Measuring international uncertainty using global vector autoregressions with drifting parameters
MMeasuring international uncertainty using global vectorautoregressions with drifting parameters
MICHAEL PFARRHOFER ∗ University of Salzburg
This paper investigates the time-varying impacts of international macroeco-nomic uncertainty shocks. We use a global vector autoregressive specificationwith drifting coefficients and factor stochastic volatility in the errors to modelsix economies jointly. The measure of uncertainty is constructed endogenouslyby estimating a scalar driving the innovation variances of the latent factors,which is also included in the mean of the process. To achieve regularization,we use Bayesian techniques for estimation, and introduce a set of hierarchicalglobal-local priors. The adopted priors center the model on a constantparameter specification with homoscedastic errors, but allow for time-variationif suggested by likelihood information. Moreover, we assume coefficients acrosseconomies to be similar, but provide sufficient flexibility via the hierarchicalprior for country-specific idiosyncrasies. The results point towards pronouncedreal and financial effects of uncertainty shocks in all countries, with differencesacross economies and over time.
JEL : C11, C55, E32, E66, G15
KEYWORDS : Bayesian state-space modeling, hierarchical priors, factorstochastic volatility, stochastic volatility in mean ∗ Salzburg Centre of European Union Studies, University of Salzburg.
Address : M¨onchsberg 2a, 5020Salzburg, Austria.
Email : [email protected].
Phone : +43 662 8044 3772. The author thanksMartin Feldkircher, Sylvia Fr¨uhwirth-Schnatter, Niko Hauzenberger, Florian Huber, Gregor Kastner andAnna Stelzer for valuable comments and suggestions. Funding from the Austrian Science Fund (FWF) forthe project “High-dimensional statistical learning: New methods to advance economic and sustainabilitypolicies” (ZK 35) is gratefully acknowledged. a r X i v : . [ ec on . E M ] D ec . INTRODUCTION Uncertainty has received a substantial amount of attention as a driving force of businesscycle fluctuations, following the experiences of economists and policy makers in the af-termath of the Great Recession. Measuring uncertainty and its impact on the economyproduced a voluminous literature, with prominent contributions including Bloom (2009),Jurado et al. (2015), Baker et al. (2016), Basu and Bundick (2017), Carriero et al. (2018b),among many others.These studies provide compelling theoretical and empirical evidence suggesting neg-ative economic consequences of uncertainty shocks. Elevated levels of uncertainty canproduce large drops in economic activity, and moreover render counteracting monetaryand fiscal policies less effective (see, e.g. Aastveit et al. , 2013; Bertolotti and Marcellino,2019). Transmission channels of uncertainty shocks to the macroeconomy relate mainly toreal phenomena such as distorted corporate decision making, while recent papers highlightthe importance of disturbances on credit and financial markets (Bloom, 2009; Alessandriand Mumtaz, 2019).The econometric literature increasingly relies on a unified framework for estimatinguncertainty and its effects jointly (see Mumtaz and Zanetti, 2013; Carriero et al. , 2015;2018b; Mumtaz and Surico, 2018). Besides many contributions relying on linear specific-ations assuming model parameters other than the residual variances to be constant, therehas been a recent focus on accounting for nonlinear relationships between uncertainty andthe real and financial economy, with potential consequences for the measurement of un-certainty (Mumtaz and Theodoridis, 2018; Alessandri and Mumtaz, 2019; Mumtaz andMusso, 2019). In addition, given the importance of international linkages in the propaga-tion of macroeconomic shocks (see, e.g. Canova and Ciccarelli, 2004; Pesaran et al. , 2004;Mumtaz and Surico, 2009; Feldkircher and Huber, 2016), multi-economy modeling frame-works have been proposed in Mumtaz and Theodoridis (2015), Berger et al. (2016), CrespoCuaresma et al. (2017), Carriero et al. (2018a) and Mumtaz and Musso (2019). A the-oretical justification for empirically assessing spillover dynamics is provided in Mumtazand Theodoridis (2017), who identify increasing globalization and trade openness as maindeterminant of international volatility comovements. Note, however, that most of thesecontributions rely on factor models or focus on specific variables, rather than providing afull systematic procedure addressing both the real and financial economic sectors acrosscountries.While some papers either consider time-variation in the relationship between uncer-tainty and the macroeconomy, or country-specific idiosyncrasies stemming from differentdomestic dynamics or spillover effects, papers addressing both features jointly are limited(with the exception of Mumtaz and Musso, 2019). The focus in Mumtaz and Musso (2019),however, is on the measurement of international, regional and country-specific uncertaintyfactors and their contribution to macroeconomic fluctuations over the cross-section, andnot directly on structural inference based on uncertainty shocks for a high-dimensional ystem of multiple economies. Motivated by these notions, we propose a multi-economymodel with drifting coefficients and factor stochastic volatility in mean to estimate uncer-tainty and its effects jointly. Allowing the volatilites to affect both the first and secondmoments of the multivariate dynamic system relates to stochastic volatility in mean models(see Koopman and Hol Uspensky, 2002; Chan, 2017).We provide two empirical and econometric contributions: First, we extend the globalvector autoregressive model of Pesaran et al. (2004) to feature time-varying parametersand residual variances that also enter the mean of the process (for a related approach,see Crespo Cuaresma et al. , 2019). Second, though this modeling framework decreasesthe number of parameters compared to unrestricted estimation, the parameter space isstill high-dimensional. As a remedy, we employ Bayesian methods and adapt global-localpriors for state-space models (see Fr¨uhwirth-Schnatter and Wagner, 2010; Belmonte et al. ,2014; Bitto and Fr¨uhwirth-Schnatter, 2019; Huber et al. , 2019) in a multi-country con-text. We center the model on a constant parameter specification with homoscedasticerrors and cross-country homogeneity, but preserve the possibility of time-variation andheterogeneous dynamics across economies via flexible hierarchical priors. The hierarchicalprior setup is designed to efficiently exploit cross-sectional information for precise infer-ence, and relates to the Bayesian treatment of panel data (Verbeke and Lesaffre, 1996;Fr¨uhwirth-Schnatter et al. , 2004).Our model is applied to monthly data for six economies (France, Germany, Italy,Japan, the United Kingdom, and the United States) for the period ranging from 1991:04to 2018:07, providing a link to the contributions of Crespo Cuaresma et al. (2017) andCarriero et al. (2018a) while generalizing the setup to allow for nonlinear relationshipsbetween uncertainty and the economy. The information set includes several recessionaryepisodes, and thus periods of economic distress when uncertainty is typically perceived toplay a major role. The endogenous measure of uncertainty is comparable to establishedproxies, and links well to events associated with high uncertainty. Impulse responsesshed light on the effects of uncertainty shocks on a set of macroeconomic and financialquantities. The responses are in line with established findings, but differ across the sixcountries in terms of magnitude and timing. Some variables show systematic declinesin their responsiveness to uncertainty shocks, while others remain comparatively stable,corroborating findings in Mumtaz and Theodoridis (2018). For selected quantities in asubset of economies, the time-varying effects of uncertainty shocks do not evolve gradually,but exhibit distinct features for specific periods (similar to Alessandri and Mumtaz, 2019).The article is organized as follows. Section 2 proposes the global vector autoregressivemodel with drifting coefficients and factor stochastic volatility in mean. Section 3 presentsthe data and discusses model specification. Section 4 investigates the uncertainty measureand provides a discussion of the empirical results. Section 5 concludes. . ECONOMETRIC FRAMEWORK2.1. Model specification
Let y it denote a k × t = 1 , . . . , T specific to country i = 1 , . . . , N . Collecting country-specific endogenous variables yields the K × y t = ( y (cid:48) t , . . . , y (cid:48) Nt ) with K = kN , while we stack the reduced form shocks to y it in a K × (cid:15) t = ( (cid:15) (cid:48) t , . . . , (cid:15) (cid:48) Nt ) (cid:48) .Following Aguilar and West (2000), we consider a factor stochastic volatility structureon the error term, (cid:15) t = Lf t + η t , f t ∼ N ( , exp( h t ) × Σ ) , η t ∼ N ( , Ω t ) . (1)Here, f t is a vector of d × d (cid:28) K ), and η t an idiosyncraticwhite noise shock vector of dimension K ×
1. Latent factors are linked to the errors by the K × d factor loadings matrix L . The factors f t are Gaussian with zero mean and commontime-varying volatility exp( h t ), scaling a diagonal matrix Σ = I d , with I d referring to a d -dimensional identity matrix.The idiosyncratic error components η t are assumed to follow a Gaussian distributioncentered on zero with K × K time-varying covariance matrix Ω t = diag(exp( ω t ) , . . . , exp( ω Kt ))For both the volatility of the factors and the variances of the idiosyncratic components,we rely on a stochastic volatility model (Jacquier et al. , 2002). Here, h t and ω ij,t for i = 1 , . . . , N and j = 1 , . . . , k follow independent random walk processes h t = h t − + ξ t , ξ t ∼ N (0 , σ h ) ω ij,t = ω ij,t − + ζ t , ζ t ∼ N (0 , σ ωij )with σ h and σ ωij denoting the state-equation innovation variances. Note that for the caseof σ h and σ ωij equal to zero, we obtain homoscedastic errors.Similar to Crespo Cuaresma et al. (2017) and Carriero et al. (2018b), we interpret h t asthe common driving force of the volatilities of all included series, and thus a measurementof uncertainty (see also Jurado et al. , 2015). Notice that Var( (cid:15) t ) = exp( h t ) LL (cid:48) + Ω t , andthe variance thus discriminates between idiosyncratic shocks and overall movements ininternational uncertainty. This feature can be linked to Mumtaz and Musso (2019), whorely on international, regional and country-specific factors. Our approach differs in thesense that we are only interested in international uncertainty h t , and the remainder ω ij,t captures both region and country-specific uncertainty. In the empirical application, the likelihood turns out to be quite flat for σ h , and we therefore impose therestriction σ h = 0 .
2. Evaluating various values for this parameter over a grid suggests this choice to beof minor importance. he dynamic evolution of y it is governed by a vector autoregressive (VAR) process withdrifting coefficients (similar to Crespo Cuaresma et al. , 2019) and features the commonvolatility of the factors in the mean: y it = α it + P (cid:88) p =1 A ip,t y it − p + Q (cid:88) q =1 B iq,t y ∗ it − q + β it h t + (cid:15) it . (2)We define the k × α it and k × k coefficient matrices A ip,t ( p = 1 , . . . , P ).To establish dynamic interdependencies between economies in the spirit of Pesaran et al. (2004), we construct a k × y ∗ it = (cid:80) Nj =1 w ij y jt . The w ij denote pre-specified weights(we let w ii = 0, w ij ≥ (cid:80) Nj =1 w ij = 1 for i, j = 1 , . . . , N ) that capture the strength ofthe linkages. The process in Eq. (2) is augmented by Q lags of these non-domestic cross-sectional averages y ∗ it , with k × k coefficient matrices B iq,t ( q = 1 , . . . , Q ). The vector β it associated with the log of the factor volatility h t is of dimension k × β it as the contemporaneous impact of uncertainty h t on the endogenous variables of country i . The structure set forth in Eqs. (1) and (2)implies that shocks to h t affect both the first and second moments of the system based oncommon shocks captured by f t . We exploit this notion for calculating impulse responsefunctions, relating to recursive identification schemes that order uncertainty indices first(see, e.g. Bloom, 2009). In general, structural identification of uncertainty shocks is achallenging task due to various reasons, as suggested in Ludvigson et al. (2019). Em-pirical evidence for the credibility of the identifying assumption employed in this paperin terms of economic interpretation is provided by Carriero et al. (2019), who find littleevidence for endogenous responses of macroeconomic uncertainty. In principle, the adop-ted setup would also allow to impose zero restrictions on the contemporaneous responsesof lower-frequency real macroeconomic quantities. However, in the empirical application,we refrain from doing so and leave it up to the data whether these variables respondcontemporaneously to uncertainty shocks.Before proceeding, we recast the model in standard regression form, y it = C it x it + (cid:15) it , (3) x it = (1 , y (cid:48) it − , . . . , y (cid:48) it − P , y ∗(cid:48) it − , . . . , y ∗(cid:48) it − Q , h t ) (cid:48) C it = ( α it , A i ,t , . . . , A iP,t , B i ,t , . . . , B iQ,t , β it ) . It is convenient to consider the j th equation of country i in Eq. (3) given by y ij,t = C (cid:48) ij,t x it + (cid:15) ij,t . Note that one may in principle also include lags of h t in the mean of the process. Doing so, however,significantly complicates the sampling algorithm and we thus solely study its contemporaneous effects.Moreover, considering the state equation of h t , its lag structure is implicitly featured in the VAR meas-urement equations subject to a set of parametric restrictions. Experiments including uncertainty asobserved variable also featuring lags (as in Bloom, 2009; Jurado et al. , 2015) do not affect the resultsfrom a qualitative viewpoint. ere, we refer to the j th row of the matrix C it by C ij,t , a vector of dimension ˜ K × K = k ( P + Q ) + 2. The state vector is assumed to follow a random walk process C ij,t = C ij,t − + u t , u t ∼ N ( , Θ ij ) , (4)with diagonal ˜ K × ˜ K variance-covariance matrix Θ ij = diag( θ ij, , . . . , θ ij, ˜ K ).As for the stochastic volatility specification, if θ ij,l equals zero in Eq. (4), the re-spective coefficient is constant over time. To test the restriction θ ij,l = 0, we introducethe non-centered parameterization set forth by Fr¨uhwirth-Schnatter and Wagner (2010),which allows to impose shrinkage priors on these innovation variances. In particular, thisapproach splits the model coefficients into a constant and a time-varying part, a featurewe exploit for achieving shrinkage in the high-dimensional multivariate system. Using a ˜ K × (cid:112) Θ ij = diag( (cid:112) θ ij, , . . . , (cid:113) θ ij, ˜ K ), the reparameterized equation is y ij,t = C (cid:48) ij, x it + ˜ C (cid:48) ij,t (cid:112) Θ ij x it + (cid:15) ij,t . (5)Let ˜ c ijl,t denote a typical element of ˜ C ij,t , then the transformation c ijl,t = c ijl, + (cid:112) θ ij,l ˜ c ijl,t yields the corresponding state equation˜ C ij,t = ˜ C ij,t − + v t , v t ∼ N ( , I ˜ K ) , with ˜ C ij, = ˜ K . This procedure moves the square root of the innovation variances to thestates into Eq. (5). The resulting state-space representation has the convenient propertythat the (cid:112) θ ij,l can conditionally be treated as standard regression coefficients.Stochastically selecting which series should feature time-variation in their respectivevolatilities can be carried out using a similar transformation (see Fr¨uhwirth-Schnatter andWagner, 2010; Kastner and Fr¨uhwirth-Schnatter, 2014). Conditional on Lf t and the fullhistory of the VAR coefficients C it , we obtain a set of unrelated heteroscedastic errorterms η t by the diagonal structure of Ω t . Here, η ij,t indicates the error term of the j thequation for country i . Squaring and taking logs and using ω ij,t = √ σ ωij ˜ ω ij,t yields˜ η ij,t = √ σ ωij ˜ ω ij,t + ν ij,t , ν ij,t ∼ ln χ (1) , ˜ ω ij,t = ˜ ω ij,t − + w ij,t , w ij,t ∼ N (0 , , again moving the square root of the innovation variances √ σ ωij from the state to themeasurement equation. The transformation allows to impose shrinkage priors on thesecoefficients, potentially pushing the model towards a homoscedastic specification if sug-gested by likelihood information. For applications of this approach in a VAR context see Bitto and Fr¨uhwirth-Schnatter (2019) and Huber et al. (2019). .2. Prior distributions
Bayesian methods are employed for estimation and inference. The panel structure of thedata allows for constructing flexible shrinkage priors that are equipped to extract bothcross-sectional information and shrink the model towards sparsity. Before proceeding westack the coefficients using c i = vec( C (cid:48) i , , . . . , C (cid:48) ik, ). In similar fashion, we collect squareroots of the innovation variances (cid:112) θ ij,l in (cid:112) θ i = ( (cid:112) θ i , , . . . , (cid:113) θ i , ˜ K , . . . , (cid:112) θ ik, , . . . , (cid:113) θ ik, ˜ K ) (cid:48) , and index the j th elements by c ij and (cid:112) θ ij , respectively, with j = 1 , . . . , k ˜ K .This article draws from the literature on the Bayesian treatment of panel data andglobal-local priors. In particular, we center the prior for all countries on a common meanthat is estimated from the data, reflecting the notion that macroeconomic dynamics acrosseconomies are typically similar. The prior setup thus relates to the random coefficients andheterogeneity model (Verbeke and Lesaffre, 1996; Fr¨uhwirth-Schnatter et al. , 2004), andrestrictions often imposed in the context of panel VARs (see Canova and Ciccarelli, 2013;Koop and Korobilis, 2016). We propose hierarchical priors akin to the Normal-Gamma(NG) shrinkage prior of Griffin and Brown (2010) recently adopted in the VAR contextby Huber and Feldkircher (2019).Since an analogous setup is applied for different parts of the parameter space, we relyon the generic indicator • for the indexes { c, θ, µ c , µ θ , σ, L } . For the constant part of thecoefficients, we assume c ij | µ cj , τ cj ∼ N ( µ cj , τ cj ) , τ cj | λ c ∼ G ( a • , a • λ c / , λ c ∼ G ( d • , d • ) . (6)Here, a key novelty is that we push all country-specific coefficients towards a commonmean µ cj . The overall degree of shrinkage is determined by a global shrinkage parameter λ c serving as a general indicator of cross-country homogeneity. To provide flexibilityfor country-specific macroeconomic dynamics, we introduce local scaling parameters τ cj .In the presence of heavy shrinkage governed by λ c , the τ cj allow for flexibly selectingidiosyncrasies in coefficients across economies. This is an innovation compared to similarapproaches (see, e.g. Malsiner-Walli et al. , 2016; Fischer et al. , 2019) who solely rely on aset of Gamma priors on these variances, disregarding a common degree of overall shrinkagetowards homogeneity.Shrinkage on the innovation variances of the states in Eq. (4) is introduced in similarvein. We follow Bitto and Fr¨uhwirth-Schnatter (2019) and stipulate a Gamma prioron these variances, which combined with a hierarchical prior relying again on Gammadistributions yields the setup they term the double Gamma prior. This is advantageous tothe often employed inverse Gamma prior, because it does not artificially pull mass awayfrom zero, a crucial feature when interest centers on stochastically shrinking the time-varying coefficients towards constancy. Fr¨uhwirth-Schnatter and Wagner (2010) show that his is equivalent to imposing a Gaussian prior on the square root of the state innovationvariances, (cid:112) θ ij | µ θj , τ θj ∼ N ( µ θj , τ θj ) , τ θj | λ θ ∼ G ( a • , a • λ θ / , λ θ ∼ G ( d • , d • ) . As in the case of the constant coefficients of the model, we introduce a common mean µ θj rather than pushing the variances towards zero a priori. This feature captures the notionthat not only the constant coefficients across countries may be similar, but also the degreeof time variation of the model parameters.The first hierarchy of priors address the notion that the dynamic coefficients of themodel might be similar over the cross-section. However, VARs with drifting coefficientsare prone to overfitting issues. We deal with this problem and induce shrinkage in thecoefficient matrices by imposing another NG prior to achieve regularization at the secondlevel of the hierarchy. On the common mean µ sj (for s ∈ { c, θ } ) we specify µ sj | τ µ s j ∼ N (0 , τ µ s j ) , τ µ s j | λ µ s ∼ G ( a • , a • λ µ s / , λ µ s ∼ G ( d • , d • ) . This setup pushes its elements towards zero, where the overall level of shrinkage is governedby the global parameter λ µ s . Similar to the first prior hierarchy, the prior allows for non-zero elements if suggested by the data via the local scalings τ µ s j .For the stochastic volatility specification we rely on analogous priors. In particular,for the state innovation variances of the stochastic volatility processes for the j th variableof country i , we impose Gamma distributed priors, translating to Gaussian priors on thesquare root of these variances. The prior is given by √ σ ωij | τ σij ∼ N (0 , τ σij ) , τ σij | λ σ ∼ G ( a • , a • λ σ / , λ σ ∼ G ( d • , d • ) , with the global shrinkage parameter λ σ pushing the model towards a homoscedastic spe-cification. The local scalings τ σij allow for non-zero state innovation variances. Intuitively,if τ σij is small, we introduce substantial prior information and the parameter is pushedtowards zero, ruling out time-variation in the respective volatility. For larger values of τ σij , the prior is less informative and allows for movements in the corresponding errorvariances.It remains to specify prior distributions on the factor loadings in L . Following Kastner(2019), we stack the elements in a vector l with typical element l j for j = 1 , . . . , R ( R = Kd ) and impose l j | τ Lj ∼ N (0 , τ Lj ) , τ Lj | λ L ∼ G ( a • , a • λ L / , λ L ∼ G ( d • , d • ) . (7)Until now we remained silent on the choices of hyperparameter values. In the empiricalspecification, and referring by • to the indexes { c, θ, µ c , µ θ , σ, L } , we follow the literatureand set d • = d • = 0 .
01 implying heavy shrinkage via the global parameter. Note that he hyperparameter a • plays a crucial role regarding shrinkage properties. In fact, setting a • = 1 would yield the Bayesian LASSO used in Belmonte et al. (2014). Given that thegeneric prior is applied to a range of different quantities of the model’s parameter space,we integrate out this hyperparameter by imposing exponential priors a • ∼ E (1).Full conditional posterior distributions obtained from combining the likelihood functionwith the priors and the corresponding estimation algorithm are discussed in AppendicesA and B. Fortunately, most distributions are of well-known form, allowing for a simpleMarkov chain Monte Carlo (MCMC) algorithm to obtain draws from the joint posteriorusing Gibbs sampling.
3. DATA AND MODEL SPECIFICATION
Our dataset consists of monthly data for the period ranging from 1991:04 to 2018:07 forsix economies: France (FRA), Germany (DEU), the United Kingdom (GBR), Italy (ITA),Japan (JPN), and the United States (USA). Consequently, the information set covers theG7 countries except Canada due to limitations of government bond yield data. The modelfeatures series on industrial production (IP, as a monthly indicator of economic activity),unemployment (UN), year-on-year consumer price inflation (PR), exports (EX) and equityprices (EQ). Industrial production, exports and equity prices enter the model in naturallogarithms. To construct the cross-sectional weights for establishing links between econom-ies, we rely on bilateral annual trade flows averaged over the sample period. Moreover, weobtain data on government bond yields at different maturities. The data are downloadedfrom the FRED database of the Federal Reserve Bank of St. Louis (fred.stlouisfed.org)and Quandl (quandl.com). A crucial determinant of business cycle fluctuations and the transmission of uncertaintyshocks to the real sector of the economy are financial markets (Gilchrist et al. , 2009;Gilchrist and Zakrajˇsek, 2012; Gilchrist et al. , 2014; Alessandri and Mumtaz, 2019). For aparsimonious representation of the full term structure of interest rates, we adopt a Nelson-Siegel type model (see Nelson and Siegel, 1987; Diebold and Li, 2006). Government bondyield curves are estimated employing a factor model denoting yields by r it ( τ ) at maturity τ , r it ( τ ) = L it + S it (cid:18) − exp( − λτ ) λτ (cid:19) + C it (cid:18) − exp( − λτ ) λτ − exp( − λτ ) (cid:19) . (8)This setup allows the factors L it , S it and C it to be interpreted as the level, (negative) slopeand curvature of the yield curve, and may be estimated using ordinary least squares. Us-ing an m × m it , we exploit the yield curve fundament-als extracted in Eq. (8) to construct the k × y it = ( m (cid:48) it , L it , S it , C it ) (cid:48) The period considered for this paper is dictated by the availability of the government bond yields atdifferent maturities. We adopt a two-stage procedure to reduce the computational burden. The factor loadings are determinedby the parameter λ = 0 . or t = 1 , . . . , T for country i = 1 , . . . , N . In the discussion of the empirical results, L it , S it and C it are labeled NSL, NSS and NSC, respectively. Thus, all dimensions of theinvolved vectors can be derived based on k = 8 and N = 6.To select the lag order of the model and the number of latent factors that drive thefull system covariance matrix, we rely on the deviance information criterion (DIC). Weestimate the model over a grid of lag and latent factor combinations, and choose thespecification minimizing the DIC. This procedure selects a model with P = Q = 2 lagsand d = 4 factors.For the empirical application, we adopt the general prior setup put forward in Section 2.In particular, to reduce influence of the prior setup on the estimated impact of uncertainty,we use a rather diffuse prior on the constant part of these coefficients with prior varianceequal to ten. The square roots of the state innovation variances of the impact vector aretightly centered on zero a priori. The latter choice mutes differences in impact reactionsover time, but improves the stability of the model.
4. EMPIRICAL RESULTS4.1.
The measure of uncertainty
Figure 1 displays the estimated measure of uncertainty: the log-volatility h t driving thecommon shocks to the system. The most striking episode of international uncertaintyoccurs during the global financial crisis and subsequent Great Recession. Several lesspronounced episodes of similarly elevated levels are worth noting. Chronologically, uncer-tainty rises in the first half of 1997, related to the Asian financial crisis. A spike in late1998 reflects the Russian financial crisis and the subsequent collapse of the U.S. hedge-fund Long-term Capital Management. Afterwards, a brief period of lower uncertainty isobservable, coming to an end with the burst of the Dot-com bubble and the 9/11 ter-ror attacks in late 2001. Sustained elevated levels, albeit declining, are observable untilthe end of 2003, a period encompassing the second Gulf War. The period between 2004and the bancruptcy of the U.S. investment bank Lehman Brothers features lower levels ofinternational uncertainty.Surging international volatilities are detected by the model starting in late 2007, cap-turing the onset of the crisis in the U.S. subprime mortgage market and first signs ofdisturbances on credit markets. After a decline of common volatilities to pre-crisis levelsaround 2010, the second highest peak of h t occurs in 2011, related to events during theEuropean sovereign debt crisis. This period of elevated uncertainty is sustained until late2013. The most recent episode of high uncertainty emerges in early 2016, indicating peaksrelated to the Brexit referendum and the election of Donald Trump as President of theUnited States in late 2016.Following this brief discussion of the measure in light of uncertainty-related events, wecompare our findings to commonly adopted proxies for uncertainty. We consider the geo-political risk (GPR) index described in Caldara and Iacoviello (2018), the global policy ussian crisis/ LTCM 9/11Gulf War II Lehman BrothersEuropean sovereign debt crisis Trump/ Brexit −10−8−6−4−2 1992 1994 1996 1998 2000 2002 2004 2006 2008 2010 2012 2014 2016 2018 h t Fig. 1:
Measurement of uncertainty depicting the log-volatility h t of the factors. Note : The thick black line depicts the posterior median, alongside the 16th and 84th posterior percentiles(thin lines). LTCM is the collapse of Long-Term Capital Management, 9/11 indicates the terror attacksof September 11, 2001. S t anda r d i z ed i nde x GPR GEPU WUI VIX Mean h t Fig. 2:
Comparison of standardized uncertainty measures over time.
Note : Measures are standardized to lie in the unit interval. The thick black line depicts the posteriormedian of h t . Uncertainty measures: Geopolitical risk (GPR), global policy uncertainty (GEPU), worlduncertainty index (WUI), CBOE volatility index (VIX). uncertainty (GEPU) index and the world uncertainty index (WUI) constructed as de-scribed in Baker et al. (2016), and complement them with the Chicago Board OptionsExchange (CBOE) volatility index (VIX). Moreover, we take the arithmetic average forall benchmark indices and label the resulting series “Mean” in corresponding visualiza-tions. To make the scales of the uncertainty measurements comparable, we standardizeall measures to lie in the unit interval.The resulting series are depicted in Fig. 2. A few points are worth noting. First, h t provides a smoother estimate of uncertainty. However, most peaks apparent in the EU FRA GBR ITA JPN USA UNIPEXPREQNSLNSSNSC w ij ,t Fig. 3:
Series-specific log-volatilities ω ij,t . Note : The thick black line depicts the posterior median, alongside the 16th and 84th posterior percentiles(thin lines). benchmark uncertainty measures are traced accurately. Differences occur mainly in themagnitude of the implied level of uncertainty. For instance, “Mean” peaks in 2003, withmost benchmark measures showing substantial uncertainty around the outbreak of thesecond Gulf War. The endogenous measure of uncertainty traces this peak, but at acomparatively lower level. The Great Recession peak in late 2008 on the other hand,exhibiting a spike in the VIX and most other measures apart from GPR, is the highest levelof uncertainty detected by h t . Maximum values of WUI are associated with elevated levelsin h t , and also the peaks of GPR and GEPU coincide with upward movements in h t . It isworth mentioning that our uncertainty measurement compares well to similar approachesdealing with the endogenous measurement of uncertainty (see Crespo Cuaresma et al. ,2017; Carriero et al. , 2018a).The discussion is complemented by the findings for idiosyncratic volatility series. Re-call that the prior setup imposes shrinkage on the idiosyncratic residual variances towardsconstancy. As evidenced by Fig. 3, the likelihood strongly suggests the necessity of astochastic volatility specification. Individual series feature pronounced heterogeneitiesboth in terms of the magnitude and the timing of peaks. This provides evidence that theapproach employed for measuring common uncertainty in this paper discriminates wellbetween country-specific events and international uncertainty-related events of signific-ance. argest differences in the magnitude of the volatilities are visible for unemployment,with Germany and France exhibiting lower residual variances. However, both feature sub-stantial higher-volatility periods in the years surrounding 2005. While ω ij,t for industrialproduction is rather homogenous for the continental European countries, the series ofthe remaining economies exhibit heterogeneities both in terms of magnitude and time-variation. The same is true, even though to a slightly lesser degree, in the case of exportvolatilities. Moreover, pronounced time-variation is clearly featured in the respective seriesrelating to country-specific inflation dynamics, and equity prices. Volatilities associatedwith the factors capturing yield curve dynamics show marked similarities across countries,reflecting international commonalities in equity markets. Dynamic responses to uncertainty shocks
The non-centered parameterization of the state-space model in principle allows to in-vestigate both shrinkage on the common mean of the time-invariant part of the modelcoefficients, and the corresponding state innovation variances. Moreover, the degree ofshrinkage towards homogeneity over the cross-section can be assessed. We find that forthe most part, the parameter space is pushed toward both cross-sectional homogeneityand time-invariance of the coefficients. This appears sensible, given the size of the sys-tem and the dicussion in Feldkircher et al. (2017) regarding time-varying coefficients oftencapturing omitted variable bias in smaller VARs. For the sake of brevity, we refrain frompresenting detailed inference on the shrinkage parameters governing homogeneity overtime and the cross-section and provide further information in Appendix C.Turning to structural inference, Fig. 4 displays an overall summary of the dynamicresponses for the periods between January 1992 and July 2017 on a biannual frequency, andreports the posterior median of the impulse response functions to the uncertainty shock.Colors refer to the respective period (red indicates early parts of the sample, blue markslater periods). Figure 5 depicts cumulative responses at the five year horizon. Units arescaled as percentages for industrial production, exports and equity prices, while consumerprice inflation, unemployment and the Nelson-Siegel factors for level, slope and curvatureare in basis points (BPs). In general, our results corroborate empirical findings from previous contributions, andboth directions and magnitudes of the responses are similar. One notable result concerningthe timing of the responses is that most react strongly on impact of the shock. We findsignificant increases of unemployment in all countries, while industrial production, exports,inflation and equity prices decrease. Timing and shape of the impulse responses for Nelson-Siegel level, slope and curvature factors indicate a flattening of the yield curve associatedwith overall decreases in interest rates at most maturities.
Unemployment.
For unemployment we detect significant peaks on impact, rangingfrom two BPs in the case of Germany, France, the United Kingdom and Italy, while Japan To preserve space, we refrain from presenting numerical results. Detailed tables are available uponrequest. EU FRA GBR ITA JPN USA UNIPEXPREQNSLNSSNSC
Impulse response horizon (in months)
Fig. 4:
Impulse responses for selected periods to an international uncertainty shock.
Note : Posterior median of the impulse response functions over time, with the shading referring to therespective period: —— —— exhibits slightly larger magnitudes. The largest responses result in the United States, withincreases up to eight BPs. The estimated effects are rather persistent, with significantpositive reactions over the impulse response horizon of five years. Figure 4 suggests onlya minor degree of time-variation, with the impacts leveling out slightly quicker in laterparts of the sample.Closer inspection of the cumulative effects in the first row of Fig. 5 yields someinteresting insights. Systematic decreases are visible for France, Italy, the United Kingdomand Japan. This notion is most pronounced for the United Kingdom. Different behavioroccurs in Germany and the United States, with substantially larger cumulative responses.Estimates for the United States gradually amplify before the global financial crisis. Industrial production.
Industrial production shows the largest declines in Italy and Ja-pan, with significant negative peak responses on impact. The remaining countries exhibitrather homogeneous responses of approximately 0 . he smallest effects, with a decline of 0 . Exports.
Exports indicate insignificant contemporaneous effects close to zero for Ger-many and the United Kingdom, with a significant peak decline around two quarters afterimpact. France and Italy exhibit positive impacts, but the responses quickly turn negat-ive. Substantial decreases are indicated for the United States and Japan. The magnitudeof the impacts is in line with findings by Crespo Cuaresma et al. (2017); they are alsointeresting when interpreted in light of Mumtaz and Theodoridis (2017), who suggest thatinternational trade plays a crucial role in the transmission of uncertainty shocks via fallingforeign demand and feedback volatility effects for the respective domestic economy.No clear time-variation patterns emerge in Fig. 4. Here, we again resort to Fig. 5, withthe third row providing evidence of substantial differences in the cumulative responses overtime. Prior to the global financial crisis, estimates fluctuate around −
10 percent acrosscountries. The largest fluctuations in the cumulative responses are observable for theUnited Kingdom. Analogous to the results for unemployment and industrial production,the consequences of uncertainty shocks on exports in the aftermath of the Great Recessionare muted in comparison to previous periods. From 2015 onwards, effects are again similarto earlier in the sample, with minor differences over the cross-section.
Consumer price inflation.
Fern´andez-Villaverde et al. (2015) identify two contradict-ing channels how uncertainty affects consumer prices: The so-called aggregate demandchannel, characterized by reducing the consumption of households and thereby leading toan overall decrease in prices; and the upward-pricing bias channel, which yields increasesin inflation based on profit-maximizing firms. In our case, the former appears to dominatethe latter, with significant decreases of inflation on impact for most economies in row fourof Fig. 5. The estimated peak effects indicate constancy in magnitudes ranging from − . − . EU FRA GBR ITA JPN USA UNIPEXPREQNSLNSSNSC
Fig. 5:
Cumulative impulse response functions to an international uncertainty shock.
Note : The thick black line depicts the posterior median, alongside the 16th and 84th posterior percentiles(thin lines). The red line marks zero.
Further inspection of the estimates in light of Fig. 5 reveals substantial heterogeneit-ies. First, we observe differences in posterior uncertainty over the sample period. Lessprecisely estimated cumulative effects mainly occur in the context of short-term interestrates hitting zero-lower bound for most economies. Inflated credible sets and differencesin the posterior median moreover occur early in the sample. Second, responses at the endof the sample period in July 2017 feature little cross-sectional heterogeneity. Finally, idio-syncratic movements for the United Kingdom are worth mentioning. After large negativeeffects early in the sample, the consequences of uncertainty shocks on inflation declinedsubstantially until late 2007. After the Great Recession, substantially larger effects aredetected.
Equity prices.
Equity prices, displayed in the fifth row of Fig. 5, prominently featuretime-variation in the dynamic responses for all countries. The responses in the UnitedKingdom are less pronounced than in the other countries, peaking after roughly one and half years at about − . − . − . Nelson-Siegel factors.
Impulse responses for the Nelson-Siegel factors are displayedin the last three rows of Fig. 4. The dynamic evolution of the level factor exhibitssubstantial heterogeneity across countries, but appears comparatively constant over timewith slight differences in the curvature of the responses. In particular, we find the largestand significant decreases on impact, coinciding with the peak response, for Germany.In general, the credible sets associated with the impulse responses of the level factorare rather large, and cover zero in most economies. The effects peter out quickly, withimpulse responses returning to zero after about two quarters. Observed heterogeneityover the cross-section may originate from international capital flows toward safer assetsin uncertain times (see, e.g. Caballero et al. , 2017). Figure 5 indicates that the posteriordistributions of the cumulative effects for the level factor cover zero for all economies overthe sample period considered, featuring detectable yet insignificant time-varying dynamicsin the responses.The slope factor detects significant positive reactions peaking instantaneously in Ger-many, the United Kingdom, and Japan. The effects for the remaining countries on impactare estimated less precisely, however, the posterior centers on positive values for all coun-tries ranging from one to five BPs. An increase in the slope factors translates to a decreasein term spreads and a flattening of the yield curve, a phenomenon that has been linkedto the emergence of recessions. This effect reverses in subsequent months, turning sig-nificantly negative between one and one and a half years after impact across countries.Given the close empirical relationship between the slope factor and central bank policy (see For interpretational clarity, recapture Eq. (8), where the loading on L it is a constant for all τ ; hence itaffects all maturities equally, and is interpreted as the long-term level of the yield curve. The loadingassociated with S it decreases rapidly in τ , and is thus closely related to the negative slope of the yieldcurve and term spreads (for details, see Diebold and Li, 2006). Consequently, an increase in S it impliesa decrease in term spreads, and thus a flattening of the yield curve. The loading of C it is hump-shaped,and loads most strongly on the middle segment of the yield curve that affects its curvature. iebold et al. , 2006), we conjecture that this pattern captures a delayed response of cent-ral banks, lowering policy rates to counteract detrimental economic effects of uncertaintyshocks.Assessing cumulative effects, we find that estimates are statistically significant early inthe sample for Germany and France. The model captures large but insignificant effects forthe remaining economies except for Japan, which is unsurprising considering the country’srecent monetary history. In general, the impact of uncertainty shocks on the slope factorappears to decrease over time, evidenced by subtle trends visible for most countries exceptthe United Kingdom and the United States. Linking this finding to less responsive equitymarkets, such dynamics may be explained based on the theoretical model in Mumtaz andTheodoridis (2018), who suggest differences in the conduct of central banking to be relatedto decreases in the effects of uncertainty shocks.The results associated with the curvature factor signal decreases for most countries.Again, we observe pronounced heterogeneity over the cross-section, but also over time.The responses peak on impact for Germany and the United Kingdom. France, the UnitedStates and Japan show only small consequences of uncertainty shocks for middle-termmaturities. In terms of cumulative responses, we find systematic declines in the magnitudeof the effects associated with inflated posterior uncertainty for Japan, dynamics that arealso visible in the case of Germany, France, the United Kingdom and the United States.Minor differences occur for selected periods after the Great Recession. Italy presentsa special case, with distinct periods featuring substantial differences in the cumulativeresponses.
5. CONCLUDING REMARKS
The obtained measure of uncertainty is comparable to established proxies, and the factorstochastic volatility structure discriminates well between events confined to individualeconomies and overall macroeconomic uncertainty. Uncertainty shocks cause downwardpressure on inflation, increase unemployment levels, decrease industrial production anddepress equity prices, with differences in timing and magnitude of the effects across eco-nomies. The term structure of interest rates generally exhibits decreases in levels at allmaturities, with an accompanying overall flattening of the yield curve. The consequencesof uncertainty shocks appear to decline gradually for some macroeconomic and financialquantities, while other variables show little variation over time. We find limited evidencefor abrupt changes in the transmission channels of uncertainty shocks. EFERENCES
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Journal of the American Statistical Association (433), 217–221. PPENDIX A: POSTERIOR DISTRIBUTIONS AND ALGORITHM
Conditional on { f t } Tt =1 and the loadings L , the full system of equations reduces to K unrelatedregression models with heteroscedastic errors. This allows for estimation of the system equation-by-equation, greatly reducing the computational burden. To see this, define ˜ y t = y t − Lf t andrefer to the j th variable of country i by ˜ y ij,t ,˜ y ij,t = C (cid:48) ij, x it + ˜ C (cid:48) ij,t (cid:112) Θ ij x it + η ij,t . Moreover, conditional on { ˜ C ij,t } Tt =1 , the innovation variances in (cid:112) Θ ij can be treated as standardregression coefficients. We define the vector d ij = ( C (cid:48) ij, , (cid:112) θ ij, , . . . , (cid:113) θ ij, ˜ K ) (cid:48) . Let • refer toconditioning on all the other parameters, latent states of the model, and the data; then the posteriordistribution of d ij is a multivariate Gaussian, d ij |• ∼ N ( ˜ µ ij , ˜ V ij ) . (A.1)The posterior moments are ˜ V ij = ( ˜ X (cid:48) ij ˜ X ij + V − ) − and ˜ µ ij = ˜ V ij ( ˜ X (cid:48) ij ˜ Y ij + V − µ ), with priormoments µ = ( µ c , . . . , µ c ˜ K , µ θ , . . . , µ θ ˜ K ) (cid:48) and V = diag( τ c , . . . , τ c ˜ K , τ θ , . . . , τ θ ˜ K ). The matrix ˜ X ij is of dimension T × K , with the t th row given by [ x (cid:48) it , ˜ C (cid:48) ij,t (cid:12) x (cid:48) it ] exp( − ω ij,t / ˜ Y ij is T × t th element ˜ y ij,t exp( − ω ij,t / µ and V . Since the results apply to the coefficients in c i and √ θ i , weagain use an indicator s ∈ { c, θ } and obtain τ sj |• ∼ GIG (cid:32) a s − N/ , N (cid:88) i =1 ( c ij − µ sj ) , a s λ s (cid:33) , λ s |• ∼ G d s + k ˜ Ka s , d s + a s k ˜ K (cid:88) j =1 τ sj , with local scalings τ sj following a generalized inverse Gaussian distribution and the global shrinkageparameter a Gamma distribution. Conditional on { c ij } Ni =1 , standard methods yield a Gaussianposterior µ sj ∼ N (˜ µ sj , ˜ V sj ) , with ˜ V sj = ( N τ − sj + τ − µ s j ) − and ˜ µ sj = ˜ V sj ( (cid:80) Ni =1 c ij τ − sj ). For the prior variance of the commonmean, τ µ s j it is straightforward to obtain τ µ s j |• ∼ GIG (cid:0) a µ s − / , µ sj , a µ s λ µ s (cid:1) , λ µ s |• ∼ G d µ s + k ˜ Ka µ s , d µ s + a µ s k ˜ K (cid:88) j =1 τ µ s j . To obtain draws from the posterior distribution of σ ωij we rely on the methods discussed inKastner and Fr¨uhwirth-Schnatter (2014). Conditional on a realization of σ ωij , we obtain τ σij |• ∼ GIG ( a σ − / , σ ωij , a σ λ σ ) , λ σ ∼ G d σ + Ka σ , d σ + a σ k (cid:88) i =1 N (cid:88) j =1 τ σij . ote that Eq. (1) conditional on the other parameters of the model is a simple linear regres-sion model with conditionally homoscedastic errors and standard formulae apply. The NG prioremployed for the factor loadings translates to the following posteriors for the corresponding globaland local shrinkage parameters: τ Lj |• ∼ GIG ( a L − / , l j , a L λ L ) , λ L ∼ G d L + Ra L , d L + a L R (cid:88) j =1 τ Lj . We proceed with the posterior distribution for the hyperparameters of the prior on the localscalings a • . For the sake of brevity we refrain from presenting all respective indices and refer tothe various possible index combinations using • . Combining likelihood and prior, the conditionalposterior for this parameter has no well-known form and we rely on a Metropolis-Hastings stepfor simulation. Given the support of a • , we propose candidate draws a ∗• from N (ln( a • ) , κ • ), with κ • denoting a tuning parameter that is updated during half of the burn-in period to achieve anacceptance rate between 0 .
15 and 0 .
35. Acceptance probabilities are given bymin (cid:20) , p ( a ∗• ) p ( a ∗• | τ • ) a ∗• p ( a • ) p ( a • | τ • ) a • (cid:21) . APPENDIX B: MCMC ALGORITHM
Employing the posterior distributions presented in Appendix A, the full MCMC algorithm cyclesthrough the following steps:1. The constant part of the coefficients and the process variances of the coefficients are simulatedequation-by-equation using Eq. (A.1).2. For { ˜ C ij,t } Tt =1 , we rely on a forward filtering backward sampling algorithm (see Carter andKohn, 1994; Fr¨uhwirth-Schnatter, 1994).3. Conditional on the country-specific coefficients, it is straightforward to obtain a draw forthe common mean µ and the associated global and local shrinkage parameters in V . Sub-sequently, given a simulated value for the common mean, we draw the global and localshrinkage parameters τ µ s j and λ s .4. Simulation of { ω ij,t } Tt =1 is carried out using the algorithm set forth in Kastner and Fr¨uhwirth-Schnatter (2014), implemented in the R -package stochvol . The package moreover draws theinnovation variances of the stochastic volatility processes. We use this draw to obtain theshrinkage parameters τ σij .5. Given { f t } Tt =1 , we simulate the factor loadings in L using standard posteriors. Conditionalon a draw of the loadings, we obtain the prior variances τ Lj .6. The full history for { h t } Tt =1 is sampled via an independence Metropolis-Hastings algorithm(Jacquier et al. , 2002). A minor adaption required by the notion of the volatility beingfeatured in the mean is accounted for in the respective acceptance probabilities.7. We update the hyperparameters a • via Metropolis-Hastings steps sketched above.For the empirical application, we iterate this algorithm 12 ,
000 times and discard the initial 6 , ,
000 resulting in a set of 2 , PPENDIX C: HOMOGENEITY AND HETEROGENEITY ACROSS COUNTRIESAND OVER TIME
The following additional results serve to provide intuition on the properties of the prior setup. Thenon-centered parameterization of the state-space model allows to investigate both shrinkage on thecommon mean of the time-invariant part of the model coefficients µ cj , and the corresponding stateinnovation variances µ θj . Figure C.1 shows the respective posterior mean of τ µ c j and τ µ θ j on thelogarithmic scale. Smaller values indicate heavier shrinkage towards zero. The first column of Fig. C.1(a) highlights the first own lag of each equation in µ cj to featuremainly non-zero coefficients, reflected in values of log( τ µ c j ) close to zero. This implies that onlylittle shrinkage towards zero is imposed on these coefficients by the resulting loose prior variance τ µ c j . Such patterns, albeit less distinctive, are also observable for the second lag of the domesticcoefficients in the second column. However, we generally detect tighter prior variances for thesecond lags, with differences depending on the respective equation. Turning to the third andfourth columns that indicate shrinkage on the foreign lags per equation, we find similar shrinkagepatterns when comparing to the first domestic lag. Non-domestic movements appear to play arole in the dynamic evolution of the Nelson-Siegel factors. In general, the results point towardsthe necessity of considering international dynamics, a feature explicitly addressed by the proposedmulti-country approach. (a) Time-invariant coefficients ll l ll l l lll l ll l l lll l ll l l lll l ll l l lll l ll l l lll l ll l l lll l ll l l lll l ll l l l ll l ll l l lll l ll l l lll l ll l l lll l ll l l lll l ll l l lll l ll l l lll l ll l l lll l ll l l l ll l ll l l lll l ll l l lll l ll l l lll l ll l l lll l ll l l lll l ll l l lll l ll l l lll l ll l l l ll l ll l l lll l ll l l lll l ll l l lll l ll l l lll l ll l l lll l ll l l lll l ll l l lll l ll l l l
Domestict−1 Domestict−2 Foreignt−1 Foreignt−2 UNIPEXPREQNSLNSSNSC UN I PEXP R E Q N S L N SS N S C UN I PEXP R E Q N S L N SS N S C UN I PEXP R E Q N S L N SS N S C UN I PEXP R E Q N S L N SS N S C −20−15−10−50−20−15−10−50−20−15−10−50−20−15−10−50−20−15−10−50−20−15−10−50−20−15−10−50−20−15−10−50 l og ( tm c j ) (b) Square root state innovation variances ll l ll l l lll l ll l l lll l ll l l lll l ll l l lll l ll l l lll l ll l l lll l ll l l lll l ll l l l ll l ll l l lll l ll l l lll l ll l l lll l ll l l lll l ll l l lll l ll l l lll l ll l l lll l ll l l l ll l ll l l lll l ll l l lll l ll l l lll l ll l l lll l ll l l lll l ll l l lll l ll l l lll l ll l l l ll l ll l l lll l ll l l lll l ll l l lll l ll l l lll l ll l l lll l ll l l lll l ll l l lll l ll l l l
Domestict−1 Domestict−2 Foreignt−1 Foreignt−2 UNIPEXPREQNSLNSSNSC UN I PEXP R E Q N S L N SS N S C UN I PEXP R E Q N S L N SS N S C UN I PEXP R E Q N S L N SS N S C UN I PEXP R E Q N S L N SS N S C −22−20−18−16−14−22−20−18−16−14−22−20−18−16−14−22−20−18−16−14−22−20−18−16−14−22−20−18−16−14−22−20−18−16−14−22−20−18−16−14 l og ( tmq j ) Fig. C.1:
Log posterior mean of τ µ s j shrinking the common mean to zero. Note : Columns refer to a countries’ own lagged “Domestic” variables in y it − p of lag t − p , while“Foreign” indicates the coefficients associated with y ∗ it − q at t − q . Figure C.1(b) provides evidence of shrinkage towards zero of the state innovation variances thatdrive time-variation in the model coefficients. Note, however, that shrinkage of the common meantowards zero does not necessarily imply constant model coefficients, due to additional flexibility For visualization purposes and the imposed prior restrictions, we do not present the corresponding priorvariances for the intercept term and the impact vector β it . a) Time-invariant coefficients ll l ll l l lll l ll l l lll l ll l l lll l ll l l lll l ll l l lll l ll l l lll l ll l l lll l ll l l l ll l ll l l lll l ll l l lll l ll l l lll l ll l l lll l ll l l lll l ll l l lll l ll l l lll l ll l l l ll l ll l l lll l ll l l lll l ll l l lll l ll l l lll l ll l l lll l ll l l lll l ll l l lll l ll l l l ll l ll l l lll l ll l l lll l ll l l lll l ll l l lll l ll l l lll l ll l l lll l ll l l lll l ll l l l
Domestict−1 Domestict−2 Foreignt−1 Foreignt−2 UNIPEXPREQNSLNSSNSC UN I PEXP R E Q N S L N SS N S C UN I PEXP R E Q N S L N SS N S C UN I PEXP R E Q N S L N SS N S C UN I PEXP R E Q N S L N SS N S C −60−40−20−60−40−20−60−40−20−60−40−20−60−40−20−60−40−20−60−40−20−60−40−20 l og ( t c j ) (b) Square root state innovation variances ll l ll l l lll l ll l l lll l ll l l lll l ll l l lll l ll l l lll l ll l l lll l ll l l lll l ll l l l ll l ll l l lll l ll l l lll l ll l l lll l ll l l lll l ll l l lll l ll l l lll l ll l l lll l ll l l l ll l ll l l lll l ll l l lll l ll l l lll l ll l l lll l ll l l lll l ll l l lll l ll l l lll l ll l l l ll l ll l l lll l ll l l lll l ll l l lll l ll l l lll l ll l l lll l ll l l lll l ll l l lll l ll l l l
Domestict−1 Domestict−2 Foreignt−1 Foreignt−2 UNIPEXPREQNSLNSSNSC UN I PEXP R E Q N S L N SS N S C UN I PEXP R E Q N S L N SS N S C UN I PEXP R E Q N S L N SS N S C UN I PEXP R E Q N S L N SS N S C −20−15−10−20−15−10−20−15−10−20−15−10−20−15−10−20−15−10−20−15−10−20−15−10 l og ( tq j ) Fig. C.2:
Log posterior mean of τ sj shrinking country-specific coefficients towards µ sj . Note : Columns refer to a countries’ own lagged “Domestic” variables in y it − p of lag t − p , while“Foreign” indicates the coefficients associated with y ∗ it − q at t − q . on the second prior hierarchy. A key finding is that the unemployment and industrial productionequations are pushed strongly towards a constant parameter specification both for domestic andforeign lags. A simliar picture is present in the inflation equation, albeit at a slightly lower overalldegree of shrinkage induced by the respective τ µ θ j . The higher value of log( τ µ θ j ) on the first owndomestic lag of inflation in the inflation equation suggests changes in the persistence of inflationdynamics.Next, we analyze the estimated prior variances τ cj and τ θj that shrink country-specific coef-ficients towards µ cj and µ θj , respectively. Again, we consider the posterior mean of log( τ cj ) andlog( τ θj ) in Fig. C.2. The scalings provide a natural measure of similarity across countries. Largenegative values on the log-scale yield a situation referred to as cross-sectional homogeneity in thepanel literature (see Canova and Ciccarelli, 2013).One notable result in Fig. C.2(a) is that all coefficients are strongly pushed towards homogen-eity, suggested by predominantly large negative values for log( τ cj ). No clear patterns of similaritiesare visible across equations for both the domestic and foreign lag structure. Note that the firstown domestic lags per equation usually feature less heavy shrinkage towards the common mean(except for inflation and equity prices), implying subtle differences in the persistence of the seriesacross countries. Particularly strong evidence of homogeneity is present for subsets of domestic andforeign lags in all equations. Figure C.2(b) displays that heavy shrinkage on the state innovationvariances is applied to all domestic and foreign lags in the unemployment and industrial productionequation. A similar picture emerges for the inflation equation, and the dynamics captured in thecontext of the Nelson-Siegel level and slope factors. However, some variables appear to requireflexibility in terms of country-specific breaks in the coefficients.Combining discussions in the context of Figs. C.1 and C.2 allows for different scenarios interms of homogeneity across countries and the degree of sparsity: First, there is the possibility of eterogeneous non-zero coefficients and state innovation variances, in cases where both τ µ s j and τ sj are comparatively large. Here, prominent examples are provided by most first own lags ofthe domestic coefficients in their respective equation. Second, if both τ µ s j and τ sj are small, theprior setup implies heavy shrinkage of the country-specific parameters towards zero, for exampleregarding state innovation variances in the equations for unemployment and industrial production.Third, for large τ µ s j and small τ sj , the prior implies homogeneous non-zero parameters featuredmainly in the context of the first autoregressive foreign lags. While no clear relationship betweenFigs. C.1 and C.2 for the constant part of the coefficients can be identified, the adverse is true forthe state innovation variances. This implies that if the common mean of the latter is non-zero onthe first hierarchy of the prior, this is typically associated with less heavy shrinkage towards thecommon mean on the second hierarchy of the prior., the prior implies homogeneous non-zero parameters featuredmainly in the context of the first autoregressive foreign lags. While no clear relationship betweenFigs. C.1 and C.2 for the constant part of the coefficients can be identified, the adverse is true forthe state innovation variances. This implies that if the common mean of the latter is non-zero onthe first hierarchy of the prior, this is typically associated with less heavy shrinkage towards thecommon mean on the second hierarchy of the prior.