A multi-country dynamic factor model with stochastic volatility for euro area business cycle analysis
AA multi-country dynamic factor model with stochastic volatilityfor euro area business cycle analysis
FLORIAN HUBER , MICHAEL PFARRHOFER , and PHILIPP PIRIBAUER University of Salzburg, Salzburg Centre of European Union Studies Vienna University of Economics and Business Austrian Institute of Economic Research
This paper develops a dynamic factor model that uses euro area (EA) country-specificinformation on output and inflation to estimate an area-wide measure of the output gap.Our model assumes that output and inflation can be decomposed into country-specificstochastic trends and a common cyclical component. Comovement in the trends isintroduced by imposing a factor structure on the shocks to the latent states. We moreoverintroduce flexible stochastic volatility specifications to control for heteroscedasticityin the measurement errors and innovations to the latent states. Carefully specifiedshrinkage priors allow for pushing the model towards a homoscedastic specification, ifsupported by the data. Our measure of the output gap closely tracks other commonlyadopted measures, with small differences in magnitudes and timing. To assess whetherthe model-based output gap helps in forecasting inflation, we perform an out-of-sampleforecasting exercise. The findings indicate that our approach yields superior inflationforecasts, both in terms of point and density predictions.
JEL : E32, C11, C32, C53
KEYWORDS : European business cycles, dynamic factor model, factor stochasticvolatility, inflation forecasting ∗ Corresponding author : Philipp Piribauer, Austrian Institute of Economic Research (WIFO), Arsenal 20, 1030 Vienna,Austria.
E-mail : [email protected]. The research carried out in this paper was supported by funds of theOesterreichische Nationalbank (Jubilaeumsfond project number: 17382).
Date : January 14, 2020. a r X i v : . [ ec on . E M ] J a n . INTRODUCTION Effective policy making in central banks such as the European Central Bank (ECB) requires accuratemeasures of latent quantities such as the output gap to forecast key quantities of interest like inflationacross euro area (EA) member states. Since using aggregate EA data potentially masks importantcountry-specific dynamics, exploiting country-level information could help in obtaining more reliableestimates of the output gap that is consequently used in Phillips curve-type models to forecast inflation.In this paper, we exploit cross-sectional information on output and inflation dynamics to constructa multi-country model for the EA. The proposed framework aims to combine the literature on outputgap modeling (see, among many others, Kuttner, 1994; Orphanides and Van Norden, 2002; Basisthaand Nelson, 2007; Planas et al. , 2008) that focuses on estimating the output gap based on data fora single country/regional aggregate, the literature on dynamic factor models (Otrok and Whiteman,1998; Kim and Nelson, 1999; Kose et al. , 2003; Breitung and Eickmeier, 2015; Jarocinski and Lenza,2018) and the literature on inflation forecasting (Stock and Watson, 1999; 2007).Our model assumes that country-specific business cycles are driven by a common latent factor,effectively exploiting cross-sectional information in the data. Moreover, we assume that output andinflation feature a non-stationary country-specific component. To control for potential comovement inthese trend terms, we assume that the corresponding shocks to the states feature a factor structure. Theresulting factor model features stochastic volatility (SV) in the spirit of Aguilar and West (2000) andthus provides a parsimonious way of controlling for heteroscedasticity. Since successful forecastingmodels typically allow for SV (Clark, 2011; Clark and Ravazzolo, 2015; Huber, 2016; Huber andFeldkircher, 2019), we also allow for time-variation in the error variances across the remaining stateinnovations and the measurement errors. One methodological key innovation is the introduction ofglobal-local shrinkage priors on the error variances of the state equations describing the law of motionof the logarithmic volatility components, effectively shrinking the system towards a homoscedasticspecification, if applicable.This increased flexibility, however, is costly in terms of additional parameters to estimate. We thusfollow the recent literature on state space modeling (Frühwirth-Schnatter and Wagner, 2010; Belmonte et al. , 2014; Kastner and Frühwirth-Schnatter, 2014; Feldkircher et al. , 2017; Bitto and Frühwirth-Schnatter, 2019) and exploit a non-centered parameterization of the model (see Frühwirth-Schnatterand Wagner, 2010) to test whether SV is supported by the data. The non-centered parameterizationallows treating the square root of the process innovation variances as standard regression coefficients,implying that conventional shrinkage priors can be used. Here we follow Griffin and Brown (2010)and use a variant of the Normal-Gamma (NG) shrinkage prior that introduces a global shrinkagecomponent that applies to all process variances simultaneously, forcing them towards zero. Localshrinkage parameters are then used to drag sufficient posterior mass away from zero even in thepresence of strong global shrinkage, allowing for non-zero process variances if required. hen applied to data for ten EA countries over the time period from 1997:Q1 to 2018:Q4, we findthat our output gap measure closely tracks other measures reported in previous studies (Planas et al. ,2008; Jarocinski and Lenza, 2018) as well as gaps obtained by utilizing standard tools commonly usedin policy institutions. We moreover perform historical decompositions to gauge the importance ofarea-wide as opposed to country-specific shocks for describing inflation movements. These measuresreveal that inflation is strongly driven by common business cycle dynamics, underlining the importanceof controlling for a common business cycle. We then turn to assessing whether there exists a Phillipscurve across EA countries by simulating a negative one standard deviation business cycle shock. Thisexercise points towards a robust relationship between the common gap component and inflation, withmagnitudes differing across countries.The main part of the empirical application applies our modeling approach to forecast inflation,paying particular attention on whether the inclusion of a common output gap improves predictivecapabilities. Since inflation across countries is driven by a term measuring trend inflation and theoutput gap, our framework can be interpreted as a New Keynesian Phillips curve, akin to Stella andStock (2013). Compared to a set of simpler alternatives that range from univariate benchmark modelsto models that use alternative ways to calculate the output gap, the proposed model yields more precisepoint and density forecasts for inflation.The remainder of the paper is structured as follows. Section 2 describes the econometric framework.After providing an overview of the model, we discuss the Bayesian prior choice and briefly summarizethe main steps involved in estimating the model. Section 3 presents the empirical application, startingwith a summary of the dataset and inspects various key features of our model. The section moreoverstudies the dynamic impact of business cycle shocks to the country-specific output and inflation series.In a forecasting exercise, Section 4 compares the out-of-sample predictive performance of our modelwith other specifications. The final section summarizes and concludes the paper.
2. ECONOMETRIC FRAMEWORK2.1.
A dynamic factor model for the euro area
In this section we describe the framework to estimate the euro area output gap using disaggregatecountry-level information. Let y it and π it denote output and inflation for country i = , . . . , N inperiod t = , . . . , T , respectively. For notational simplicity, we define k ∈ { y , π } .Country-specific output and inflation are driven by unobserved common non-stationary trendcomponents τ kit that aim to capture low-frequency movements, while a common cyclical component g t tracks mid- to high-frequency fluctuations in inflation and output. These unobserved (latent) uantities are related to the observed quantities through a set of measurement equations: y it = τ yit + α i g t + (cid:15) yit , (1) π it = τ π it + β i g t + (cid:15) π it , (2) (cid:15) kit ∼ N ( , e h kit ) . (3)These equations imply that the trend components can loosely be interpreted as country-specific trendinflation and potential output for the i th country, respectively. Moreover, the stationary component ofoutput and inflation depends on the common cycle g t through a set of idiosyncratic factor loadings α i and β i and measurement errors that feature time-varying variances e h kit . It is worth stressing thatEq. (2) represents a country-specific Phillips curve that establishes a relationship between inflationand the area-wide output gap g t . One key goal of this paper is to assess whether there exists a Phillipscurve across EA countries by inspecting β i and functions thereof.The country-specific components in Eq. (1) and Eq. (2) are stacked in τ yt = ( τ y t , . . . , τ yNt ) (cid:48) and τ π t = ( τ π t , . . . , τ π Nt ) (cid:48) and evolve according to a VAR(2) process given by the state equation τ yt τ π t g t (cid:124)(cid:123)(cid:122)(cid:125) f t = I N . . . ... I N ... . . . φ (cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) Φ τ yt − τ π t − g t − (cid:124) (cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32) (cid:125) f t − + . . . ... . . . ... . . . φ (cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) Φ τ yt − τ π t − g t − (cid:124) (cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32) (cid:125) f t − + η yt η π t η gt (cid:124)(cid:123)(cid:122)(cid:125) η t . (4)By defining Φ = ( Φ , Φ ) and F t = ( f (cid:48) t − , f (cid:48) t − ) (cid:48) , Eq. (4) can be written more compactly as f t = Φ F t + η t , η t ∼ N ( , Σ t ) . Here, η t denotes the stacked error terms that follows a multivariate Gaussian distribution with zeromean and time-varying variance covariance matrix Σ t that is specified below.For the unrestricted AR(2) parameters φ and φ we follow Planas et al. (2008) and reparameterizethe state equation coefficients of g t using polar coordinates imposing complex roots, g t = Q cos ( π / γ ) g t − − Q g t − + η gt . Hereby, Q determines the amplitude and γ the frequency of the cycle. The parameterization hasthe convenient property that available information on the duration and intensity of business cycles canbe introduced with relative ease. Incorporating such information using normally distributed priors iscomplicated, since autoregressive coefficients are more difficult to interpret in terms of the intensity andfrequency of the time series. Moreover, allegedly weakly informative Gaussian priors could introduceinformation on functions of the parameters, potentially placing too much prior weight on dynamics that o not fit observed behavior of output at business cycle frequencies (for a more detailed discussion,see Planas et al. , 2008).Turning to the state equation errors, we assume the elements of η t in Eq. (4) to be blockwiseorthogonal and achieve this by employing a flexible factor stochastic volatility structure (see, e.g.,Aguilar and West, 2000), η yt = Λ y z yt + ε yt , z yt ∼ N ( , Υ yt ) , ε yt ∼ N ( , Ω yt ) , (5) η π t = Λ π z π t + ε π t , z π t ∼ N ( , Υ π t ) , ε π t ∼ N ( , Ω π t ) , (6) η gt ∼ N ( , e ω gt ) . (7)Here, z kt denotes a q -dimensional vector of normally distributed latent factors (for k ∈ { y , π } ) withdiagonal q × q -dimensional variance-covariance matrix Υ kt = diag ( e υ k t , . . . , e υ kqt ) , and Λ k is an N × q matrix of factor loadings. The idiosyncratic error term ε kt is also Gaussian, with zero mean anddiagonal N × N variance-covariance matrix Ω kt = diag ( e ω k t , . . . , e ω k N t ) . It is noteworthy that anycommon movements in the innovations determining potential output and trend inflation is purely drivenby the latent factors. The presence of ε kt implies that our model is flexible to allow for country-specificdeviations.The factor model on the shocks to the states is a parsimonious way of modeling a time-varyingvariance-covariance matrix since q (cid:28) N . To see this, consider Υ t = Υ yt Υ π t
00 0 , Ω t = Ω yt Ω π t
00 0 e ω gt , Λ = Λ y Λ π
00 0 . (8)Using Eq. (8), the M × M time-varying variance covariance matrix (with M = N +
1) of η t in Eq. (4)is given by Σ t = ΛΥ t Λ (cid:48) + Ω t . Consequently, Σ t is block-diagonal, allowing for non-zero covariances of the trend components foroutput and inflation across countries, respectively, while we impose orthogonality on the trend andcycle components τ yt , τ π t and g t across variable types (similar to the assumption introduced by Stockand Watson, 1999; 2007, in the context of single-country output gap estimation). For convenience, wedefine z t = ( z (cid:48) yt , z (cid:48) π t , ) .The law of motion imposed on the variances in Eq. (3) and Eq. (4) remains to be specified. Herewe assume that the logarithmic volatilities in Υ t , Ω t , and h kit follow independent AR(1) processes.Specifically, the log-volatility in the measurement equations is given by h kit = µ hki + (cid:37) hki ( h kit − − µ hki ) + ν kit , ν kit ∼ N ( , ϑ hki ) . (9) sing l = , . . . , q and j = , . . . , M to indicate the corresponding diagonal element in Υ t and Ω t , the log of the variances in the state equation evolve according to: υ lt = µ υ l + (cid:37) υ l ( υ lt − − µ υ l ) + ν lt , ν lt ∼ N ( , ϑ υ l ) , (10) ω jt = µ ω j + (cid:37) ω j ( ω jt − − µ ω j ) + ν jt , ν jt ∼ N ( , ϑ ω j ) . (11)To simplify notation in the following, we let • denote a placeholder for the various possible combinationsof indices. The autoregressive parameters are given by (cid:37) • , while the means of the log-volatilityprocesses are denoted by µ • . Finally, the state innovation variances are given by ϑ • . It is worth notingthat if a given ϑ • equals zero, the corresponding variance in the measurement or state equation isconstant. Selecting whether equations exhibit time variation in the error variances can thus be carriedout efficiently using the techniques stipulated in Frühwirth-Schnatter and Wagner (2010). Bayesian inference
The model outlined above is quite flexible but also heavily parameterized. This calls for regularizationvia Bayesian shrinkage priors. We start by outlining a general strategy to shrink our proposedfactor model towards a simpler specification when it comes to deciding which components shouldfeature conditional heteroscedasticity. The prior setup on the remaining free coefficients of the modelcompletes this subsection.In the following we describe how to flexibly select which equations should feature time variationin the variances by shrinking innovation variances in the stochastic volatility specifications to zero.Shrinkage to homoscedasticity in the observation equation is achieved in a similar manner. We startby substituting Eq. (5) and Eq. (6) in Eq. (4) and then proceed by squaring and taking logs of the r th equation ( r = , . . . , M ) to obtain the non-centered parameterization of the state space model(Frühwirth-Schnatter and Wagner, 2010; Kastner and Frühwirth-Schnatter, 2014),˜ ε rt = µ ω r + (cid:112) ϑ ω r ˜ ω rt + v rt , v rt ∼ ln χ ( ) (12)˜ ω rt = (cid:37) ω r ˜ ω rt − + w rt , w rt ∼ N ( , ) , (13)˜ ω rt = ω rt − µ ω r √ ϑ ω r , (14)with ˜ ε rt = ln ( f rt − Φ r • F t −[ Λ z t ] r • ) , Φ r • selecting the r th row of the matrix Φ , and [ Λ z t ] r • indicatingthe r th row of Λ z t . Equation (13) implies that the process variance ϑ ω r as well as the unconditionalmean µ ω r is moved from the stochastic volatility state equation into Eq. (4). Conditional on the fullhistory of the normalized log-volatilities and employing a mixture approximation to render Eq. (13)conditionally Gaussian (Kim et al. , 1998), the process variances and parameters can be obtained byestimating an otherwise standard Bayesian linear regression model. his implies that standard shrinkage priors can be specified on √ θ ω r . We adopt a flexible global-local shrinkage prior proposed in Griffin and Brown (2010) that was recently adopted for state spacemodels in Bitto and Frühwirth-Schnatter (2019). Here, ϑ ω r ∼ G (cid:18) , B ω r (cid:19) ⇔ (cid:112) ϑ ω r ∼ N ( , B ω r ) , with a local shrinkage hyperparameter B ω r ∼ G( κ ω , κ ω ξ ω / ) , ξ ω ∼ G( c , c ) .ξ ω is the global shrinkage parameter that pushes √ ϑ ω = (√ ϑ ω , . . . , √ ϑ ω M ) (cid:48) towards the prior mean.The hyperparameters κ ω and c , c are specified by the researcher. Intuitively, the global shrinkageparameter exerts shrinkage towards the zero vector, while B ω r serves to pull elements of √ ϑ ω awayfrom zero when ξ ω is large (i.e. heavy global shrinkage is introduced) if supported by likelihoodinformation. We choose an analogous setup for the innovations driving the variances of the latentfactors in z t , (cid:112) ϑ υ l ∼ N ( , B υ l ) , B υ l ∼ G( κ υ , κ υ ξ υ / ) , ξ υ ∼ G( d , d ) . The same prior choice is also employed for the process innovation variances in the log volatilityequations for the measurement errors, (cid:112) ϑ hki ∼ N ( , B hki ) , B hki ∼ G( κ h , κ h ξ h / ) , ξ h ∼ G( e , e ) . Notice that the common parameter ξ h pools information on error variances in the log-volatilities acrossall output and inflation equations, effectively introducing global shrinkage across variable types. Bittoand Frühwirth-Schnatter (2019) refer to this prior as a double Gamma prior. Consistent with theliterature, we set κ ω = κ υ = κ h = . c = c = d = d = e = e = .
01. This choice introducesheavy shrinkage on all process variances while maintaining heavy tails in the underlying marginalprior, and completes the specifcation to stochastically select the innovation variances that potentiallyresult in heteroscedastic errors in both the observation and state equations.Priors on the factor loadings in Eq. (1), Eq. (2) and Eq. (8), the free autoregressive coefficient inEq. (4), and the stochastic volatility parameters remain to be specified. Following Planas et al. (2008),we specify a Beta distributed prior on the amplitude Q of the business cycle, Q ∼ B( a Q , b Q ) , (15) ith a Q = .
82 and b Q = .
45 denoting hyperparameters chosen specifically for euro area businesscycles. For the frequency γ we also adopt a Beta prior with γ − γ L γ H − γ L ∼ B( a γ , b γ ) . (16)This prior restricts the support of γ by specifying a minimum wave length γ L , which is set equal totwo, and a maximum length γ H set equal to T . The parameters a γ = .
96 and b γ = . . et al. , 2008; Jarocinski and Lenza, 2018).For the remaining parameters of Eqs. (9) to (11) we follow Kastner and Frühwirth-Schnatter(2014) and use a weakly informative Gaussian prior on the unconditional means, µ • ∼ N ( , ) aswell as a Beta prior on the persistence parameter (cid:37) • ∼ B( , ) . On the factor loadings α ki and β ki that reflect the sensitivity of country-specific output and inflation measures to the cycle components,we use a sequence of independent Gaussian priors with α ki ∼ N ( , ) and β ki ∼ N ( , ) . For thefactor loadings in Λ k governing the covariance structure for the trend components across countries,with λ k • indicating the elements, we use tight independent Gaussian priors λ k • ∼ N ( , . ) . Finally,we specify the priors on the initial state f and the log-volatilities to be fairly uninformative with eachelement being normally distributed with zero mean and variance 10 .Notice that some parts of the parameter space of the model specified above are not econometricallyidentified. In the measurement equation, to identify the scale and sign of the output gap, we normalizethe loading for the first country using the restriction α =
1. Moreover, we restrict the factor loadingsmatrices Λ k following Aguilar and West (2000) by setting the respective upper q × q blocks equal tolower triangular matrices with ones on the main diagonals.These priors are then combined with the likelihood to obtain the posterior distribution. Since thejoint posterior is intractable, we employ a Markov chain Monte Carlo (MCMC) algorithm detailed inAppendix A. This algorithm samples all coefficients and latent quantities from their full conditionalposterior distributions to obtain, after a potentially large number of iterations, valid draws from thejoint posterior density. The algorithm is repeated 50 ,
000 times with the first 25 ,
000 draws discardedas burn-in. Convergence and mixing of most model parameters appear to be satisfactory. However,we find a substantial degree of autocorrelation for the factor loadings in selected countries. To assessthe sensitivity of our findings, we thus re-estimated the model a moderate number of times based ondifferent initial values. The corresponding findings appear to be remarkably robust. . IN-SAMPLE FEATURES OF THE MODEL3.1. Data overview
For the empirical application, we use quarterly data for economic output, measured in terms of realgross domestic product (RGDP, seasonally adjusted), and the harmonized index of consumer prices(HICP, in year-on-year growth rates), respectively. To obtain a measure of the output gap in percent,we transform the output variable by applying the transformation 400 log ( RGDP ) . We choose q = N = Euro area output gap estimates
In this subsection we present some key in-sample results of our proposed model. We start by comparingthe estimated output gap with other competing measures, which are depicted in Fig. 1. The black linein Fig. 1 shows the posterior median of the euro are output gap using the model specification sketchedabove (DFM-SV). To assess whether using cross-sectional information on prices and output leadsto significantly different conclusions, we include a model similar to the one proposed but exclusivelyrelying on aggregate data for the EA (labeled UCP-SV). This model is closely related to the multivariateunobserved components model proposed in Stella and Stock (2013). Furthermore, to inspect whetherour state evolution specification yields different dynamics in the gap component, we also include twomodel specifications that replace g t with a plug-in estimate ˆ g t . As estimators for g t , we use theapproach proposed in Hamilton (2018) (labeled Hamilton) and the well-known Hodrick-Prescott filter(HP, Hodrick and Prescott 1997) as a means to dissecting economic output series into a trend and acyclical component. These gap terms are computed based on aggregated data and then included in themodel described in Section 2.Figure 1 indicates that the output gap obtained from our multi-country framework closely tracksthe output gap measure obtained by estimating a bivariate unobserved components model based onaggregate data, especially in the beginning of the sample. During the GFC, we observe a slightdecoupling in terms of gap estimates between DFM-SV and the UCP-SV. Comparing both output gapmeasures with estimates arising from a model based on the approach proposed in Hamilton (2018) anda standard HP filter yields several interesting insights.First, the Hamilton gap indicates that in the end of the 1990s, output in the EA has been consistentlybelow potential output until the early 2000s. This pronounced negative gap is not visible for any of O u t pu t G ap DFM−SVUCP−SVHamiltonHP
Fig. 1:
Competing approaches to measuring the euro area output gap.
Notes : Dynamic factor model with stochastic volatility (DFM-SV) is the approach set forth in this paper exploiting cross-sectional information; unobserved components model with stochastic volatility (UCP-SV) refers to a standard specificationbased on aggregate euro area data. Hamilton denotes the approach set forth in Hamilton (2018), while the remainingspecification is the Hodrick-Prescott filter (HP, Hodrick and Prescott, 1997). Lines indicate the respective estimatedposterior median. the remaining three approaches. Second, the Hamilton and the HP measure indicate a strong positivedeviation of output from trend output in the run-up to the GFC with a slightly delayed but sharp dropin the final quarter of 2008. By contrast, our proposed measure already drops in the first half of 2008while a turning point in the business cycle is visible from mid 2009 onwards. At a first glance, it seemsthat this earlier drop in the output gap and the more timely rebound in real activity can be traced backto the fact that cross-sectional information is efficiently exploited. However, it is noteworthy that themeasure based on the UCP-SV model also tends to react faster compared to Hamilton and HP. Since thismodel, as opposed to DFM-SV, is not exploiting cross-sectional information explicitly, we conjecturethat the more timely reaction might come from modeling real activity and prices jointly. Third, andfinally, notice that both measures based on unobserved components models exhibit a significantlysmaller volatility and appear to be smoother. This effect is mainly due to our prior setup that softlyintroduces smoothness as well as additional information on the length and intensity of the cycle.We close this subsection by reporting prior and posterior summary statistics of the amplitude Q and frequency γ , depicted in Table 1. The table shows means and standard deviations associatedwith the prior and posterior of Q and γ , respectively. This comparison allows us to assess how muchinformation on the shape of the output gap is contained in the likelihood and, in addition, enables acomparison to the results reported in Planas et al. (2008). Considering the posterior mean and standarddeviation of γ suggests that the average length of the cycle is about 6.5 years. For data spanning fromthe 1980s to the early 2000s, Planas et al. (2008) report significantly longer cycles. Since our sampleperiod covers the GFC as well as the EA periphery crisis, this finding is not surprising since both able 1: Prior and posterior moments of the AR(2)-process parameters.
Prior PosteriorMean SD Mean SDFrequency γ Q Notes : SD – Standard deviation. Summary statistics refer to the priormoments in Eqs. (15) to (16). Posterior indicates the measures ob-tained from the posterior draws. shocks lead to abrupt downward movements in the business cycle. Comparing the prior and posteriordispersion indicates that the information contained in the prior is not reducing estimation uncertaintysignificantly.Next, we discuss the intensity of business cycle movements by considering the amplitude Q .Compared to previous studies, our estimate appears to be slightly lower. Since Planas et al. (2008) relyon aggregate data, the lower value of Q can be explained by the fact that our aggregate gap measurestrikes a balance between capturing the higher business cycle variance of EA peripheral countriessuch as Greece and Spain while capturing information on more stable business cycles found in, e.g.,Germany and Austria. Note that the prior and posterior mean are close to each other but the prior andposterior standard deviations differ strongly. This highlights that the introduction of prior informationhelps in reducing posterior uncertainty. The role of stochastic volatility in modeling the output gap
In the next step, we ask whether the volatility of the shocks driving the area-wide output gap is time-varying. To this end, the left panel in Fig. 2 displays the posterior median of the stochastic volatilitycomponent of the euro area output gap of our proposed multi-country model DFM-SV along withthe lower 16th and upper 84th percentile of the credible interval (grey shaded area). Considering theposterior quantiles in Fig. 2 provides some limited evidence in favor of heteroscedasticity. We observeslight increases during the burst of the dot-com bubble as well as during the period of the GFC.One way of assessing the likelihood that heteroscedasticity in the business cycle shocks is presentis to consider the posterior distribution of the square root of ϑ ω M up to a random sign switch. In caseof homoscedasticity, the corresponding marginal posterior would be unimodal and centered on zero.Consideration of the right panel of Fig. 2 corroborates the discussion above, namely that evidencefor heteroscedasticity is, at best, limited. While the marginal posterior is clearly not unimodal, mostposterior mass is located around zero.To assess how the presence of stochastic volatility in the unobserved components impacts theestimate of the output gap, Figure 3 shows the posterior median of the output gap under our baselinespecification (in solid black) alongside the 16th and 84th percentiles (dark shaded area) for the DFMand the UCP model. The dashed black line represents the posterior median of the output gap obtained Time e w g t ± J w M D en s i t y Fig. 2:
Stochastic volatility of the euro area output gap.
Notes : The left panel depicts the posterior variance of the output gap component over time. The solid line indicates theestimated posterior median, with grey shaded areas covering the area between the 16th and 84th percentile. The right panelshows a kernel density estimate of the posterior distribution of the signed square root of the innovation variance to thestochastic volatility process of the output gap. The dotted line marks zero. by estimating the model without stochastic volatility for all latent components, with the light grayarea denoting the 16th and 84th percentiles. One key finding of this figure is that for the DFM,switching off SV yields a similar measure of the output gap that is quite close to the one obtained underthe DFM with SV. The main differences concern the magnitude and variability of the gap measure.Put differently, comparing the posterior median across the two specifications points towards morepronounced movements in g t obtained from the model without stochastic volatility. This finding isclosely related to the critique raised by Sims (2001) and Stock (2001) in response to the work ofCogley and Sargent (2001), who estimate a time-varying parameter model without stochastic volatility.Ignoring stochastic volatility, within the framework of a time-varying parameter model, is expectedto exaggerate movements in the regression coefficients and potentially creating transient variations infiltered estimates. Dissecting euro area business cycle movements
In the following, we provide information on the quantitative contributions of shocks to trend, cyclicaland idiosyncratic components to the observed series of inflation over time. Here we use an approachsimilar to a standard historical time series decomposition. Notice that the non-stationary nature of thetrend components in Eq. (4) implies that shocks to these quantities are persistent and do not peter out.In fact, instead of becoming less important over time, the relative importance of shocks to the trendcomponents increases by construction. As a consequence, we focus on the contributions of the shocksat each point in time. Combining Eqs. (2), (4) and (6), we can decompose inflation across countries FM UCP2000 2005 2010 2015 2000 2005 2010 2015−10−50510 O u t pu t gap heteroscedastichomoscedastic Fig. 3:
Dynamic factor and unobserved component models with and without stochastic volatility.
Notes : Dynamic factor model (DFM) is the approach set forth in this paper exploiting information across euro area countries.Unobserved components model (UCP) refers to a standard specification based on aggregate euro area data. Solid and dashedlines indicate the estimated posterior median, with grey shaded areas covering the area between the 16th and 84th percentile. in terms of their shocks and lagged states: π it = τ π it − + β i φ g t − + β i φ g t − + [ Λ π z π t ] i • + [ ε π t ] i + β i η gt + (cid:15) π it . The decomposition yields three individual shocks of interest, with [ Λ π z π t ] i • and β i η gt reflecting jointarea-wide dynamics, while [ ε π t ] i and (cid:15) π it capture idiosyncratic shocks. In particular, [ Λ π z π t ] i • arisesfrom the factor stochastic volatility structure and indicates common euro area trend component shocks(subsequently labeled Euro area trend shocks ). The contribution of the gap component is given by β i η gt (indicated as Gap shocks in the following). The quantity [ ε π t ] i is a country-specific shockto the trend component, while (cid:15) π it is the idiosyncratic measurement error (labeled Country shocks and considered jointly in what follows). To ease visualization, Fig. 4 shows the posterior median ofperiod-specific shocks exclusively based on ˜ π it = π it − τ π it − − β i φ g t − − β i φ g t − .Figure 4 reveals a set of interesting results for the shock decomposition of inflation across countries.First, the most striking observation is that Euro area trend shocks do not play a role in driving observedinflation series. This finding results from an almost diagonal variance-covariance structure betweencountry-specific trends of inflation, with most covariances rather close to zero. In terms of the modelingsetup, this implies that one may safely impose orthogonality on the errors for the trend inflation stateequations.Second, we find substantial evidence for the existence of a Phillips curve relationship across theEA countries given by
Gap shocks . Notice that the sensitivity of country-specific inflation series toarea-wide output gap shocks is governed by the factor loadings β i . Here, we find that the slope ofthe Phillips curve exhibits heterogeneity, with the Netherlands and Finland providing examples of lesssensitive countries. By contrast, the area-wide output gap shocks appear to be particularly important L PTGR ITFI FRDE ESAT BE2000 2005 2010 2015 2000 2005 2010 2015−2−101−2−101−2−101−2−101−2−101 S ho cks Country shocks Euro area trend shocks Gap shocks
Fig. 4:
Decomposing shocks shaping inflation across countries.
Notes : Shocks refer to the posterior median of the estimated shocks of the fitted model.
Country shocks are shocks specific toall countries and thus include the idiosyncratic component of the factor stochastic volatility specification and the measurementerrors. The remaining quantities arise from joint euro area dynamics;
Euro area trend shocks refer to shocks identified basedon common euro area factors underlying country-specific potential output, while
Gap shocks for country i arise solely fromthe gap component. The dotted line marks zero. for the dynamic evolution of inflation in Belgium, Spain and Greece. This result implies that almostall comovements in inflation across countries arises from the joint gap component rather than shocksto country-specific trend inflation.Finally, we assess the importance of country-level shocks. Recall that these shocks depict bothshocks to idiosyncratic trends, but also the measurement errors. It is worth mentioning that measure-ment errors play only a minor role in shaping the observed inflation series over the cross-section, andthe contributions labeled Country shocks mainly feature shocks to the trend components. The highestimportance of such country-level shocks is apparent for the cases of Greece, Italy and Portugal in thefive year period after 2010, while inflation in the Netherlands appears to be shaped to a large extent byidiosyncratic shocks throughout the observed period. Impulse response horizon
Fig. 5:
Impulse response function of a negative one standard deviation shock to the output gap.
Notes : The solid black line depicts the median response alongside the 16th and 84th percentiles shaded in grey. The dottedline marks zero.
Responses of output and inflation to business cycle shocks
This subsection aims at studying the dynamic effect of business cycle shocks to inflation across the euroarea. Such a common shock is of interest for policy makers in order to assess the sensitivity of theirrespective countries to common adverse movements in an area-wide business cycle. In our framework,a business cycle shock is defined as an unexpected decrease in η gt by one standard deviation. Thisyields dynamic reactions of g t + h ( h = , . . . , H ) that are then transformed into dynamic reactions of y it + h and π it + h by using the factor loadings α i and β i . These impulse response functions (IRFs) thusprovide not only information on the specific time profile of the output gap reactions but also on thesensitivity of a given country and variable to such changes.Figure 5 depicts the posterior distribution of the IRF of the common output gap to a (negative) onestandard deviation business cycle shock. The black line in the figure represents the median responsesover time along with lower 16th and upper 84th percentiles of the posterior distribution. The figurepresents a negative and immediate impact on the common gap component, with a peak decline inthe output gap of around 1.5 percentage points. This peak is reached after around three quarters andrapidly dies out afterwards. After around 2.5 years, the effect on the output gap is zero.It is worth noting that Fig. 5 only measures the dynamic impact to the latent gap component.However, policy makers might be particularly interested in how changes in the common cycle impactinflation across countries. Since the dynamics of π it are proportional to movements in g t , we reportpeak effects that are reached after around three quarters (see Fig. 5).Inspection of the maximum responses of inflation in Fig. 6 reveals that a common business cycleshock translates into a drop in inflation across all countries under scrutiny. This drop in inflation rangesfrom about − . − . Country P ea k r e s pon s e Fig. 6:
Peak responses of a euro area business cycle shock for inflation across countries.
Notes : Boxplots refer to the posterior draws at the peak response. Boxes cover the area between the 16th and 84th percentile,with the solid black line depicting the posterior median. The dotted line marks zero. provide strong evidence that a Phillips curve relationship exists in the EA. This corroborates recentevidence reported in Moretti et al. (2019), who find a relationship between inflation and real activitybased on aggregate EA data. However, notice that there exist considerable differences in peak inflationreactions across countries, which could explain findings in Barigozzi et al. (2014) and Peersman(2004), who report asymmetric responses of macroeconomic quantities to common monetary policyshocks in the euro area, given that the link between demand-sided policies and inflation differ acrossEA member states.
4. FORECASTING EVIDENCE
Up to this point we have focused on in-sample results to illustrate the key features of our proposedmodeling approach. However, a successful model that could be useful for policy analysis should alsobe able to predict well. To investigate the predictive capabilities, this section builds on the literatureon inflation forecasting (see Stock and Watson, 1999; 2007; Jarocinski and Lenza, 2018; Koop andKorobilis, 2018, among others) and uses our model to forecast aggregate inflation for the EA andacross individual member states up to four quarters ahead.
Design of the forecasting exercise and competing models
To evaluate forecast performance for both the EA and individual countries, we split the sample into aninitial estimation period that ranges from 1997:Q2 to 2008:Q3 (47 observations) and use the remaining40 observations as a hold-out period. The forecasting design adopted is recursive, implying that afterobtaining a set of predictive densities, we increase the length of the initial observation period by onequarter until we reach the end of the hold-out period. ifferences in predictive accuracy are gauged by relying on root mean squared errors (RMSEs) andlog predictive scores (LPSs, see Geweke and Amisano, 2010). RMSEs are obtained by consideringthe differences between the posterior median of the predictive distribution and the realized values of π it for each model and across the hold-out period. Analogously, LPSs are computed by evaluating therealized values under the predictive density of a given model, summed over the hold-out.We benchmark the proposed DFM-SV model against a range of competing models that differin several respects. First, we distinguish between models that exploit cross-sectional information(labeled Multi-country ) versus specifications that utilize only country-specific information (labeled
Single-country ). In the case of aggregate euro area inflation forecasts, we use the abbreviation
EA-level to indicate that predictions are based on observations of output and inflation that are aggregatedfrom country-level data prior to estimation to yield a measure of EA-level inflation and output. Here,aggregate refers to taking the arithmetic mean over the respective country-specific series. Second, weconsider a range of alternative measures of the output gap to assess differences between treating theoutput gap as a latent quantity as opposed to using an observed measure. Third, we gauge the accuracygains from stochastic volatility by also including homoscedastic variants of all competing models. The model set we consider is comprised of the following benchmarks:(i) Unobserved components model (UCP): This specification refers to a modeling approach in thespirit of Stella and Stock (2013). For
Single-country , this implies that we introduce country-specific gap components and estimate individual country-specifications with orthogonal errorterms. This yields a model setup per country that estimates three unobserved components. Bycomparing the out-of-sample predictive performance of the UCP specifications with forecastsproduced by DFM, the inclusion of these specifications serves to assess the merits of consid-ering multi-country information as a means to improving both country-specific and aggregatepredictions. In the case of
EA-level , we aggregate country-level series a priori and estimate themodel using three latent factors.(ii) Hamilton (
Ham ): These specifications rely on a plugin-estimate ˆ g t as the measure of the outputgap in the framework proposed in this paper. For Ham , we follow recent work by Hamilton(2018) as a means to estimating the gaps. We calculate forecasts for
Single-country (extractinggaps for each country individually) and
Multi-country (aggregating a priori and using EA-levelinformation).(iii) Hodrick-Prescott (HP): Similar to the strategy employed for for
Ham , these specifications usethe well-known HP filter (Hodrick and Prescott 1997) to produce euro area output gap estimates.For the forecast comparison, again both multi-country and single-country specifications for HPare implemented. Here, homoscedasticity implies that we assume constant error variances in the state and observation equations. iv) AR(1): A standard homoscedastic autoregressive process of order one used to forecast aggregateeuro area inflation, and country-specific inflation series independently.In what follows, all models are benchmarked against the AR(1) model. Here, we consider relativeRMSEs and differences in LPSs of all specifications versus the AR(1) model. RMSEs below 100 thusreflect that the respective model outperforms the benchmark in terms of point predictions, while LPSsexceeding zero indicate superior performance for density forecasts vis-á-vis the AR(1) specification. Aggregate euro area inflation forecasts
In this subsection we assess whether our model yields competitive forecasts for aggregate data. Out-of-sample performance for aggregate euro area inflation is evaluated for the one-quarter up to thefour-quarter ahead horizon. Table 2 reports relative RMSEs and differences in LPSs, benchmarked tothe AR(1) model.
Table 2:
Forecast evaluation for euro area inflation.
Multi-country EA-levelnon-SV SV non-SV SVDFM Ham HP DFM Ham HP UCP Ham HP UCP Ham HP
LPS . . . . . . − . − . − . − . − . − . . . . . . . − . − . − . − . − . − . . . . . . . − . − . − . − . − . − . . . . . . . − . − . − . − . − . − . RMSE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Notes : Multi-country indicates that cross-sectional information from individual countries is used. Single-country refers to independent indi-vidual models for all countries. SV indicates the specification allowing for heteroscedastic errors, while non-SV assumes homoscedasticity.DFM – dynamic factor model; Ham – Hamilton’s approach (Hamilton, 2018); HP – Hodrick Prescott filter. LPS – log predictive score;RMSE – root mean squared error. to refer to the forecast horizon by quarter between one-quarter to one-year. LPS and RMSE arepresented relative to independent homoscedastic univariate AR(1) processes. For LPS, the maximum value is indicated in bold, for RMSEs(in percent), the minimum is in bold, indicating the best performing specification. Overall, our proposed multi-country framework DFM-SV appears to produce highly competitiveout-of-sample predictions, outperforming most competing models. This finding holds true for bothpoint and density predictive performance. Accuracy improvements in terms of LPS tend to be sub-stantial, irrespective of whether g t is estimated alongside the remaining model parameters and statesor whether we rely on other measures of the output gap. Considering relative RMSEs reveals thatwhile our DFM-SV specification improves upon the benchmark model, these improvements appear tobe muted and range from three percent (in the case of the one-step-ahead horizon) to 7.5 percent (forthe four-quarter-ahead forecast). Only in two cases our proposed DFM-SV is slightly outperformedby multi-country versions where the latent gap component is replaced by estimates obtained using the amilton (for point forecasts) and the HP (for LPS) approach and with SV turned on. In both cases,however, DFM-SV displays the second best performance.Comparing the out-of-sample performance of models that utilize cross-sectional information tothe ones that rely solely on aggregate EA data points towards accuracy gains of the multi-countrymodels. Models that utilize only aggregate data generally appear to be inferior to the AR(1) modelin terms of density forecasts while being slightly superior to the univariate benchmark in some cases.Specifically, the UCP model slightly improves upon the AR(1) in terms of RMSEs. These resultsconfirm and corroborate findings in Marcellino et al. (2003), who report that the inclusion of country-specific information improves out-of-sample predictions even if interest centers on predicting aggregatequantities of interest.To sum up, Table 2 suggests that, when interest centers on forecasting euro area inflation, ourproposed model framework yields strong density and point forecasts. These accuracy improvementsare especially pronounced when compared to models that rely exclusively on aggregate information,highlighting the necessity to take a cross-sectional stance when forecasting inflation. Forecasts for individual countries
The previous subsection provided an overall gauge on how our model performs in predicting inflation.Next, we take a cross-sectional perspective and assess whether there exist interesting cross-countrydifferences in forecast performance. For the sake of brevity, we focus on one-quarter-ahead forecastsin Table 3 and one-year-ahead predictions in Table 4. These tables include marginal LPS obtained byintegrating out the remaining elements of the joint predictive density.Starting with the one-step-ahead marginal LPS, Table 3 suggests that the homoscedastic variantof our proposed DFM outperforms all competing specification by large margins for most countriesconsidered, both in terms of point and density forecasts. Only for the Netherlands, Austria and Finland,we observe that single-country models yield more precise density prediction whereas point forecastsfor the Netherlands are most precise when single-country models are adopted. We conjecture thatthis stems from the fact that these countries tend to share a common business cycle and thus using allavailable cross-section information and a single factor potentially translates into a misspecified model.Considering accuracy gains from controlling for heteroscedasticy shows that for most countries, ex-plicitly allowing for SV translates into weaker point and density forecasts relative to the homoscedasticcounterparts. This result is in contrast to the findings based on using the full predictive distribution ofinflation reported in Table 2 and the literature on inflation forecasting (Stock and Watson, 2007; Stellaand Stock, 2013; Jarocinski and Lenza, 2018). The reasons for this slightly inferior performance ofSV specifications in terms of marginal LPS are at least threefold. First, the hold-out period coversthe beginning of the global financial crisis, implying that error variances are already tilted upwards.Second, our DFM-SV specification constitutes a parsimonious means of modeling a large dimensionaltime-varying variance-covariance matrix. Thus, if interest centers on capturing the potentially time- able 3: Forecast evaluation for inflation at the one-quarter ahead forecast horizon.
Multi-country Single-countrynon-SV SV non-SV SVDFM Ham HP DFM Ham HP UCP Ham HP UCP Ham HP
LPS DE . . − . . − . − . . . . . − . − . FR . . . . . . . . . . . . IT . − . − . . . − . − . − . − . − . − . − . ES . − . − . . − . − . − . − . − . − . − . − . NL . − . − . − . . − . − . − . . − . − . − . BE . . . . . . − . − . − . . . . AT . − . − . . − . − . . − . . . . . FI . . . . . . . . . . − . . PT . − . − . . − . − . . − . . . − . − . GR . − . − . − . − . − . . − . − . − . − . − . RMSE DE . . . . . . . . . . . . FR . . . . . . . . . . . . IT . . . . . . . . . . . . ES . . . . . . . . . . . . NL . . . . . . . . . . . . BE . . . . . . . . . . . . AT . . . . . . . . . . . . FI . . . . . . . . . . . . PT . . . . . . . . . . . . GR . . . . . . . . . . . . Notes : Multi-country indicates that cross-sectional information from individual countries is used. Single-country refers to independent indi-vidual models for all countries. SV indicates the specification allowing for heteroscedastic errors, while non-SV assumes homoscedasticity.DFM – dynamic factor model; Ham – Hamilton’s approach (Hamilton, 2018); HP – Hodrick Prescott filter. LPS – log predictive score;RMSE – root mean squared error. LPS and RMSE are presented relative to independent homoscedastic univariate AR(1) processes. ForLPS, the maximum value is indicated in bold, for RMSEs (in percent), the minimum is in bold, indicating the best performing specification. varying nature of covariances (which is relevant if the full predictive density is evaluated), predictivegains in terms of density forecasts tend to increase with the dimension of the model (Kastner, 2019).Third, and contrasting accuracy differences between models that treat the gap component as observedas opposed to latent, we generally find that multi-country models profit from explicitly controllingfor estimation uncertainty surrounding g t . This premium in predictive accuracy stems from the factthat integrating out g t from the predictive density translates into a heavy-tailed marginal predictivedistribution that is capable of handling outlying values well. This lowers the necessity to explicitlycontrol for stochastic volatility, especially for data at quarterly frequency.Turning attention to the one-year-ahead forecast horizon, Table 4 shows similar results to thosereported for the one-quarter-ahead horizon. For this longer forecast horizon, the homoscedastic DFMsetup appears to be particularly successful in terms of producing accurate point predictions, which isnot surprising given the fact that for higher-order forecasts, the log-volatilities approach their stationarydistribution. The predictive performance in terms of point forecasts of the DFM model is comparableto its heteroscedastic counterpart. Unlike the remaining alternative models, both DFM and DFM-SValso manage to notably outperform the AR(1) benchmark for the one-year-ahead horizon. In terms ofdensity forecasts, however, the predictive dominance of DFM appears less distinctive. able 4: Forecast evaluation for inflation at the one-year-ahead forecast horizon.
Multi-country Single-countrynon-SV SV non-SV SVDFM Ham HP DFM Ham HP UCP Ham HP UCP Ham HP
LPS DE . − . − . . − . − . − . . . . . − . FR . − . − . . − . − . − . − . − . − . − . − . IT − . − . − . − . − . − . − . − . − . − . − . − . ES . − . − . . − . − . − . − . − . − . − . − . NL − . − . − . . − . − . − . − . − . . − . − . BE . − . − . . . − . − . − . − . . . − . AT . − . − . − . − . − . − . − . − . . . − . FI − . − . − . − . − . − . . − . . − . . − . PT . − . − . − . − . − . − . − . − . − . − . − . GR . − . − . − . − . − . − . − . − . − . − . − . RMSE DE . . . . . . . . . . . . FR . . . . . . . . . . . . IT . . . . . . . . . . . . ES . . . . . . . . . . . . NL . . . . . . . . . . . . BE . . . . . . . . . . . . AT . . . . . . . . . . . . FI . . . . . . . . . . . . PT . . . . . . . . . . . . GR . . . . . . . . . . . . Notes : Multi-country indicates that cross-sectional information from individual countries is used. Single-country refers to independent indi-vidual models for all countries. SV indicates the specification allowing for heteroscedastic errors, while non-SV assumes homoscedasticity.DFM – dynamic factor model; Ham – Hamilton’s approach (Hamilton, 2018); HP – Hodrick Prescott filter. LPS – log predictive score;RMSE – root mean squared error. LPS and RMSE are presented relative to independent homoscedastic univariate AR(1) processes. ForLPS, the maximum value is indicated in bold, for RMSEs (in percent), the minimum is in bold, indicating the best performing specification.
An overall comparison between multi-country and single-country models for the one-year-aheadhorizon again reveals no clear pattern. However, this is particularly due to the strong performance ofour proposed multi-country frameworks DFM and DFM-SV. Without these two specifications, Table4 shows that single-country models appear to be preferable compared to the multi-country setups.However, for one-year-ahead predictions, the table again highlights the importance for including cross-sectional information to produce accurate point forecasts for inflation in Portugal and Greece.
5. CONCLUDING REMARKS
In this paper, we develop a multivariate Bayesian dynamic factor model with stochastic volatility foranalyzing euro area business cycles. The multi-country framework decomposes country-specific outputand inflation series into idiosyncratic non-stationary trends and a joint stationary cyclical component.This enables us to exploit cross-sectional information and obtain an EA-wide measure of the output gapused for structural analysis and inflation forecasting. A key model feature is to allow for heteroscedasticerror terms and comovements in the trends using a flexible factor stochastic volatility structure. Thesetup is completed by considering time variation also in the variances of the measurement equations. he proposed Bayesian model alleviates concerns of overparameterization via global-local shrinkagepriors that push the model towards a homoscedastic specification, but allows for time-varying variancesif necessary.In an empirical section, we study both in-sample features and out-of-sample predictive performanceof the proposed model. We compare the obtained measure of the output gap to a set of competingapproaches for estimation and discuss the role of time variation in error variances. The analysis iscomplemented by an empirical assessment regarding the slope of the Philips curve across EA memberstates. In a forecasting exercise, the paper provides evidence that accounting for a common euro areaoutput gap component produces competitive forecasts for inflation both on the aggregate EA, but alsothe country level. REFERENCES A GUILAR O, AND W EST
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Journal of Monetary Economics (2), 293–335.——— (2007), “Why has US inflation become harder to forecast?” Journal of Money, Credit and Banking (S1), 3–33. . FULL CONDITIONAL POSTERIOR DISTRIBUTIONS It is worth noting that the joint posterior distribution of the model parameters and the set of latent states isintractable. Fortunately, the full conditional posterior distributions for most quantities are of a simple formand thus amenable to standard Gibbs updating. In order to obtain a draw from the joint posterior we design astraightforward Markov chain Monte Carlo (MCMC) algorithm that cycles through the following steps:(i) Simulate the full history of { f t } Tt = using a forward filtering backward sampling algorithm (Carter andKohn 1994, Frühwirth-Schnatter 1994).(ii) Draw the sequence of log-volatilities { h kit } Tt = , { υ lt } Tt = , { ω rt } Tt = , for all i , k , l , r as well as the parametersin the corresponding state equations independently using the algorithm proposed in Kastner and Frühwirth-Schnatter (2014).(iii) Conditional on the unobserved components, we simulate the loadings α ki and β ki by estimating 2 N independent regression models with heteroscedastic innovations(iv) Conditional on the other parameters of the model, we simulate the history of the factors { z t } Tt = drivingthe covariances between the country-specific trend components based on the regressions and quantitiesgiven in Eq. (5) and Eq. (6).(v) The free elements in Λ conditionally on knowing the full history of the factors z t can be sampled on anequation-by-equation basis involving a sequence of standard linear regression models with heteroscedasticerrors (see also Aguilar and West, 2000; Kastner, 2019).(vi) The parameters Q and γ are updated in a block by using a standard random walk Metropolis Hastingsalgorithm.(vii) Sample ξ h , ξ ω and ξ υ from a Gamma distributed conditional posterior distribution.Steps (i) to (v) are standard and easily executed. Steps (vi) and (vii) deserve more attention. The full conditionalposterior distributions of B hki , B ω r , and B υ l are similar, and we thus only present specifics for one of them, B ω r . The conditional posterior of this parameter follows a GIG distribution that is obtained by combining theconditional density p (√ ϑ ω r | B ω r ) with the conditional prior p ( B ω r | ξ ω ) , B ω r |• ∼ GIG( κ ω − / , ϑ ω r , ξ ω κ ω ) , (A.1)where • denotes conditioning on all remaining quantities of the model.To obtain the full conditional posterior distribution for the global scaling parameters that is again similarfor ξ h , ξ ω and ξ υ , we combine the joint density (cid:206) Mr = p ( B ω r | ξ ω ) with the prior p ( ξ ω ) . This yields a Gammadistributed conditional posterior distribution, ξ ω |• ∼ G (cid:32) c + κ ω M , c + κ ω M (cid:213) r = B ω r (cid:33) . (A.2)This setup completes the full simulation-based algorithm. We iterate the steps above for 50 ,
000 times with aburn-in period of the first 25 ,
000 cycles. The obtained results provide evidence for satisfactory convergenceproperties of the MCMC algorithm.000 cycles. The obtained results provide evidence for satisfactory convergenceproperties of the MCMC algorithm.