An Output Gap Measure for the Euro Area: Exploiting Country-Level and Cross-Sectional Data Heterogeneity

Finance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C. An Output Gap Measure for the Euro Area: Exploiting Country-Level and Cross-Sectional Data Heterogeneity Manuel Gonz´alez-Astudillo 2018-040 Please cite this paper as: Gonz´alez-Astudillo, Manuel (2018). “An Output Gap Measure for the Euro Area: Exploiting Country-Level and Cross-Sectional Data Heterogeneity,” Finance and Economics Discussion Series 2018-040. Washington: Board of Governors of the Federal Reserve System, https://doi.org/10.17016/FEDS.2018.040. NOTE: Staff working papers in the Finance and Economics Discussion Series (FEDS) are preliminary materials circulated to stimulate discussion and critical comment. The analysis and conclusions set forth are those of the authors and do not indicate concurrence by other members of the research staff or the Board of Governors. References in publications to the Finance and Economics Discussion Series (other than acknowledgement) should be cleared with the author(s) to protect the tentative character of these papers.

An Output Gap Measure for the Euro Area: Exploiting Country-Level and Cross-Sectional Data Heterogeneity Manuel Gonzalez-Astudillo† manuel.p.gonzalez-astudillo@frb.gov Board of Governors of the Federal Reserve System Washington, D.C., USA June 22, 2018 Abstract This paper proposes a methodology to estimate the euro-area output gap by taking advantage of two types of data heterogeneity. On the one hand, the method uses information on real GDP, inflation, and the unemployment rate for each member state; on the other hand, it jointly considers this information for all the euro-area countries to extract an area-wide output gap measure. The setup is an unobserved components modelthattheorizesacommoncycleacrosseuro-areaeconomiesinadditiontocountryspecific cyclical components. I estimate the model with Bayesian methods using data forthe19countriesoftheeuroareafrom2000:Q1through2017:Q2andperformmodel comparisons across different specifications of the output trend. The estimation of the model preferred by the data indicates that, because of negative shocks to trend output during the global financial crisis, output remained slightly above potential in that period, but an output gap of about negative 31⁄ percent emerged during the European 2 debt crisis. At the end of the sample period, output is estimated to be about 1 percent above potential. Keywords: Unobserved components model, euro area, Okun’s law, Phillips curve, output gap JEL Classification Numbers: C13, C32, C52 ∗Theviewsexpressedinthispaperaresolelytheresponsibilityoftheauthorandshouldnotbeinterpreted as reflecting the views of the Board of Governors of the Federal Reserve System. †I thank Stephanie Aaronson, Jose-Luis Mass´on-Guerra, and John Roberts for insightful comments on earlier versions of this document. Paul Yoo and Philip Coyle provided speedy and accurate research assistance.

1 Introduction There is an ongoing debate among central bankers about the true degree of economic slack in their respective economies and about the sensitivity of inflation to the estimated levels of economic slack. In the case of the euro area, the European Central Bank (ECB) President, Mario Draghi, states the following (see Draghi, 2017): Weseegrowthabovetrendandwelldistributedacrosstheeuroarea, butinflation dynamics remain more muted than one would expect on the basis of output gap estimates and historical patterns. An accurate diagnosis of this apparent contradiction is crucial to delivering the appropriate policy response. Inparticular, PresidentDraghiemphasizestheuncertaintysurroundingthesizeoftheoutput gap. In this paper, I take advantage of the heterogeneity in the country-level information on real GDP, inflation, and the unemployment rate, as well as the heterogeneity of this information across euro-area member states to estimate the area-wide output gap, potential output, and natural rate of unemployment, along with the uncertainty around these estimates. Specifically, I set up an unobserved components (UC) model with a common component in the output gap across countries in addition to idiosyncratic country-level counterparts. The same setup takes place for potential output and the natural rate of unemployment. In the model, output, inflation, and the unemployment rate at the country level inform the estimation of each country’s output gap, potential output, and natural rate of unemployment. At the same time, these country-level macroeconomic variables also inform the estimation of the common components by jointly considering the information of all member states. I aggregate the common and country-level idiosyncratic components in a suitable way to obtain estimates for the euro area. Moreover, by estimating the features of each member economy, the model allows one to determine the effect of area-wide policies on each country’s unemployment and inflation rates. Hence, this paper aims to offer information that can be relevant to the monetary policy decision process in the euro area. Given the single currency among euro-area member states and the uniform monetary policy implementation by the ECB, one could expect that at least some portion of the variation in the country-level degrees of slack can be attributed to the variation in the area-wide output gap. Of course, there may be features particular to each country that make the evolution of every economy different for what could be considered a pattern of synchronization of business cycles. In that regard, a modeling framework that takes into account the possibility of diverse degrees of strength in the relationship between the countryspecific output gaps and their area-wide counterpart can be beneficial to understand how monetarypolicy,forexample,couldpropagateatthenationallevel,whichinturncanprovide information about the effects of these policies on the area-wide indicators. I intend to use this feedback mechanism between the area-wide business cycle and those of the member states to offer an alternative to the estimation of the output gap produced from approaches that use aggregate data at the euro-area level. Even though there have been multiple attempts and approaches in the literature to estimate the output gap of the euro area as a whole, none of them has considered combining the two frameworks I suggest—namely a multivariate perspective at the country level along 1

withthecross-sectionalinformationacrosscountries. Byimposingacommonfactorstructure across macroeconomic variables and countries to estimate the area-wide output gap, a UC model can potentially gain efficiency and reduce the uncertainty around the estimated degree of slack. One of the studies that contemplates the multivariate framework for the euro area is done by Lenza and Jarocin´ski (2016), who investigate (through Bayesian methods) the performance of several UC models to estimate the output gap and accurately forecast inflation in the euro area. The models vary according to the information set used to estimate the output gap and the trend specification for the variables involved. The results show that a model that assumes local linear trends and includes inflation expectations as an observable variable is the most favored by the data, according to its log marginal likelihood. However, a model in which the trends are modeled as random walks with drift and which includes inflation expectations offers the best forecast performance of inflation. The cross-sectional framework to obtain the euro-area output gap has at its core the assumption of different degrees of business cycle synchronization among countries and between countries and the area as a whole. Lee (2012) estimates a time-varying parameters dynamic factor model for the 11 countries that originally joined the Economic and Monetary Union (EMU), plus Greece. A variance decomposition shows that the area-wide factor explains about 70 percent of the growth rate of real GDP per capita between the mid-1990s and the late 2000s, up from about 50 percent in the early 1980s. This evidence suggests greater business cycle synchronization across European countries in the run-up to the third stage of the EMU. The modeling approach I propose in this paper combines the two frameworks previously described. That is, it incorporates information on several macroeconomic variables, as had been suggested by Ru¨nstler (2002), Azevedo, Koopman and Rua (2006), Basistha and Startz (2008), andFleischmanandRoberts(2011), amongothers,toimprovetheprecisionofoutput gap estimates and, at the same time, it includes cross-sectional data in a common factor structure, as originally put forward by Kose, Otrok and Whiteman (2003) and Del Negro and Otrok (2008), among others. The setup is similar to what Gonzalez-Astudillo (2017) proposes to estimate the output gap of the United States using state-level data on real GDP and the unemployment rate. The main difference in the present paper is that I include the inflation rate to explicitly address how inflation can inform the estimation of the output gap and how it responds to the degree of slack in the economy. Usingdatafromthefirstquarterof2000tothesecondquarterof2017forthe19countries of the euro area, I estimate the proposed model using Bayesian methods. It turns out that an output trend specified as a random walk with drift outperforms in sample a local linear trend or an integrated random walk specification. As mentioned before, the structure of the UC model allows one to estimate the common components across countries, such as the euro-area output gap, potential output, and the natural rate of unemployment. Yet it also provides estimates of these indicators at the country level, although the paper does not analyze the economic fundamentals driving the evolution of these country-specific indicators. The model estimates that the euro-area level of output was about 51⁄ 2 percent above trend right before the financial crisis and then declined to a level just above potential at the end of 2009. In the coming years, output increases to almost 3 percent above its potential level in 2011 and then declines again during the European debt crisis to about 31⁄ 2 percent 2

below in late 2013. At the end of the sample period, in the second quarter of 2017, the model estimates that the euro-area output is about 1 percent above trend. In the same vein, the model estimates that the area’s potential output was significantly affected during the financial crisis, with a decline of about 13⁄ 4 percent in its level between the first quarter of 2008 and the first quarter of 2009. This decline is more than enough to prevent the appearance of an output gap during this period. All in all, the estimate of the growth rate of potential output is about 11⁄ 4 percent per year, on average. The estimates also indicate that the euro-area natural rate of unemployment started around 11 percent in 2000 and remained there until late 2006 when it starts to increase to reach a peak of about 12.5 percent in mid-2009; it declines thereafter to about 7.7 percent in the second quarter of 2017. Regarding the estimation of the sensitivity of inflation to economic slack, the crosssectional average of the slopes of the Phillips curves implies that inflation increases by 0.14 percentage point for every percentage increase in output above potential. However, there is heterogeneity across countries in this sensitivity. For example, the model estimates that inflation in Estonia and Latvia would react much more strongly with respect to their respective output gaps than in Germany or Slovakia. I emphasize that this heterogeneity needs to be considered when measuring the impact of economic policies designed to affect the euro area as whole, such as those usually put in place to curb inflation pressures. The results also show that the model with a random walk specification in the output trends of the countries performs better at forecasting out of sample the three macroeconomic variables compared with other UC models and a benchmark vector autoregressive (VAR) model. Additionally, estimating the euro-area output gap using a UC model with countrylevel data decreases the uncertainty around the degree of economic slack compared with a similar model that uses aggregate data at the euro area level for the same macroeconomic variables. I structure the rest of the paper as follows: Section 2 provides a more in-depth revision of the literature regarding the estimation of the output gap as well as the synchronization of business cycles across the euro area. Section 3 describes the model proposed. In Section 4, I lay out the Bayesian estimation strategy. Section 5 makes reference to the data used, while the results appear in Section 6. The next two sections, 7 and 8, investigate the forecast performance of the model and the efficiency gains in estimating the output gap compared with a model that uses aggregate data at the euro area level. Finally, Section 9 concludes. 2 Connections with the Literature This paper relates to two main strands of the literature. Clearly, the paper connects to the existent research on the estimation of the output gap for the euro area using UC models. The second topic relates to the literature on synchronization of business cycles within Europe and the euro area using factor models and other statistical techniques. Next, I make an overview of both points separately. 3

2.1 Estimation of the Output Gap in the Euro Area One of the first attempts to estimate the euro-area output gap is Ubide and Ross (2001), who used four different perspectives, as follows: statistical properties, economic properties, survey data, and “thick” estimates. Under the economic properties approach, there are UC models based on the Phillips curve. The authors estimate four variants of such a model for the euro area including real GDP, the unemployment rate, and consumer price inflation. They find that the best UC model to forecast inflation includes hysteresis effects in which the current NAIRU reacts to the lagged unemployment rate gap. Ru¨nstler (2002) examines the reliability of real-time estimates of the output gap for the euro area. He finds that a multivariate UC model that adds capacity utilization and factor inputs greatly reduces the filter uncertainty compared to a baseline bivariate model of inflation and output, the latter being uninformative to a great extent because of large standard errors and significant biases. Among the models, one based on the output-capital ratio and total factor productivity performs best, and its conditional forecast performance with respect to inflation is also satisfactory. AlongwithVAR-basedBeveridge-Nelsondecompositions,Camba-MendezandRodriguez- Palenzuela (2003) specify three variants of a trivariate UC model that includes real GDP, the unemployment rate, and consumer price inflation for the euro area, exploiting Okun’s law and the Phillips curve. One variant assumes that the trend components of each of the three observable variables is a random walk with drift. Another variant specifies local linear trends in each of the three variables, whereas the last variant assumes smooth trend components. Results show that UC models perform less well than VAR models in forecasting output, the unemployment rate, and inflation. Fabiani and Mestre (2004) also exploit Okun’s law and the Phillips curve to estimate the NAIRU for the euro area using UC models with local linear trends in the unemployment rate and output, but not on inflation. Four different specifications of such models are estimated in which the timing convention of the unemployment rate and output cycles, the causality channel, and the variable defining inflation in the Phillips curve are altered from the baseline model. The authors find that the area-wide NAIRU estimates are robust to changes in the underlying models, as long as the models belong to the UC class. To check if inflation improves the reliability of real-time estimates of the output gap for the EU-11 countries, Planas and Rossi (2004) introduce a Phillips curve relationship in a bivariate UC model for decomposing output into trend and cycle. The results show that the bivariate model that includes inflation does not significantly improve the uncertainty around the real-time estimate of the output gap compared to a univariate model. Proietti,MussoandWestermann(2007),underfourvariantsoftrendoutputspecification— random walk with drift, local linear, damped slope, and integrated random walk—estimate a bivariate UC model with inflation and output to obtain the output gap for the euro area. Additionally, using a production function approach, they estimate a multivariate UC model that includes inflation, output, the labor force participation rate, a proxy of the unemployment rate, capacity utilization, and the consumer price index to obtain another estimate of the output gap also under the four variants of trends mentioned before. The forecasting exercises results indicate that the UC model based on the production function approach outperforms the bivariate model, in general. Furthermore, the models with higher forecasting 4

accuracy include a trend specification with a damped slope. The first attempt to estimate the European Monetary Union output gap with Bayesian methods was done by Planas, Rossi and Fiorentini (2008), who use a UC model with real GDP and the inflation rate as observable variables. The results show that the modal period ofthecycleisaboutnineyears. Inaddition, theresultsindicateasmallbutpositivedegreeof correlation between the innovations to the cycle and to inflation. The authors also illustrate the effect of a perturbation to the output gap on inflation both in the short and in the long run. Marcellino and Musso (2011) perform an evaluation of the reliability of real-time output gap estimates for the euro area using a real-time data set. The technique used to estimate the output gap is through UC models in the multivariate context of Proietti, Musso and Westermann (2007). The findings indicate that real-time estimates of the output gap tend to be characterized by a high degree of uncertainty both on the magnitude and sign dimensions. The uncertainty is mostly due to parameter instability and model uncertainty, rather than data revisions. Gurin,MaurinandMohr(2015)estimateunivariateandbivariateUCmodelsthatinclude output and inflation to obtain a measure of the euro-area output gap. The models vary in terms of the output trend specifications, which include the usual random walk with drift, a Markov-switching trend intercept, and Markov-switching in the intercept of the trend and in the variance of the trend shock. The inflation specification includes, besides the output gap, the nominal exchange rate, the price of oil, and a time-varying intercept that follows a random walk. The authors then combine the output gap estimates obtained from each of the different models to get a global estimate of the output gap. The forecasting exercise for inflation shows that the output gap measures help forecasting inflation over most of the sample, but fail dramatically since the last recession. Under a different approach and with a different goal than to measure the output gap but rather to obtain insights about the sensitivity of inflation to the degree of slack in the economy, Blanchard, Cerutti and Summers (2015) estimate a time-varying coefficients Phillips curve that includes inflation expectations for 20 countries, including several belonging to the euro area. Results show that the median slope of the Phillips curve across countries has dropped substantially and has remained relatively constant since the early 1990s at around one-fifth of the value reached in the late 1970s, while there is evidence of a rise since the 1970s in the anchoring to long-term inflation expectations. 2.2 Common Business Cycles and Synchronization in Europe The literature on common components of business cycles in Europe relates to the present paper in that both assume the existence of underlying factors to economic activity and let the data determine the strength of the connection between the economic activity of each country and those factors. For example, Forni et al. (2000) estimate a generalized dynamic factor model for 10 countries of the euro area, taking into account seven macroeconomic variables for each country. After obtaining the common component of GDP for each separate country, a weighted average of the common components, using the GDP levels as weights, is constructed and labeled as the coincident indicator for the euro area. The authors then obtain the proportion of the variance of each variable that is explained by the common 5

component, which reaches 85 percent for the euro-area aggregate of GDP. In a somewhat different analysis, Artis, Krolzig and Toro (2004) estimate a Markovswitching VAR to identify a common cycle in Europe. Results show that there is evidence of a common unobserved component governing European business cycle dynamics, suggesting the existence of a common business cycle. The authors also explore the contribution of the European business cycle to the individual country cycles and undertake an impulse response analysis to investigate the response of each individual country to European expansions and recessions. Another way to analyze the business cycles of a group of countries as a whole is by determining the degree of synchronization among the member states. Aguiar-Conraria and Soares (2011) use wavelet analysis to study the business cycle synchronization of the EU-12 countries. The wavelet analysis allows one to compare the wavelet spectra of two countries to check if the contribution of cycles at each frequency to the total variance is similar between both countries, if this contribution happens at the same time or not, and, finally, if the ups and downs of each cycle occur simultaneously. Results show that business cycles dissimilarities are highly correlated with geographical physical distances. In particular, France and Germany appear to be the most synchronized countries with the rest of Europe, while Portugal, Greece, Ireland, and Finland do not show statistically relevant degrees of synchronization with the other European countries. Some countries show a French accent, like Spain, while others have a German accent, like Austria. However, Camacho, Perez-Quiros and Saiz (2006) warn that assuming that a European cycle exists and that it coincides either with the cycle of a leading European economy, the cycle of a weighted average of several European economies, or the cycle of a common factor, should be done cautiously. The authors use cluster analysis to find out if there is a “European business cycle” that links the European economies. The results show that the degree of business cycle synchronization within the group of old European Union members is higher than across the recently acceded countries, but synchronization across old members hasnotsignificantlyincreasedsincetheestablishmentofthecommoncurrency. Inparticular, the results do not favor evidence of some distinct euro-economy attractor. 3 The Model Toestimatetheeuro-areaoutputgap, IadoptthesetupinBasisthaandNelson(2007)for each country of the monetary union and propose the following UC model that incorporates country-level information of output, the unemployment rate, and inflation: y = τy +c , (1) it it it u it = τ i u t +θ 1i c it +θ 2i c i,t−1 , (2) π it = β 0i +β 1i π t e +(1−β 1i )π i,t−1 +κ i c it +η i π t , (3) where y , u , and π are (the log of) output, the unemployment rate, and the inflation it it it rate, respectively, for country i = 1,2...,n with n being the number of countries and t = 1,2,...,T, with T being the sample size. In equation (1), output is the sum of a trend, τy, and a cyclical component, c . Similarly, the unemployment rate for each country in it it 6

equation (2) is the sum of a trend, τu and a cycle which is a linear combination of the it cyclical component of output in a way resembling Okun’s law with coefficients θ and θ . 1i 2i Inflation is central to the notion of potential GDP and, hence, to that of the output gap. The New Keynesian Phillips curve with forward-looking expectations implies that the output gap drives the dynamics of inflation relative to expected inflation. Equation (3) is a ‘hybrid’ Phillips curve that allows one to estimate the output gap using UC models. The specified Phillips curve is both forward- and backward-looking as a way to incorporate possibleexpectationmismeasurements,non-rationaladaptiveexpectationformation, orprice rigidities not fully captured by a pure forward-looking Phillips curve, which can affect the estimation of the output gap. In said equation, π it is the inflation rate of country i, π i,t−1 is its lagged value, πe denotes the one-year ahead (survey) euro-area inflation expectations, and t ηπ is a composite unobserved variable that plays a role in both expected inflation and supply it shocks. Long-run neutrality is ensured by restricting β ∈ [0,1] for all i. The parameter 1i κ represents the slope of the Phillips curve for country i, which theory suggests is positive. i Lenza and Jarocin´ski (2016) also consider inflation expectations in a UC model that includes a Phillips curve-style equation to estimate the output gap. Their results show that the data favor models in which the inflation trend is informed by inflation expectations data.1 In equation (1), following Gonzalez-Astudillo (2017), I assume that each country’s output trend is a function of a trend that is common to all the countries in the monetary union and of an idiosyncratic trend, as follows: τy = δyτy +ξy, (4) it i t it where τy and ξy are the common and idiosyncratic trends, respectively. The coefficient δy t it i determines how intensely a country’s i output trend is affected by the common output trend, which is a random walk with drift, as shown below: τy = µ+τy +ηy. (5) t t−1 t Then, I take an approach similar to Lenza and Jarocin´ski (2016) to specify the idiosyncratic output trend as one of the following three alternatives:2 • Random walk (RW): ξy = µ +ξy +ηy. (6) it i i,t−1 it • Local linear trend (LLT): ξ i y t = µ i,t−1 +ξ i y ,t−1 +η i y t , (7) µ it = µ i,t−1 +ν it . 1This setup takes a stand on the way to model the trend inflation. Clark and Doh (2014) assess the inflation forecasting ability of different inflation trend specifications and find that the accuracy of the local linear trend setup of Stock and Watson (2007) or the survey-based specification are about equal. 2In Lenza and Jarocin´ski (2016), the three alternatives are proposed for the trend of the euro area aggregate output instead of for that of each country. 7

• Integrated random walk (IRW): ξ i y t = µ i,t−1 +ξ i y ,t−1 , (8) µ it = µ i,t−1 +ν it . The reasoning behind having these three alternatives is that it may be too restrictive to assume that the conditional average growth rate of trend output has remained constant over time, as it happens with the RW specification. Both the LLT and the IRW specifications allow the conditional average growth rate of trend output to change over time. These variations can be potentially useful to incorporate the possibility of periods with sustained low growth rates of output like those experienced during episodes of secular stagnation (see Summers, 2015).3 In a similar manner as for the trend of output, the trend of the unemployment rate is modeled as a function of a common trend with a loading coefficient that determines the strength of the feedback to the country-specific trend, and an idiosyncratic unemployment rate trend that follows a random walk without drift, as shown below: τu = δuτu +ξu, (9) it i t it ξu = ξu +ηu, (10) it i,t−1 it where the common unemployment rate trend follows a random walk as well, as specified below: τu = τu +ηu. (11) t t−1 t In this specification, neither the common nor the idiosyncratic unemployment trends include a drift term. These assumptions are based on the reasoning that the unemployment rate would not be a trending variable, hence I set the drifts to zero.4 This type of specification is used by Clark (1989), Sinclair (2009), Basistha and Nelson (2007), and Gonzalez-Astudillo and Roberts (2016), among others, for the U.S. economy. Each country’s cycle is a linear combination of a euro-area common cycle and an idiosyncratic cyclical component, as follows: c = α c +υ , (12) it i t it where the loading coefficient α determines if a country is procyclical, acyclical, or counteri cyclical with respect to the common cycle, and the strength of the cyclicality. The idiosyn- 3There are two other possibilities for incorporating a time-varying intercept in the output trends. The first is to adopt the three variants on the common output trend in equation (5) and leave the idiosyncratic trend as a random walk specification as in (6). I prefer to have time-varying intercepts in the idiosyncratic trends because of the more sensible model identification restrictions to be described below with respect to the intercepts of the output trends. The second possibility is to allow time-varying intercepts in both the common and the idiosyncratic output trends. However, the model-fit comparisons would have become much more cumbersome in that case. 4Lenza and Jarocin´ski (2016) find that the posterior mean of the intercept of their unemployment trend is undistinguishable from zero. 8

cratic component follows an AR(1) process, as indicated below: υ it = ρ i υ i,t−1 +ζ it , (13) with |ρ | < 1, whereas the common cyclical component is specified as a stationary AR(2) i process, as follows: c t = φ 1 c t−1 +φ 2 c t−2 +ε t . (14) Finally, the error terms ε ,ηy,ηu,ν ,ηy,ηu,ηπ,ζ ,ν are assumed to be white noise, unt t t t it it it it it correlated with each other and normally distributed.5 Appendix A presents the state-space models in matrix form. 4 Estimation Strategy To obtain the euro-area output gap, I estimate with Bayesian methods the three UC models in equations (1)-(14), which are distinguished by the three idiosyncratic output trend specifications (RW, LLT, and IRW). I need to impose some identification conditions to estimate the coefficients of the model and the output gap. First, I set to one the variances of the error terms of every common component, that is, σ2 = σ2 = σ2 = 1. These restrictions are necessary to identify the scale ε ηy ηu of the loading coefficients, α , δy and δu, respectively. I set µ = 0 to allow the unrestricted i i i estimation of the country-specific drifts, µ , in the case of the random walk specification, i or their initial values, µ , in the cases of the local linear and the integrated random walk i0 trends. Generally speaking, to avoid having country-level GDP and unemployment trends that diverge in expectation with respect to their common counterparts, I restrict the loading coefficients, δy and δu, to be positive. The common cycle, c , is identified only up to its i i t sign, therefore I restrict one of the loading coefficients α > 0 for i = Germany, without loss i of generality. Lastly, to be consistent with the theoretical aspects of the Phillips curve, I restrict β ∈ [0,1] and κ > 0 for all i. Basistha and Nelson (2007) adopt the first restriction 1i i to guarantee long-run neutrality while Matheson and Stavrev (2013) and Blanchard, Cerutti and Summers (2015) restrict the slope of the Phillips curve to be positive in a time-varying coefficient setup; I incorporate said constraint in the constant coefficient setup proposed. The model has 268 coefficients to be estimated when the identification restrictions are imposed. I use the Gibbs sampler to obtain sequential draws from the posterior distribution between latent states, using the Durbin and Koopman (2002) simulation smoother, and coefficients. Appendix C describes in detail the sampling procedure. 5Morley, Nelson and Zivot (2003), Basistha and Nelson (2007), and Sinclair (2009), among others, allow for correlation between the trend and cycle perturbations. In the current context, that type of specification wouldincreasesignificantlythenumberofestimatedcoefficients,increasingtheparameteruncertaintyofthe model. Lenza and Jarocin´ski (2016) argue that, by allowing more flexible specifications in the output trend, as done here, it is possible to account for the consequences of the trend-cycle correlation assumption. 9

5 Data Iemploydataataquarterlyfrequencyfromthefirstquarterof2000tothesecondquarter of 2017 for the 19 countries of the euro area.6 Real GDP is expressed in chained 2010 million euro. The unemployment rate is the quarterly average of the percentage of unemployed persons aged 15 to 74. The inflation rate corresponds to the annualized quarterly percent change in the overall harmonized index of consumer prices (HICP). Eurostat provides all the data, which are seasonally adjusted either directly by the statistical agency or manually using conventional seasonal adjustment techniques. Inflation expectations correspond to the mean point estimate of the year-over-year percentage change of the HICP one year ahead obtained from the survey of professional forecasters. The ECB provides this information. Appendix B describes the data and their transformations in more detail. 6 Model Estimation Results This section lays out the estimation results of the UC models of equations (1)-(8). It briefly describes the choice of prior distributions for the parameters of the models. Then, the section discusses which of the three specifications for the idiosyncratic trend (RW, LLT, or IRW) is preferred by the data using marginal data density measures. Once the preferred model is chosen according to this criterion, the section describes the features of the euroarea output gap, potential output, and natural rate of unemployment, as well as the features of the country-specific counterparts without discussing the economic reasons behind their features because they are beyond the scope of the present study. 6.1 Prior Distributions The prior distributions used are consistent with the application of the Gibbs sampler for linear regression models with independent conditional mean and variance components. That is, I assume that the coefficients of the conditional means of both the observation and transition equations of each state-space model have a normal prior distribution, whereas the variance coefficients of the shocks have inverse gamma prior distributions. Appendix D describes the choice of hyperparameters. 6.2 Estimate of the Euro Area Output Gap The Gibbs sampler produced 300,000 draws. After burning in the first 100,000 and thinning every 100th draw, I kept 2,000 draws from the posterior distribution. Appendix E shows the posterior means and standard deviations of the parameters, as well as the convergence diagnostics of the sampler. 6Iconsiderabalancedpanelofinformationtoestimatethemodel. Implicitly,Iassumethatthecountries that joined the euro area after its original inception, namely Cyprus, Greece, Estonia, Latvia, Lithuania, Malta, Slovakia, and Slovenia were sharing at least part of the business cycle properties with the rest of euro-area members before joining. 10

Figure 1: SmoothedEstimatesoftheEuro-AreaOutputGapsUsingCountrylevel Data for Three Specifications of the Output Trend 10 9 8 7 6 5 4 3 2 1 0 -1 -2 -3 -4 -5 2002 2004 2006 2008 2010 2012 2014 2016 PDG laitnetop morf noitaived % 10 Random Walk 9 Local Linear Trend 8 Integrated Random Walk 7 6 5 4 3 2 1 0 -1 -2 -3 -4 -5 Note:TimeseriesaretheaveragesoftheposteriordrawsoftheeuroareaoutputgapusingtheDurbinandKoopman (2002)simulatorsmoother. To obtain the euro-area output gap, I average the smoothed estimate of the countryspecific output gaps, c in equation (12), using the nominal GDP weights in each period. it The smoothed estimates of the euro-area output gaps for each of the three output trend specifications appear in Figure 1. As expected, the estimated output gaps from the LLT and IRW trend specifications look very similar, whereas the output gap estimated from the RW specification shows significant differences with respect to the other two. For example, the LLT- and IRW-implied output gaps indicate that the euro-area output started the 2000s about 5 percent above potential, while the RW-implied output gap indicates that the economy was only about 3 percent above trend. Another difference occurs around the period of the global financial crisis when the LLT and IRW trend models estimate output stayed around 2.5 percent above potential, whereas the RW trend model estimates that real and potential GDP were almost at the same level. The other difference occurs at the end of the sample, in the second quarter of 2017, when the first two trend assumptions imply that output is almost 3 percent above potential while the RW specification yields an estimate of only 1 percent. In general, the RW specification estimates deviations of output from trend that are less positive and more negative than those of the LLT and IRW specifications. Regardless of the trend specification, all the models estimate that output remained above potentialaroundtheonsetoftheglobalfinancialcrisisandthatanegativeoutputgapstarted toappearonlyaroundtheeventsoftheEuropeandebtcrisis. Throughthelensofthemodels, this situation means that the estimates of potential output were severely affected during the financial crisis, causing output not to go below its potential counterpart. Figure 2 shows the evolution of the smoothed common output trend shock, ηy, for each of the three trend t 11

Figure 2: Smoothed Estimates of the Aggregate Output Trend Perturbations 2 1 0 -1 -2 2002 2004 2006 2008 2010 2012 2014 2016 egnahc % retrauq-revo-retrauq 2 Random Walk Local Lineal Trend Integrated Random Walk 1 0 -1 -2 Note: Timeseriesareaveragesoftheposteriordrawsofthecommonoutputtrendshocks, ηt y, usingtheDurbin andKoopman(2002)simulatorsmootherweightedbytheposteriormeansoftheoutputtrendloadings, δ i y, and nominalGDPofeachcountry. specifications. As the figure illustrates, under any of the three trend output assumptions, the model estimates that all the aggregate output trends suffered significantly negative shocks (the largest was about negative 1.5 percent) between the second quarter of 2008 and the first quarter of 2009. The effect of these shocks on the estimate of potential implies that, even though real GDP declined markedly during the global financial crisis, a euro-area output gap did not emerge.7 To determine which of the three output trend specifications is more favored by the data, I perform a marginal data density analysis. I use the method outlined in Chib (1995) to obtain the marginal likelihood of the data, which encompasses real GDP, the unemployment rate, and the HICP inflation rate for each of the 19 countries of the euro area. The results are as follows: • Random walk output trends: -5,615.2 • Local linear output trends: -6,016.0 • Integrated random walk output trends: -6,048.1 7The results about the shocks to the output trend are consistent with the differences among the output gaps from the three idiosyncratic trend specifications. In particular, the common output trend under the RW specification experiences upward level perturbations between 2001 and 2006 that are larger than under the IRW and LLT alternatives. As a consequence, the common output trend in the RW case is higher than the other two cases, implying a lower estimated euro-wide output gap throughout. 12

Figure 3: Smoothed Estimate of the Euro-Area Output Gap Using Country- Level Data and Other Estimates 10 8 6 4 2 0 -2 -4 -6 -8 -10 2000 2002 2004 2006 2008 2010 2012 2014 2016 PDG laitnetop morf noitaived % 10 IMF OECD 8 European Commission This paper 6 4 2 0 -2 -4 -6 -8 -10 Source: WorldEconomicOutlookDatabaseoftheInternationalMonetaryFund(IMF),EconomicOutlookofthe OrganisationforEconomicCo-operationandDevelopment(OECD),AnnualMacro-economicDatabaseoftheEuropeanCommission(EuropeanCommission),andauthor’scalculations(Thispaper). Shadedareasdenotethe90 percentconfidencesets. Thefrequencyisannualandthequarterlyestimatedoutputgapisaveragedoverthefour quartersofeachyear. Thelastdatapointis2016. According to the results, the data favor a specification in which the idiosyncratic output trend follows a random walk.8 Based on these results, the forthcoming discussion focuses on theresultsforrealGDP,theunemploymentrate,andinflationobtainedfromthespecification in which the idiosyncratic trends follow a random walk specification. To further put in perspective the results of the model proposed in this paper, Figure 3 shows the smoothed estimate of the output gap for the euro area under the preferred output trend specification (RW), along with the estimates by the International Monetary Fund, the Organisation for Economic Co-operation and Development, and the European Commission. Although the 90 percent confidence set of the estimated deviation of output from potential includes those of the three institutions over most of the sample, there is a stark difference around the onset of the global financial crisis in 2009 through 2010. The three institutions estimate a large output gap (around negative 3 to negative 4 percent) during this period, whereas the estimate in this paper indicates that output was still above potential, even though the confidence set includes zero. The difference is likely that the aforementioned institutions have not estimated as negative a shock to potential output as the one I obtain using the model proposed. Other than that, the estimates of the euro-area output gap at the end of 2016 for the model and the institutions are relatively close. This paper estimates the output gap to be negative 1⁄ 2 percent, whereas the European Commission, the IMF, and the 8In contrast, the marginal likelihood measures in Lenza and Jarocin´ski (2016) indicate that, for a UC model with aggregate data and several macroeconomic variables, a specification with local linear trends in the observables is the most favored by the data. 13

OECD estimate output gaps of about negative 1 percent, negative 11⁄ 4 percent, and negative 13⁄ 4 percent, respectively. 6.3 Estimates of Country-level Output Gaps and Potential Outputs As a byproduct of the model proposed, one can estimate the country-level output gaps andpotentialoutputs. Inaddition,themodelallowsonetoestimatehowconnectedthestatelevel cycles and trends are with their common counterparts. Figure 4 shows the estimated output gaps for three groups of countries. The first group is composed of the countries considered to be weaker economically following the global financial crisis: Greece, Ireland, Italy, Portugal, and Spain. In the second group, I include the countries that joined the euro area later than its original inception in 1999 and which are not in the first group: Cyprus, Estonia, Latvia, Lithuania, Malta, Slovakia, and Slovenia. Finally, the third group is formed by the countries that are original members of the euro area and which are not in the first group: Austria, Belgium, Finland, France, Germany, Luxembourg, and Netherlands. Theresultsshowthatthethreegroupsofcountriescanhaveverydifferentfeaturesintheir output gaps. For instance, some of the countries in the first group experienced output above potential significantly beyond than what is estimated for the euro area (labeled “Eurozone” in each panel) since 2000 and prior to the global financial crisis: Greece’s output was about 10 percent above potential from 2000 to 2010, Ireland’s output hovers around 5 percent above trend from 2003 to 2007, whereas Spain has a degree of over utilization between 5 percent and 10 percent from 2000 to 2009. This situation reverses around late 2011 for most countries, the exception being Ireland which started to experience large negative output gaps as early as 2008. In recent years, all the countries in this group have estimated output gaps that are mostly larger than the euro-area estimate, with Greece going through an estimated output gap of about negative 10 percent in the second quarter of 2017, and Ireland being affected by positive idiosyncratic perturbations since late 2014 that bring its output gap close to the euro-wide estimate at the end of the sample. The second group of countries seems to have converged to the euro-area estimate of the output gap more rapidly than the first group. There are countries such as Estonia and Latvia with output significantly above potential estimates in the run-up to the financial crisis between late 2004 and late 2007. The same is true for Cyprus, Lithuania, Slovakia, and Slovenia, but to a lesser extent. Starting in 2013, countries such as Cyprus and Slovenia begin to experience output gaps larger than the euro-area as a whole. Slovenia has virtually closed its output gap by the second quarter of 2017, but Cyprus still maintains an elevated degree of economic slack of the order of about 6 percent. The most homogeneous group appears to be the third one. Apart from Germany and Finland, the cyclical positions of the other countries in this group are mostly inside the confidence set of the estimate of the euro area. Germany presents a relatively large degree of slack that reaches 5 percent between 2004 and 2006. The output gap then materially closes by mid-2010 and starts to deviate from the euro-wide estimate starting in 2014 to put output about 6 percent above potential in the second quarter of 2017. Regarding Finland, its estimated cyclical position is outside the confidence set of that for the euro-area between 14

Figure 4: Smoothed Estimates of the Country- Level Output Gaps 20 15 10 5 0 -5 -10 -15 -20 -25 2002 2004 2006 2008 2010 2012 2014 2016 PDG laitnetop morf noitaived % Greece Ireland Italy Portugal Spain Eurozone 20 15 10 5 0 -5 -10 -15 -20 -25 30 25 20 15 10 5 0 -5 -10 -15 2002 2004 2006 2008 2010 2012 2014 2016 PDG laitnetop morf noitaived % Cyprus Estonia Latvia Lithuania Malta Slovakia Slovenia Eurozone 30 25 20 15 10 5 0 -5 -10 -15 15 10 5 0 -5 -10 2002 2004 2006 2008 2010 2012 2014 2016 PDG laitnetop morf noitaived % Figure 5: Variance Decomposition of Country- Level Output Gaps ± Finland Estonia Latvia Lithuania Ireland Netherlands BelgiumGermany Luxembourg Slovakia France Austria Slovenia Italy PortugalSpain Greece Malta Cyprus Austria Belgium Finland France Germany Luxembourg Netherlands Eurozone x < 50% 15 50% ≤ x < 70% 70% ≤ x < 90% 10 90% ≤ x 5 0 Note: “x”istheproportionofthevarianceofthecountry’soutputgap thatisexplainedbythevarianceofthecommoncyclicalcomponent. Coefficientsareevaluatedattheposteriormean. -5 -10 Note: Timeseriesaretheaveragesoftheposteriordrawsoftheeuro area output gap using the Durbin and Koopman (2002) simulator smoother. Shadedareasdenotethe90percentconfidencesets. 15

2005 and 2008 when output reaches about 10 percent above trend. After that, this economy experiences less tight resource utilization, but still has output more than 5 percent above potential in 2011. By the end of the sample period, Finland’s output gap is estimated to have closed. Another way to characterize the economies of the member states is to break down their output cyclical and trend components in their respective common and idiosyncratic counterparts. Recall that for each country, its cycle is a linear combination of common cyclical and idiosyncratic components. The same is true for the output trend. Figure 5 depicts the contribution of the variation in the common cyclical component to the variance of each country’s output gap. This contribution is a measure of how close a member state’s business cycle has been to that of the euro area in the analyzed period. InlinewiththeresultsshowninFigure4, themapshowsthat, ontheonehand, thecountries whose cyclical positions are least influenced by the common euro-area cycle are Greece, Ireland, Spain, Latvia, andGermany. Thesecountriesexperienceidiosyncraticperturbations that make their economies more unique and less connected with the cyclical position of the euro area as a whole. On the other hand, the member states that more strongly co-move with the common cycle are Estonia, Lithuania, Slovenia, and Finland. Few countries are estimated to have experienced a negative output gap during the years oftheglobalfinancialcrisis. Amongthem, thereareIreland, Malta, andGermany, indicating that the model estimates negative shocks to the output trend for most countries during the crisis. Figure 6 illustrates the evolution of the estimated output trends for the euro area and for individual countries divided in the three groups described previously. The results indicate that the euro-area potential output (labeled “Eurozone” in each panel) increased about twenty percent between early 2000 and early 2017, slightly below 1.2 percent per year, on average. As evidenced before, negative shocks to the common output trend between the second quarter of 2008 and the first quarter of 2009 make potential output to decline during that period by about 1.7 percent. In fact, apart from Ireland and Spain, all countries are estimated to have experienced a decline in their potential outputs during the global financial crisis. The first panel shows that the member states with the lowest potential output growth rates are Italy and Portugal, reaching approximately 7 percent and 13 percent, respectively, since2000. ThefastestgrowingpotentialoutputisestimatedtobeinIreland,whereitalmost doubles since 2000. As can be inferred from the middle panel, the countries that joined the euro area in the most recent years have experienced higher potential output growth rates, on average, than the other member states, with Lithuania having an estimated increase of about 70 percent compared with its level in 2000. In the third group of countries, Luxembourg has reached an increase of 50 percent in its potential output with respect to the level in 2000. All the other countries are clustered with potential output growth rates between 20 percent and 30 percent, except Germany, which is estimated to have experienced a growth rate of only 17 percent (roughly 1 percent per year, on average). Figure 7 shows the proportion of the variation in potential output at the country level that has been related to the variation of the common trend for each member state during the period of analysis. The variations in the potential outputs of Ireland, Spain, Malta, and Cyprus are the least affected by the variations of the trend common to the euro area, meaning that these countries have idiosyncratic output trend perturbations that make them 16

Figure 6: Smoothed Estimates of the Country- Level Potential Outputs 90 80 70 60 50 40 30 20 10 0 2002 2004 2006 2008 2010 2012 2014 2016 2Q:0002 morf noitaived % Greece Ireland Italy Portugal Spain Eurozone 90 80 70 60 50 40 30 20 10 0 70 60 50 40 30 20 10 0 2002 2004 2006 2008 2010 2012 2014 2016 2Q:0002 morf noitaived % Cyprus Estonia Latvia Lithuania Malta Slovakia Slovenia Eurozone 70 60 50 40 30 20 10 0 50 40 30 20 10 0 2002 2004 2006 2008 2010 2012 2014 2016 2Q:0002 morf noitaived % Figure 7: Variance Decomposition of Country- Level Potential Outputs ± Finland Estonia Latvia Lithuania Ireland Netherlands BelgiumGermany Luxembourg Slovakia France Austria Slovenia Italy PortugalSpain Greece Malta Cyprus Austria Belgium Finland France Germany Luxembourg Netherlands Eurozone x < 50% 50% ≤ x < 70% 50 70% ≤ x < 90% 90% ≤ x 40 30 20 Note: “x”istheproportionofthevarianceofthecountry’spotential outputthatisexplainedbythevarianceofthecommontrendcompo- 10 nent. Coefficientsareevaluatedattheposteriormean. 0 Note: Timeseriesaretheaveragesoftheposteriordrawsoftheeuro area output gap using the Durbin and Koopman (2002) simulator smoother. Shadedareasdenotethe90percentconfidencesets. 17

Figure 8: Variance Decomposition of Country-Level Real GDP Growth Rates ± Finland Estonia Latvia Lithuania Ireland Netherlands BelgiumGermany Luxembourg Slovakia Austria France Slovenia Italy PortugalSpain Greece Malta Cyprus x < 60% 60% ≤ x < 70% 70% ≤ x < 80% 80% ≤ x Note: “x”istheproportionofthevarianceofthecountry’srealGDPgrowthratethatisexplainedbythevariance ofitscyclicalcomponent. Coefficientsareevaluatedattheposteriormean. distinct from a euro-wide aggregate. In contrast, Estonia, Lithuania, Slovenia, and Finland show variations in their trend outputs that are highly affected by variations in the common component of trend output. 6.3.1 A Further Look at the Characteristics of the Country-Level Output Gaps Finally, the model also allows a more detailed exploration to decompose the variations of real GDP growth for each country into variations that are due to either the country-specific cycle or trend. There may be countries that are “more cyclical” than others in that they can experience relatively more perturbations in the growth of real GDP that can be thought of transitory rather than permanent, which have a more lasting effect on real GDP growth. Figure 8 illustrates the results. 18

The countries that can be referred to as more cyclical in the euro area are Portugal, Spain, Cyprus, and Austria, whereas the countries that are relatively more influenced in their real GDP growth rates by perturbations to the trend are Greece, Estonia, Lithuania, and Slovakia. 6.3.2 A Further Look at the Characteristics of the Euro-Area Output Gap With the information at the country level about cycles and trends, and considering the covariance between potential output and the output gap within and across countries, one can estimate the proportion of the euro-area real GDP growth that is due to changes in either its cyclical or trend components (see Gonzalez-Astudillo, 2017, for details on the derivation). Using the posterior means of the parameters, the model implies that about 69 percent of the variations in real GDP growth are due to the cycle and the remainder to potential output. This percentage is somewhat above estimates for the United States that put the contribution of the cycle at about 60 to 65 percent (see Gonzalez-Astudillo and Roberts, 2016). 6.4 Estimates of the Euro Area and Country-Level Natural Rates of Unemployment Themodelpresentedalsoallowstheestimationofthenaturalunemploymentrateforeach country. I use the labor market size of each country to construct an estimate of the euroarea natural rate of unemployment. Figure 9 depicts the estimated natural unemployment rates for the three groups of countries described before and the respective euro-area estimate (labeled “Eurozone” in each panel). The euro-area natural unemployment rate is estimated to have started around 11 percent in 2000 and remained there until late 2006 when it starts to increase to reach a peak of about 12.5 percent in mid-2009; it declines thereafter to about 7.7 percent in the second quarter of 2017 with a 90 percent confidence set between 6.1 percent and 9.4 percent.9 The results show again very distinctive patterns for the three groups of countries. The group in the first panel has estimated natural unemployment rates higher than the area-wide equivalent, whereas the second group is closer to the euro-area estimate, and the third group has, in general, lower estimated rates than the euro area-wide measure. In the first group, Greece and Spain stand out with estimates of the natural unemployment rate around 15 percent, reaching almost 20 percent in the case of Greece and surpassing that level in the case of Spain in the years following the global financial crisis. Italy shows a relatively stable natural unemployment rate that is around 11 percent at the end of the sample. Portugal and Ireland show more volatility around the years of the global financial crisis, but not as much as Greece and Spain, and have gone back roughly to pre-crisis levels. 9TheOECDoffersameasureoftheNAIRUorequilibriumunemploymentrate(notshowninthefigure) that is much less variable than the estimate offered by the proposed model in this paper. The OECD estimates the NAIRU of the sixteen euro-area countries that belong to the organization to have started around 8.8 percentin 2000, increaseto almost 9.3 percent in 2005 and then decline slightly until 2008, when it begins to increase to reach almost 9.2 percent in 2010. The NAIRU estimated by the OECD declines thereafter to end at about 8.7 percent in 2017, its lowest level since 2000, which is qualitatively in line with the results in this paper. 19

Figure 9: Estimates of the Euro-area Natural Unemployment Rate 25 20 15 10 5 0 2002 2004 2006 2008 2010 2012 2014 2016 % 25 20 15 10 5 0 % Greece Ireland Italy Portugal Spain Eurozone 25 20 15 10 5 0 2002 2004 2006 2008 2010 2012 2014 2016 % 25 20 15 10 5 0 % Cyprus Estonia Latvia Lithuania Malta Slovakia Slovenia Eurozone 15 10 5 0 2002 2004 2006 2008 2010 2012 2014 2016 % 15 10 5 0 % Figure 10: VarianceDecompositionofCountry- Level Natural Unemployment Rates ± Finland Estonia Latvia Lithuania Ireland Netherlands BelgiumGermany Luxembourg Slovakia France Austria Slovenia Italy PortugalSpain Greece Malta Cyprus Austria Belgium Finland France Germany Luxembourg Netherlands Eurozone x < 20% 20% ≤ x < 40% 40% ≤ x < 60% 60% ≤ x Note: “x” is the proportion of the variance of the country’s natural unemploymentratethatisexplainedbythevarianceofthecommon trendcomponent. Coefficientsareevaluatedattheposteriormean. Note: TimeseriesaretheaveragesoftheposteriordrawsoftheeuroareaunemploymenttrendusingtheDurbinandKoopman(2002)simulatorsmoother. Shadedareasdenotethe90percentconfidencesets. 20

In the second group, Estonia, Latvia, Lithuania, and Slovakia show downward trends in their estimated natural unemployment rates from very high levels until 2008 when they start to increase significantly until late 2009 or early 2010, at which moment they trend downward again to end in a range between 7 percent and 9 percent. Cyprus, Malta, and Slovenia show less volatile natural unemployment rate estimates and seem to have returned to pre-crisis levels or even lower, except for Cyprus whose natural rate remains elevated at the end of the sample at a level near 8.5 percent, about 2.5 percentage points higher than in 2000. The countries in the third group show very stable estimates of the natural rate of unemployment, except for Germany, whose estimate mostly trends downward from an original level above 8 percent in 2000 to about 5 percent in 2017. This latter level is around which the estimates of Austria, Luxembourg, and Netherlands have hovered for most of the sample period. In contrast, Belgium, Finland, and France show natural unemployment rate estimates that fluctuate around a relatively higher level of about 9 percent. One can further analyze how closely related the structural features of the country-level labor markets have been to the structure of the common labor market, in particular with respect to the natural rate of unemployment during the period analyzed. Figure 10 shows the proportions of the variation in the natural rates of unemployment of each country that are due to variations in the common component of the unemployment rate trend. A higher proportion means that the trend component of the country-level unemployment rate is more affected by changes in the common component. The results indicate that only about a fourth of the member states have their natural ratesofunemploymentsignificantlyconnectedwiththearea-widenaturalrate. Thecountries in this group are: Ireland, Spain, Estonia, Latvia, and Lithuania. The rest of the countries have a very low proportion of the variation in their natural unemployment rates explained by variations in the common unemployment rate trend, with about half of the member states having a proportion below 10 percent. These findings would imply that the structural component of the country-level unemployment rates cannot be easily affected by policies at the euro area level designed to affect the common structural unemployment rate.10 6.4.1 Characterizing the Cyclical Properties of Country-Level Labor Markets Based on the estimates of the Okun’s law coefficients, θ and θ , the model allows one 1i 2i to characterize how cyclical the labor markets of the member states are. Figure 11 shows the countries whose unemployment rates react more strongly to changes in their cyclical positions of output for given unemployment rate trends. I plot the sum of the Okun’s law coefficients whose interpretation is the decline in percentage points in the unemployment rate for an increase of one percentage point in the level of output above potential. The results show that the countries with most cyclical labor markets are Spain and Greece with Okun’s law coefficients larger (in absolute value) than 0.5, followed by Cyprus and Portugal with coefficients above 0.4 (in absolute value). In contrast, the least cyclical labor markets would be in Malta, Ireland, Luxembourg, and Lithuania, all of them with coefficients below 0.1 (in absolute value). 10Labor market or social insurance policies can vary significantly across the country members. These differences could explain why structural unemployment remains so country-specific even as potential output seems to be affected by euro-area wide policies. 21

Figure 12: Impulse-Response Analysis for the Figure 11: Okun’s Law Coefficient Estimates Cyclical Component of the Unemployment Rate ± 0 -0.2 -0.4 -0.6 Finland -0.8 -1 0 2 4 6 8 10 12 14 16 18 20 quarters Estonia Latvia Lithuania Ireland Netherlands BelgiumGermany Luxembourg Slovakia France Austria Slovenia Italy PortugalSpain Greece Malta Cyprus θ < -0.5 -0.5 ≤ θ < -0.3 -0.3 ≤ θ < -0.1 -0.1 ≤ θ Note: θinthefigureisthesumθ1+θ2foreachcountryevaluatedat theposteriormean. stniop egatnecrep Greece Ireland Italy Portugal Spain 0 -0.2 -0.4 -0.6 -0.8 -1 0 -0.2 -0.4 -0.6 -0.8 -1 0 2 4 6 8 10 12 14 16 18 20 quarters stniop egatnecrep Cyprus Estonia Latvia Lithuania Malta Slovakia Slovenia 0 -0.2 -0.4 -0.6 -0.8 -1 0 -0.2 -0.4 -0.6 -0.8 -1 0 2 4 6 8 10 12 14 16 18 20 quarters stniop egatnecrep Austria Belgium Finland France Germany Luxembourg Netherlands 0 -0.2 -0.4 -0.6 -0.8 -1 Note: Response to a 1⁄2 percentage point increase in the euro-area outputgap. Coefficientsareevaluatedattheposteriormean. 22

One can also use the model to simulate the response of the country-level unemployment ratesafterashocktotheeuro-wideoutputgap, keepingconstanttheidiosyncratictrendsand cycles as well as the common trend. This simulation is a way to understand how economic policies (monetary or fiscal) designed to affect the business cycle of the euro area would propagate geographically. The shock is an initial widening of the euro-area output gap by 1⁄ 2 percentage point. The responses of the country-level unemployment rates appear in Figure 12. In line with the estimated Okun’s law coefficients, the impulse-response analysis shows that the cyclical component of the unemployment rates of Greece and Spain would respond more significantly to an area-wide increase of output above trend. In the same vein, those countries with smaller coefficients mentioned before have more muted unemployment rate responses from the aggregate shock. However, there can be countries with relatively high estimated Okun’s law coefficients, but with only moderate unemployment rate responses to a common cycle shock. That is the case of France.11 In this case, the unemployment rate cycle would respond more to the idiosyncratic component of the cycle of output than to its common counterpart. 6.5 Estimates of Country-Level Phillips Curves A concept central to that of potential output and, hence, to that of the output gap is the Phillips curve. Understanding the dynamics of inflation with respect to the degree of slack in the economy is central to the design and implementation of economic policies in general, and of monetary policy in particular. This section explores the cyclicality of inflation in the euro-area member states. Figure 13 classifies the countries according to the posterior mean estimates of the slope of the Phillips curve, κ , for a given level of inflation expectations and lagged inflation. i The higher the coefficient, the stronger the response of the country’s inflation to its cyclical position. In general, the posterior mean estimates of the slope of the Phillips curve for the countries are centered around a value close to 0.14. However, there are countries with slopes larger than twice this value and others with coefficients about half the cross-sectional average. The countries with the largest Phillips curve slope are estimated to be Estonia and Latvia, whereas those with the smallest response of inflation to the output gap would be Germany and Slovakia. As with the country-level unemployment rates, one can simulate the propagation of the effects of common cyclical component changes on the inflation rates of the euro-area country members. Figure 14 depicts the impulse-response analysis of an initial deviation of output 1⁄ 2 percentage point above trend, keeping inflation expectations fixed.12 The first 11Recall that under the model specification, a perturbation to the common cycle affects each country’s cycle depending on the loading coefficients, αi. The response of each country’s unemployment rate to the country-specific cycle on impact, in turn, depends on the Okun’s law coefficients, θ1i and θ2i. 12Recall that under the model specification, a perturbation to the common cycle affects each country’s cycle depending on the loading coefficients, αi. The response of each country’s inflation rate to the countryspecificcycleonimpact,inturn,dependsonthePhillipscurveslopecoefficients,κi. Thedynamicresponseof inflationdependsonthepersistencecoefficient,1−β1i. Thehigherthiscoefficient,thelowerthedependence of inflation on inflation expectations. 23

Figure 13: Phillips Curve Slope Coefficient Es- Figure 14: Impulse-Response Analysis for the timates Cyclical Component of the Inflation Rate ± 5 4 3 2 1 Finland 0 0 2 4 6 8 10 12 14 16 18 20 quarters Estonia Latvia Lithuania Ireland Netherlands BelgiumGermany Luxembourg Slovakia France Austria Slovenia Italy PortugalSpain Greece Malta Cyprus κ ≤ 0.1 0.1 < κ ≤ 0.2 0.2 ≤ κ Note: Coefficientsareevaluatedattheposteriormean. stniop egatnecrep Greece Ireland Italy Portugal Spain 5 4 3 2 1 0 5 4 3 2 1 0 0 2 4 6 8 10 12 14 16 18 20 quarters stniop egatnecrep Cyprus Estonia Latvia Lithuania Malta Slovakia Slovenia 5 4 3 2 1 0 5 4 3 2 1 0 0 2 4 6 8 10 12 14 16 18 20 quarters stniop egatnecrep Austria Belgium Finland France Germany Luxembourg Netherlands 5 4 3 2 1 0 Note: Response to a 1⁄2 percentage point increase in the euro-area outputgap. Coefficientsareevaluatedattheposteriormean. 24

panel shows that, except for Italy because of a highly persistent inflation, those countries that experienced weaker economies following the global financial crisis have very uniform responses of inflation, which are relatively muted. The group of countries in the middle panel has more varied responses, with the inflation rates of Latvia and Estonia reacting more strongly and persistently to the common shock, whereas Slovakia’s inflation shows one of the least significant responses. The countries in the last group show some variability in the responses of inflation, with Germany at the bottom because of its very low estimated Phillips curveslopecoefficientandNetherlandsatthetop, albeitwitharesponsestillrelativelysmall. 7 Evaluating the Forecasting Performance of the Model Even though the marginal data densities favor the model in which the country output trends are modeled as random walks, an additional consideration to take into account to choose the best specification and to diagnose the features of a chosen model against other alternatives is to evaluate its forecasting capabilities. This section performs a pseudo out-ofsample forecasting exercise in which I use each of the specifications proposed in Section 3 to predict the three variables of interest (real GDP growth, unemployment rate, and inflation rate) at the euro area level. The forecast horizon is one and four quarters ahead. The forecasting exercise is also useful for determining whether there is any gain in using disaggregated data at the country level for real GDP growth, the unemployment rate, and the inflation rate to obtain the euro-area output gap compared to using euro-area aggregate data for the three variables of interest. A priori, there is no reason to believe that using disaggregate data would offer an advantage in terms of forecasting capabilities, so it is worth to explore how the model proposed in this paper performs. To that end, I set up a UC model for real GDP growth, the unemployment rate, and the inflation rate at the aggregate euro area level. The structure is basically the same as for each country in the model of equations (1)-(8) and resembles the model proposed by Basistha and Nelson (2007): output is the sum of a trend and a cycle, the unemployment rate is the sum of a trend and a cycle that is a function of the cycle of output following an Okun’s law, and the inflation rate is a ‘hybrid’ Phillips curve. However, I setup the aggregate output trend under the three specifications used before: random walk (RW), local linear trend (LLT), and integrated random walk (IRW). The description of the models and the estimation results appear in Appendix G. In the exercise, I assume that the data, both at the country as well as the aggregate euro arealevel,areavailablethroughthefirstquarterof2009,sothattheout-of-sampleforecasting exercise starts in the second quarter of 2009. At each subsequent quarter, I re-estimate the model with the new data and obtain 10,000 draws from the posterior distribution and the Durbin and Koopman (2002) simulator smoother to construct the forecasts. To forecast the three euro-area aggregate variables (real GDP growth, unemployment rate, and inflation rate), I first forecast the country-level variables using the UC models proposed (with the threeoutputtrendspecifications—RW,LLT,andIRW—)andthenaggregatethemtoobtain the euro-area-level equivalent (these models are labeled “Country-level”).13 These forecasts are then compared with those of (i) three UC models corresponding to the three output 13Toaggregatethecountry-levelforecaststotheeuro-areaaggregatesofrealGDPgrowth,theunemployment rate, and inflation, I use nominal GDP, the labor force, and HICP weights, respectively. 25

trend specifications based on the three aggregate variables (labeled “Aggregate”), and (ii) a VAR(4) model for the three aggregate variables. Table 1 shows the mean squared forecast errors for all the models and their standard deviations across posterior draws. In general, according to the results, the UC models either with aggregate or country-level data forecast any of the three variables better than the VAR model, one or four quarters ahead. Between the UC models that use aggregate or country-level data, there is no significant difference in terms of forecasting inflation at any horizon. However, the models with country-level data, generally speaking, have more forecast accuracy in terms of real GDP growth and the unemployment rate than the models with aggregate data. Additionally, almost everywhere, the UC models with output trends specified as random walks have the lowest mean squared forecast errors. In particular, among the UC models with countrylevel data, the RW specification in the idiosyncratic output trends forecasts any of the three variable at least as well as the other models at all horizons. More specifically, at the onequarter-ahead horizon, the model with RW idiosyncratic output trends is significantly better at forecasting real GDP growth and the unemployment rate than the other two trend alternatives, given the small standard deviations and the relatively large differences in the mean squared forecast errors. That does not happen with inflation, in which case the differences are rather small. The significantly better performance of the RW specification at forecasting all the variables four quarters ahead is even more noticeable. These results, along with those about the marginal data density, favor the country-level UC model with RW idiosyncratic output trends as the best modeling choice. As a way to visualize how well the RW specification forecasts the variables of interest, Figure 15 illustrates its four-quarter-ahead forecasts. Each of the panels shows the observed macroeconomic variable since the first quarter of 2010 in the black line, the median of model forecasts in the thin blue line, and the 90 percent confidence set in the light blue shaded area. The first panel shows that the confidence sets of the model forecast include the realized real GDP growth rate almost everywhere, except for the four quarters of 2011 and the second and fourth quarters of 2012. The unemployment rate forecast appears in the second panel, which shows the model has a relatively hard time forecasting the increase in the unemployment rate around the events of the European debt crisis between mid-2011 and early 2013. After that, the forecast misses to the upside the downward trajectory of the observed series. Finally, the third panel evidences that the inflation forecast is much less volatile than the actual series. The reason is that the predictable component of inflation puts more weight on expected inflation (a much less volatile variable) four quarters ahead than on lagged realized inflation for each country. In fact, the model forecast of inflation is qualitatively very similar to the observed core HICP inflation rate (not shown) in the period of analysis. That means that the unpredictable component, through the lens of the model, may be given by perturbations to the more volatile categories of energy and unprocessed food price inflation. The performances of the other two trend specifications (LLT and IRW) appear in Appendix F. 26

Table 1: Mean Squared Forecast Errors for Out-Of-Sample Forecasts Real GDP growth rate Model 1 quarter ahead 4 quarters ahead VAR(4) 0.74 ( — ) 1.12 ( — ) Aggregate (RW) 0.51 (0.07) 0.62 (0.09) Aggregate (LLT) 0.68 (0.08) 0.87 (0.12) Aggregate (IRW) 0.67 (0.08) 0.88 (0.12) Country-level (RW) 0.43 (0.06) 0.36 (0.05) Country-level (LLT) 0.57 (0.08) 0.65 (0.10) Country-level (IRW) 0.62 (0.10) 0.69 (0.10) Unemployment rate Model 1 quarter ahead 4 quarters ahead VAR(4) 0.23 ( — ) 1.08 ( — ) Aggregate (RW) 0.32 (0.02) 1.02 (0.09) Aggregate (LLT) 0.29 (0.02) 0.90 (0.08) Aggregate (IRW) 0.28 (0.02) 0.89 (0.08) Country-level (RW) 0.17 (0.01) 0.65 (0.04) Country-level (LLT) 0.22 (0.01) 0.77 (0.05) Country-level (IRW) 0.22 (0.01) 0.76 (0.06) Inflation rate Model 1 quarter ahead 4 quarters ahead VAR(4) 2.15 ( — ) 2.19 ( — ) Aggregate (RW) 1.54 (0.06) 1.39 (0.09) Aggregate (LLT) 1.54 (0.06) 1.38 (0.07) Aggregate (IRW) 1.54 (0.06) 1.39 (0.07) Country-level (RW) 1.58 (0.03) 1.34 (0.04) Country-level (LLT) 1.57 (0.03) 1.44 (0.07) Country-level (IRW) 1.56 (0.03) 1.47 (0.08) Note: Aggregatedenotesthemodelestimatedwitheuro-area-leveldataonrealGDP,theunemploymentrate,and theinflationrate. Country-leveldenotesthemodelestimatedwithcountry-leveldata. RWdenotestheoutputtrend followsarandomwalkprocess. LLTdenotestheoutputtrendisspecifiedasalocallineartrend. IRWdenotesthe outputtrendfollowsandintegratedrandomwalk. Numbersinparenthesisarethestandarddeviationofthemean squareforecasterroracrossposteriordraws. TheVAR(4)isestimatedwithclassicalmethods. 27

Figure 15: Model Forecasts under the RW Specification and Observed Variables 2% 90% confidence set Euro-area real GDP growth t/t-1 Model forecast four quarters ahead 1% 0% -1% -2% 2010 2011 2012 2013 2014 2015 2016 2017 14% 13% 12% 11% 10% 9% 8% 90% confidence set 7% Euro-area unemployment rate Model forecast four quarters ahead 6% 2010 2011 2012 2013 2014 2015 2016 2017 4% 90% confidence set 3% Euro-area inflation rate t/t-4 Model forecast four quarters ahead 2% 1% 0% -1% -2% -3% 2010 2011 2012 2013 2014 2015 2016 2017 Note: Themodelforecastfourquartersaheadisthemedianofthemodelforecastacrossdraws. Source: Eurostat andauthor’scalculations. 28

Figure 16: Output Gap Estimates Comparison: Country-Level vs. Aggregate Data Note:Timeseriesaretheaveragesoftheposteriordrawsoftheeuro-areaoutputgapusingtheDurbinandKoopman (2002)simulatorsmoother. Theaggregateestimateisobtainedwithdatafrom2001:Q2becauseofdataavailability. 8 Does Introducing Country-Level Data Reduce the Uncertainty around the Estimate of the Output Gap? AccordingtothereasoningprovidedbyStockandWatson(2016), theuncertaintyaround the estimate of the euro-area output gap would have two opposing forces at work when one uses country-level as opposed to aggregate data. First, a model with heterogenous units provides more precise estimates of a given common component among units and, hence, makes the signal extraction uncertainty smaller compared with a model that uses a single unit in which the data are aggregated. Second, a model with several heterogenous units has many more parameters to estimate than an equivalent model with aggregate data, which makes the parameter uncertainty larger in the former. It is not clear which of the two forces dominates in a general framework. Figure 16 compares the estimates of the output gap using the model with country-level data of equations (1)-(8) with that of the aggregate data model used for the forecast comparisons of the previous section along with the corresponding confidence sets. Both estimates correspond to the models with RW output trends. Based on the results, the use of country-level data somewhat reduces the uncertainty around the estimate of the euro-area output gap compared with the estimate obtained using aggregate data. The confidence sets in the model proposed in this paper are 5 percent narrower, on average, than those from the UC model with aggregate data. Hence, one can conclude that, in this case, the signal extraction uncertainty channel dominates the parameter uncertainty one, and it is beneficial to use disaggregated data. 29

9 Conclusion This paper proposes a method to estimate the output gap of the euro area using data on real GDP, the unemployment rate, and the inflation rate of the member states through the Bayesian estimation of an unobserved components model. A key feature of the model is the assumption of common cycle and trend components in real GDP and the unemployment rate across countries. This feature allows one to obtain measures of the output gap, potential output, and the natural unemployment rate of the euro area. Additionally, the framework proposed also permits the characterization of the degree of slack, potential output, and natural rate of unemployment for each member state. This characterization is important for understanding how policies that affect the area as a whole, for example with the goal to curb inflation pressures, propagate across countries. Among other results, the model’s estimation of the data-preferred specification indicates that output was about 5 percent above potential prior to the financial crisis at the end of 2007, then declined to almost its potential level at the end of 2009. After this period, output surpassed its potential counterpart by about 3 percent in 2011 and then declined to nearly 4 percent below trend at the end of 2013, during the European debt crisis. In the second quarter of 2017, it is estimated that output stands at about 1 percent above potential. A negative output gap did not emerge during the global financial crisis because the model estimates potential output to have declined about 13⁄ 4 percent in that period, which is more than enough to prevent real GDP to fall below trend. In addition, the estimate of the natural unemployment rate is 7.7 percent in the second quarter of 2017, after having reached about 12.5 percent in mid-2009. 30

Appendix A State-Space Model in Matrix Form The model in Equations (1)-(14) can be written in matrix form as follows: z it = C(Θ mi )+H(Θ mi )x it +w it , w it |F t−1 ∼ iid N (0,R(Θ mi )) (15) x it = F(Θ si )x i,t−1 +Gv it , v it |F t−1 ∼ iid N (0,Q(Θ si )), (16) where F t−1 is the sigma-field containing information until period t−1, with ε t   ∆y ηy ηy  it   it t z = ∆u , w = ηu , v = ηu. it it it it it  t  ∆π it −β 1i (π t e −π i,t−1 ) η i π t    ν it   ζ  it For the RW specification, we have the following matrices: c t   c t−1   c t−2   µ +δyµ  ηy   i i  x =  t , C(Θ ) = 0 , it  ηu  mi  t   β   υ  0i it     υ i,t−1   υ i,t−2  α −α 0 δy 0 1 −1 0  i i i  H(Θ ) = α θ α (θ −θ ) −α θ 0 δu θ θ −θ −θ , mi i 1i i 2i 1i i 2i i 1i 2i 1i 2i α κ 0 0 0 0 κ 0 0  i i i 1 0 0 0 0 φ φ 0 0 0 0 0 0    1 2  0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0     0 1 0 0 0 0 0 0 0 0 0 0 0   F(Θ ) =  , G = 0 0 1 0 0. si 0 0 0 0 0 0 0 0     0 0 0 0 0 0 0 0 0 0 ρ 0 0    i  0 0 0 0 1 0 0 0 0 0 1 0 0     0 0 0 0 0 0 0 0 0 0 0 1 0   0 0 0 0 0 31

For the LLT and RW specifications, the matrices are the following: c t   c t−1  c t−2     ηy   t  δyµ  ηu   i  x =  t , C(Θ ) = 0 , it  µ  mi  it  β  µ i,t−1  0i    υ   it    υ i,t−1   υ i,t−2  α −α 0 δy 0 0 1 1 −1 0  i i i  H(Θ ) = α θ α (θ −θ ) −α θ 0 δu 0 0 θ θ −θ −θ , mi i 1i i 2i 1i i 2i i 1i 2i 1i 2i α κ 0 0 0 0 0 0 κ 0 0  i i i φ φ 0 0 0 0 0 0 0 0 1 0 0 0 0 1 2     1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0     0 0 0 0 0 0 0 0 0 0 0 1 0 0 0     0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 F(Θ ) =  , G =  . si 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0     0 0 0 0 0 1 0 0 0 0 0 0 0 0 0     0 0 0 0 0 0 0 ρ 0 0 0 0 0 0 1  i    0 0 0 0 0 0 0 1 0 0 0 0 0 0 0     0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 For all specifications, the variance matrices are as follows: σ2 0 0 0 0 ε  σ η 2 i y 0 0   0 σ η 2 y 0 0 0  R(Θ )) = 0 σ2 0 , Q(Θ ) = 0 0 σ2 0 0 , vi  0 0 η i u σ2  vi  0 0 0 ηu σ2 0    η i π  0 0 0 0 νi σ2   ζi where for the IRW trend setup, σ2 = 0. Finally, ηy i Θ = {µ,µ ,α ,δy,σ2 ,θ ,θ ,δu,σ2 ,β ,β ,κ ,σ2 }, mi i i i ηy 1i 2i i ηu 0i 1i i ηπ i i i Θ = {φ ,φ ,ρ ,σ2,σ2 ,σ2 ,σ2,σ2 }, si 1 2 i e ηy ηu ζi νi for i = 1,...,n. 32

B Data Details The information on real GDP, the unemployment rate, the HICP inflation rate at the country- and euro-level come from Eurostat. The Survey of Professional Forecasters (SPF) HICP inflation rate expectations come from the ECB. The data are as follows: • Real GDP: Both at the country and at the euro area level, I use GDP in million 2010 euroseasonallyandcalendaradjustedfrom2000:Q1to2017:Q2. Slovakiadoesnothave this information available, so I perform an X-12 seasonal adjustment to the available data for this country in million 2010 euro not seasonally adjusted. The growth rate of real GDP is obtained as the quarterly percent change. • Unemployment rate: It is the percentage of active population (15 - 74 years old) that is unemployed. Both at the country and at the euro-level, the data are the quarterly average of the monthly indicator and seasonally adjusted but not calendar adjusted from 2000:Q1 to 2017:Q2. • HICP:Itistheharmonizedindexofconsumerpriceswithbaseyear2015atthemonthly frequency which I seasonally adjust using an X-12 procedure and then pick the last value of every quarter to calculate the annualized quarterly rate of inflation. At the country level, the monthly data runs from 1999:M12 to 2017:M06. At the euro area level, the monthly data runs from 2000:M12 to 2017:M06. • SPF HICP inflation rate expectations: It is the mean point estimate of the yearover-year percentage change one year ahead in the HICP. The data cover 2000:Q1 to 2017:Q2. • GDP, labor market, and price inflation weights: To obtain the euro-area real GDP growth rate, I weight the real GDP growth rates across countries using nominal GDP weights at the country level, where GDP is million current euro not seasonally adjusted, which I seasonally adjust using an X-12 procedure. To obtain the euro-area unemployment rate from the country-level rates, I aggregate the unemployment rates from the countries using the shares of active population over the total of the euro area asweights. Thesedataareinanannualfrequency, whichIassumeremainsconstantevery quarter of a given year at the annual figure. Finally, to obtain the aggregate HICP inflation rate, I construct the euro-area inflation rate by aggregating the country-level rates weighted by the HICP weights for each country. These data are also in annual frequencyandthesametreatmentastheactivepopulationisgivenwithineachquarter. C Details on the Gibbs Sampler Let z , x , Θ , and Θ for i = 1,2,...,n, be defined as in Appendix A. Let Z = it it mi si T {˜z ,˜z ,...,˜z } denote the observed data and let X = {x˜ ,x˜ ,...,x˜ }. Here, ˜z = 1 2 T T 1 2 T t n n {z ,z ,...,z } and x˜ = {x ,x ,...,x }. Denote Θ = Θ and Θ = Θ . 1t 2t nt t 1t 2t nt m mi s si S S i=1 i=1 33

Partition Θ = Θ1 Θ2 Θ3, with si s si si S S Θ1 = {φ ,φ }, s 1 2 Θ2 = {ρ ,σ2,σ2 }, si i ζi νi Θ3 = {σ2,σ2 ,σ2 }. si e ηy ηu Notice that the identification conditions imply that Θ3 is fixed. si Also, partition Θ = Θ1 Θ2 Θ3 , with mi mi mi mi S S Θ1 = {µ ,α ,δy,σ2 }, mi i i i ηy i Θ2 = {θ ,θ ,δu,σ2 }, mi 1i 2i i ηu i Θ3 = {β ,β ,κ ,σ2 }, mi 0i 1i i ηπ i where µ has been excluded because it is fixed under the identification conditions. For the RW output trend specification there is no sampling of σ2 , whereas for the LLT specification νi there is no sampling of the parameters µ , and for the IRW specification there is no sampling i of either µ or σ2 . i ηy i The Gibbs sampler operates as follows: 1. Start with initial values for the model’s parameters, Θ = Θ Θ . m s S 2. Draw X from p(X |Z ,Θ ,Θ ) using the Durbin and Koopman (2002) simulation T T T m s smoother. 3. Draw Θ1 from p(Θ1|Z ,X ,Θ ,Θ2,Θ3) using the conditional distributions implied s s T T m si si by the independent normal-inverse-gamma prior. 4. For i = 1,2,...,n, sample as follows: (a) Draw Θ2 from p(Θ2|Z ,X ,Θ ,Θ1,Θ2) using the conditional distributions imsi si T T m s si plied by the independent normal-inverse-gamma prior. (b) Draw Θ1 from p(Θ1 |Z ,X ,Θ2 ,Θ3 ,Θ ,Θ ) using the conditional distrimi mi T T mi mi m(/i) s butions implied by the independent normal-inverse-gamma prior. Repeat similarly for Θ2 and Θ3 and sample from mi mi • p(Θ2 |Z ,X ,Θ1 ,Θ3 ,Θ ,Θ ) and mi T T mi mi m(/i) s • p(Θ3 |Z ,X ,Θ1 ,Θ2 ,Θ ,Θ ), mi T T mi mi m(/i) s respectively. 5. Return to step 2. D Choice of Prior Distributions The choice of prior distributions of the parameters of the UC models appears in Table 2. For convenience, I have included the equation to which each of the parameters belongs. 34

Table 2: Prior Distributions of the Parameters - Country Level Data Parameter Distribution Mean Sd. Equation φ Truncated Normal 1.7 0.2 1 ct=φ1ct−1+φ2ct−2+εt φ Truncated Normal -0.75 0.2 2 ct=φ1ct−1+φ2ct−2+εt µ i Normal 0.5 0.5 ξ i y t =µi+ξ i y ,t−1+η i y t α Normal 1 1 i cit=αict+υit δ i y Truncated Normal 1 1 τ i y t=δ i yτt y+ξ i y t δ i u Truncated Normal 1 1 τi u t=δi uτt u+ξi u t θ 1i Normal -0.2 0.2 uit=τi u t+θ1icit+θ2ici,t−1 θ 2i Normal -0.2 0.2 uit=τi u t+θ1icit+θ2ici,t−1 β 0i Normal 0 0.5 πit=β0+β1πt e+(1−β1)πi,t−1+κcit+ηi π t β 1i Normal 0.5 0.5 πit=β0+β1πt e+(1−β1)πi,t−1+κcit+ηi π t κ i Normal 0.1 0.1 πit=β0+β1πt e+(1−β1)πi,t−1+κcit+ηi π t ρ Truncated Normal 0 1 i υit=ρiυi,t−1+ζit σ η 2 i y Inverse Gamma 0.25 Inf ξ i y t =µi+ξ i y ,t−1+η i y t , var(η i y t )=σ η 2 i y σ η 2 i u Inverse Gamma 0.25 Inf ξi u t=ξi u ,t−1+ηi u t, var(ηi u t)=σ η 2 i u σ η 2 i π Inverse Gamma 1 Inf πit=β0+β1πt e+(1−β1)πi,t−1+κcit+ηi π t, var(ηi π t)=σ η 2 i π σ ζ 2 i Inverse Gamma 10.145 Inf υit=ρiυi,t−1+ζit, var(ζit)=σζ 2 i σ ν 2 i Inverse Gamma 1 Inf µit=µi,t−1+νit, var(νit)=σν 2 i For several parameters, I choose prior means roughly consistent with the posterior mean and variance results in Lenza and Jarocin´ski (2016) for the comparable parameters, although I chose larger standard deviations. That is the way in which I specify the prior distributions of the mean growth rate of trend GDP, µ , in the RW case, the parameters of the common i cycle, φ and φ , the Okun’s law coefficients, θ and θ , and the slope of the Phillips curve, 1 2 1i 2i κ .14 The latter parameter choice is also informed by the results in Blanchard, Cerutti and i Summers(2015). RegardingtheremainingparametersofthePhillipscurve,β andβ ,given 0i 1i the lack of previous estimates for this parameters, I choose a somewhat agnostic distribution in which the mean of the intercept is zero and the persistence coefficient is centered at 0.5. Under the same criteria, I set the persistence coefficient of the idiosyncratic component of the country’s cycle, ρ , to be centered at zero. I choose a truncated distribution in some of i the cases given the stationary or the identification restrictions. The prior distributions of the variance parameters is inverse gamma in all cases. I set the means as follows: For the variances of the shocks of the idiosyncratic trends, σ2 and σ2 , I η i y η i u specify that 80 percent of the variation of the output or unemployment trend of a particular country is explained by the common respective trend and the rest by the idiosyncratic component. I apply the same criterion to the proportion of the variation in the country’s cycle that is due to the common cycle; that is how the mean of the prior distribution of σ2 ζi is picked. Additionally, given the absence of previous estimates in the literature, I assume σ2 and σ2 to have a prior mean of one. In all cases, the prior distributions of the variances η i π νi do not have a well defined variance. 14Gurin,MaurinandMohr(2015)findsmallerestimatedcoefficientsforφ1 andφ2 inaMarkov-switching specification of the output trend than the choice made in this paper. 35

Finally, I set the prior distribution of the loading coefficients α , δy, and δu to be normally i i i distributed and centered at one with unity standard deviation as a way to incorporate the lack of prior knowledge about these parameters. E Parameter Results In this appendix, I lay out the full results from the Bayesian estimation showing the estimates of the posterior mean and standard deviation of the parameters of the model, as well as the first and fiftieth order autocorrelation coefficient, the relative numerical efficiency (RNE)usinga4percenttaper, andthep-valueoftheGeweke(1991)convergencediagnostics using a 4 percent tapper as well in which the null hypothesis considers equality of the means of the first 20 percent of draws with that of the last 50 percent. Following Lenza and Jarocin´ski (2016), I include the value of the output gap at the end of the sample among the parameters for which convergence diagnostics are reported. Given the large number of parameters, the results appear in Tables 3-9. In the tables, the parameters µ , α , δy, δu, i i i i θ , θ , β , β , κ , ρ , σ2, σ2 , σ2 , σ2 are numbered from i = 1,...,19 according to 1i 2i 0i 1i i i ζi η i y η i u η i π the number of countries in the euro area in the following order: Austria, Belgium, Cyprus, Estonia, Finland, France, Germany, Greece, Ireland, Italy, Latvia, Lithuania, Luxembourg, Malta, Netherlands, Portugal, Slovakia, Slovenia, and Spain. The estimated value of the output gap at the end of the sample period is denoted c . φ and φ are the parameters of T 1 2 the AR(2) specification of the common component of the cycle. Recall that the draws from the posterior distribution used to produce the results of the model are based on 300,000 draws after burning in the first 100,000 and thinning every 100th draw, which left us with 2,000 draws from the posterior distribution. The results of the diagnostics tests show that these 2,000 draws do not evidence significant autocorrelation offirstorderandalmostnoautocorrelationoforderfifty. Apartfrom2outof268coefficients, the p-values of the test of equality of means between the fist 20 percent of the draws and the last 50 percent are all above 1 percent, which indicate that the null hypothesis is not rejected for any of the parameters and the sampler has converged. An analysis of the results indicates that the weighted average mean growth rate of trend output is about 1.2 percent per year.15 F Four-Quarters-Ahead Forecasts for the Local Linear Trend and Integrated Random Walk Trend Models Figures 17 and 18 show the four-quarter ahead forecasts of the macroeconomic variables for the LLT and IRW specifications, respectively. Each of the panels in the figures shows the observed macroeconomic variable since the first quarter of 2010 in the black line, the median of model forecasts in the thin blue line, and the 90 percent confidence set in the light blue shaded area. The first panel refers to real GDP growth, the second panel depicts the unemployment rate, and the third panel, the inflation rate. 15This estimate is the result of taking the posterior mean estimates of µi for all i and weight them by the average relative importance of each country on the euro-area nominal GDP over the sample period. 36

Table 3: Posterior Distribution Estimates and Convergence Diagnostics Parameter Mean Sd. auto1 auto50 RNE p-value φ 1.686 0.100 0.032 0.003 0.956 0.460 1 φ -0.713 0.100 0.034 0.009 0.957 0.441 2 c 1.039 1.808 0.005 0.011 1.460 0.900 T µ 0.387 0.058 0.104 0.048 0.924 0.287 1 µ 0.375 0.052 0.081 0.006 0.933 0.540 2 µ 0.503 0.084 -0.000 -0.017 1.310 0.561 3 µ 0.819 0.156 0.160 0.026 0.635 0.630 4 µ 0.329 0.094 0.136 0.033 0.813 0.129 5 µ 0.314 0.048 0.069 0.001 0.837 0.022 6 µ 0.220 0.086 0.100 0.017 0.814 0.666 7 µ 0.187 0.151 0.123 0.067 1.135 0.172 8 µ 1.224 0.226 0.116 0.036 0.956 0.900 9 µ 0.071 0.064 0.068 0.050 1.090 0.202 10 µ 0.909 0.123 0.094 0.013 0.688 0.001 11 µ 0.944 0.176 0.155 0.026 0.903 0.152 12 µ 0.709 0.094 0.024 0.025 1.067 0.551 13 µ 0.780 0.122 0.000 -0.015 1.120 0.135 14 µ 0.375 0.070 0.141 0.030 0.633 0.029 15 µ 0.169 0.067 0.047 0.047 1.100 0.084 16 µ 0.889 0.163 0.123 0.022 0.793 0.274 17 µ 0.590 0.087 0.104 0.032 0.935 0.270 18 µ 0.508 0.060 0.022 -0.016 1.288 0.517 19 α 0.343 0.090 0.204 -0.010 0.633 0.274 1 α 0.283 0.084 0.140 0.036 0.682 0.971 2 α 0.528 0.134 0.133 0.013 0.729 0.403 3 α 0.868 0.227 0.271 0.001 0.558 0.166 4 α 0.631 0.140 0.302 0.040 0.464 0.789 5 α 0.269 0.077 0.183 0.052 0.885 0.620 6 α 0.423 0.119 0.232 0.002 0.671 0.332 7 α 0.541 0.199 0.188 0.047 0.598 0.084 8 α 0.717 0.318 0.198 0.047 0.496 0.855 9 α 0.446 0.097 0.244 0.029 0.623 0.750 10 α 0.748 0.235 0.200 0.022 0.673 0.463 11 α 0.864 0.210 0.304 -0.006 0.488 0.974 12 α 0.457 0.146 0.174 0.045 0.727 0.895 13 α 0.303 0.159 0.101 -0.021 1.206 0.453 14 α 0.443 0.103 0.266 0.031 0.481 0.893 15 α 0.440 0.114 0.208 0.043 0.728 0.551 16 α 0.632 0.178 0.248 0.018 0.649 0.118 17 α 0.794 0.148 0.366 0.073 0.499 0.629 18 α 0.442 0.097 0.190 0.033 0.979 0.719 19 Notes: Statisticsare based on 2,000draws from the Gibbs sampler. Mean: average across draws; Sd.: standard deviationacrossdraws;auto1: firstorderautocorrelationacrossdraws;auto50: fiftiethorderautocorrelationacross draws;RNE:relativenumericalefficiencyusinga4%taper;p-value: p-valueofthenullhypothesisthatthemean ofthefirst20%ofdrawsisequaltothemeanofthelast50%ofdrawsusinga4%taperforthestandarderror. cT denotestheoutputgapin2017:Q2. 37

Table 4: Posterior Distribution Estimates and Convergence Diagnostics Parameter Mean Sd. auto1 auto50 RNE p-value δy 0.288 0.109 0.105 0.022 0.816 0.757 1 δy 0.293 0.109 0.066 0.021 0.836 0.231 2 δy 0.338 0.154 0.084 0.050 0.743 0.232 3 δy 1.335 0.334 0.202 -0.019 0.657 0.844 4 δy 0.745 0.169 0.141 0.003 0.678 0.368 5 δy 0.285 0.088 0.066 0.030 1.463 0.939 6 δy 0.539 0.132 0.082 0.008 0.913 0.909 7 δy 1.207 0.226 0.098 0.014 1.049 0.351 8 δy 0.436 0.308 0.053 0.015 0.906 0.285 9 δy 0.441 0.114 0.091 -0.003 0.747 0.024 10 δy 0.773 0.328 0.198 -0.011 0.850 0.488 11 δy 1.707 0.305 0.245 0.029 0.928 0.920 12 δy 0.578 0.204 0.081 0.062 1.042 0.217 13 δy 0.261 0.178 0.008 -0.022 1.566 0.881 14 δy 0.547 0.109 0.034 -0.012 0.706 0.296 15 δy 0.333 0.123 0.008 0.032 0.758 0.599 16 δy 1.103 0.232 0.085 -0.029 0.924 0.274 17 δy 0.648 0.181 0.217 -0.005 1.067 0.917 18 δy 0.153 0.100 0.070 0.048 0.927 0.884 19 δu 0.040 0.029 0.029 -0.013 1.492 0.454 1 δu 0.112 0.063 0.027 0.035 0.920 0.175 2 δu 0.123 0.077 -0.007 -0.004 0.760 0.192 3 δu 0.575 0.216 0.207 -0.010 0.529 0.228 4 δu 0.065 0.038 0.236 -0.017 0.874 0.956 5 δu 0.063 0.030 -0.006 -0.006 0.906 0.928 6 δu 0.019 0.017 0.038 -0.027 1.012 0.666 7 δu 0.246 0.094 -0.007 -0.025 1.131 0.953 8 δu 0.461 0.099 0.058 0.001 0.783 0.499 9 δu 0.045 0.034 0.074 -0.002 1.271 0.962 10 δu 0.706 0.214 0.197 -0.033 0.523 0.711 11 δu 0.725 0.229 0.247 -0.008 0.454 0.627 12 δu 0.017 0.015 -0.008 0.028 1.312 0.675 13 δu 0.108 0.061 -0.020 -0.001 1.284 0.380 14 δu 0.029 0.024 0.050 0.003 1.225 0.076 15 δu 0.144 0.071 0.055 -0.001 1.008 0.977 16 δu 0.313 0.105 0.111 -0.049 1.187 0.597 17 δu 0.081 0.055 -0.006 -0.021 1.111 0.944 18 δu 0.322 0.098 0.025 0.002 1.054 0.430 19 Notes: Statisticsare based on 2,000draws from the Gibbs sampler. Mean: average across draws; Sd.: standard deviationacrossdraws;auto1: firstorderautocorrelationacrossdraws;auto50: fiftiethorderautocorrelationacross draws;RNE:relativenumericalefficiencyusinga4%taper;p-value: p-valueofthenullhypothesisthatthemean ofthefirst20%ofdrawsisequaltothemeanofthelast50%ofdrawsusinga4%taperforthestandarderror. 38

Table 5: Posterior Distribution Estimates and Convergence Diagnostics Parameter Mean Sd. auto1 auto50 RNE p-value θ -0.035 0.055 -0.010 0.006 1.191 0.520 1,1 θ -0.034 0.109 0.029 0.018 1.043 0.827 1,2 θ -0.304 0.068 0.021 -0.004 1.043 0.832 1,3 θ -0.115 0.060 0.159 -0.001 0.716 0.812 1,4 θ -0.052 0.022 0.056 -0.013 0.637 0.194 1,5 θ -0.138 0.056 -0.006 0.007 1.054 0.643 1,6 θ -0.122 0.036 0.014 -0.000 1.186 0.647 1,7 θ -0.315 0.047 0.035 0.026 1.148 0.173 1,8 θ -0.038 0.035 0.227 0.004 0.486 0.728 1,9 θ -0.158 0.065 -0.000 0.030 1.026 0.826 1,10 θ -0.095 0.064 0.065 -0.016 0.532 0.930 1,11 θ -0.045 0.090 0.182 0.013 0.434 0.701 1,12 θ -0.040 0.019 -0.021 -0.009 1.016 0.260 1,13 θ -0.026 0.045 0.016 0.008 0.820 0.766 1,14 θ -0.099 0.042 0.034 0.022 0.841 0.442 1,15 θ -0.222 0.065 0.053 0.009 1.598 0.515 1,16 θ -0.126 0.055 0.036 0.024 0.970 0.130 1,17 θ -0.044 0.056 0.002 -0.011 1.075 0.980 1,18 θ -0.494 0.125 -0.011 0.003 1.896 0.717 1,19 θ -0.197 0.054 0.018 0.010 0.871 0.831 2,1 θ -0.190 0.106 0.044 0.010 0.979 0.951 2,2 θ -0.162 0.067 -0.002 -0.014 0.874 0.468 2,3 θ -0.102 0.062 0.135 -0.006 0.579 0.765 2,4 θ -0.078 0.021 0.035 0.048 1.051 0.980 2,5 θ -0.180 0.054 -0.018 0.046 1.336 0.125 2,6 θ -0.115 0.035 0.013 0.029 1.403 0.481 2,7 θ -0.214 0.043 0.021 0.036 0.722 0.961 2,8 θ -0.038 0.032 0.237 -0.033 0.637 0.079 2,9 θ -0.119 0.061 -0.022 0.004 1.371 0.211 2,10 θ -0.105 0.071 0.151 -0.017 0.699 0.955 2,11 θ -0.032 0.082 0.186 -0.028 0.903 0.976 2,12 θ -0.037 0.019 0.012 0.014 1.119 0.060 2,13 θ -0.046 0.047 0.000 -0.010 1.209 0.291 2,14 θ -0.208 0.043 0.045 0.003 0.862 0.233 2,15 θ -0.205 0.064 0.002 0.010 0.980 0.367 2,16 θ -0.135 0.057 0.093 0.024 1.002 0.558 2,17 θ -0.148 0.056 0.007 -0.029 1.139 0.149 2,18 θ -0.179 0.120 0.002 -0.022 1.229 0.842 2,19 Notes: Statisticsare based on 2,000draws from the Gibbs sampler. Mean: average across draws; Sd.: standard deviationacrossdraws;auto1: firstorderautocorrelationacrossdraws;auto50: fiftiethorderautocorrelationacross draws;RNE:relativenumericalefficiencyusinga4%taper;p-value: p-valueofthenullhypothesisthatthemean ofthefirst20%ofdrawsisequaltothemeanofthelast50%ofdrawsusinga4%taperforthestandarderror. 39

Table 6: Posterior Distribution Estimates and Convergence Diagnostics Parameter Mean Sd. auto1 auto50 RNE p-value β 0.051 0.228 0.000 -0.013 1.362 0.193 0,1 β 0.095 0.279 -0.008 0.024 1.288 0.285 0,2 β -0.160 0.386 0.015 0.008 1.066 0.409 0,3 β 0.353 0.464 -0.010 -0.010 1.350 0.401 0,4 β -0.277 0.286 0.033 0.016 0.714 0.950 0,5 β -0.230 0.240 0.008 -0.005 1.217 0.630 0,6 β -0.123 0.228 -0.028 -0.025 1.593 0.716 0,7 β 0.147 0.397 0.017 0.014 1.241 0.178 0,8 β -0.000 0.366 0.027 -0.047 1.064 0.621 0,9 β -0.048 0.271 0.020 0.027 1.202 0.674 0,10 β 0.016 0.469 -0.001 0.006 0.976 0.399 0,11 β -0.047 0.378 0.026 0.004 1.163 0.257 0,12 β 0.223 0.310 0.010 -0.001 0.895 0.979 0,13 β 0.250 0.278 0.043 -0.049 0.847 0.619 0,14 β 0.207 0.312 0.049 0.017 0.849 0.762 0,15 β 0.178 0.317 0.016 0.019 0.960 0.206 0,16 β 0.340 0.350 0.065 0.033 1.300 0.521 0,17 β 0.287 0.360 0.014 -0.005 1.038 0.763 0,18 β 0.103 0.345 -0.043 0.009 1.450 0.305 0,19 β 0.888 0.078 0.006 0.021 0.865 0.133 1,1 β 0.927 0.048 0.012 0.009 1.197 0.919 1,2 β 0.987 0.011 0.002 0.026 1.081 0.006 1,3 β 0.954 0.026 0.025 -0.002 1.196 0.332 1,4 β 0.921 0.045 0.029 -0.035 0.837 0.064 1,5 β 0.977 0.018 -0.006 0.005 1.346 0.965 1,6 β 0.918 0.064 0.016 0.039 0.740 0.867 1,7 β 0.852 0.077 0.051 0.017 0.663 0.051 1,8 β 0.763 0.110 0.032 -0.021 0.739 0.951 1,9 β 0.992 0.006 -0.007 -0.031 0.927 0.483 1,10 β 0.904 0.040 -0.002 0.028 1.481 0.243 1,11 β 0.574 0.124 -0.007 -0.017 1.574 0.573 1,12 β 0.927 0.060 -0.017 -0.014 1.442 0.915 1,13 β 0.832 0.085 -0.017 -0.030 1.324 0.732 1,14 β 0.906 0.064 0.022 -0.041 1.317 1.000 1,15 β 0.870 0.088 0.013 -0.015 1.046 0.957 1,16 β 0.556 0.109 -0.016 -0.035 1.256 0.539 1,17 β 0.696 0.109 -0.004 0.001 1.180 0.029 1,18 β 0.914 0.055 -0.013 -0.033 1.032 0.512 1,19 Notes: Statisticsare based on 2,000draws from the Gibbs sampler. Mean: average across draws; Sd.: standard deviationacrossdraws;auto1: firstorderautocorrelationacrossdraws;auto50: fiftiethorderautocorrelationacross draws;RNE:relativenumericalefficiencyusinga4%taper;p-value: p-valueofthenullhypothesisthatthemean ofthefirst20%ofdrawsisequaltothemeanofthelast50%ofdrawsusinga4%taperforthestandarderror. 40

Table 7: Posterior Distribution Estimates and Convergence Diagnostics Parameter Mean Sd. auto1 auto50 RNE p-value κ 0.098 0.058 0.011 0.001 1.481 0.974 1 κ 0.180 0.074 -0.004 0.029 0.888 0.274 2 κ 0.132 0.050 0.036 0.015 0.616 0.187 3 κ 0.216 0.057 0.024 -0.015 0.885 0.419 4 κ 0.111 0.052 -0.018 -0.020 1.273 0.370 5 κ 0.146 0.064 0.006 -0.006 0.973 0.226 6 κ 0.047 0.037 0.030 -0.015 0.932 0.321 7 κ 0.113 0.024 -0.026 0.030 0.891 0.284 8 κ 0.089 0.049 0.234 0.056 0.616 0.415 9 κ 0.167 0.047 -0.084 0.001 1.392 0.082 10 κ 0.289 0.053 0.013 0.002 1.105 0.835 11 κ 0.156 0.072 0.035 -0.017 0.837 0.947 12 κ 0.140 0.065 -0.038 -0.023 1.101 0.395 13 κ 0.096 0.073 0.152 -0.010 1.006 0.917 14 κ 0.181 0.052 0.002 0.001 0.860 0.133 15 κ 0.154 0.055 0.016 -0.029 1.178 0.710 16 κ 0.069 0.052 0.035 -0.010 0.967 0.523 17 κ 0.140 0.053 0.005 0.002 1.142 0.551 18 κ 0.120 0.039 -0.031 -0.016 1.517 0.643 19 ρ 0.918 0.078 -0.009 -0.013 1.150 0.852 1 ρ 0.912 0.076 0.013 -0.009 0.966 0.235 2 ρ 0.978 0.020 0.009 0.003 1.279 0.037 3 ρ 0.914 0.108 0.050 0.020 0.987 0.865 4 ρ 0.891 0.107 -0.006 0.012 1.452 0.802 5 ρ 0.912 0.085 -0.001 0.022 1.568 0.432 6 ρ 0.985 0.014 -0.007 -0.003 1.124 0.752 7 ρ 0.984 0.013 -0.014 0.017 1.410 0.259 8 ρ 0.939 0.106 0.041 -0.013 1.048 0.522 9 ρ 0.917 0.075 -0.004 -0.021 1.368 0.548 10 ρ 0.963 0.028 0.037 -0.024 0.993 0.003 11 ρ 0.827 0.149 0.080 -0.005 0.949 0.829 12 ρ 0.619 0.190 -0.006 -0.040 1.368 0.635 13 ρ 0.694 0.357 0.235 -0.019 0.496 0.709 14 ρ 0.961 0.035 0.025 0.025 0.808 0.963 15 ρ 0.936 0.057 -0.019 0.059 1.138 0.945 16 ρ 0.943 0.085 0.012 0.022 1.143 0.364 17 ρ 0.897 0.105 0.023 -0.015 1.106 0.327 18 ρ 0.983 0.015 0.025 -0.007 0.941 0.446 19 Notes: Statisticsare based on 2,000draws from the Gibbs sampler. Mean: average across draws; Sd.: standard deviationacrossdraws;auto1: firstorderautocorrelationacrossdraws;auto50: fiftiethorderautocorrelationacross draws;RNE:relativenumericalefficiencyusinga4%taper;p-value: p-valueofthenullhypothesisthatthemean ofthefirst20%ofdrawsisequaltothemeanofthelast50%ofdrawsusinga4%taperforthestandarderror. 41

Table 8: Posterior Distribution Estimates and Convergence Diagnostics Parameter Mean Sd. auto1 auto50 RNE p-value σ2 0.487 0.088 -0.037 0.018 1.597 0.199 ζ1 σ2 0.449 0.082 0.033 0.014 1.265 0.301 ζ2 σ2 0.904 0.182 -0.006 0.001 1.250 0.900 ζ3 σ2 2.672 0.557 -0.014 -0.003 1.175 0.060 ζ4 σ2 0.830 0.168 0.025 0.003 1.582 0.987 ζ5 σ2 0.393 0.070 -0.015 0.019 0.668 0.561 ζ6 σ2 0.644 0.124 -0.063 0.006 1.257 0.654 ζ7 σ2 1.932 0.437 0.004 0.081 0.959 0.029 ζ8 σ2 7.364 2.892 0.488 0.062 0.271 0.112 ζ9 σ2 0.479 0.087 0.023 0.049 1.109 0.400 ζ10 σ2 3.088 0.636 0.020 -0.052 1.564 0.287 ζ11 σ2 1.545 0.358 0.043 -0.015 1.088 0.379 ζ12 σ2 1.803 0.409 0.001 0.016 1.459 0.852 ζ13 σ2 1.763 0.587 0.174 -0.016 0.508 0.259 ζ14 σ2 0.498 0.092 -0.037 -0.048 0.793 0.434 ζ15 σ2 0.648 0.126 -0.002 0.020 0.758 0.005 ζ16 σ2 1.400 0.313 0.021 0.027 1.042 0.837 ζ17 σ2 0.613 0.120 -0.014 -0.005 0.855 0.005 ζ18 σ2 0.495 0.096 -0.003 0.007 1.093 0.662 ζ19 σ2 0.065 0.027 -0.016 -0.000 0.993 0.048 ηy 1 σ2 0.061 0.023 0.039 0.041 1.090 0.513 ηy 2 σ2 0.130 0.074 0.030 -0.011 0.958 0.394 ηy 3 σ2 0.217 0.194 0.046 -0.002 0.722 0.004 ηy 4 σ2 0.100 0.053 0.010 0.022 1.014 0.352 ηy 5 σ2 0.056 0.021 0.021 -0.014 1.128 0.452 ηy 6 σ2 0.077 0.033 0.021 -0.019 0.967 0.200 ηy 7 σ2 0.401 0.261 0.051 -0.012 1.021 0.333 ηy 8 σ2 2.044 2.831 0.622 0.077 0.264 0.284 ηy 9 σ2 0.068 0.029 0.003 0.016 0.807 0.012 ηy 10 σ2 0.263 0.262 0.014 0.000 1.527 0.443 ηy 11 σ2 0.163 0.126 -0.003 0.001 1.326 0.921 ηy 12 σ2 0.318 0.227 -0.007 -0.017 2.000 0.962 ηy 13 σ2 0.600 0.464 0.274 0.024 0.670 0.565 ηy 14 σ2 0.063 0.024 0.001 -0.001 1.262 0.878 ηy 15 σ2 0.083 0.039 0.004 -0.022 1.122 0.921 ηy 16 σ2 0.272 0.227 0.022 -0.009 1.326 0.717 ηy 17 σ2 0.085 0.041 0.009 -0.018 0.807 0.323 ηy 18 σ2 0.085 0.041 0.018 0.023 1.103 0.057 ηy 19 Notes: Statisticsare based on 2,000draws from the Gibbs sampler. Mean: average across draws; Sd.: standard deviationacrossdraws;auto1: firstorderautocorrelationacrossdraws;auto50: fiftiethorderautocorrelationacross draws;RNE:relativenumericalefficiencyusinga4%taper;p-value: p-valueofthenullhypothesisthatthemean ofthefirst20%ofdrawsisequaltothemeanofthelas4t250%ofdrawsusinga4%taperforthestandarderror.

Table 9: Posterior Distribution Estimates and Convergence Diagnostics Parameter Mean Sd. auto1 auto50 RNE p-value σ2 0.056 0.011 0.005 -0.027 1.109 0.800 ηu 1 σ2 0.179 0.032 0.021 -0.029 0.738 0.460 ηu 2 σ2 0.217 0.043 -0.012 0.026 0.980 0.816 ηu 3 σ2 0.396 0.082 0.017 0.002 0.810 0.247 ηu 4 σ2 0.024 0.005 0.006 -0.010 1.189 0.078 ηu 5 σ2 0.025 0.005 -0.003 -0.026 0.718 0.962 ηu 6 σ2 0.032 0.007 0.024 -0.000 1.256 0.747 ηu 7 σ2 0.089 0.031 0.043 0.003 0.998 0.099 ηu 8 σ2 0.139 0.041 0.085 0.009 0.964 0.297 ηu 9 σ2 0.070 0.013 0.009 -0.001 0.881 0.020 ηu 10 σ2 0.477 0.108 0.017 -0.019 0.672 0.156 ηu 11 σ2 0.378 0.114 0.180 0.012 0.585 0.650 ηu 12 σ2 0.039 0.007 0.009 0.014 1.255 0.919 ηu 13 σ2 0.148 0.027 -0.026 0.034 1.234 0.157 ηu 14 σ2 0.027 0.006 -0.003 -0.006 1.164 0.490 ηu 15 σ2 0.128 0.025 -0.019 0.022 1.000 0.661 ηu 16 σ2 0.167 0.042 0.038 0.000 0.717 0.748 ηu 17 σ2 0.130 0.024 0.042 0.032 0.768 0.395 ηu 18 σ2 0.134 0.038 0.059 -0.002 1.224 0.931 ηu 19 σ2 1.981 0.342 -0.021 0.003 1.399 0.679 ηπ 1 σ2 3.189 0.561 -0.035 -0.010 1.261 0.861 ηπ 2 σ2 10.123 1.758 -0.030 -0.022 0.849 0.738 ηπ 3 σ2 9.089 1.607 0.026 -0.011 1.008 0.085 ηπ 4 σ2 2.627 0.467 -0.005 -0.000 0.763 0.225 ηπ 5 σ2 1.874 0.331 -0.003 -0.012 0.685 0.694 ηπ 6 σ2 1.895 0.327 0.010 0.011 0.932 0.214 ηπ 7 σ2 3.141 0.539 -0.020 -0.001 0.804 0.906 ηπ 8 σ2 4.167 0.782 0.014 -0.001 1.380 0.372 ηπ 9 σ2 1.453 0.258 -0.054 -0.012 1.481 0.850 ηπ 10 σ2 9.731 1.775 -0.021 -0.039 1.359 0.096 ηπ 11 σ2 7.484 1.367 0.042 0.016 1.257 0.208 ηπ 12 σ2 6.782 1.170 0.013 0.016 1.301 0.504 ηπ 13 σ2 4.239 0.735 -0.033 0.035 1.017 0.765 ηπ 14 σ2 2.099 0.385 0.020 -0.005 0.805 0.264 ηπ 15 σ2 4.166 0.746 0.013 -0.014 1.091 0.421 ηπ 16 σ2 9.164 1.599 -0.011 -0.005 0.869 0.085 ηπ 17 σ2 7.636 1.344 0.011 0.028 1.004 0.066 ηπ 18 σ2 4.822 0.818 -0.001 -0.004 1.077 0.550 ηπ 19 Notes: Statisticsare based on 2,000draws from the Gibbs sampler. Mean: average across draws; Sd.: standard deviationacrossdraws;auto1: firstorderautocorrelationacrossdraws;auto50: fiftiethorderautocorrelationacross draws;RNE:relativenumericalefficiencyusinga4%taper;p-value: p-valueofthenullhypothesisthatthemean ofthefirst20%ofdrawsisequaltothemeanofthelast50%ofdrawsusinga4%taperforthestandarderror. 43

The confidence sets for the real GDP growth forecast in both specifications are wider than in the RW case due to the time-varying nature of the expected growth rate of potential output and both cover almost everywhere the observed series. The unemployment rate forecasts are not very different across the three specifications, but the inflation rate forecast is. In particular, at the beginning of the forecasting exercise, the densities of the inflation rate forecasts cover negative grounds. G Unobserved Components Models with Aggregate Data This appendix describes the UC models with aggregate data and the results obtained from their estimation over the same sample period. The model is given by the following equations: y = τy +c , t t t u t = τ t u +θ 1 c t +θ 2 c t−1 , π t = β 0 +β 1 π t e +(1−β 1 )π t−1 +κc t +η t π, τu = τu +ηu, t t−1 t c t = φ 1 c t−1 +φ 2 c t−2 +ε t , with trend output following one of the three specifications listed below: • RW: τy = µ+τy +ηy, t t−1 t • LLT: τ t y = µ t−1 +τ t y −1 +η t y, µ t = µ t−1 +ν t , • IRW: τ t y = µ t−1 +τ t y −1 , µ t = µ t−1 +ν t , and where the error terms ε , ηy, ηu, and ν are assumed to be white noise, uncorrelated t t t t with each other, and normally distributed. The prior distributions of the coefficients are similar to those in the specification with data at the country level. However, in this case it is possible to estimate the variances of the aggregate cycle and trend shocks, σ2, σ2 , and σ2 . The sources for the priors, as before, ε ηy ηu are the results in Lenza and Jarocin´ski (2016) for real GDP and the unemployment rate and Blanchard, Cerutti and Summers (2015) for inflation. Prior distributions and posterior distribution results appear in Tables 10-12. Convergence diagnostics (not shown) indicate the Gibbs sampler has reached a stable posterior distribution. The estimated output gaps appears in Figure 19. Regardlessofthechoiceofthespecificationoftheoutputtrend, thethreemodelsestimate that an output gap emerged starting in 2009, when the global financial crisis hit, and reached between negative 2 percent and negative 3 percent around 2010. The double-dip recession consequence of the European debt crisis causes the output gap to become even further negative around 2013. However, the model with a RW output trend reaches only about negative 5 percent, compared with about negative 7 percent to negative 8 percent for the LLT and IRW specifications. At the end of the sample period, all three specifications yield estimated output gaps close to zero. 44

Figure 17: Model Forecasts and Observed Variables (LLT) 3% 90% confidence set Euro-area real GDP growth t/t-1 2% Model forecast four quarters ahead 1% 0% -1% -2% 2010 2011 2012 2013 2014 2015 2016 2017 14% 13% 12% 11% 10% 9% 8% 90% confidence set 7% Euro-area unemployment rate Model forecast four quarters ahead 6% 2010 2011 2012 2013 2014 2015 2016 2017 4% 90% confidence set 3% Euro-area inflation rate t/t-4 Model forecast four quarters ahead 2% 1% 0% -1% -2% -3% 2010 2011 2012 2013 2014 2015 2016 2017 Note: Thefour-quarters-aheadforecastisthemedianofthemodelforecastacrossdrawsobtainedfromtheDurbin andKoopman(2002)simulationsmoother. Source: Eurostatandauthor’scalculations. 45

Figure 18: Model Forecasts and Observed Variables (IRW) 3% 90% confidence set Euro-area real GDP growth t/t-1 2% Model forecast four quarters ahead 1% 0% -1% -2% 2010 2011 2012 2013 2014 2015 2016 2017 14% 13% 12% 11% 10% 9% 8% 90% confidence set 7% Euro-area unemployment rate Model forecast four quarters ahead 6% 2010 2011 2012 2013 2014 2015 2016 2017 4% 90% confidence set 3% Euro-area inflation rate t/t-4 Model forecast four quarters ahead 2% 1% 0% -1% -2% -3% 2010 2011 2012 2013 2014 2015 2016 2017 Note: Thefour-quarters-aheadforecastisthemedianofthemodelforecastacrossdrawsobtainedfromtheDurbin andKoopman(2002)simulationsmoother. Source: Eurostatandauthor’scalculations. 46

Table 10: Parameter Prior Distributions and Posterior Distribution Results (RW) Parameter Prior Distribution Prior Mean Prior SD Posterior Mean [5th,95th] φ Truncated Normal 1.7 0.2 1.74 [1.61,1.87] 1 φ Truncated Normal -0.75 0.2 -0.78 [-0.91,-0.63] 2 µ Normal 0.5 0.5 0.30 [0.10,0.50] θ Normal -0.2 0.2 -0.22 [-0.32,-0.12] 1 θ Normal -0.2 0.2 -0.21 [-0.31,-0.11] 2 β Normal 0 0.5 0.11 [-0.22,0.46] 0 β Truncated Normal 0.5 0.5 0.23 [0.05,0.41] 1 κ Truncated Normal 0.1 0.1 0.11 [0.03,0.19] σ2 Inverse Gamma 0.1 Inf 0.11 [0.06,0.17] ε σ2 Inverse Gamma 0.1 Inf 0.10 [0.06,0.16] ηy σ2 Inverse Gamma 0.1 Inf 0.01 [0.01,0.02] ηu σ2 Inverse Gamma 0.25 Inf 1.48 [1.13,1.96] ηπ σ2 Inverse Gamma 1 Inf — — ν According to the forecasting results in Table 1, the three output trend specifications perform almost equally well with respect to the inflation rate. However, the RW specification forecasts real GDP growth significantly better than theother two specifications, while it does not perform significantly worse with respect to the unemployment rate. Hence, I pick the random walk specification for the output trend as the benchmark model and interpret the results according to this choice, which are shown in Table 10. The posterior mean estimate of the drift of the output trend indicates that the average growth rate of the euro-area potential GDP would be around 1.2 percent per year, almost exactly the same value as with country-level data. Taking the estimates of the AR(2) coefficients of the common cycle at their posterior mean, one can estimate the period of the cycle, which would be about 11 years for the euro area as whole. The Okun’s law coefficients are in line with those usually cited in the literature, in which the sum of the two coefficients would be around -0.5 (see Abel, Bernanke and Croushore, 2013, for example), implying that for every percentage point increase in with respect to potential, the unemployment rate would be reduced a little more than 0.4 percentage point. The posterior mean estimate of the inflation expectations anchoring coefficient of the hybrid Phillips curve implies that realized inflation would be about 0.5 percentage point above expected inflation in the long run under a zero output gap scenario. The estimated slope of the hybrid Phillips curve indicates that inflation would increase about 0.1 percentage point for every percentage point increase in output with respect to potential. This estimate is well in line with the results in the literature (see Blanchard, Cerutti and Summers, 2015, for example). In general, the estimated residual variance of the inflation equation is the largest, indicating that forecasting inflation for these type of models can be harder, relatively speaking, than forecasting the other two variables of interest. 47

Table 11: Parameter Prior Distributions and Posterior Distribution Results (LLT) Parameter Prior Distribution Prior Mean Prior SD Posterior Mean [5th,95th] φ Truncated Normal 1.7 0.2 1.74 [1.53,1.88] 1 φ Truncated Normal -0.75 0.2 -0.76 [-0.90,-0.57] 2 µ Normal 0.5 0.5 — — θ Normal -0.2 0.2 -0.23 [-0.35,-0.13] 1 θ Normal -0.2 0.2 -0.16 [-0.28,-0.06] 2 β Normal 0 0.5 0.13 [-0.18,0.48] 0 β Truncated Normal 0.5 0.5 0.24 [0.05,0.44] 1 κ Truncated Normal 0.1 0.1 0.09 [0.02,0.17] σ2 Inverse Gamma 0.1 Inf 0.14 [0.06,0.27] ε σ2 Inverse Gamma 0.1 Inf 0.03 [0.01,0.06] ηy σ2 Inverse Gamma 0.1 Inf 0.01 [0.01,0.02] ηu σ2 Inverse Gamma 0.25 Inf 1.49 [1.13,1.96] ηπ σ2 Inverse Gamma 1 Inf 0.12 [0.08,0.18] ν Table 12: Parameter Prior Distributions and Posterior Distribution Results (IRW) Parameter Prior Distribution Prior Mean Prior SD Posterior Mean [5th,95th] φ Truncated Normal 1.7 0.2 1.74 [1.57,1.87] 1 φ Truncated Normal -0.75 0.2 -0.76 [-0.90,-0.59] 2 µ Normal 0.5 0.5 — — θ Normal -0.2 0.2 -0.21 [-0.32,-0.13] 1 θ Normal -0.2 0.2 -0.16 [-0.25,-0.07] 2 β Normal 0 0.5 0.14 [-0.20,0.50] 0 β Truncated Normal 0.5 0.5 0.24 [0.06,0.44] 1 κ Truncated Normal 0.1 0.1 0.08 [0.02,0.16] σ2 Inverse Gamma 0.1 Inf 0.16 [0.08,0.28] ε σ2 Inverse Gamma 0.1 Inf — — ηy σ2 Inverse Gamma 0.1 Inf 0.01 [0.01,0.02] ηu σ2 Inverse Gamma 0.25 Inf 1.49 [1.11,1.99] ηπ σ2 Inverse Gamma 1 Inf 0.13 [0.08,0.18] ν 48

Figure 19: SmoothedEstimateoftheEuro-AreaOutputGapUsingAggregate Data for Three Specifications of the Output Trend 5 4 3 2 1 0 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 2002 2004 2006 2008 2010 2012 2014 2016 PDG laitnetop morf noitaived % 5 4 3 2 1 0 -1 -2 -3 -4 -5 -6 Random Walk -7 Local Linear Trend -8 Integrated Random Walk -9 -10 Note:TimeseriesaretheaveragesoftheposteriordrawsoftheeuroareaoutputgapusingtheDurbinandKoopman (2002)simulatorsmoother. 49

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