A multi-country dynamic factor model with stochastic volatility for euro area business cycle analysis

This paper develops a dynamic factor model that uses euro area (EA) country-specific information 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-specific stochastic trends and a common cyclical component. Comovement in the trends is introduced by imposing a factor structure on the shocks to the latent states. We moreover introduce flexible stochastic volatility specifications to control for heteroscedasticity in the measurement errors and innovations to the latent states. Carefully specified shrinkage priors allow for pushing the model towards a homoscedastic specification, if supported by the data. Our measure of the output gap closely tracks other commonly adopted measures, with small differences in magnitudes and timing. To assess whether the model-based output gap helps in forecasting inflation, we perform an out-of-sample forecasting exercise. The findings indicate that our approach yields superior inflation forecasts, both in terms of point and density predictions.


INTRODUCTION
Effective policy making in central banks such as the European Central Bank (ECB) requires accurate measures of latent quantities such as the output gap to forecast key quantities of interest like inflation across euro area (EA) member states. Since using aggregate EA data potentially masks important country-specific dynamics, exploiting country-level information could help in obtaining more reliable estimates 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 construct a multi-country model for the EA. The proposed framework aims to combine the literature on output gap modeling (see, among many others, Kuttner, 1994;Orphanides and Van Norden, 2002;Basistha and Nelson, 2007;Planas et al., 2008) that focuses on estimating the output gap based on data for a 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;. 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 and inflation feature a non-stationary country-specific component. To control for potential comovement in these trend terms, we assume that the corresponding shocks to the states feature a factor structure. The resulting factor model features stochastic volatility (SV) in the spirit of Aguilar and West (2000) and thus provides a parsimonious way of controlling for heteroscedasticity. Since successful forecasting models typically allow for SV (Clark, 2011;Clark and Ravazzolo, 2015;Huber, 2016;Huber and Feldkircher, 2019), we also allow for time-variation in the error variances across the remaining state innovations and the measurement errors. One methodological key innovation is the introduction of global-local shrinkage priors on the error variances of the state equations describing the law of motion of the logarithmic volatility components, effectively shrinking the system towards a homoscedastic specification, if applicable.
This increased flexibility, however, is costly in terms of additional parameters to estimate. We thus follow 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-Schnatter and Wagner, 2010) to test whether SV is supported by the data. The non-centered parameterization allows 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 shrinkage component that applies to all process variances simultaneously, forcing them towards zero. Local shrinkage parameters are then used to drag sufficient posterior mass away from zero even in the presence of strong global shrinkage, allowing for non-zero process variances if required.
When applied to data for ten EA countries over the time period from 1997:Q1 to 2018:Q4, we find that 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 used in policy institutions. We moreover perform historical decompositions to gauge the importance of area-wide as opposed to country-specific shocks for describing inflation movements. These measures reveal that inflation is strongly driven by common business cycle dynamics, underlining the importance of controlling for a common business cycle. We then turn to assessing whether there exists a Phillips curve across EA countries by simulating a negative one standard deviation business cycle shock. This exercise points towards a robust relationship between the common gap component and inflation, with magnitudes 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 predictive capabilities. Since inflation across countries is driven by a term measuring trend inflation and the output gap, our framework can be interpreted as a New Keynesian Phillips curve, akin to Stella and Stock (2013). Compared to a set of simpler alternatives that range from univariate benchmark models to models that use alternative ways to calculate the output gap, the proposed model yields more precise point 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 summarize the main steps involved in estimating the model. Section 3 presents the empirical application, starting with a summary of the dataset and inspects various key features of our model. The section moreover studies 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 model with other specifications. The final section summarizes and concludes the paper.

A dynamic factor model for the euro area
In this section we describe the framework to estimate the euro area output gap using disaggregate country-level information. Let y it and π it denote output and inflation for country i = 1, . . . , N in period t = 1, . . . , T, respectively. For notational simplicity, we define k ∈ {y, π}.
Country-specific output and inflation are driven by unobserved common non-stationary trend components 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) quantities are related to the observed quantities through a set of measurement equations: These equations imply that the trend components can loosely be interpreted as country-specific trend inflation and potential output for the ith country, respectively. Moreover, the stationary component of output 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 k i t . It is worth stressing that Eq.
(2) represents a country-specific Phillips curve that establishes a relationship between inflation and the area-wide output gap g t . One key goal of this paper is to assess whether there exists a Phillips curve across EA countries by inspecting β i and functions thereof.
The country-specific components in Eq.
For the unrestricted AR(2) parameters φ 1 and φ 2 we follow Planas et al. (2008) and reparameterize the state equation coefficients of g t using polar coordinates imposing complex roots, Hereby, Q determines the amplitude and γ the frequency of the cycle. The parameterization has the convenient property that available information on the duration and intensity of business cycles can be introduced with relative ease. Incorporating such information using normally distributed priors is complicated, since autoregressive coefficients are more difficult to interpret in terms of the intensity and frequency of the time series. Moreover, allegedly weakly informative Gaussian priors could introduce information on functions of the parameters, potentially placing too much prior weight on dynamics that do 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 blockwise orthogonal and achieve this by employing a flexible factor stochastic volatility structure (see, e.g., Aguilar and West, 2000), η gt ∼ N (0, e ω gt ).
Here, z kt denotes a q-dimensional vector of normally distributed latent factors (for k ∈ {y, π}) with diagonal q × q-dimensional variance-covariance matrix Υ kt = diag(e υ k1t , . . . , e υ k q t ), and Λ k is an N × q matrix of factor loadings. The idiosyncratic error term ε kt is also Gaussian, with zero mean and diagonal N × N variance-covariance matrix Ω kt = diag(e ω k1t , . . . , e ω k N t ). It is noteworthy that any common movements in the innovations determining potential output and trend inflation is purely driven by the latent factors. The presence of ε kt implies that our model is flexible to allow for country-specific deviations.
The factor model on the shocks to the states is a parsimonious way of modeling a time-varying variance-covariance matrix since q N. To see this, consider Using Eq. (8), the M × M time-varying variance covariance matrix (with M = 2N + 1) of η t in Eq. (4) is given by Consequently, Σ t is block-diagonal, allowing for non-zero covariances of the trend components for output and inflation across countries, respectively, while we impose orthogonality on the trend and cycle components τ yt , τ πt and g t across variable types (similar to the assumption introduced by Stock and Watson, 1999;, in the context of single-country output gap estimation). For convenience, we define z t = (z yt , z πt , 0).
The law of motion imposed on the variances in Eq.
(3) and Eq. (4) remains to be specified. Here we 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 Using l = 1, . . . , 2q and j = 1, . . . , 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: To simplify notation in the following, we let • denote a placeholder for the various possible combinations

Bayesian inference
The model outlined above is quite flexible but also heavily parameterized. This calls for regularization via Bayesian shrinkage priors. We start by outlining a general strategy to shrink our proposed factor model towards a simpler specification when it comes to deciding which components should feature conditional heteroscedasticity. The prior setup on the remaining free coefficients of the model completes this subsection.
In the following we describe how to flexibly select which equations should feature time variation in 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 start by substituting Eq. (5) and Eq. (6) in Eq. (4) and then proceed by squaring and taking logs of the rth equation (r = 1, . . . , M) to obtain the non-centered parameterization of the state space model (Frühwirth-Schnatter and Wagner, 2010;Kastner and Frühwirth-Schnatter, 2014), with a local shrinkage hyperparameter ξ ω is the global shrinkage parameter that pushes The hyperparameters κ ω and c 0 , c 1 are specified by the researcher. Intuitively, the global shrinkage parameter exerts shrinkage towards the zero vector, while B ωr serves to pull elements of √ ϑ ω away from zero when ξ ω is large (i.e. heavy global shrinkage is introduced) if supported by likelihood information. We choose an analogous setup for the innovations driving the variances of the latent The same prior choice is also employed for the process innovation variances in the log volatility equations for the measurement errors, Notice that the common parameter ξ h pools information on error variances in the log-volatilities across all output and inflation equations, effectively introducing global shrinkage across variable types. Bitto we specify a Beta distributed prior on the amplitude Q of the business cycle, with a Q = 5.82 and b Q = 2.45 denoting hyperparameters chosen specifically for euro area business cycles. For the frequency γ we also adopt a Beta prior with This prior restricts the support of γ by specifying a minimum wave length γ L , which is set equal to two, and a maximum length γ H set equal to T. The parameters a γ = 2.96 and b γ = 10.7 are fixed hyperparameters again set specifically according to prior research on business cycles in the euro area.
These choices are weakly informative and imply a periodicity of around eight years and a contraction factor of 0.8 (Gerlach and Smets, 1999;Planas 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 (0, 10 2 ) as well as a Beta prior on the persistence parameter • ∼ B(25, 5). 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 (0, 1) and β ki ∼ N (0, 1). For the factor 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 (0, 0.1). Finally, we specify the priors on the initial state f 0 and the log-volatilities to be fairly uninformative with each element being normally distributed with zero mean and variance 10 2 .
Notice that some parts of the parameter space of the model specified above are not econometrically identified. In the measurement equation, to identify the scale and sign of the output gap, we normalize the loading for the first country using the restriction α 1 = 1. Moreover, we restrict the factor loadings matrices Λ k following Aguilar and West (2000) by setting the respective upper q × q blocks equal to lower triangular matrices with ones on the main diagonals.
These priors are then combined with the likelihood to obtain the posterior distribution. Since the joint posterior is intractable, we employ a Markov chain Monte Carlo (MCMC) algorithm detailed in Appendix A. This algorithm samples all coefficients and latent quantities from their full conditional posterior distributions to obtain, after a potentially large number of iterations, valid draws from the joint posterior density. The algorithm is repeated 50, 000 times with the first 25, 000 draws discarded as 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 assess the sensitivity of our findings, we thus re-estimated the model a moderate number of times based on different initial values. The corresponding findings appear to be remarkably robust.

Data overview
For the empirical application, we use quarterly data for economic output, measured in terms of real gross 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 = 1 latent factors for both the potential output for all countries and trend inflation measures to capture the covariances between the country-specific quantities.

Euro area output gap estimates
In this subsection we present some key in-sample results of our proposed model. We start by comparing the estimated output gap with other competing measures, which are depicted in Fig. 1. The black line in Fig. 1 shows the posterior median of the euro are output gap using the model specification sketched above (DFM-SV). To assess whether using cross-sectional information on prices and output leads to significantly different conclusions, we include a model similar to the one proposed but exclusively relying on aggregate data for the EA (labeled UCP-SV). This model is closely related to the multivariate unobserved components model proposed in Stella and Stock (2013). Furthermore, to inspect whether our state evolution specification yields different dynamics in the gap component, we also include two model specifications that replace g t with a plug-in estimateĝ t . As estimators for g t , we use the approach proposed in Hamilton (2018)   to the fact that cross-sectional information is efficiently exploited. However, it is noteworthy that the measure based on the UCP-SV model also tends to react faster compared to Hamilton and HP. Since this model, as opposed to DFM-SV, is not exploiting cross-sectional information explicitly, we conjecture that the more timely reaction might come from modeling real activity and prices jointly. Third, and finally, notice that both measures based on unobserved components models exhibit a significantly smaller volatility and appear to be smoother. This effect is mainly due to our prior setup that softly introduces 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   shocks lead to abrupt downward movements in the business cycle. Comparing the prior and posterior dispersion indicates that the information contained in the prior is not reducing estimation uncertainty significantly.
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) rely on aggregate data, the lower value of Q can be explained by the fact that our aggregate gap measure strikes a balance between capturing the higher business cycle variance of EA peripheral countries such 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 and posterior standard deviations differ strongly. This highlights that the introduction of prior information helps 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 timevarying. To this end, the left panel in Fig. 2  One way of assessing the likelihood that heteroscedasticity in the business cycle shocks is present is to consider the posterior distribution of the square root of ϑ ωM up to a random sign switch. In case of 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 evidence for heteroscedasticity is, at best, limited. While the marginal posterior is clearly not unimodal, most posterior mass is located around zero.
To assess how the presence of stochastic volatility in the unobserved components impacts the estimate of the output gap, Figure 3 shows by estimating the model without stochastic volatility for all latent components, with the light gray area 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 under the 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 more pronounced movements in g t obtained from the model without stochastic volatility. This finding is closely related to the critique raised by Sims (2001)

Dissecting euro area business cycle movements
In the following, we provide information on the quantitative contributions of shocks to trend, cyclical and idiosyncratic components to the observed series of inflation over time. Here we use an approach similar to a standard historical time series decomposition. Notice that the non-stationary nature of the trend 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 trend components increases by construction. As a consequence, we focus on the contributions of the shocks at each point in time. Combining Eqs.
(2), (4) and (6) 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 dashed lines 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:

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 observed inflation series. This finding results from an almost diagonal variance-covariance structure between country-specific trends of inflation, with most covariances rather close to zero. In terms of the modeling setup, this implies that one may safely impose orthogonality on the errors for the trend inflation state equations.
Second, we find substantial evidence for the existence of a Phillips curve relationship across the EA countries given by Gap shocks. Notice that the sensitivity of country-specific inflation series to area-wide output gap shocks is governed by the factor loadings β i . Here, we find that the slope of for the dynamic evolution of inflation in Belgium, Spain and Greece. This result implies that almost all comovements in inflation across countries arises from the joint gap component rather than shocks to country-specific trend inflation.
Finally, we assess the importance of country-level shocks. Recall that these shocks depict both shocks to idiosyncratic trends, but also the measurement errors. It is worth mentioning that measurement errors play only a minor role in shaping the observed inflation series over the cross-section, and the contributions labeled Country shocks mainly feature shocks to the trend components. The highest importance of such country-level shocks is apparent for the cases of Greece, Italy and Portugal in the five year period after 2010, while inflation in the Netherlands appears to be shaped to a large extent by idiosyncratic shocks throughout the observed period.

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 euro area. Such a common shock is of interest for policy makers in order to assess the sensitivity of their respective 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. This yields dynamic reactions of g t+h (h = 1, . . . , 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) thus provide not only information on the specific time profile of the output gap reactions but also on the sensitivity of a given country and variable to such changes. 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 impact inflation across countries. Since the dynamics of π it are proportional to movements in g t , we report peak effects that are reached after around three quarters (see Fig. 5).
Inspection of the maximum responses of inflation in Fig. 6

FORECASTING EVIDENCE
Up to this point we have focused on in-sample results to illustrate the key features of our proposed modeling approach. However, a successful model that could be useful for policy analysis should also be able to predict well. To investigate the predictive capabilities, this section builds on the literature on inflation forecasting (see Stock and Watson, 1999;Jarocinski and Lenza, 2018;Koop and Korobilis, 2018, among others) and uses our model to forecast aggregate inflation for the EA and across 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 an initial estimation period that ranges from 1997:Q2 to 2008:Q3 (47 observations) and use the remaining 40 observations as a hold-out period. The forecasting design adopted is recursive, implying that after obtaining a set of predictive densities, we increase the length of the initial observation period by one quarter until we reach the end of the hold-out period.
Differences in predictive accuracy are gauged by relying on root mean squared errors (RMSEs) and log predictive scores (LPSs, see Geweke and Amisano, 2010). RMSEs are obtained by considering the 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 the realized 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 differ in 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 EAlevel to indicate that predictions are based on observations of output and inflation that are aggregated from 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, we consider a range of alternative measures of the output gap to assess differences between treating the output gap as a latent quantity as opposed to using an observed measure. Third, we gauge the accuracy gains from stochastic volatility by also including homoscedastic variants of all competing models.1 The model set we consider is comprised of the following benchmarks: terms. This yields a model setup per country that estimates three unobserved components. By comparing the out-of-sample predictive performance of the UCP specifications with forecasts produced by DFM, the inclusion of these specifications serves to assess the merits of considering multi-country information as a means to improving both country-specific and aggregate predictions. In the case of EA-level, we aggregate country-level series a priori and estimate the model using three latent factors.
(ii) Hamilton (Ham): These specifications rely on a plugin-estimateĝ t as the measure of the output gap 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 (extracting gaps for each country individually) and Multi-country (aggregating a priori and using EA-level information).
(iii) Hodrick-Prescott (HP): Similar to the strategy employed for for Ham, these specifications use the 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 HP are implemented.
1 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 aggregate euro area inflation, and country-specific inflation series independently.
In what follows, all models are benchmarked against the AR(1) model. Here, we consider relative RMSEs and differences in LPSs of all specifications versus the AR(1) model. RMSEs below 100 thus reflect that the respective model outperforms the benchmark in terms of point predictions, while LPSs exceeding 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. Outof-sample performance for aggregate euro area inflation is evaluated for the one-quarter up to the four-quarter ahead horizon. Table 2 reports relative RMSEs and differences in LPSs, benchmarked to the AR(1) model. Notes: Multi-country indicates that cross-sectional information from individual countries is used. Single-country refers to independent individual 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. 1-Qt to 4-Qt refer to the forecast horizon by quarter between one-quarter to one-year. LPS and RMSE are presented 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 competitive out-of-sample predictions, outperforming most competing models. This finding holds true for both point and density predictive performance. Accuracy improvements in terms of LPS tend to be substantial, irrespective of whether g t is estimated alongside the remaining model parameters and states or whether we rely on other measures of the output gap. Considering relative RMSEs reveals that while our DFM-SV specification improves upon the benchmark model, these improvements appear to be muted and range from three percent (in the case of the one-step-ahead horizon) to 7.5 percent (for the four-quarter-ahead forecast). Only in two cases our proposed DFM-SV is slightly outperformed by multi-country versions where the latent gap component is replaced by estimates obtained using the Hamilton (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 to the ones that rely solely on aggregate EA data points towards accuracy gains of the multi-country models. Models that utilize only aggregate data generally appear to be inferior to the AR(1) model in 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 results confirm and corroborate findings in Marcellino et al. (2003), who report that the inclusion of countryspecific information improves out-of-sample predictions even if interest centers on predicting aggregate quantities of interest.
To sum up, Table 2 suggests that, when interest centers on forecasting euro area inflation, our proposed model framework yields strong density and point forecasts. These accuracy improvements are 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-country differences in forecast performance. For the sake of brevity, we focus on one-quarter-ahead forecasts in Table 3 and one-year-ahead predictions in Table 4. These tables include marginal LPS obtained by integrating out the remaining elements of the joint predictive density.
Starting with the one-step-ahead marginal LPS, Table 3 suggests that the homoscedastic variant of our proposed DFM outperforms all competing specification by large margins for most countries considered, 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 forecasts for the Netherlands are most precise when single-country models are adopted. We conjecture that this stems from the fact that these countries tend to share a common business cycle and thus using all available 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, explicitly allowing for SV translates into weaker point and density forecasts relative to the homoscedastic counterparts. This result is in contrast to the findings based on using the full predictive distribution of inflation reported in Table 2   Notes: Multi-country indicates that cross-sectional information from individual countries is used. Single-country refers to independent individual 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. For LPS, 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), predictive gains 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 observed as opposed to latent, we generally find that multi-country models profit from explicitly controlling for estimation uncertainty surrounding g t . This premium in predictive accuracy stems from the fact that integrating out g t from the predictive density translates into a heavy-tailed marginal predictive distribution that is capable of handling outlying values well. This lowers the necessity to explicitly control for stochastic volatility, especially for data at quarterly frequency.
Turning attention to the one-year-ahead forecast horizon, Table 4 shows similar results to those reported for the one-quarter-ahead horizon. For this longer forecast horizon, the homoscedastic DFM setup appears to be particularly successful in terms of producing accurate point predictions, which is not surprising given the fact that for higher-order forecasts, the log-volatilities approach their stationary distribution. The predictive performance in terms of point forecasts of the DFM model is comparable to its heteroscedastic counterpart. Unlike the remaining alternative models, both DFM and DFM-SV also manage to notably outperform the AR(1) benchmark for the one-year-ahead horizon. In terms of density forecasts, however, the predictive dominance of DFM appears less distinctive. Notes: Multi-country indicates that cross-sectional information from individual countries is used. Single-country refers to independent individual 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. For LPS, 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-ahead horizon again reveals no clear pattern. However, this is particularly due to the strong performance of our proposed multi-country frameworks DFM and DFM-SV. Without these two specifications, Table   4 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 crosssectional information to produce accurate point forecasts for inflation in Portugal and Greece.

CONCLUDING REMARKS
In this paper, we develop a multivariate Bayesian dynamic factor model with stochastic volatility for analyzing euro area business cycles. The multi-country framework decomposes country-specific output and 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 gap used for structural analysis and inflation forecasting. A key model feature is to allow for heteroscedastic error terms and comovements in the trends using a flexible factor stochastic volatility structure. The setup is completed by considering time variation also in the variances of the measurement equations.

The proposed Bayesian model alleviates concerns of overparameterization via global-local shrinkage
priors that push the model towards a homoscedastic specification, but allows for time-varying variances if necessary.
In an empirical section, we study both in-sample features and out-of-sample predictive performance of the proposed model. We compare the obtained measure of the output gap to a set of competing approaches for estimation and discuss the role of time variation in error variances. The analysis is complemented by an empirical assessment regarding the slope of the Philips curve across EA member states. In a forecasting exercise, the paper provides evidence that accounting for a common euro area output gap component produces competitive forecasts for inflation both on the aggregate EA, but also the country level.