 -0.96 0.28 0.63 -0.17 0.87 -0.67 -0.27 0.81 -0.70 0.37 -0.08 0.48 0.72

2.36 3.18 2.94 0.07 1.54 2.24 0.81 2.32 0.28 0.77 1.83 0.34 1.31 0.74 1.42 1.52

0.45 0.45 0.42 0.42 0.41 0.40 0.40 0.40 0.40 0.39 0.38 0.38 0.36 0.36 0.35 0.34

Notes: This table lists the posterior mean of the elasticities produced by the probit model, along with the 16th and 84th percentiles. The first column denotes the mnemonics (see Appendix B for a definition) and the last column the posterior inclusion probabilities.

24

Table 3: Sign restrictions imposed on the impact responses Groups of macroeconomic time series ID Federal Funds Rate 1 Industrial production & components 3; 10-23 Unemployment rate 26 Housing 52-56 Monetary quantities 64-71; 73 Selected interest rates 79; 81-87 Selected price indices 7; 105-108

Monetary policy shock − + − + + − +

Notes: Restrictions on different groups of macroeconomic time series are imposed on the average response of the time series in a given group.

25

1.0

Transition probabilities

0.8

0.6

0.4

0.2

0.0 1960

1970

1980

1990

2000

2010

Notes: Posterior mean of transition probabilities: P rob(St = 1|St−1 = 0) in grey (lower part of the figure) and, P rob(St = 0|St−1 = 1) in black (upper part of the figure). Shaded areas refer to recessions dated by the Business Cycle Dating Committee of the National Bureau of Economic Research (www.nber.org). Results are based on 35,000 posterior draws.

Fig. 1: Posterior mean of transition probabilities over time

26

Fig. 2: Posterior distribution of concordance statistics of our model in comparison to the MS-VAR, and the MS-FAVAR with fixed transition probabilities, all relative to the NBER reference cycle

27

Expansion

Difference

Recession (a) INDPRO

8.00 5.14 2.28 −0.58 −3.44 0

5

10 15 20 25 30 35 40 45 50 55 Months

0

4

8

12

20

28 Months

36

44

52

0

4

8

12

20

28 Months

36

44

52

52

0

4

8

12

20

28 Months

36

44

52

52

0

4

8

12

20

28 Months

36

44

52

52

0

4

8

12

20

28 Months

36

44

52

52

0

4

8

12

20

28 Months

36

44

52

52

0

4

8

12

20

28 Months

36

44

52

(b) UNRATE 0.27 0.04 −0.20 −0.43 −0.66 0

5

10 15 20 25 30 35 40 45 50 55 Months

0

4

8

12

20

28 Months

36

44

(c) BAAFFM 5.66 1.25 −3.16 −7.58 −11.99 0

5

10 15 20 25 30 35 40 45 50 55 Months

0

4

8

12

20

28 Months

36

44

(d) T10YFFM 3.63 0.93 −1.78 −4.48 −7.18 0

5

10 15 20 25 30 35 40 45 50 55 Months

0

4

8

12

20

28 Months

36

44

(e) FEDFUNDS 7.15 4.15 1.16 −1.84 −4.84 0

5

10 15 20 25 30 35 40 45 50 55 Months

0

4

8

12

20

28 Months

36

44

(f ) CPIAUCSL 2.62 1.53 0.43 −0.67 −1.77 0

5

10 15 20 25 30 35 40 45 50 55 Months

0

4

8

12

20

28 Months

28

36

44

Expansion

Difference

Recession (g) HOUST

0.96 0.60 0.24 −0.12 −0.48 0

5

10 15 20 25 30 35 40 45 50 55 Months

0

4

8

12

20

28 Months

36

44

52

0

4

8

12

20

28 Months

36

44

52

52

0

4

8

12

20

28 Months

36

44

52

(h) S&P 500 0.08 0.05 0.02 −0.01 −0.04 0

5

10 15 20 25 30 35 40 45 50 55 Months

0

4

8

12

20

28 Months

36

44

Notes: The solid black line is the median response and the grey shaded area represents the 16th and 84th percentiles. The first column of the figure relates to expansions, the second to recessions, while the final to the difference between the posterior distributions of both. The dashed line indicates the zero line. For the definition of the variables, see Appendix B.

Fig. 3: Dynamic responses of selected macroeconomic and financial quantities to a 100 basis points monetary policy shock

29

(a) P rob(St = 1|St−1 = 0)

(b) P rob(St = 0|St−1 = 1)

4.3

4.3

−2.6

2.9

−9.4

1.5

−16.3

0.1

−23.2

−1.3 0

5

10 15 20 25 30 35 40 45 50 55

0

Months

5

10 15 20 25 30 35 40 45 50 55 Months

Notes: The solid black line is the median response and the grey shaded area represents the 16th and 84th percentiles. Panel (a) relates to the changes in the probability of entering a recession and panel (b) to get out of a recession. The dashed line indicates the zero line.

Fig. 4: Dynamic responses of endogenous transition probabilities to a negative 100 basis points monetary policy shock: (a) to enter a recession, and (b) to get out of a recession

30

Appendix A

Posterior distributions

This appendix provides details on the corresponding posterior distributions and how to simulate them. In what follows, π t = (π 01 , . . . , π 0t )0

(A.1)

denotes the entire history of a generic vector π up to time t, and Πt = (vec(Π1 )0 , . . . , vec(ΠT )0 )0

(A.2)

the history of a generic matrix Π up to time T . Moreover, let us use the following notation to indicate estimates of some random quantity χ based on information available at time t, χt|t = E(χt |It ), (A.3) with It denoting a generic information set. Accordingly, we denote a forecast of χ by χt+1|t = E(χt+1 |It ).

(A.4)

Conditional posterior distributions for the state equation The (conditional) posterior distributions of the parameters in Eq. (2.6) take a particularly simple form, vec(ASt )|ΞT , DT ∼ N (vec(ASt ), ΣSt ⊗ V ASt ) T

T

ΣεSt |Ξ , D ∼ IW(ΨSt , v St ),

(A.5) (A.6)

where ΞT stores the remaining parameters, regime indicators and latent factors, while DT denotes the available data up to time T . The posterior moments for ASt are given by 0

0

ASt = (D St D St )−1 D St Z St

(A.7)

0 (D St D St )−1

(A.8)

V St =

with D St = (D 0St , D 0St )0 and Z St = (Z 0St , Z 0St )0 . The posterior scale matrix of ΣεSt , ΨSt is given by ΨSt = (Z St − D 0St ASt )0 (Z St − D 0St ASt ). (A.9) The posterior degrees of freedom are v St = TSt + v.

31

The matrices Z St and D St are so-called “dummy”-observations constructed as follows (see Ba´ nbura et al., 2010),   diag(a1 σ1 , . . . , aR σR )/θSt       0R(Q−1)×R    (A.10) Z St =      diag(σ , . . . , σ ) 1 R     01×R  D St

J Q ⊗ diag(σ1 , . . . , σR )/θSt 0RQ×1

  =  0R×RQ  01×RQ

0R×1 $−1/2

   ,  

(A.11)

with J Q = (1, . . . , Q)0 and $ denoting a hyperparameter that controls the tightness of the prior on the intercept term. Loosely speaking, the first part of the matrices in Eqs. (A.10) and (A.11) implement the prior on the coefficients associated with the lags of z t while the second block the prior on ΣεSt and the final block the prior on the intercept term. For the shrinkage hyperparameter θSt we use a standard random walk MetropolisHastings algorithm with proposal distribution given by ln(θS∗ t ) ∼ N (ln(θSj t ), rSt ).

(A.12)

Here θS∗ t is the proposed value, θSj t the last accepted value and rSt is a scaling factor that is specified to result in an acceptance rate of 20 to 40 percent by adjusting it on-line based on the first 10% of the burn-in stage. We accept a candidate for θS∗ t with probability, P rob(θS∗ t , θSj t )

=

p(Z St |θSt = θS∗ t ) p(θSt = θS∗ t ) θS∗ t p(Z St |θSt = θSj t ) p(θSt = θSj t ) θSj t

,

(A.13)

where p(Z St |θSt = θS∗ t ) is a regime-specific marginal likelihood and the term θS∗ t /θSj t stems from the non-symmetry of the proposal density.

32

Conditional posterior distributions for the probit model The parameters of the latent regression model obey posterior distributions which are of a well-known form (George and McCulloch, 1993), namely a normal distribution for ˜ = (γi0 , γ 0 )0 , γ ˜ |ΞT , DT ∼ N (γ, V γ ), γ

(A.14)

˜ 0 H) ˜ −1 Vγ = (w ˜0w ˜+H

(A.15)

where

γ = Vγ (w ˜ 0 r).

(A.16)

Consistent with the notation used above w ˜ and r are the corresponding full-data coun˜ = diag(h1 , . . . , hG , ϕ0 , ϕ1 ) be the prior terparts of w ˜ t = (w0t , 1, St )0 and r t . We let H variance-covariance matrix augmented with the prior variances on the regime-specific intercepts. The prior variances on the slope coefficients, hg , are set such that ( τ02 if δg = 1 (A.17) hg = τ12 if δg = 0. The posterior of δg follows a Bernoulli distribution, δg ∼ Bernoulli (pg ), with the corresponding posterior probability given by   1 1 γg 2 pg exp − ( ) τ0 2 τ0     pg = . 1 1 γg 2 1 1 γg 2 exp − 2 ( τ0 ) pg + τ1 exp − 2 ( τ1 ) (1 − pg ) τ0

(A.18)

(A.19)

The posterior of rt takes a particularly simple distributional form, namely a truncated standard normal distribution as described in Albert and Chib (1993). Conditional posterior distributions for the observation equation Since we assume that the variance-covariance matrix associated with the innovations in Eq. (2.1) is diagonal and in light of the restrictions described in subsection 2.3, the conditional posterior for Λ is described exclusively in terms of the remaining M − K rows of Λ, Λj• |ΞT , DT ∼ N (Λj• , V Λj• ), (A.20)

33

where Λj• selects the jth row of Λ for K < j ≤ M . The corresponding posterior moments are given by V Λj• = (ςj−1 f 0 f + L0j Lj )−1 0

Λj• = V Λj• (ςj−1 f Y •j ).

(A.21) (A.22)

Here, f = (f 1 , . . . , f T )0 , Lj denotes the block of L = diag(l1 , . . . , lM (K+N ) ) associated with the coefficients of the jth row in Eq. (2.1), and Y •j selects the jth column of a T × M matrix Y = (y 1 , . . . , y T )0 . The elements of L, lj (j = 1, . . . , M (K + N )) are defined as ( %20 if ιj = 1 lj = (A.23) %21 if ιj = 0. The posterior of ιj is Bernoulli distributed with the corresponding posterior probability ρj given by   1 1 ιj 2 exp − 2 ( %0 ) ρj %0     ρj = . (A.24) 1 1 ιj 2 1 1 ιj 2 exp − ( ) ρ + exp − ( ) (1 − ρ ) j j %0 2 %0 %1 2 %1 Sampling the latent factors f t The latent factors are obtained by using the well-known algorithm put forth in Carter and Kohn (1994) and Fr¨ uhwirth-Schnatter (1994). The density of f t can be factored as T

T

T

T

T

p(f |Ξ , D ) = p(f T |Ξ , D )

T −1 Y

p(f t |f t+1 , ΞT , DT ),

(A.25)

t=1

where the moments are given by f t |f t+1 , ΞT , DT ∼ N (f t|t+1 , Ωt|t+1 )

(A.26)

f t|t+1 = E(f t |f t+1 , ΞT , DT )

(A.27)

Ωt|t+1 = V ar(f t |f t+1 , ΞT , DT ).

(A.28)

If f t|t+1 and Ωt|t+1 are available, the full history of the latent factors can be sampled in a straightforward fashion from N (f t|t+1 , Ωt|t+1 ). f t|t+1 and Ωt|t+1 are obtained using Kalman filtering and the corresponding backward recursions. More specifically, let us assume without loss of generality that Q equals one and no observable quantities are included. Then Eq. (2.6) can be rewritten as f t = A1St f t−1 + εt .

34

(A.29)

In addition, the observation equation (2.1) collapses to y t = Λf f t + et .

(A.30)

Conditional on f 0|0 and Ω0|0 , the Kalman filter produces f t|t−1 = A1St f t−1|t−1 Ωt|t−1 = Kt =

(A.31)

A1St Ωt−1|t−1 A01St + Σεt Ωt|t−1 (Λf )0 (Λf Ωt|t−1 (Λf )0

(A.32) + Σe )

−1

(A.33)

f t|t = f t|t−1 + K t (y t − Λf f t|t−1 )

(A.34)

Ωt|t = Ωt|t−1 − K t Λf K 0t Ωt|t−1 .

(A.35)

Note that at time t = T we obtain f T |T and ΩT |T , which permits us to sample f T . This draw of f T , in conjunction with f T |T and ΩT |T is then used to obtain f t|t+1 and Ωt|t+1 until time t = 0 is reached. The corresponding recursions are given by f t|t+1 = f t|t + Ωt|t A01St Ω−1 t+1|t (f t+1 − A1St f t|t )

(A.36)

−1 0 Ωt|t+1 = Ωt|t − Ωt|t A01St Ω−1 t+1|t A1St Ωt|t .

(A.37)

Sampling the regime indicators St Following Kim and Nelson (1999) and Amisano and Fagan (2013) we obtain the filtered and predicted probabilities, pˆjt|t = P rob(St = j|Ξt , Dt ) and pˆit+1|t = P rob(St = i|Ξt , Dt ) through a standard filter (Kim and Nelson, 1999). The corresponding prediction and updating probabilities are given by pˆjt+1|t =

2 X

pij,t|t pˆjt|t

(A.38)

i=1

pˆjt+1|t+1 = P2

pˆjt+1|t p(z t+1 |A1St+1 =j , ΣεSt+1 =j)

h=1

pˆht+1|t p(z t+1 |A1St+1 =h , ΣεSt+1 =h )

.

(A.39)

The filtered probabilities are then used in the next step to sample the full history of regime indicators S T . Similar to the decomposition of the joint conditional density of the latent factors, it is possible to use the following factorization, T

T

T

T

T

p(S |Ξ , D ) = p(ST |Ξ , D )

T −1 Y

p(St |St+1 , ΞT , DT ),

(A.40)

t=1

where p(ST |f T , ΞT , DT ) is obtained from the final iteration of the Hamilton (1989) filter. St conditional on St+1 and the remaining parameters can be obtained in a straightfor-

35

ward fashion by noting that p(St |St+1 , ΞT , DT ) ∝ p(St+1 |St )p(St |ΞT , DT ).

(A.41)

The first term on the right hand side refers to the transition probability and the second term is obtained from the Hamilton filter. Thus p(St |St+1 , ΞT , DT ) can be obtained by iterating backwards until time t = 0 is reached. To be more precise, pˆit|t pij,t+1 P rob(St = i|St+1 = j, ΞT , DT ) = P2 . ˆht|t phj,t+1 h=1 p

(A.42)

Finally, the corresponding transition probabilities pij,t are obtained straightforwardly through Eq. (2.4).

36

Appendix B

Data description

The time series used to construct the vectors xt and yt are presented in Tables B.1 and B.2, respectively. The format is as follows: series number, transformation code, series mnemonic and brief series description. The transformation codes are 1=no transformation, 2=first difference, 3=second difference, 4=logarithm, 5=first difference of logarithm, 6=second difference of logarithm, and 7=change in percentage change.

Table B.1: Data series used for xt in Eq. (2.1) ID

tcode

Fred

Description

1 2 3 4 5 6 7

2 1 5 1 6 5 6

FEDFUNDS BAAFFM INDPRO AWHMAN CES2000000008 DPCERA3M086SBEA CPIAUCSL

Effective Federal Funds Rate Moody’s Baa Corporate Bond Minus FEDFUNDS IP Index Avg. Weekly Hours: Manufacturing Avg. Hourly Earnings: Construction Real Personal Consumption Expenditures CPI: All Items

Table B.2: Data series used for yt in Eq. (2.1) ID Output 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

tcode

Fred

and income 5 RPI 5 W875RX1 5 IPFPNSS 5 IPFINAL 5 IPCONGD 5 IPDCONGD 5 IPNCONGD 5 IPBUSEQ 5 IPMAT 5 IPDMAT 5 IPNMAT 5 IPMANSICS 5 IPB51222s 5 IPFUELS 1 NAPMPI 2 CUMFNS

Labor market 24 5 25 5 26 2 27 2 28 5 29 5 30 5 31 5

CLF16OV CE16OV UNRATE UEMPMEAN UEMPLT5 UEMP5TO14 UEMP15OV UEMP15T26

Description Real Personal Income Real Personal Income Less Transfer Receipts IP: Final Products and Nonindustrial Supplies IP: Final Products (Market Group) IP: Consumer Goods IP: Durable Consumer Goods IP: Nondurable Consumer Goods IP: Business Equipment IP: Materials IP: Durable Materials IP: Nondurable Materials IP: Manufacturing (SIC) IP: Residential Utilities IP: Fuels ISM Manufacturing: Production Index Capacity Utilization: Manufacturing

Civilian Labor Force Civilian Employment Civilian Unemployment Rate Average Duration of Unemployment (Weeks) Civilians Unemployed - Less Than 5 Weeks Civilians Unemployed for 5-14 Weeks Civilians Unemployed - 15 Weeks & Over Civilians Unemployed for 15-26 Weeks

37

Table B.2 ctd. ID

tcode

Fred

Description

32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51

5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 1 2 1 6 6

UEMP27OV CLAIMSx PAYEMS USGOOD CES1021000001 USCONS MANEMP DMANEMP NDMANEMP SRVPRD USTPU USWTRADE USTRADE USFIRE USGOVT CES0600000007 AWOTMAN NAPMEI CES0600000008 CES3000000008

Civilians Unemployed for 27 Weeks and Over Initial Claims All Employees: Total Nonfarm All Employees: Goods-Producing Industries All Employees: Mining and Logging: Mining All Employees: Construction All Employees: Manufacturing All Employees: Durable Goods All Employees: Nondurable Goods All Employees: Service-Providing Industries All Employees: Trade, Transportation & Utilities All Employees: Wholesale Trade All Employees: Retail Trade All Employees: Financial Activities All Employees: Government Avg. Weekly Hours: Goods-Producing Avg. Weekly Overtime Hours: Manufacturing ISM Manufacturing: Employment Index Avg. Hourly Earnings: Goods-Producing Avg. Hourly Earnings: Manufacturing

HOUST HOUSTNE HOUSTMW HOUSTS HOUSTW

Housing Housing Housing Housing Housing

Housing 52 4 53 4 54 4 55 4 56 4

Starts: Starts, Starts, Starts, Starts,

Total New Privately Owned Northeast Midwest South West

Consumption, orders and inventories 57 5 RETAILx 58 1 NAPM 59 1 NAPMNOI 60 1 NAPMSDI 61 1 NAPMII 62 5 AMDMNOx 63 5 AMDMUOx

Retail and Food Services Sales ISM: PMI Composite Index ISM: New Orders Index ISM: Supplier Deliveries Index ISM: Inventories Index New Orders for Durable Goods Unfilled Orders for Durable Goods

Money 64 65 66 67 68 69 70 71 72 73

M1 Money Stock M2 Money Stock Real M2 Money Stock St. Louis Adjusted Monetary Base Total Reserves of Depository Institutions Reserves of Depository Institutions Commercial and Industrial Loans Real Estate Loans at All Commercial Banks MZM Money Stock Securities in Bank Credit at All Commercial Banks

and credit 6 M1SL 6 M2SL 5 M2REAL 6 AMBSL 6 TOTRESNS 7 NONBORRES 6 BUSLOANS 6 REALLN 6 MZMSL 6 INVEST

Interest 74 75 76 77 78 79

and exchange rates 2 CP3Mx 2 TB3MS 2 TB6MS 2 GS1 2 GS5 2 GS10

3-Month AA Financial Commercial Paper Rate 3-Month Treasury Bill 6-Month Treasury Bill 1-Year Treasury Rate 5-Year Treasury Rate 10-Year Treasury Rate

38

Table B.2 ctd. ID

tcode

Fred

Description

80 81 82 83 84 85 86 87 88 89 90 91 92

2 2 1 1 1 1 1 1 1 5 5 5 5

AAA BAA COMPAPFFx TB3SMFFM TB6SMFFM T1YFFM T5YFFM T10YFFM AAAFFM EXSZUSx EXJPUSx EXUSUKx EXCAUSx

Moody’s Seasoned Aaa Corporate Bond Yield Moody’s Seasoned Baa Corporate Bond Yield 3-Month Commercial Paper Minus FEDFUNDS 3-Month Treasury C Minus FEDFUNDS 6-Month Treasury C Minus FEDFUNDS 1-Year Treasury C Minus FEDFUNDS 5-Year Treasury C Minus FEDFUNDS 10-year Treasury C Minus FEDFUNDS Moody’s Aaa Corporate Bond Minus FEDFUNDS Switzerland / U.S. Foreign Exchange Rate Japan / U.S. Foreign Exchange Rate U.S. / U.K. Foreign Exchange Rate Canada / U.S. Foreign Exchange Rate

Prices 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108

6 6 1 6 6 6 6 6 6 6 6 6 6 6 6 6

OILPRICEx PPICMM NAPMPRI CPIAPPSL CPITRNSL CPIMEDSL CUSR0000SAC CUUR0000SAD CUSR0000SAS CPIULFSL CUUR0000SA0L2 CUSR0000SA0L5 PCEPI DDURRG3M086SBEA DNDGRG3M086SBEA DSERRG3M086SBEA

Crude Oil, Spliced WTI and Cushing PPI: Metals and Metal Products: ISM Manufacturing: Prices Index CPI: Apparel CPI: Transportation CPI: Medical Care CPI: Commodities CPI: Durables CPI: Services CPI: All Items Less Food CPI: All items Less Shelter CPI: All items Less Medical Care Personal Consumption Expend.: Chain Index Personal Consumption Expend.: Durable Goods Personal Consumption Expend.: Nondurable Goods Personal Consumption Expend.: Services

S&P 500 S&P: Indust

S&P’s Common Stock Price Index: Composite S&P’s Common Stock Price Index: Industrials

Stock Market 109 5 110 5

39

Online Appendix Mixing and convergence properties of the Markov chain Monte Carlo algorithm In order to assess how well the Markov chain mixes, we consider the inefficiency factors for the posterior estimates of all the different sets of coefficients. The inefficiency factor (IF) is defined as the inverse of the relative numerical efficiency measure of Geweke (1992). We follow Primiceri (2005) to calculate the estimate in this application using a four percent tapered window for the estimation of the spectral density at frequency zero. The table below summarizes the distribution of the inefficiency factors for the posterior estimates of the different sets of coefficients. Inefficiency factors below 20 are generally considered to be satisfactory. Note that this holds true for most quantities with a few outlying observations. The IFs are remarkably low for the free elements of the factor loading matrix (λ) and the VAR coefficients in A0 .

Inefficiency factors

Required number of runs

Low0.1

Median

High0.9

Low0.1

Median

High0.9

A0 A1 λ γ

1.46 2.36 1.18 9.35

3.73 11.10 1.38 15.96

11.20 77.29 2.23 32.36

3,700 3,742 3,689 11,844

3,802 3,865 3,782 22,044

3,929 8,280 4,017 42,173

Σε0 Σε1 Σe

2.63 5.92 1.03

7.12 19.36 1.08

35.47 80.46 1.16

3,740 3,761 3,680

3,823 3,908 3,761

4,084 11,667 3,844

Notes: Aj (j = 0, 1) denotes the VAR coefficients in state equation (2.2), λ are the free elements of the factor loadings matrix in Eq. (2.1), γ are the coefficients of the probit regression Eq. (2.5), Σεj (j = 0, 1) are the coefficients of the variance-covariance matrix of the state innovations and Σe are the elements of the variance-covariance matrix of the measurement errors. Low0.1 and High0.9 denote the 10th and 90th percentiles, respectively.

As a check for the convergence we consider the Raftery and Lewis (1992) diagnostic of the total number of runs required to achieve a certain precision, and specify the parameters of this diagnostic according to Primiceri (2005) as follows: quantiles equal to 0.025, the desired degree of accuracy is 0.025, the required probability of attaining the required accuracy is 0.95. The table indicates that the required number of iterations is far below the total number of iterations in almost all cases (except γ). In face of the very high dimensionality of the problem at hand the convergence statistics appear to be satisfactory. 40