Measuring the Euro Area Output Gap

Finance and Economics Discussion Series Federal Reserve Board, Washington, D.C. ISSN 1936-2854 (Print) ISSN 2767-3898 (Online) Measuring the Euro Area Output Gap Matteo Barigozzi, Claudio Lissona, and Matteo Luciani 2024-099 Please cite this paper as: Barigozzi, Matteo, Claudio Lissona, and Matteo Luciani (2024). “Measuring the Euro Area Output Gap,” Finance and Economics Discussion Series 2024-099. Washington: Board of Governors of the Federal Reserve System, https://doi.org/10.17016/FEDS.2024.099. 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.

Measuring the Euro Area Output Gap∗ Matteo Barigozzi Claudio Lissona Matteo Luciani University of Bologna University of Bologna Federal Reserve Board matteo.barigozzi@unibo.it claudio.lissona2@unibo.it matteo.luciani@frb.gov Last update: October 29, 2024 Abstract WemeasuretheEuroArea(EA)outputgapandpotentialoutputusinganon-stationarydynamic factor model estimated on a large dataset of macroeconomic and financial variables. From 2012 to 2023, we estimate that the EA economy was tighter than the European Commission and the International Monetary Fund estimate, suggesting that the slow EA growth is the result of a potential output issue, not a business cycle issue. Moreover, we find that credit indicators are crucial for pinning down the output gap, as excluding them leads to estimating a lower output gap in periods of debt build-up and a higher gap in periods of deleveraging. 1 Introduction The decomposition of GDP in potential output and the output gap—defined as the percentage deviation of GDP from its potential—is a fundamental task for policymakers. These quantities are essential for the common Euro Area (EA) monetary policy and the fiscal policy of individual countries. The output gap helps assess the cyclical position of the economy and, thus, potential inflationary pressures (e.g., Ban´bura and Bobeica, 2023). Potential output and the output gap are among the main pillars of the EA fiscal surveillance framework, ultimately affecting the fiscal capacity of each member country (European Commission, 2018). However, since both quantities are unobserved, policymakers need a model that can extract them from the data. This paper proposes a new measure of potential output and the output gap for the EA based on a non-stationary dynamic factor model estimated on a large dataset of EA macroeconomic and financial variables that incorporates relevant EA data features. Compared to the prevailing literature, which focuses on theoretical structural models with few variables of interest, we adopt a ∗WewouldliketothankforhelpfulcommentTravisBerge,DaniloCascaldiGarcia,AntonioConti,ThiagoFerreira, Manuel Gonzalez-Astudillo, Michele Lenza, Giovanni Pellegrino, and Riccardo Trezzi. This paper has benefited also fromdiscussionswithseminarparticipantsattheFederalReserveBoardandwithparticipantsofseveralconferences. M.BarigozziandC.LissonagratefullyacknowledgefinancialsupportfromMIUR(PRIN2020,Grant2020N9YFFE). Of course, any error is our responsibility. Disclaimer: the views expressed in this paper are those of the authors and do not necessarily reflect the views and policies of the Board of Governors or the Federal Reserve System.

distinct approach because we let the data speak by leveraging a large information set conditional on a few key macroeconomic priors—for example, the long-run slowdown in output growth (Cette et al., 2016). There are at least two reasons why using a rich information set is crucial when estimating the output gap. First, cross-sectional aggregation of a large number of series allows us to consistently disentangle the co-movements in the data from idiosyncratic dynamics. Second, it is now widely recognized that potential output and the output gap, along with many other macroeconomic latent policy variables, commonly referred to as “stars,” are driven by a large number of shocks; hence, we need a large information set to obtain meaningful estimates of these quantities (Buncic and Pagan, 2022). To the best of our knowledge, this is the first paper that estimates the EA output gap with such a rich information set. We conduct our analysis on a new large-dimensional dataset comprising 119 EA economic indicators from 2001:Q1-2023:Q4. Three main results emerge from our analysis: first, our output gap estimate is in line with those published by the European Commission (EC) and the International Monetary Fund (IMF) in that the dating of the turning points perfectly coincides. The three estimates closely align until the 2011–2012 Sovereign Debt Recession (henceforth, SDR), after which our output gap measure suggests that the EA economy was tighter than estimated by the EC and the IMF. Moreover, we estimate that potential output growth decelerated after the 2008–2009 Global Financial Crisis (henceforth, GFC), the SDR further compounded this deceleration, and as of the end of 2023, potential output growth has yet to return to the pre-GFC pace. In other words, our results suggest that the EA has a potential output issue, not a business cycle issue. Hence, if the goal is to achieve better economic conditions in the EA, European countries should implement supply-side structural reforms that have long-run effects, while policies aiming at stimulating aggregate demand will have only short-term effects at best. Second, we find that our model takes considerable signal for estimating the output gap from labor market and inflation variables, as, on average, it satisfies the Okun’s law relationship and exhibits Phillips correlation, despite the fact we do not impose either of them. Moreover, we find that our output gap measure better predicts inflation than those obtained using commonly adopted methods. These results confirm that our data-driven measure is economically meaningful. Third, in line with the argument of Borio et al. (2017), who argue that incorporating financial indicators in the dataset is necessary to obtain meaningful estimates of the business cycle, we find that credit indicators (mainly household liabilities) are crucial for pinning down the cyclical positionoftheeconomy. Specifically, wefindthatexcludinghouseholdliabilitiesleadstoestimating a lower output gap in periods of debt build-up (e.g., before the GFC) and a higher gap in periods of deleveraging (e.g., from 2013 to 2016), as the boost to GDP from increased leveraging is offset in periods of deleveraging. In other words, our results show that growth financed through household debt is not sustainable in the long run. To measure potential output and the output gap, we first estimate the non-stationary dynamic factor model by Quasi Maximum Likelihood using the EM algorithm (Doz et al., 2012; Barigozzi and Luciani, 2024). Then, we extract a common trend from the estimated common factors running a second EM algorithm compute the cyclical component by subtracting the common trend from the common factors. In practice, however, to account for the Covid shock, we run this procedure twice, 2

as we first estimate the model using only pre-Covid data. Then, we estimate the effects induced by the Covid shock (level-shift and increased volatility). Lastly, we re-estimate the model on the full dataset after purging the data from Covid-induced dynamics. Having estimated the common trend and the common cyclical component, we measure potential output as the part of GDP explained by the common trends and the output gap as the part of GDP explained by the common cyclical component. How to estimate potential output and the output gap has been a hotly debated topic for several decades. The literature has proposed two main approaches: a theoretical approach and a statistical approach. The theoretical approach uses theoretical models, such as production-function-based models used by the EC (Havik et al., 2014) and the IMF (De Masi, 1997) or New-Keynesian DSGE models (Justiniano et al., 2013; Burlon and D’Imperio, 2020; Furlanetto et al., 2021). The statisticalapproachuses(univariateormultivariate)statisticalmodels, sometimespairedwithsome macroeconomic relationships of interest, e.g., the Phillips curve. For example, many papers rely on univariate models (e.g., Morley et al., 2003; Kamber et al., 2018; Hamilton, 2018; Phillips and Jin, 2021;PhillipsandShi,2021;Hartletal.,2022),whilefewothersemploymultivariatenon-stationary methods, whichareeitherlow-dimensionalmodels(Jarocin´skiandLenza,2018;Gonz´alez-Astudillo, 2019; T´oth, 2021; Hasenzagl et al., 2022), or medium-size, but stationary, models (Aastveit and Trovik, 2014; Morley and Wong, 2020; Morley et al., 2023). Most of these works focus on the US, while only a few of them focus on the EA, the most recent being Morley et al. (2023). Our model is large-dimensional and non-stationary, thus allowing us to capture well-established co-movements in macroeconomic variables while retaining data in levels. Moreover, being an enhanced version of the model Barigozzi and Luciani (2023) used to measure the US output gap— enhanced because it does not require a long sample to identify the trend and deals with the Covid pandemic—our model follows the statistical approach and belongs to the class of unobserved component models. This class of models is among the most indicated ones for extracting the transitory component of GDP (Canova, 2022). The rest of the paper is structured as follows. In Section 2, we present the data used. In Section 3, we present the model, the estimation strategy, and the model setup. Section 4 presents the main results. Next,Sections5and6shedlightonfromwhichvariablesthemodeltakesmoresignalforthe output gap estimation, with Section 5 focusing on labor market and inflation variables, and Section 6 on credit indicators. In Section 7, we assess the ability of our output gap estimate to forecast inflation. Section 8 concludes. This paper also contains an online appendix providing additional details and robustness analysis. Specifically, Appendix A provides full details on the dataset, while Appendixes B, D, and E provide additional details about the model. Next, Appendixes F, H, and I provide robustness analysis. Lastly, Appendixes C, G, and J provide additional results that did not make the cut to be in the main text. 2 A large Euro Area dataset We construct a large macroeconomic dataset of n = 119 EA series, observed from 2001:Q1 to 2023:Q4, (T = 92). The dataset contains a wide range of macroeconomic indicators, including national account statistics, industrial production and turnover indicators, labor market and compensation indicators, price indexes, oil prices, natural gas prices, house prices, exchange rates, 3

interest rates, a stock market index, monetary aggregates, non-financial assets and liabilities, and confidence indexes. In terms of broad categories of data, we include in the dataset the usual suspects normally considered for high-dimensional macroeconomic analysis (see, for example, McCracken and Ng, 2016, 2020). Intermsofwhichandhowmanyseriestoincludeforeachcategory,weuseamixofeconomicand statistical reasoning. On the one hand, to identify the common factors driving the co-movement in the data, it is crucial to pool information from many indicators; hence, a larger information set should be preferred. On the other hand, a key assumption of the model is the presence of mild cross-sectional correlation among the idiosyncratic components: violating this assumption leads to a deterioration of the model’s performance (Boivin and Ng, 2006; Luciani, 2014). Thus, when building a dataset for factor analysis, we face a trade-off between the need for a larger information set and the risk of introducing too much idiosyncratic correlation. For this reason, we selected the variables to include in the dataset to maximize economic signal while limiting the noise, with exceptions motivated by economic reasoning. For instance, we include GDP and its components since their informational content justifies a relatively high level of idiosyncratic correlations. Similarly, consumption and employment are decomposed according to durability and sectoral composition, respectively, while assets and liabilities are decomposed by ownership. In contrast, we keep the consumer price index for energy while dropping the producer price index for energy, as they carry the same signal (their correlation is greater than 0.95). Likewise, we drop the consumer price index for industrial goods because it has a correlation greater than 0.95 with the goods consumer price index. As for the treatment of the series, we take logarithms for all variables except for confidence indicators and those already expressed in percentage points. We keep all variables in levels except for price indicators, for which we take first differences; i.e., we work with inflation rates. This is a common approach used in the literature to avoid spurious dynamics resulting from the potential I(2)behaviorinpriceindexes(StockandWatson,2016;McCrackenandNg,2016,2020). Appendix A provides the complete list of variables included in the dataset, along with their sources and treatment. 3 Methodology In Section 3.1, we outline the model and its main features—we discuss all the details, formal assumptions, and further comments in Appendix B. In Section 3.2, we sketch how we estimate the model, while referring the reader to Appendix D for a step-by-step guide on how estimation is carried out in practice, and to Appendix E for the bootstrap procedure used to measure uncertainty around our estimates. Finally, in Section 3.3 we discuss the model setup. 3.1 The model We denote the observed i-th time series at a given quarter t as y , with 1 ≤ i ≤ 119 and 2001 : it Q1 ≤ t ≤ 2023 : Q4. In our non-stationary dynamic factor model, each variable is the sum of: (i) a secular component D , which is treated either as deterministic or stochastic, (ii) q common factors it f = (f ···f )′, which capture the macroeconomic long- and short-run co-movements and have a t 1t qt 4

dynamics governed by a VAR, (ii) an idiosyncratic component ξ , which captures local dynamics it or measurement errors and is possibly correlated across i and t. We partition the n series according to two features. First, according to the nature of the secular component, that is whether D is a stochastic or a deterministic process. Second, according to the it nature of the idiosyncratic component, that is whether ξ has a stochastic trend or it is stationary. it In particular, we model D as a local linear trend for GDP to account for the well-documented it slowdown in productivity (Cette et al., 2016) and for households’ financial liabilities (HHLB) and households’ long-term loans (HHLB.LLN), which are more than 80% of total household’s liabilities, whose average growth rate has slowed down consistently since the GFC. Moreover, we model D it as a local level model for the unemployment rate (UNETOT) to account for relevant labor market features, such as the reallocation of employees across sectors, which contributed to the slowdown in the EA productivity growth (Cette et al., 2016), and for overall and core inflation (HICPOV and HICPNEF, respectively) to account for the slowdown in inflation occurred after the GFC.1 For all other series, D is either a linear trend with a constant slope, in which case we say that i ∈ I , or it b D is just a constant equal to D . it i0 As for the idiosyncratic components, if ξ ∼ I(1), then we say that i ∈ I and we model ξ as a it 1 it random walk, while if ξ ∼ I(0), we say that i ∈/ I and we leave its dynamics unspecified to avoid it 1 over-parametrization of the model. Furthermore, we capture the effect of the Covid shock, which generated a large shift both in the levels(Ng,2021;Marozetal.,2021),andinthevolatility(Carrieroetal.,2022;LenzaandPrimiceri, 2022) of most macroeconomic EA series, through an additional common factor g (Maroz et al., t 2021), and a scalar s scaling the conditional volatility of the latent factors (Lenza and Primiceri, t 2022). While the former has an impact on all series only in 2020 and 2021, the latter has an effect that persists even after the recovery from the pandemic. This parametrization is different from the one normally used for the US, where the Covid factor’s effect persists only through the summer of 2020(Marozetal.,2021),andtheincreaseinvolatilitylastsforashorteramountoftime(Lenzaand Primiceri, 2022). We use this different parametrization because mobility restrictions and lockdowns intheEAhavebeenonandoffuntilearly2022, whileintheUSwereenforcedonlyatthebeginning of the pandemic. Formally, the model reads as follows: y it = D it +λ′ i f t +γ i g t I 2020:Q1≤t≤2021:Q4 +ξ it , 1 ≤ i ≤ 119, 2001:Q1 ≤ t ≤ 2023:Q4, (1) D = D +b I +ϵ , ϵ i. ∼ i.d. (0,σ2I ), (2) it it−1 i,t−1 i∈I b it it ϵi i=UNETOT,HICPOV,HICPNEF b = b +η , η i. ∼ i.d. (0,σ2 I ), (3) it it−1 it it ηi i=GDP,HHLB,HHLB.LLN p−1 f = X A f +{s I +I )}u , u i. ∼ i.d. (0,Σ ), (4) t j t−j t t≥2020:Q1 t<2020:Q1 t t u j=0 ξ = ξ I +e , e ∼ (0,σ2), (5) it it−1 i∈I1 it it ei where I = 1, if A is true, and I = 0, otherwise. A A Furthermore, we assume that one common trend, τ , which we model as a random walk, drives t 1Appendix I shows robustness results when removing the time-variation in the secular trends. 5

the non-stationarity in the factors f (Del Negro et al., 2007). We remain agnostic on the law of t motion of the residual, i.e., the stationary cyclical component, which we denote as ω . Specifically, t we consider the decomposition: f = ψτ +ω , ω ∼ (0,Σ ), (6) t t t t ω τ = τ +ν , ν ∼ (0,σ2). (7) t t−1 t t ν Wecanthenidentifythecommontrendτ byproperlyinitializingitsvarianceσ2,whichisanalogous t ν tocontrollingthesignal-to-noiseratio. ThisapproachisequivalenttoMorleyetal.(2023)approach, as it has the same goal (smoothing the trend) but achieves it differently: we smooth the trend using theKalmansmootherproperlyinitializedbysettingσ2; Morleyetal.(2023)smooththetrendusing ν the estimated parameters of an ARIMA model. A comparison between the two approaches is in Appendix F. Inaddition,byproperlyinitializingitsvarianceσ2,ourestimationstrategyforthecommontrend ν is consistent with the results of Kim and Kim (2022), who show that extracting a common trend modeled with an unconstrained random walk component overfits GDP, thus producing a potential output that fluctuates too much and generating the so-called “pile-up” problem. Moreover, Kim and Kim (2022) show that the common practice of modeling the stationary component as an AR(2) poses identification problems since the same state-space representation can also be obtained with an ARMA(2,1) specification. For this reason, we do not impose any parametric model for ω . t The model we just described is a modified version of the model Barigozzi and Luciani (2023) (BL) used to estimate the output gap in the US. We modified BL’s model to overcome two important limitations. First, their model relies on estimating cointegrating relationships to identify the common trends, thus requiring long time series to get a reliable estimate. As such, BL’s model can be estimated only on US macroeconomic data for which more than 50 years of quarterly data are available. Second, BL estimate their model on pre-Covid data; thus, to incorporate more recent observations, some modification is needed to handle the different co-movements brought about by the Covid pandemic. In this paper, we solve both limitations by introducing (6)-(7), which we can estimate even on short samples and by incorporating recently proposed methods to handle the Covid period in the estimation strategy. Combining Equations (1) and (6), we obtain the decomposition of each observed variable: y = D +λ′ψτ +λ′ω +γ g I +ξ . it it i t i t i t 2020:Q1≤t≤2021:Q4 it Focusing on GDP, we define potential output, PO , and the output gap, OG , as: t t PO = D +λ′ ψτ , (8) t GDP,t GDP t OG = λ′ ω . (9) t GDP t Hence, in our framework, potential output is the sum of the time-varying secular trend of GDP (D ), which captures the long-run decline in EA output growth, and the part of GDP driven GDP,t by the common trend component (τ ); the output gap is the part of GDP driven by the stationary t cyclical component (ω ). What is left out are the idiosyncratic component, ξ and the Covid t GDP,t component γ g . While the idiosyncratic component is likely to be just a measurement error GDP t 6

(Aruoba et al., 2016), hence, it is clear why we are leaving it out; the exclusion of the Covid shock deserves an explanation. The Covid component represents the co-movements from 2020:Q1 to 2021:Q4 that neither potential output nor the output gap captures. In principle, this component could be allocated to the output gap, which would be equivalent to assuming that the productive capacity of the EA “froze” due to the lockdowns. While this view is commonly accepted by European institutions (Thum- Thysen et al., 2022), it is still unclear whether, and by what amount, the EA productive capacity has been affected by the Covid shock. Thus, we remain agnostic on the allocation of the Covid component between potential output and the output gap, and we will present it as a standalone component. In summary, our setup lets the data speak by leveraging a large number of variables. Moreover, it allows us to reconcile a purely statistical factor model with some key macroeconomic priors that are likely important to obtain meaningful estimates of potential output and the output gap. Lastly, it allows us to go beyond the common practice of pre-transforming data to work with stationary and centered variables, thus being able to capture essential features that would be inevitably lost if we were to difference the data to achieve stationarity (Ng, 2018). 3.2 Estimating the model To estimate the model in (1)-(5), we need to extract the latent states f , g , D , D , t t GDP,t HHLB,t D , D , D , D , and ξ (if i ∈ I ), and estimate the parameters λ , γ , HHLB.LLN,t UNETOT,t HICPOV,t HICPNEF,t it 1 i i A , s , Σ , σ2, a , and b . To do so, we use a three-step estimation procedure that we summarize j t u ei i i below. Step 1: Estimate the model up to 2019:Q4 (pre-Covid step). We obtain a preliminary estimate of the parameters using non-stationary PCAs (Bai and Ng, 2004; Barigozzi et al., 2021; OnatskiandWang,2021). Then, weruntheEMalgorithm, jointlywiththeKalmansmoother, as describedinDozetal.(2012)andBarigozziandLuciani(2024)inthehigh-dimensionalstationary case, and reconsidered in the non-stationary case by Barigozzi and Luciani (2019, 2023). Step 2: Estimate the Covid factor and volatility (Covid step). Using the parameter estimated over the pre-Covid period, we apply the Kalman filter and smoother to extract the latent states using data up to the end of the sample. To address the influence of the Covid outliers on the estimates, we truncate the Kalman Smoother at 2020:Q1 and then we continue the backward iterations by re-initializing the Kalman smoother using the Kalman filter estimate for 2019:Q4. This truncation avoids any backward spurious effect from the Covid period to the pre-Covid estimates, but creates a structural break in the estimated secular components in 2020:Q1. Forthisreason,weadjustthelevelofthesmoothedsecularcomponentsbyjudgmentally allocating the break in 2020:Q1 to the idiosyncratic component, as suggested by Ahn and Luciani (2024).2 2Analternativeapproachwecouldhavetakenconsistsinadjustingforoutliersonaseries-by-seriesbasis. However, wedidnotpursuethisoptionbecausemuchoftheadditionalvolatilityduringtheCovidshockis“economic”volatility induced by the pandemic, not measurement error (Ng, 2021). A univariate outlier adjustment method cannot distinguish between the two, so it likely removes economically relevant information. Indeed, if we do univariate outlier adjustment, we get that the Covid factor is essentially zero and that the output gap is flat as if nothing happened. 7

At this point, we have an estimate of the states over the entire sample given the information set prior to the Covid shock. This implies that the co-movements in the Covid period are left unaccounted for, and are captured by the idiosyncratic component ξ . Thus, since Covid was a it common shock affecting most of (if not all) the series in the dataset, we estimate the Covid factor g and its loadings γ by PCA on the variance-covariance matrix of the idiosyncratic component bt bi for the period 2020:Q1-2021:Q4, following Maroz et al. (2021). Lastly, we estimate the Covid volatilities s for the period 2020:Q1-2023:Q4 by maximum bt likelihood and by using the factors extracted using pre-Covid parameters. We find that s jumps bt from 1 to about 3.5 at the onset of the Covid pandemic, and then remain larger than 2 until the end of 2022, thus justifying our choice of imposing a time-varying volatility until the end of the sample. In Appendix H, we show how our measures would change if we do not model the effect of Covid explicitly, or if we use the exponential decay parametrization proposed by Lenza and Primiceri (2022). Step 3: Full sample estimation. We estimate all the parameters and latent states up to the end of the sample, by using data net of the Covid component, i.e., with y −γ g . Specifically, it bibt by using the factors estimated over the whole sample in Step 2 rescaled by s in the last part of bt the sample, we estimate the parameters, λbi , Abj , Σbu , σ be 2 i , a bi , and bb i , by maximizing the expected likelihood. Finally,withtheestimatedparametersinhand,weobtainafinalestimateofthestates, bf t , D bi,t , andξbit , throughtheKalmansmootheragaintruncatedin2020:Q1andreinitializedbefore iterating backward, as explained in Step 2. Havingestimatedthemodelparametersandunobservedstates,wecannowestimatethecommon trend. Tothisend,weestimatethestate-spacemodelin(6)-(7)usingtheEMalgorithmbyreplacing the true factors with the estimated factors. At convergence of the EM algorithm, we obtain a final estimate of the parameters, ψb , Σbω , and σ2, and using these estimates we have a final estimate of the trend τ and of the cyclical component bν bt ω bt = bf t −ψbτ bt , obtained through the Kalman smoother. Given the estimates of the common trend and the cyclical common component, we compute our final estimates of potential output and the output gap according to (8) and (9), respectively. The estimation procedure we just outlined delivers consistent estimators of all parameters and of the factors, provided that n and T grow to infinity. Furthermore, we neither have to impose the gaussianity assumption, nor we have to require uncorrelatedness of the idiosyncratic components (ξ if i ∈/ I or e if i ∈ I ) across i or t. Rather, we just have to impose mild moment conditions it 1 it 1 (see Doz et al., 2012, Bai and Li, 2016, and Barigozzi and Luciani, 2019, 2024, for details). 3.3 Setting-up the model Before estimating the model, we need to choose the number of common factors q, the number of common trends and the number of lags p in the VAR for the common factors. As for the number of lags p, we select p = 2 based on the BIC criterion for a VAR on the estimated factors. Assumptions (1)-(3), and (7) imply that the covariance matrix of the differenced data ∆y = t (∆y ···∆y )′ has at most q eigenvalues diverging as n → ∞, with all the others staying bounded. 1t nt Furthermore, the q-largest eigenvalues of the spectral density of ∆y diverge as n → ∞ at all t 8

frequencies except the zero-frequency, where the number of diverging eigenvalues corresponds to the number of common trends, while all the others stay bounded. This allows us to consistently recover the numbers of common factors and common trends via the information criteria by (Bai and Ng, 2002; Hallin and Liˇska, 2007; Barigozzi et al., 2021). We find evidence of q = 4 common factors and one common trend. Details are in Appendix C. Finally, we need to choose which idiosyncratic component to model as random walk and for which variable to include a linear trend. For the idiosyncratic component, we employ the test proposed by Bai and Ng (2004) for the null hypothesis of an idiosyncratic unit root. To determine for which variable to include a linear trend, we test the significance of the sample mean of ∆y . it The results of these tests can be found in Appendix A. 4 Potential output and output gap of the Euro Area Figure 1 presents our potential output estimate, both in 100×log levels (left plot) and in yearon-year (YoY) growth rates (right plot). The right plot also shows estimates of potential output growthfromECandIMF.ComparingourestimateswiththoseoftheECandIMFisusefulbecause our approach differs from that of these institutions, as our estimate of the output gap is primarily data-driven, whereas the EC’s and IMF’s estimates are based mainly on theoretical macroeconomic models. Specifically, both the EC and IMF derive the output gap and potential output according to the so-called “production function approach” (Kiley, 2013): after having specified a neoclassical productionfunctionendowedwithaPhillipscurveandanOkun’slaw, thetrendcomponentofGDP is extracted, and the output gap is defined as the deviation from the trend (Havik et al., 2014). Three main results emerge from Figure 1. First, potential output growth decelerated after the GFC, and the SDR further compounded this deceleration. This result aligns with the prevailing institutional perspective, according to which the GFC had a persistent negative effect on economic capacity (Thum-Thysen et al., 2022)—potential output growth was 1.9% right before the GFC, and the subsequent peak was 1.4% right before the Covid pandemic.3 Second, while the EC and IMF also estimate a slowdown in potential growth after the GFC (as shown by blue and red lines in the right plot), they do not estimate any effects of the SDR on potential growth. Third, in contrast with the GFC, the Covid recession had only a transitory effect on potential output growth, with potential growth averaging 1.6% in the second half of 2022 before slipping to 0.7% at the end of the sample. Figure 2 presents the estimated output gap, together with the estimates from the EC and the IMF.4 Our output gap estimate looks similar to those of the EC and IMF in that the dating of the turning points perfectly coincides. Moreover, the three estimates closely align until the SDR, as they all suggest a substantial overheating of the economy in the pre-GFC period, followed by a persistently negative output gap during the two recessions. However, our measure increased 3Our result is also consistent with Schm¨oller and Spitzer’s (2021) interpretation that the decline in total factor productivity resulting from the GFC induced hysteresis effects which can explain the decline in potential growth. 4In Appendix G, we show alternative estimates of the output gap based on: the HP filter, the filter by Hamilton (2018), the boosted HP filter Phillips and Shi (2021), the Butterworth filter as recommended by Canova (2022), and thelargeBayesianVARapproachbyMorleyetal.(2023)—overall,theButterworthfilterestimateoftheoutputgap seems to be the most similar to ours and the most stable one. Moreover, in Appendix J, we assess the real-time reliability of our output gap estimate. 9

Figure 1: Potential output. 100×log levels Year-on-year growth rates 1490 4 1485 2 1480 0 1475 -2 1470 GDP 1465 -4 BLL GDP EC PO IMF 1460 -6 2001 2003 2005 2007 2009 2011 2013 2015 2017 2019 2021 2023 2002 2004 2006 2008 2010 2012 2014 2016 2018 2020 2022 Notes: In all charts, the black solid line is our estimate of potential output, the grey shaded areas are the 68% and 84% confidence bands, and the dashed black line is GDP—we truncated the y-axis in the right chart for readability. In the rightchart, theblueandredlinesarethepotentialoutputestimatespublishedbytheEuropeanCommissionandtheIMF, respectively. TheIMFestimateofYoYpotentialoutputgrowthreportedintherightchartistheresultofourowncalculation. Indeed, the IMF publishes only an estimate of the output gap from which we backed out potential output. Thus, the blue lineintherightchartdoesnotaccountforanyadjustmentforCovidthattheIMFmighthavedone. after the SDR, hovering at about 2% from 2017 to the Covid pandemic, thus signaling a much tighter economy than the EC or the IMF. As we will discuss in Section 6, household liabilities, whose growth rate picked up in 2014 after declining for seven years in a row, are key drivers of the output gap after the GFC (see also Borio et al., 2017).5 This result highlights how considering many variables beyond the usual suspects (i.e., inflation indexes and labor market indicators) is important to estimate the cyclical position of the economy (Buncic and Pagan, 2022). Figure 2: Output gap. Levels Year-on-year growth rates 10 6 8 4 6 4 2 2 0 0 -2 -2 -4 -4 -6 BLL BLL -6 EC -8 EC IMF IMF -10 2001 2003 2005 2007 2009 2011 2013 2015 2017 2019 2021 2023 2002 2004 2006 2008 2010 2012 2014 2016 2018 2020 2022 Notes: The black line is our estimate of the output gap (OG) in levels (left plot) and YoY growth rates (right plot)—the leveloftheoutputgapisthepercentagedeviationfrompotential,theYoYgrowthratesisOGt−OGt−4. Eachblackmarker denotesoneyear(fourquarters),startingfrom2001:Q1. Thegreyshadedareasarethe68%and84%confidencebands. The redandbluelinesaretheoutputgapestimatespublishedbytheEuropeanCommissionandtheIMF,respectively. Figure 3 decomposes GDP growth. During the Covid pandemic, the output gap subtracted 7.3 percentage points (p.p.) from YoY GDP growth in 2020:Q2. At a YoY GDP growth rate of -15.3%, a -7.3 p.p. contribution from the output gap seems dubious. However, as explained in Section 3.2, 5In line with our result, Gambetti and Musso (2017) find that loan supply shocks have a significant effect on the EA business cycle. 10

our output gap estimate captures only standard co-movements, while the Covid factor captures the additional co-movements brought about by the Covid shock. As shown by the green bars in the right plot in Figure 3, the Covid factor accounts for an additional -5.5 p.p. of the YoY GDP plunge in 2020:Q2. Figure 3: Decomposition of GDP growth Annual year-on-year growth rate Quarterly year-on-year growth rates Notes: The black line with dot markers is GDP growth. The bars represent the contribution of each component to GDP growthrateofGDP.TheleftplotshowsYoYgrowthforthefullyear,whiletherightplotshowsYoYgrowthineachquarter. In 2021 and 2022, GDP growth was driven by the output gap and the Covid factor, while in 2023, GDP growth was sustained by potential output. However, as shown in the left plot, potential output growth reached only about half its pre-GFC average pace of around 1.6%. This result, coupled with the left plot in Figure 2 showing the output gap at 3 p.p. above potential at the end of 2023, suggests that there is a potential output issue in the EA, not a business cycle issue. Consequently, European countries should implement supply-side structural reforms with long-term effects to boost growth, as policies aimed solely at stimulating aggregate demand are likely to yield short-lived effects. 5 What about the Okun’s law and the Phillips curve? Our estimate of the output gap has a different meaning than the production-function-based model estimate. In production-function-based models, the output and unemployment gaps are related through the Okun’s law. Thus, in these models, the labor and the goods and services markets are tightly related. As a result, the unemployment gap decreases whenever the output gap increases, and vice-versa. Likewise, in these models, the output gap is related to inflation through the Phillips curve. As a result, these models take considerable signal from inflation data to estimate the output gap: the absence of inflation typically suggests a negative output gap, while high inflation indicates a positive gap; similarly, a decrease in inflation generally corresponds to a reduction in the output gap, and vice-versa. In our model (like any statistical model), there is no Okun’s law or Phillips curve, just the data. The output gap is the part of GDP driven by the common cyclical component, and as such, it is about whether or not the recent GDP growth pace is sustainable in the long run. Thus, when we compareourestimatewiththoseoftheIMFortheEC,wearenotreallycomparingapplestoapples. Nonetheless, this comparison is useful because whenever these measures differ, inspecting why they 11

differ can be very useful to better understand what happened (or is happening) in the economy. Thus, in this section and the next, we try to open the black box to understand what signals the model takes from the data and how these signals affect our estimate of the output gap. In this section, we will focus on labor market indicators (and so, the Okun’s law) and on price inflation indicators (and so, the Phillips curve); in the next section, we will focus on credit variables, which production-function-based models usually do not consider. 5.1 The Okun’s law Figure 4 shows what our output gap estimate would be if we remove all 18 labor market indicators included in our dataset. As can be seen, our output gap estimate would be much lower than our baseline estimate between 2015 and the end of the sample. Indeed, since 2012, the unemployment rate has constantly decreased from just over 12% to just under 7%, except between 2020 and 2021 during the Covid lockdowns and mobility restrictions. If not fed with this information, the model would have concluded that the economy was less tight than our benchmark estimate. Figure 4: Output gap and potential output growth when excluding labor market indicators Potential output growth Output gap 6 4 4 2 2 0 0 -2 -2 -4 -4 GDP alldata -6 alldata noEMP noEMP -6 2003 2005 2007 2009 2011 2013 2015 2017 2019 2021 2023 2001 2003 2005 2007 2009 2011 2013 2015 2017 2019 2021 2023 Notes: Theblacksolidlineisourbenchmarkestimateandthegreyshadedareasarethe68%and84%confidencebands,the black dashed line is GDP YoY growth rate, the red lines are estimates obtained by excluding labor market indicators from thedataset. To further corroborate the intuition that there is a tight relationship between our output gap estimate and labor market indicators, the left chart in Figure 5 shows that in our model, the unemployment rate gap and the output gap are negatively correlated; that is, our model captures the Okun’s law relationship in the data. As shown by the slope of the least squares fit line in the chart, on average, for every percentage point increase in the output gap, the unemployment gap decreases 0.6 p.p. Moreover, as shown by the similar slope of the pre-Covid (blue) and post- Covid(red)leastsquaresfitlines,thisrelationshiphasremainedstableovertime. Wereachasimilar conclusionevenwhenweestimateanOkun’slawregressionoftheoutputgapontheunemployment rate using an expanding window, as shown in the right chart.6 Lastly, Figure 6 shows the Generalized Impulse Response Functions (GIRFs) of the common component of the unemployment rate, GDP, potential output, and the output gap to a 1 p.p. shock 6Wealsoestimatedtherelationshipbetweentheoutputgapandhoursworkedgapandfindapositiverelationship in line with the results of Morley et al. (2023). Moreover, we find this relation to be stable in time. 12

Figure 5: Okun’s Law Unemployment rate gap vs. output gap Okun’s law (Expanding windows estimation) 4 -0.45 3 -0.5 2 -0.55 1 -0.6 0 -1 -0.65 -2 -0.7 -3 -0.75 -4 -5 -4 -3 -2 -1 0 1 2 3 4 5 2015Q4 2016Q4 2017Q4 2018Q4 2019Q4 2020Q4 2021Q4 2022Q4 2023Q4 Notes: TheleftchartshowstheOkun’slawrelationshipwiththeoutputgaponthehorizontalaxisandtheunemployment rate gap on the vertical axis. Each circle correspond to an output gap - unemployment gap pair at time t. The blue circles refertopre-Covidobservations,theredcirclesrefertopost-Covidobservations. Thedotted/blue/redlinesaretheleastsquares fitlines. The right chart shows the least squares estimate, based on an expanding window starting from 2015:Q1, of the Okun’s law slope β given by the regression (URt−DUR,t) = α+βOGt+εUR,t, where DUR,t is the time-varying mean of the unemploymentratedefinedin(2). Eachdotisanestimateofαwhilethewhiskersare±oneHACstandarderrors. to the common component of the unemployment rate (Crump et al., 2021).7 Results confirm that our model, on average, associates an unemployment rate increase with an output gap decrease. The common component of the unemployment rate remains 1 p.p. (or more) above the baseline for about a year and a half before decreasing and slowly returning to zero. In response, GDP decreases and keeps decreasing, reaching a through a year after the shock; then, it slowly returns to baseline. The model attributes most of the GDP response to movements in the output gap, while potential output slightly decreases only after a few quarters. The shock is fully absorbed in about 4 years. Figure 6: Generalized Impulse Response Functions to a shock to the unemployment rate Common component: UR Common component: GDP Potential output and output gap 0.5 0.5 1 0 0 0.8 0.6 -0.5 -0.5 0.4 -1 -1 0.2 -1.5 -1.5 PotentialOutput 0 OutputGap 0 4 8 12 16 20 0 4 8 12 16 20 0 4 8 12 16 20 Notes: Theblacksolid/dashedlinesaretheGIRFstoa1p.p. shocktothecommoncomponentoftheunemploymentrate. Themajorticksinthex-axisrepresentquartersaftertheshock. 5.2 The Phillips curve Figure7showsourestimatedoutputgapafterremovingall15inflationindicators(i.e.,priceindexes, and natural gas and the oil prices) from the dataset. As can be seen, our output gap estimate would 7Thelag-hGIRFofallvariablesisobtainedbycomputingthedifferencesbetweentheh-stepaheadforecastoftheir common component conditional on a shock to a given variable at time T +1 minus the h-step ahead unconditional forecast of the common component, i.e., when no shock is imposed. Both forecasts are computed conditional on all information available at time T (the last observation in our sample) by means of the Kalman filter (Ban´bura et al., 2015). 13

have indicated a much tighter economy after the GFC and a less tight economy in 2022 and 2023 hadwenotincludedinflationindicators. Inotherwords, ourmodelinterpretsthelowinflationafter the GFC as a signal that there is slack in the economy (see also Jarocin´ski and Lenza, 2018) and the high inflation in 2022 and 2023 as a signal that the economy is tight, even accounting for the surge in oil and natural gas prices. Figure 7: Output gap and potential output growth when excluding price indexes Potential output growth Output gap 6 4 4 2 2 0 0 -2 -2 -4 -4 GDP alldata -6 alldata noPRC noPRC -6 2003 2005 2007 2009 2011 2013 2015 2017 2019 2021 2023 2001 2003 2005 2007 2009 2011 2013 2015 2017 2019 2021 2023 Notes: Theblacksolidlineisourbenchmarkestimateandthegreyshadedareasarethe68%and84%confidencebands,the blackdashedlineisGDPYoYgrowthrate,theredlinesareestimatesobtainedbyexcludingpriceindexesfromthedataset. Tofurthercorroboratetheintuitionthatthereisarelationshipbetweenouroutputgapestimate and inflation indicators, the left plot in Figure 8 shows that in our model, the core inflation rate gap (i.e., the cyclical common component of core inflation) and the output gap are positively correlated; that is, there is Phillips correlation in the data and our model captures it. As shown by the slope of the least squares fit line in the chart, on average, for every percentage point increase in the output gap, the core inflation gap increases 4 basis points. Moreover, as shown by the difference between the slope of the pre-Covid (blue) and post-Covid (red) least squares fit lines, this correlation has increased significantly after Covid (the slope of the fit line increases from 0.017 to 0.062). The right plot in Figure 8, which shows expanding window estimates of the slope of the Phillips Curve, together with the respective standard errors, confirms that the relationship between inflation and the output gap has strengthened after Covid. Lastly, Figure 9 shows the GIRFs of the common component of core inflation, GDP, potential output, and the output gap to a 0.5 p.p. shock to the common component of core inflation. Results confirm that our model, on average, associates an increase in inflation with an output gap increase. The GIRF of the common component of core inflation peaks one quarter after the shocks before decreasing and slowly returning to zero. In response, GDP initially increases, but then after about a year, it starts decreasing, reaching a trough about 2 years after the shock—the shock is fully absorbed in 5 years. The response of potential output is negative and persistent. The output gap initially increases, then decreases, and increases again before returning to zero. 6 The role of credit indicators Borio et al. (2017) argue that incorporating financial indicators in the dataset is necessary to obtain meaningful estimates of the business cycle because credit expansions tend to overheat the economy, 14

Figure 8: Phillips Curve Core Inflation gap vs Output gap Slope of the Phillips Curve over time 0.5 0.25 0.4 0.2 0.3 0.2 0.15 0.1 0.1 0 -0.1 0.05 -0.2 0 -0.3 -5 -4 -3 -2 -1 0 1 2 3 4 5 2016Q4 2017Q4 2018Q4 2019Q4 2020Q4 2021Q4 2022Q4 2023Q4 Notes: TheleftchartshowsthePhillipsCurverelationshipwiththeoutputgaponthehorizontalaxisandthecoreinflation gapontheverticalaxis. Eachcirclecorrespondtoanoutputgap-coreinflationgappairattimet. Thebluecirclesreferto pre-Covid observations, the red circles refer to post-Covid observations. The dotted/blue/red lines are the least squares fit lines. Therightchartshowstheleastsquaresestimate, basedonanexpandingwindowstartingfrom2015:Q1, oftheslope ofthe Phillips Curve given by the following expectation-augmented specification (e.g., Conti, 2021): πt = c+αOGt+βπt−1+ γEπ t+k +επ,t, where πt is core inflation and Eπ t+k are the long-run (5-year ahead) inflation expectations in the Survey of ProfessionalForecasters. Eachdotisanestimateofαwhilethewhiskersare±oneHACstandarderrors. Figure 9: Generalized Impulse Response Functions to a shock to core inflation Common component: core HICP Common component: GDP Potential output and output gap 3 3 0.5 2 2 0.4 1 1 0.3 0 0 0.2 -1 -1 0.1 -2 -2 0 PotentialOutput -3 -3 OutputGap 0 4 8 12 16 20 0 4 8 12 16 20 0 4 8 12 16 20 Notes: The black solid/dashed lines are the GIRF to a 0.5 p.p. shock to the common component of core inflation. The majorticksinthex-axisrepresentquartersaftertheshock. particularly from the late 1990s onward. In support of Borio et al.’s results, Berger et al. (2022) find that a large share of the US economy overheating in the build-up of the financial crisis was due to financial imbalances in the credit and housing market. Moreover, Claessens et al. (2012), Ru¨nstler and Vlekke (2018), and Winter et al. (2022) find that the business and financial cycles are correlated and co-move in the medium run, thus supporting the idea of a “medium-term” business cycle (Comin and Gertler, 2006). Our dataset includes various indicators of financial conditions, including credit indicators, monetary aggregates, and house prices, totaling 26 variables. What is the role of those financial variables (and in particular, credit indicators) in our model? Does our model support the view that the financial and business cycles are correlated? Out of its intrinsic interest, answering this question is very important because it may explain why our output gap estimate differs from those of the IMF and EC, which do not consider credit indicators. As shown in the left plot in Figure 10, removing all 26 financial variables from our dataset yields significant consequences. Is this outcome primarily due to removing many variables from the dataset? Or is there a specific sub-group of financial variables that carries more weight? The 15

right plot in Figure 10 conclusively shows that household credit indicators are the primary driver of this result. Additionally, the left plot in Figure 11 clarifies that it is household liabilities that matter for the output gap estimation, while household assets do not have any significant effect. The results in Figures 10 and 11 provide further evidence that household leverage is an important driver of the business cycle, a trend that emerged in the early 2000s in many advanced economies (Mian et al., 2017). Indeed, while in the 1990s, non-financial corporations were the main driver of the financial cycle, households were behind both the pre-GFC excess leverage, which boosted households’ demand, and the subsequent deleveraging, which curtailed household’s demand (Mian and Sufi, 2018; Plagborg-Møller et al., 2020; Reichlin et al., 2020). Figure 10: Output gap excluding financial financial assets/liabilities or their sub-groups No financial assets/liabilities No financial assets/liabilities by sector 6 6 4 4 2 2 0 0 -2 -2 -4 -4 alldata noHH -6 alldata -6 noNFC noFIN noGOV 2001 2003 2005 2007 2009 2011 2013 2015 2017 2019 2021 2023 2001 2003 2005 2007 2009 2011 2013 2015 2017 2019 2021 2023 Notes: Theblacklineisourestimateoftheoutputgapinlevels,thegreyshadedareasareits68%and84%confidencebands, andthered/blue/greenlinesaretheestimatesobtainedomittingalltheinformationfrom: financialassets/liabilities(FIN); assets/liabilitiesofhouseholds(HH);assets/liabilitiesofNon-FinancialCorporations(NFC);assets/liabilitiesofGovernment (GOV). Figure 11: Potential output and output gap excluding households assets/liabilities Potential output – no liabilities Output gap – no assets and liabilities 6 4 4 2 2 0 0 -2 -2 -4 -4 GDP alldata alldata -6 noHH.liabilities noHH.assets noHH.assets -6 2003 2005 2007 2009 2011 2013 2015 2017 2019 2021 2023 2001 2003 2005 2007 2009 2011 2013 2015 2017 2019 2021 2023 Notes: In the left chart, the black line is our estimate of the output gap in levels, the grey shaded areas are its 68% and 84% confidence bands, the red line is the estimate obtained omitting the information from all liabilities of households (HH.liabilities)andthebluelineistheestimateobtainedomittingtheinformationfromallassetsofhouseholds(HH.assets). Intherightchart,theblacksolidlineisourestimateofyear-on-yearpotentialoutputgrowth,thegreyshadedareasarethe 68% and 84% confidence bands, the black dashed line is GDP YoY growth rate, and the red line is the estimate obtained omittingalltheinformationfromallliabilitiesofhouseholds(HH.liabilities). In summary, credit indicators, notably household liabilities, have a crucial role when estimating 16

the output gap—this is confirmed by the inflation forecasting exercise presented in Section 7. When we exclude these variables, in periods of debt build-up (e.g., before the GFC), we estimate a lower output gap, while in periods of deleveraging (e.g., from 2013 to 2016), we estimate a higher output gap. The scenario analysis in Figure 12 confirm this interpretation of the results in Figure 11. Figure 12: Scenario analysis Household liabilities Household liabilities (Data and counterfactual) (Conditional and unconditional forecast) GDP Potential output and output gap (Scenario dynamic effects) (Scenario dynamic effects) Notes: Intheupper-leftchart,theblacklineisthedata(in100×log-levels),andtheredlineisalinearpathstartingfrom thevalueofhouseholdliabilitiesin2003:Q1andendingin2011:Q4. Intheupper-rightchart,theblacklinearethedata,the blue line is the scenario we simulate, and the red line is the forecast of household liabilities when no alternative scenario is imposed. Inthelowercharts,theblacksolid/dashedlinesarethedynamiceffectsofthesimulatedscenario. Figure 12 shows the impact of a scenario in which household liabilities increase faster than in the baseline for about 31⁄ years—reaching a level about 61⁄ p.p. higher than in the baseline—and 2 4 then return to baseline after about 8 years. To calibrate this scenario, we looked at the difference between the actual times series of household liabilities between 2003:Q1 and 2011:Q4 against the counterfactualhadhouseholdliabilitiesgrewlinearlyinthisperiod(upper-leftchartofFigure12),8 and then smoothed it with a 5-th order degree polynomial. We then added this smoothed path to the unconditional forecast of household liabilities (the red line in the upper-right chart in Figure 8Between 2003:Q1 and 2011:Q4, household liabilities grew at an average annualized growth rate of 5.4%. This is the result of period of increased leverage between 2003 and 2008:Q1, when household liabilities grew at an average annualized growth rate of 7.6%, and a period of slower growth between 2008:Q1 and 2011:Q4, when household liabilities grew at an average annualized growth rate of 2.5%. 17

12), which gives us a path for household liabilities (blue line) conditional on which we can obtain forecasts for all the variables in the model. The lower charts in Figure 12 show the dynamic effects of this scenario on the log level of GDP and on the output gap and potential output (Crump et al., 2021).9 Specifically, GDP increases for a little over three years, reaching a peak at 3 p.p. above the baseline. Since then it starts declining and after six years turns negative. The response of the output gap mimics that of GDP, but it is a little faster. Potential output slowly increases for the first five years and then returns to the baseline. These results show that growth financed through household debt is not sustainable in the long run. 7 Has our output gap measure predictive power for inflation? The output gap is often seen as an indicator of current/future inflationary pressure (or easing). Thus, predicting inflation is considered a must-have property for any output gap measure. This is the case, even though, as opposed to production-function-based model estimates of the output gap, our measure is not directly tied to inflation and was not designed to be an inflation gauge. To assess the forecasting properties of our output gap estimate, we replicate the analysis conducted in Ban´bura and Bobeica (2023), and we employ a simple autoregressive distributed lag model: π = απ +βOG +v , (10) t+4 t t t+4 where π = 100log(P /P ) is the quarter-on-quarter inflation rate in quarter t, P is the harmot t t−1 t nized consumer price index (either headline or core), π = P4 π is year-on-year inflation in t+4 i=1 t+i quarter t + 4, and OG is the output gap. Ban´bura and Bobeica (2023) labeled model (10) the t “benchmark model,” and they show that, despite being very simple, it delivers decent forecasts compared to more complex alternative models. Our exercise compares the inflation forecast obtained using the benchmark model (10) with a forecastobtainedbyreplacingourestimateoftheoutputgapwithdifferentunivariateandmultivariatestatisticalmodels(seeAppendixGforadescriptionofthealternativemodels). Thiscomparison is carried out through an expanding window exercise, where the first window is a 60-quarter window. We look at the forecasting performance of the different output gap measures over two distinct samples: a pre-Covid sample, 2015:Q4–2019:Q4, and a post-Covid sample, 2022:Q1–2023:Q4. Table 1 compares the forecasting performance of the different output gap measures in terms of relative Root Mean Squared Error (RMSE)—numbers lower than one indicate a better forecasting performancewhenusingouroutputgapestimate. Asshowninrows(1)-(8),inthepre-Covidperiod, when inflation was low and stable, our output gap measure performed better than all the other alternatives in forecasting headline inflation and better than most alternatives when forecasting core inflation. However, in the post-Covid period, when inflation surged and then declined, our model outperformed all the other measures, sometimes substantially. Row(9)inTable1comparestheforecastingperformanceoftwodifferentestimatesoftheoutput 9The dynamic effects of the simulated scenario at a lag h on all variables are computed as the difference between theh-stepaheadforecastoftheircommoncomponentconditionalonthesimulatedpathofhouseholdliabilitiesminus theh-stepaheadunconditionalforecastofthecommoncomponent,i.e., whennoalternativepathissimulated. Both forecasts are computed conditional on all information available at time T (the last observation in our sample) by means of the Kalman filter (Ban´bura et al., 2015). 18

gap obtained with our model: the benchmark estimate and the one obtained excluding all credit indicators. The results in Table 1 add evidence to our claim that including credit variables in the dataset when estimating the output gap is crucial, as the model including credit indicators outperforms the model excluding credit indicators. Table 1: 4-quarter ahead year-over-year inflation forecasting Relative Root Mean Squared Errors 2015:Q4-2019:Q4 2022:Q1-2023:Q4 Output Gap Measure Headline Core Headline Core (1) HP Filter (λ=1600) 0.91 1.02 0.81 0.96 (2) HP Filter (λ=51200) 0.97 0.97 0.90 0.97 Hamilton Filter 0.88 0.83 0.89 0.85 (3) (4) Boosted HP Filter (λ=1600) 0.88 0.95 0.71 0.76 (5) Boosted HP Filter (λ=51200) 0.88 1.00 0.74 0.80 Christiano-Fitzgerald Filter 0.89 1.00 0.85 0.83 (6) Butterworth Filter 0.93 0.95 0.83 0.78 (7) Multivariate Beveridge-Nelson 0.90 0.97 - - (8) No household 0.95 0.98 0.93 0.94 (9) Notes: ThetableshowstherelativeRMSEofforecastingyear-on-yearinflationusing(10),whereOGt iseitherouroutputgap estimate, or an alternative output gap estimate. Our benchmark output gap estimate is always the numerator of the RMSE, thus numbers lower than 1 indicate a better forecasting performance when using our benchmark output gap estimate. Rows (1)–(8) compare our benchmark estimate with alternative models, while row (9) compares our benchmark estimate with the oneobtainedwithourmodelbutexcludingallcreditindicators. Inrow(8)forecastsareobtainedwiththeoutputgapmeasure by Morley et al. (2023) which is available only until 2021:Q3. Therefore, we only present results for the pre-Covid forecasting exercise. Figure 13: Conditional forecasts of inflation HICP: Overall HICP: Core data data 10 commoncomponent:estimationsample 6 commoncomponent:estimationsample conditionalforecast conditionalforecast 8 5 6 4 4 3 2 2 0 1 -2 0 2001 2003 2005 2007 2009 2011 2013 2015 2017 2019 2021 2023 2001 2003 2005 2007 2009 2011 2013 2015 2017 2019 2021 2023 Notes: In both charts, the black line is the quarter-on-quarter annualized inflation, the red line is the common component estimatedin-sample,andthebluelineistheconditionalforecast. Thedashedlinesarethe84%confidenceintervals. To conclude, having established that our model captures the Phillips correlation in the data and is capable of forecasting inflation, we use our model to understand how much of the inflation surge and deceleration is driven by macroeconomic co-movement. To this end, we produce Kalman filterbasedforecastsofthecommoncomponentofinflationconditionalonobservingateachpointin timeallthevariablesinthemodelexceptpriceindicators(Ban´buraetal.,2015).10 Theconditional forecasts in Figure 13 show that the macroeconomic co-movement can only account for some of the 10Thechoiceofwheretostarttheexerciserequiressomeexplanation. Thefirsttwoconsecutivequarter-on-quarter (qoq) at an annual rate (a.r.) core inflation reading above 2% occurred in the second half of 2021. After that, from 19

inflation acceleration and very little of the inflation deceleration. Thus, according to our model, the 2023 inflation deceleration was not due to the aggregate demand cooling down (as indicated by our almost flat output gap estimate in 2023) but to other factors, such as the expansion of aggregate supply or the normalization of inflation expectations. 8 Conclusions This paper proposes a new measure of potential output and the output gap for the EA based on letting a large number of macroeconomic and financial indicators speak. To do so, we estimate a large-dimensional non-stationary dynamic factor model, which allows us to capture co-movements across series while incorporating relevant macroeconomic priors, such as the long-run decline in output growth. Our model is a modified version of the model Barigozzi and Luciani (2023) used to estimate the output gap in the US, which we upgraded so that it can also be estimated on short samples and can handle the anomalous dynamics induced by the Covid pandemic. Our output gap estimate is in line with those published by the EC and the IMF in most of the sample. However, our estimate diverges significantly after the 2011–2012 sovereign debt recession when our output gap measure suggests that the EA economy was tighter than estimated by the EC and the IMF. This result suggests that the EA has a potential output issue, not a business cycle issue. Hence, if the goal is to achieve better economic conditions in the EA, its member countries should implement supply-side structural reforms that have long-run effects, while policies aiming at stimulating aggregate demand will have only short-term effects at best. Moreover, we find that incorporating financial indicators in the dataset, particularly household liabilities, is necessary to pin down the output gap. Excluding these variables leads to estimating a lower output gap in periods of debt build-up (e.g., before the GFC) and a higher gap in periods of deleveraging(e.g., from2013to2016), asgrowthfinancedthroughhouseholddebtisnotsustainable in the long run. References Aastveit, K. A. and T. Trovik (2014). Estimating the output gap in real time: A factor model approach. The Quarterly Review of Economics and Finance 54, 180–193. Ahn, H. J. and M. Luciani (2024). Common and idiosyncratic inflation. FEDS 2020-024r1, Board of Governors of the Federal Reserve System. Aruoba, S. B., F. X. Diebold, J. Nalewaik, F. Schorfheide, and D. Song (2016). Improving GDP measurement: A measurement-error perspective. Journal of Econometrics 191, 384–397. Bai, J. and K. Li (2016). Maximum likelihood estimation and inference for approximate factor models of high dimension. Review of Economics and Statistics 98(2), 298–309. Bai, J. and S. Ng (2002). Determining the number of factors in approximate factor models. Econometrica 70, 191–221. 2022:Q1,wehadsixconsecutiveqoqa.r. coreinflationreadingabove4%. Asaresult,wechosetostartourexerciseat thebeginningof2022. Qualitatively,theconclusionremainsunchangedifwebegintheexercisein2022:Q3. However, quantitatively,thereisadifference,asourpeakforecastisabout1p.p. lower. Nevertheless,thisdifferencecanalmost entirely be attributed to the different starting points of the two forecasts.. 20

Bai, J. and S. Ng (2004). A PANIC attack on unit roots and cointegration. Econometrica 72, 1127–1177. Ban´bura, M. and E. Bobeica (2023). Does the Phillips curve help to forecast euro area inflation? International Journal of Forecasting 39, 364–390. Ban´bura, M., D. Giannone, and M. Lenza (2015). Conditional forecasts and scenario analysis with vector autoregressions for large cross-sections. International Journal of Forecasting 31(3), 739–756. Barigozzi, M., M.Lippi, andM.Luciani (2021). Large-dimensionaldynamicfactormodels: Estimation of impulse-response functions with I(1) cointegrated factors. Journal of Econometrics 221, 455–482. Barigozzi, M. and M. Luciani (2019). Quasi maximum likelihood estimation of Non-Stationary Large Approximate Dynamic Factor Model. arXiv 1910.09841. Barigozzi, M.andM.Luciani(2023). MeasuringtheOutputGapusingLargeDatasets. The Review of Economics and Statistics 105, 1500–1514. Barigozzi, M. and M. Luciani (2024). Quasi maximum likelihood estimation and inference of large approximate dynamic factor models via the EM algorithm. arXiv 1910.03821v5. Berger, T., J. Richter, and B. Wong (2022). A unified approach for jointly estimating the business and financial cycle, and the role of financial factors. Journal of Economic Dynamics and Control 136, 104315. Boivin, J. and S. Ng (2006). Are more data always better for factor analysis? Journal of Econometrics 132, 169–194. Borio,C.,P.Disyatat,andM.Juselius(2017). Rethinkingpotentialoutput: embeddinginformation about the financial cycle. Oxford Economic Papers 69, 655–677. Buncic, D. and A. Pagan (2022). Discovering stars: Problems in recovering latent variables from models. SSRN working paper 4220302. Burlon, L. and P. D’Imperio (2020). Reliable real-time estimates of the euro-area output gap. Journal of Macroeconomics 64, 103191. Canova, F. (2022). FAQ: How do I estimate the output gap? mimeo. Carriero, A., T. E. Clark, M. Marcellino, and E. Mertens (2022). Addressing COVID-19 outliers in BVARs with stochastic volatility. The Review of Economics and Statistics. available online. Cette, G., J. Fernald, and B. Mojon (2016). The pre-Great Recession slowdown in productivity. European Economic Review 88, 3–20. Claessens,S.,M.A.Kose,andM.E.Terrones(2012). Howdobusinessandfinancialcyclesinteract? Journal of International Economics 87, 178–190. Comin, D. and M. Gertler (2006). Medium-term business cycles. The American Economic Review 96. Conti, A. M. (2021). Resurrecting the Phillips curve in low-inflation times. Economic Modelling 96, 172–195. 21

Crump, R. K., S. Eusepi, D. Giannone, E. Qian, and A. M. Sbordone (2021). A large bayesian var of the united states economy. FRB of New York Staff Report (976). De Masi, P. (1997). IMF estimates of potential output: Theory and practice. IMF Working Paper 177. Del Negro, M., F. Schorfheide, F. Smets, and R. Wouters (2007). On the fit of new Keynesian models. Journal of Business & Economic Statistics 25, 123–143. Doz, C., D. Giannone, and L. Reichlin (2012). A quasi-maximum likelihood approach for large, approximate dynamic factor models. The Review of Economics and Statistics 94, 1014–1024. European Commission (2018). Staff working document on the review of the flexibility under the Stability and Growth Pact. COM(2018) 335. Furlanetto, F., P. Gelain, and M. T. Sanjani (2021). Output gap, monetary policy trade-offs, and financial frictions. Review of Economic Dynamics 41, 52–70. Gambetti, L. and A. Musso (2017). Loan supply shocks and the business cycle. Journal of Applied Econometrics 32, 764–782. Gonz´alez-Astudillo, M. (2019). An output gap measure for the euro area: Exploiting country-level and cross-sectional data heterogeneity. European Economic Review 120, 103301. Hallin, M. and R. Liˇska (2007). Determining the number of factors in the general dynamic factor model. Journal of the American Statistical Association 102, 603–617. Hamilton, J. D. (2018). Why you should never use the Hodrick-Prescott filter. The Review of Economics and Statistics 100, 831–843. Hartl, T., R. Tschernig, and E. Weber (2022). Solving the unobserved components puzzle: A fractional approach to measuring the business cycle. mimeo. Hasenzagl, T., F. Pellegrino, L. Reichlin, and G. Ricco (2022). A Model of the Fed’s View on Inflation. The Review of Economics and Statistics 104, 686–704. Havik, K., K. Mc Morrow, F. Orlandi, C. Planas, R. Raciborski, W. R¨oger, A. Rossi, A. Thum- Thysen, V. Vandermeulen, et al. (2014). The production function methodology for calculating potential growth rates & output gaps. Economic Papers 535, Directorate General Economic and Financial Affairs (DG ECFIN), European Commission. Jarocin´ski, M. and M. Lenza (2018). An inflation-predicting measure of the output gap in the euro area. Journal of Money, Credit and Banking 50, 1189–1224. Justiniano, A., G. Primiceri, and A. Tambalotti (2013). Is there a trade-off between inflation and output stabilization? American Economic Journal: Macroeconomics 5, 1–31. Kamber, G., J. Morley, and B. Wong (2018). Intuitive and reliable estimates of the output gap from a Beveridge-Nelson filter. The Review of Economics and Statistics 100, 550–566. Kiley, M. T. (2013). Output gaps. Journal of Macroeconomics 37, 1–18. Kim, C.-J. and J. Kim (2022). Trend-cycle decompositions of real GDP revisited: Classical and Bayesian perspectives on an unsolved puzzle. Macroeconomic Dynamics 26, 394–418. Lenza, M. and G. E. Primiceri (2022). How to estimate a vector autoregression after March 2020. Journal of Applied Econometrics 37, 688–699. 22

Luciani, M.(2014). Forecastingwithapproximatedynamicfactormodels: Theroleofnon-pervasive shocks. International Journal of Forecasting 30, 20–29. Maroz, D., J. H. Stock, and M. W. Watson (2021). Comovement of economic activity during the Covid recession. mimeo. McCracken, M. and S. Ng (2020). FRED-QD: A quarterly database for macroeconomic research. NBER working paper 26872. McCracken, M.W.and S.Ng (2016). FRED-MD: Amonthly databasefor macroeconomic research. Journal of Business & Economic Statistics 34, 574–589. Mian, A. and A. Sufi (2018). Finance and business cycles: The credit-driven household demand channel. Journal of Economic Perspectives 32, 31–58. Mian, A., A. Sufi, and E. Verner (2017). Household debt and business cycles worldwide. The Quarterly Journal of Economics 132, 1755–1817. Morley, J., C. R. Nelson, and E. Zivot (2003). Why are the Beveridge-Nelson and unobservedcomponents decompositions of GDP so different? The Review of Economics and Statistics 85, 235–243. Morley, J., D. Rodr´ıguez-Palenzuela, Y. Sun, and B. Wong (2023). Estimating the euro area output gap using multivariate information and addressing the COVID-19 pandemic. European Economic Review 153. Morley, J., T. D. Tran, and B. Wong (2023). A simple correction for misspecification in trend-cycle decompositions with an application to estimating r. Journal of Business & Economic Statistics. available online. Morley, J. and B. Wong (2020). Estimating and accounting for the output gap with large Bayesian vector autoregressions. Journal of Applied Econometrics 35, 1–18. Ng, S. (2018). Comments on the cyclical sensitivity in estimates of potential output. Brookings Papers on Economic Activity 49, 412–423. Ng, S. (2021). Modeling macroeconomic variations after COVID-19. NBER working paper 29060. Onatski, A. and C. Wang (2021). Spurious factor analysis. Econometrica 89, 591–614. Phillips, P.C.andS.Jin(2021). Businesscycles, trendelimination, andtheHPfilter. International Economic Review 62, 469–520. Phillips, P. C. and Z. Shi (2021). Boosting: Why you can use the HP filter. International Economic Review 62, 521–570. Plagborg-Møller, M., L. Reichlin, G. Ricco, and T. Hasenzagl (2020). When is growth at risk? Brookings Papers on Economic Activity Spring 2020, 167–229. Reichlin, L., G. Ricco, and T. Hasenzagl (2020). Financial variables as predictors of real growth vulnerability. Deutsche Bundesbank Discussion Papers 05/2020. Ru¨nstler, G. and M. Vlekke (2018). Business, housing, and credit cycles. Journal of Applied Econometrics 33. Schm¨oller, M. E. and M. Spitzer (2021). Deep recessions, slowing productivity and missing (dis-) inflation in the euro area. European Economic Review 134, 103708. 23

Stock, J. H. and M. W. Watson (2016, January). Dynamic factor models, factor-augmented vector autoregressions, and structural vector autoregressions in macroeconomics. In J. B. Taylor and H. Uhlig (Eds.), Handbook of Macroeconomics, Volume 2, pp. 415–525. Elsevier. Thum-Thysen, A., F. Blondeau, F. d’Auria, B. D¨ohring, A. Hristov, and K. Mc Morrow (2022). Potential output and output gaps against the backdrop of the COVID-19 pandemic. Quarterly Report on the Euro Area 21, 21–30. T´oth, M. (2021). A multivariate unobserved components model to estimate potential output in the Euro Area: A production function based approach. ECB working paper 2523. Winter, J. d., S. J. Koopman, and I. Hindrayanto (2022). Joint decomposition of business and financial cycles: Evidence from eight advanced economies. Oxford Bulletin of Economics and Statistics 84, 57–79. 24

Supplementary material for the paper: Measuring the Euro Area Output Gap Supplementary material for the paper: Measuring the Euro Area Output Gap Matteo Barigozzi Claudio Lissona Matteo Luciani University of Bologna University of Bologna Federal Reserve Board matteo.barigozzi@unibo.it claudio.lissona2@unibo.it matteo.luciani@frb.gov Summary In Appendix A, we provide full details on the dataset. In Appendix B, we formally state all assumptions of our model and provide motivation and comments for each of them. In Appendix D, we provide details of all estimation steps, and Appendix E explains how we compute the confidence bandstomeasuretheuncertaintyaroundouroutputgapestimate. Next, AppendixFcomparesour estimate of the output gap with the one obtained when identifying the common trend as suggested by Morley et al. (2023), and Appendix G compares it with alternative methodologies. In Appendix H, we show how our measures would change if we did not model the effect of Covid explicitly, and we also compare our measure of the output gap with the ones obtained by using a different estimate of the Covid factor or when modeling the Covid induced volatility as suggested by Lenza and Primiceri (2022). In Appendix I, we show the effect of not allowing some parameters to be time-varying. In Appendix J, we assess the reliability of our output gap estimate. M. Barigozzi and C. Lissona gratefully acknowledges financial support from MIUR (PRIN2020, Grant 2020N9YFFE). Disclaimer: the views expressed in this paper are those of the authors and do not necessarily reflect the views and policies of the Board of Governors or the Federal Reserve System. Page 1 of 26

Supplementary material for the paper: Measuring the Euro Area Output Gap A Data Description Table A2 provides a brief description for each of the 119 series in our dataset. Moreover, for each variable, TableA2indicatesthesource, theunitofmeasure, theseasonaladjustmenttreatment, the transformation (if any), model for the deterministic component for the idiosyncratic component. Table A1 presents a glossary to proper understand the data description presented in Table A2. AlltheserieswereretrievedstartingfromJanuary2000uptoJune2023. Afterdroppingmissing valuesandtransformingthevariables, theactualstartingpointfortheanalysisis2001:Q1. Monthly series, which constitute around one-third of the dataset, are aggregated at the quarterly level by simple averages, hence, the sample used for the analysis is 2001:Q1-2023:Q4 (T = 92). Most of the series in the dataset are available already seasonally adjusted from the source, while others, e.g., financial variables and producer price indexes, are only available not seasonality adjusted. In these cases, we deseasonalize the series using a simple dummy variable approach. Table A1: Glossary Source Unit EUR = Eurostat CLV= Chain-linked volumes OECD = Organization for Economic Co-operation and Development 1000-ppl= Thousands of persons ECB = European Central Bank 1000-U = Thousands of Units FRED = Federal Reserve Economic Data CP = Current Prices SA F Trans Trend Idio NSA=NoSeasonalAdjustment Q=Quarterly 0=NoTransformation 0=NoTrend 0=I(0) SA=SeasonalAdjustment M=Monthly 1=FirstDifferences 1=DeterministicTrend 1=I(1) SCA=SeasonalandCalendarAdjustment 2=Log-Transformations 2=Time-VaryingTrend MSA=Manualadjustment 3=First-DifferencesinLogs 3=Time-varyingMean Page 2 of 26

Supplementary material for the paper: Measuring the Euro Area Output Gap Table A2: Data Description: Euro Area N ID Series Unit SA F Source Trans Trend Idio (1)NationalAccounts 1 GDP RealGrossDomesticProduct CLV(2015) SCA Q EUR 2 2 0 2 EXPGS RealExportGoodsandservices CLV(2015) SCA Q EUR 2 1 0 3 IMPGS RealImportGoodsandservices CLV(2015) SCA Q EUR 2 1 0 4 GFCE RealGovernmentFinalconsumptionexpenditure CLV(2015) SCA Q EUR 2 1 1 5 HFCE RealHouseholdsconsumptionexpenditure CLV(2015) SCA Q EUR 2 1 1 6 CONSD RealHouseholdsconsumptionexpenditure: DurableGoods CLV(2015) SCA Q EUR 2 1 1 7 CONSND RealHouseholdsconsumptionexpenditure: Non-DurableGoodsandServices CLV(2015) SCA Q EUR 2 1 1 8 GCF RealGrosscapitalformation CLV(2015) SCA Q EUR 2 0 1 9 GCFC RealGrossfixedcapitalformation CLV(2015) SCA Q EUR 2 0 1 10 GFACON RealGrossFixedCapitalFormation: Construction CLV(2015) SCA Q EUR 2 1 1 11 GFAMG RealGrossFixedCapitalFormation: MachineryandEquipment CLV(2015) SCA Q EUR 2 1 1 12 AHRDI AdjustedHouseholdsRealDisposableIncome %change SCA Q EUR 0 0 0 13 AHFCE ActualFinalConsumptionExpenditureofHouseholds %change SCA Q EUR 0 0 0 14 GNFCPS GrossProfitShareofNon-FinancialCorporations Percent SCA Q EUR 0 0 0 15 GNFCIR GrossInvestmentShareofNon-FinancialCorporations Percent SCA Q EUR 0 0 0 16 GHIR GrossInvestmentRateofHouseholds Percent SCA Q EUR 0 0 0 17 GHSR GrossHouseholdsSavingsRate Percent SCA Q EUR 0 0 0 (2)LaborMarketIndicators 18 TEMP TotalEmployment(domesticconcept) 1000-ppl SCA Q EUR 2 1 1 19 EMP Employees(domesticconcept) 1000-ppl SCA Q EUR 2 1 1 20 SEMP SelfEmployment(domesticconcept) 1000-ppl SCA Q EUR 2 0 1 21 THOURS HoursWorked: Total 2015=100 SCA Q EUR 2 1 0 22 EMPAG QuarterlyEmployment: Agriculture,Forestry,Fishing 1000-ppl SCA Q EUR 2 1 0 23 EMPIN QuarterlyEmployment: Industry 1000-ppl SCA Q EUR 2 0 0 24 EMPMN QuarterlyEmployment: Manufacturing 1000-ppl SCA Q EUR 2 1 0 25 EMPCON QuarterlyEmployment: Construction 1000-ppl SCA Q EUR 2 0 1 26 EMPRT QuarterlyEmployment: Wholesale/Retailtrade,transport,food 1000-ppl SCA Q EUR 2 1 1 27 EMPIT QuarterlyEmployment: InformationandCommunication 1000-ppl SCA Q EUR 2 1 1 28 EMPFC QuarterlyEmployment: FinancialandInsuranceactivities 1000-ppl SCA Q EUR 2 0 1 29 EMPRE QuarterlyEmployment: RealEstate 1000-ppl SCA Q EUR 2 0 0 30 EMPPR QuarterlyEmployment: Professional,Scientific,Technicalactivities 1000-ppl SCA Q EUR 2 1 1 31 EMPPA QuarterlyEmployment: PA,education,healthadsocialservices 1000-ppl SCA Q EUR 2 1 1 32 EMPENT QuarterlyEmployment: Artsandrecreationalactivities 1000-ppl SCA Q EUR 2 1 0 33 UNETOT Unemployment: Total %active SA M EUR 0 3 0 34 UNEO25 Unemployment: Over25years %active SA M EUR 0 0 1 35 UNEU25 Unemployment: Under25years %active SA M EUR 0 0 0 36 RPRP RealLabourProductivity(person) 2015=100 SCA Q EUR 2 1 1 37 WS Wagesandsalaries CP SCA Q EUR 2 1 0 38 ESC Employers’SocialContributions CP SCA Q EUR 2 1 0 (3)CreditAggregates 39 TAS.SDB TotalEconomy-Assets: Short-TermDebtSecurities MLNe MSA Q EUR 2 1 1 40 TAS.LDB TotalEconomy-Assets: Long-TermDebtSecurities MLNe MSA Q EUR 2 1 0 41 TAS.SLN TotalEconomy-Assets: Short-TermLoans MLNe MSA Q EUR 2 1 0 42 TAS.LLN TotalEconomy-Assets: Long-TermLoans MLNe MSA Q EUR 2 1 1 43 TLB.SDB TotalEconomy-Liabilities: Short-TermDebtSecurities MLNe MSA Q EUR 2 0 1 44 TLB.LDB TotalEconomy-Liabilities: Long-TermDebtSecurities MLNe MSA Q EUR 2 1 0 45 TLB.SLN TotalEconomy-Liabilities: Short-TermLoans MLNe MSA Q EUR 2 1 1 46 TLB.LLN TotalEconomy-Liabilities: Long-TermLoans MLNe MSA Q EUR 2 1 1 47 NFCAS Non-FinancialCorporations: TotalFinancialAssets MLNe MSA Q EUR 2 1 0 48 NFCAS.SLN Non-FinancialCorporations-Assets: Short-TermLoans MLNe MSA Q EUR 2 1 1 49 NFCAS.LLN Non-FinancialCorporations-Assets: Long-TermLoans MLNe MSA Q EUR 2 1 1 50 NFCLB Non-FinancialCorporations: TotalFinancialLiabilities MLNe MSA Q EUR 2 1 1 51 NFCLB.SLN Non-FinancialCorporations-Liabilities-Short-TermLoans MLNe MSA Q EUR 2 1 1 52 NFCLB.LLN Non-FinancialCorporations-Liabilities-Long-TermLoans MLNe MSA Q EUR 2 1 1 53 GGAS GeneralGovernment: TotalFinancialAssets MLNe MSA Q EUR 2 1 0 54 GGAS.SLN GeneralGovernment-Assets: Short-TermLoans MLNe MSA Q EUR 2 1 0 55 GGAS.LLN GeneralGovernment-Assets: Short-TermLoans MLNe MSA Q EUR 2 0 1 56 GGLB GeneralGovernment: TotalFinancialLiabilities MLNe MSA Q EUR 2 1 1 57 GGLB.SLN GeneralGovernment-Liabilities: Short-TermLoans MLNe MSA Q EUR 2 1 0 58 GGLB.LLN GeneralGovernment-Liabilities: Long-TermLoans MLNe MSA Q EUR 2 0 1 59 HHAS Households: TotalFinancialAssets MLNe MSA Q EUR 2 1 1 60 HHAS.SLN Households-Assets: Short-TermLoans MLNe MSA Q EUR 2 1 0 61 HHAS.LLN Households-Assets: Long-TermLoans MLNe MSA Q EUR 2 1 0 62 HHLB Households: TotalFinancialLiabilities MLNe MSA Q EUR 2 2 3 63 HHLB.SLN Households-Liabilities: Short-TermLoans MLNe MSA Q EUR 2 0 1 64 HHLB.LLN Households-Liabilities: Long-TermLoans MLNe MSA Q EUR 2 2 3 Inabsenceofavailabledataondurableandnon-durablegoodsfortheEuroArea,wefollowCasalisandKrustev(2022)andbuildtheaggregate seriesofdurableconsumption(CONSD)aggregatingthedataforthe20individualEuroAreacountries. Sincedataforservicesandnon-durable goodsareunavailableformanyindividualcountriesaswell,webuildanaggregatemeasureofnon-durablegoods(CONSND)whichalsoincludes semi-durablegoodsandservices. Page 3 of 26

Supplementary material for the paper: Measuring the Euro Area Output Gap Table A2: Data Description: Euro Area N ID Series Unit SA F Source Trans Trend Idio (4)LaborCosts 65 ULCIN NominalUnitLaborCosts: Industry 2016=100 SCA Q EUR 2 1 0 66 ULCMQ NominalUnitLaborCosts: MiningandQuarrying 2016=100 SCA Q EUR 2 1 1 67 ULCMN NominalUnitLaborCosts: Manufacturing 2016=100 SCA Q EUR 2 1 0 68 ULCCON NominalUnitLaborCosts: Construction 2016=100 SCA Q EUR 2 1 0 69 ULCRT NominalUnitLaborCosts: Wholesale/RetailTrade,Transport,Food,IT 2016=100 SCA Q EUR 2 1 0 70 ULCFC NominalUnitLaborCosts: FinancialActivities 2016=100 SCA Q EUR 2 1 0 71 ULCRE NominalUnitLaborCosts: RealEstate 2016=100 SCA Q EUR 2 1 0 72 ULCPR NominalUnitLaborCosts: Professional,Scientific,Technicalactivities 2016=100 SCA Q EUR 2 1 0 (5)ExchangeRates 73 REER42 RealExchangeRate(42mainindustrialcountries) 2010=100 NSA M EUR 2 0 0 74 ERUS ExchangeRate(USdollar) 2010=100 NSA M EUR 2 0 0 (6)InterestRates 75 IRT3M 3-MonthsInterestRates Percent NSA M EUR 0 0 1 76 IRT6M 6-MonthsInterestRates Percent NSA M EUR 0 0 1 77 LTIRT Long-TermInterestRates(EMUCriterion) Percent NSA M EUR 0 1 1 (7)IndustrialProductionandTurnover 78 IPMN IndustrialProductionIndex: Manufacturing 2021=100 SCA M EUR 2 0 1 79 IPCAG IndustrialProductionIndex: CapitalGoods 2021=100 SCA M EUR 2 0 1 80 IPCOG IndustrialProductionIndex: ConsumerGoods 2021=100 SCA M EUR 2 1 0 81 IPDCOG IndustrialProductionIndex: DurableConsumerGoods 2021=100 SCA M EUR 2 1 0 82 IPNDCOG IndustrialProductionIndex: NonDurableConsumerGoods 2021=100 SCA M EUR 2 1 0 83 IPING IndustrialProductionIndex: IntermediateGoods 2021=100 SCA M EUR 2 0 1 84 IPNRG IndustrialProductionIndex: Energy 2021=100 SCA M EUR 2 1 0 85 TRNMN TurnoverIndex: Manufacturing 2021=100 SCA M EUR 2 1 1 86 TRNCAG TurnoverIndex: CapitalGoods 2021=100 SCA M EUR 2 1 1 87 TRNCOG TurnoverIndex: ConsumerGoods 2021=100 SCA M EUR 2 1 1 88 TRNDCOG TurnoverIndex: DurableConsumerGoods 2021=100 SCA M EUR 2 0 0 89 TRNNDCOG TurnoverIndex: NonDurableConsumerGoods 2021=100 SCA M EUR 2 1 0 90 TRNING TurnoverIndex: IntermediateGoods 2021=100 SCA M EUR 2 0 0 91 TRNNRG TurnoverIndex: Energy 2021=100 SCA M EUR 2 0 1 (8)Prices 92 PPICAG ProducerPriceIndex: CapitalGoods 2021=100 MSA M EUR 3 0 0 93 PPIDCOG ProducerPriceIndex: DurableConsumerGoods 2021=100 MSA M EUR 3 0 0 94 PPINDCOG ProducerPriceIndex: NonDurableConsumerGoods 2021=100 MSA M EUR 3 0 0 95 PPIING ProducerPriceIndex: IntermediateGoods 2021=100 MSA M EUR 3 0 0 96 PPIFD ProducerPriceIndex: Food 2021=100 MSA M EUR 3 0 1 97 HICPOV HarmonizedIndexofConsumerPrices: OverallIndex 2010=100 SCA M ECB 3 0 1 98 HICPNEF HarmonizedIndexofConsumerPrices: AllItems: noEnergy&Food 2010=100 SCA M ECB 3 0 0 99 HICPG HarmonizedIndexofConsumerPrices: Goods 2010=100 SCA M ECB 3 0 1 100 HICPSV HarmonizedIndexofConsumerPrices: Services 2010=100 SCA M ECB 3 0 0 101 HICPNG HarmonizedIndexofConsumerPrices: Energy 2010=100 MSA M EUR 3 0 1 102 HICPFD HarmonizedIndexofConsumerPrices: Food 2010=100 MSA M EUR 3 0 1 103 DFGDP RealGrossDomesticProductDeflator 2015=100 SCA Q EUR 3 0 0 104 HPRC ResidentialPropertyPrices(BIS) MLNe SCA Q FRED 3 0 0 105 POIL CrudeOilPrices: Brent-Europe e/barrel MSA Q FRED 3 0 0 106 PNGAS GlobalpriceofNaturalgas,EU e/MMbtu MSA Q FRED 3 0 0 (9)ConfidenceIndicators 107 ICONFIX IndustrialConfidenceIndicator Index SA M EUR 0 0 1 108 CCONFIX ConsumerConfidenceIndicator Index SA M EUR 0 0 0 109 ESENTIX EconomicSentimentIndicator Index SA M EUR 0 0 1 110 KCONFIX ConstructionConfidenceIndicator Index SA M EUR 0 0 1 111 RTCONFIX RetailConfidenceIndicator Index SA M EUR 0 0 0 112 SCONFIX ServicesConfidenceIndicator Index SA M EUR 0 0 1 113 BCI Cyclically-AdjustedBusinessConfidenceIndex 2010=100 SA M OECD 0 0 1 114 CCI Cyclically-AdjustedConsumerConfidenceIndex 2010=100 SA M OECD 0 0 0 (10)MonetaryAggregates 115 CURR MoneyStock: Currency MLNe SCA M ECB 2 1 1 116 M1 MoneyStock: M1 MLNe SCA M ECB 2 1 1 117 M2 MoneyStock: M2 MLNe SCA M ECB 2 1 1 (11)Others 118 SHIX SharePrices 2010=100 SA M OECD 2 0 1 119 CAREG Passenger’sCarsRegistrations 1000-U SCA M ECB 2 0 1 Page 4 of 26

Supplementary material for the paper: Measuring the Euro Area Output Gap B Assumptions In this section, we state all the formal assumptions underlying the model outlined in Section 3.1, and we provide both econometric and economic justifications for these assumptions. Throughout the text, we let t , t and t denote 2019:Q4, 2020:Q1 and 2021:Q4, respectively. 19Q4 20Q1 21Q4 Identifying assumptions for the space spanned by the factors (1) The number of factors q is such that q < n and is independent of n. (2) The q-dimensional vector f is such that E[∆f ] = 0 and E[∆f ∆f′] = I. t t t t (3) Then×qmatrixΛ = (λ ···λ )′,withλ = (λ ···λ )′,1 ≤ i ≤ n,issuchthatlim n−1Λ′Λ = 1 n i i1 iq n→∞ H positive definite. (4) The scalar g is such that E[∆g ] = 0 and E[(∆g )2] = 1. t t t (5) The n-dimensional vector γ = (γ ···γ )′, is such that lim n−1γ′γ > 0. 1 n n→∞ (6) The q-dimensional vector u is such that E[u ] = 0, E[u u′] = Σ is positive definite, and t t t t u E[u u′ ] = 0 for k ̸= 0. Moreover, s is deterministic with s > 0. t t−k t t (7) The idiosyncratic innovations e , 1 ≤ i ≤ n, are such that E[e ] = 0, E[e2] = σ2 > 0 for all t. it it it ei Moreover, there exist finite constants M independent of t and 0 < ρ < 1 independent of t, i, ij and j such that |E[e e ]| ≤ M ρ|k| for all k ∈ Z, with Pn M ≤ M and Pn M ≤ M it j,t−k ij i=1 ij j=1 ij for some finite constant M > 0 independent of n. (8) E[e u ] = 0, for all i,t,s. it s Assumptions (1)-(3) require the q factors f to be pervasive so that they have a non-negligible t effect on the variables of interest (Bai and Ng, 2004; Barigozzi et al., 2021). Following Maroz et al. (2021), these assumptions are extended in parts (4) and (5) to the Covid factor, g , where t pervasivenessstemsfromthecommonnatureoftheCovidshockaffectingmostoftheseriesincluded in the dataset. Including both 2020 and 2021 in the Covid period is consistent with the evolution of the pandemic in Europe. Ng (2021) has proposed an alternative approach that uses external information from health statistics. Assumption (6) assumes white noise innovations whose volatility changes over time after the Covid shock—time-varying volatility is not a prominent feature in the pre-2020 sample (Jarocin´ski and Lenza, 2018). Accounting for the change in volatility due to Covid has proven to be fundamental both for estimation and forecasting, and here we adopt an approach similar to Lenza and Primiceri (2022) by introducing a scaling term s modeled independently for each period starting t from 2020:Q1. Lenza and Primiceri (2022) analyze monthly US data and impose an exponential decay for s starting in June 2020. In contrast, we estimate one parameter for each period starting t in 2020:Q1 because many series exhibit large variation even after the first half of 2020, which is not surprising given that (i) mobility restriction measures in the EA were much more restrictive than in the US, lasted for longer, and were also implemented in 2021, and (2) the Russia-Ukraine war had a much larger impact on Europe by pushing natural gas prices (and gasoline prices to a lesser Page 5 of 26

Supplementary material for the paper: Measuring the Euro Area Output Gap extent) to the roof, and in creating a lot of macro-financial uncertainty. Moreover, as Morley et al. (2023) pointed out, quarterly data do not allow for a sharp identification of the decay parameter. Assumption (7) allows the idiosyncratic innovations to be mildly cross-sectionally correlated and serially correlated with summable autocovariances, thus compatible with stationary ARMA dynamics(BaiandNg,2004;Barigozzietal.,2021). Last, Assumption(8)requirestheidiosyncratic and factor innovations to be uncorrelated at all leads and lags, a requirement consistent with the idea of global macroeconomic shocks being unrelated to local dynamics. Assumptions on the dynamic specifications on the non-stationary idiosyncratic components and the secular components (9) Let I be the set of indexes such that ξ ∼ I(1) if i ∈ I , then n = #{i : i ∈ I } is such that 1 it 1 I 1 0 < n < n. I (10) Let I be the set of indexes such that b ̸= 0 if i ∈ I , then n = #{i : i ∈ I } is such that b it b B b 0 < n < n. Moreover, D = a ̸= 0, for all i. B i0 i (11) For i = GDP, HHLB, HHLB.LLN , E[η it ] = 0 and σ η 2 i = (400σ b∆ 2 yi )−1, while for i = UNETOT HICPOV , HICPNEF ,E[ϵ it ] = 0andσ ϵ 2 i = (400σ by 2 i )−1 whereσ b∆ 2 yi andσ y 2 i arethesamplevariances of ∆y and y respectively, computed for 1 ≤ t ≤ t and t +1 ≤ t ≤ T. it it 19Q4 21Q4 Assumption(9)allowstheidiosyncraticcomponenttobeI(1)forsome, butnotall, oftheseries. This assumption is crucial when estimating the model on a large dataset. Imposing the assumption of all idiosyncratic components being I(0) would be overly restrictive, as it implies cointegration for any q-dimensional vector of series (Barigozzi et al., 2021). While cointegration may hold for certain series, it is highly unlikely to hold for many others. To accommodate potential cointegration, we allow only a limited number n of variables to possess a non-stationary idiosyncratic component. I Our dataset, where only n = 57 out of n = 119 series exhibit a non-stationary idiosyncratic I component, supports this assumption. Assumption (10) allows for a non-stationary secular component for some, but not all, of the variables in the dataset. This modeling choice is coherent with the properties of a standard macroeconomic dataset. Specifically, variables related to the real sector of the economy, such as consumption or investments, commonly display a distinct (upward) trend. Conversely, this may not hold for other variables, such as inflation rates or interest rates, for example. This intuition finds support in the empirical data, where only n = 58 out of n = 119 series exhibit a linear trend. B Specifically, letting a = D , Assumptions (10) and (11) imply that GDP, households’ financial i i0 liabilities and long-term loans, have a secular component given by the local linear trend model D it = a i +b it t, b it = b i,t−1 +η it , i = GDP, LBHH, HHLB.LLN. (B1) We introduce a local-linear trend for GDP to capture the gradual drift in the secular decline in long-run output growth documented both for the US and the EA (Cette et al., 2016; Antolin- Diaz et al., 2017; Gordon, 2018). The literature has identified several factors contributing to this slowdown, with particular emphasis on declining productivity growth. This decline has been more pronounced in the EA due to heterogeneity between core and peripheral countries, as peripheral Page 6 of 26

Supplementary material for the paper: Measuring the Euro Area Output Gap countries are experiencing a larger misallocation of economic resources (Cette et al., 2016). Therefore, it is crucial to accurately account for these features to assess GDP’s long-run dynamics. This assessment is essential for estimating potential output, as it avoids spuriously inflating the output gap with unexplained predictable variation (Ng, 2018). We introduce a local-linear trend for household financial liabilities and long-term loans, which constitute about 85% of total household liabilities, to capture the slowdown in their average growth rates that occurred since the GFC. Assumption (11) implies that the unemployment rate, overall and core inflation have a secular component given by the local level model D it = a it , a it = a i,t−1 +ϵ it , i = UNETOT,HICPOV,HICPNEF. (B2) This specification captures relevant labor and demographic factors that may affect the unemployment rate secular trend, such as, for example, the aging of the population and the misallocation of resources in the labor market due to “soft budget constraints” or stringent labor market policies can lead to a mismatch between employers’ needs and the skill-set of the unemployed (Cette et al., 2016). Similarly, this specification also allows us to account for the slowdown in inflation occurred after the GFC. All the other variables in the dataset have either a deterministic linear trend or a constant mean, i.e, D = a +b t if i ∈ I , or D = a otherwise. Although it is technically possible to model it i i b it i a time-varying component for all the variables in the dataset, such an approach would introduce complexities in the estimation framework, with the number of latent states increasing linearly with the number of series. In Assumption (11) we fix the variances of the stochastic secular components following Del Negro et al. (2019) in order to effectively capture the gradual and persistent nature of the secular trends. This specification implies that the standard deviation of the secular trend, over a century, is approximately 1%, a choice consistent with the notion of a slow-moving secular component. Assumptions on the dynamics of the factors, trend, and cycles (12) The polynomial det(I−Pp−1A zj+1) = 0 has 1 root in z = 1 and the remaining q−1 roots j=0 j in |z| > 1. (13) The q-dimensional vector ψ is such that β′ψ = 0, where β is the q×(q−1) matrix having as columns the cointegrating vectors of f , i.e., such that β′f is weakly stationary. t t (14) The q-dimensional vector ω is weakly stationary and such that E[ω ] = 0 and E[ω ω′] = Σ t t t t ω is positive definite. (15) The scalar ν is such that E[ν ] = 0 and E[ν2] = σ2 > 0. t t t ν (16) E[ν ω ] = 0 for all t. t t Assumption (12) imposes that 1 common trend drives the non-stationarity in the common factors, hence, that the factors are cointegrated with q −1 cointegrating relations—our data provide Page 7 of 26

Supplementary material for the paper: Measuring the Euro Area Output Gap strong support for the presence of just one common trend. This is a standard assumption in the literature,whichoftenassumesthatcommonproductivitytrendisthesoledriveroflong-runeconomic growth (see, e.g., Del Negro et al., 2007). Assumptions (13) and (14) imply that ω , defined in (6), belongs to the cointegration space t of the common factors. This view is consistent with theoretical models assuming that the output gap represents deviations from long-run equilibria determined by a common productivity trend (Del Negro et al., 2007; T´oth, 2021). Assumption(15)assumesthatν isastochasticprocess—henceτ isacommonstochastictrend— t t but it does not constraint ν to be a white noise—hence τ to be a random walk. Indeed, our t t estimates suggest that ν is autocorrelated, in line with the theoretical arguments by Lippi and t Reichlin (1994). Finally, Assumption (16) implies contemporaneous orthogonality between potential output and the output gap, which is also assumed in the non-parametric approaches used by Barigozzi and Luciani (2023). C Number of Common Factors In this section, we present the results of various information criteria employed to select the number of common factors. Specifically, we look at the log-information criteria IC of Bai and Ng (2002) (BN), Hallin and Liˇska (2007) (HL), and Alessi et al. (2010) (ABC), as well at the eigenvalue-ratio criterion by Ahn and Horenstein (2013) (AH), and the test proposed by Onatski (2009) (ON). In order to avoid spurious effects from the Covid period, we employ standardized and de-meaned first-differenced data up to 2019:Q4. Table C1: Number of common factors ABC AH BN HL ON q 4 1 4 4 1 Notes: ABC=Alessietal.(2010)criteria;AH=AhnandHorenstein (2013)test;BN=BaiandNg(2002)criteria,HL=HallinandLiˇska (2007)criteria,ON=Onatski(2009). As shown in Table C1, the ABC, BN, and HL criteria suggest q = 4, while the AH criteria and the ON test suggest q = 1. We picked q = 4 because (i) the criteria of Ahn and Horenstein (2013) may underperform if there is a significant difference in the explanatory power of the different factors, which is the case in our dataset where the first factor has a much larger explanatory power than the others; and (ii) the Onatski (2009) may not perform well when T is small compared to n. D Estimation in detail In this section, we provide details on the estimation procedure described in Section 3.2. Then, the extended state-space form of the model is given by: y = D +λ′f +γ g I +ζ +z , z i. ∼ i.d. (0,R ), 1 ≤ i ≤ n, 1 ≤ t ≤ T, (D1a) it it i t i t t20Q1≤t≤t21Q4 it i,t it i p−1 f = X A f +{s I +(1−I )}u , u i. ∼ i.d. (0,Σ ), (D1b) t j t−j t t≥t20Q1 t≥t20Q1 t t u j=0 Page 8 of 26

Supplementary material for the paper: Measuring the Euro Area Output Gap  D =  a a i i +b i t i i f f i i ∈ ∈ I I a b , , (D1c) it  D D it−1 + + b ϵ it−1 , b it = b it−1 +η it i i f f i i = = GDP , HHL , B , HHLB , .LLN , , η ϵ it i i . . ∼ i ∼ i . . d d . . ( ( 0 0 , , σ σ η 2 2 i ) ) , , it−1 i,t UNETOT HICPOV HICPNEF i,t ϵi ( ζ +e if i ∈ I , e i. ∼ i.d. (0,σ2) ζ it = 0 it it if i ∈/ I 1 , it ei (D1d) 1 ( σ2 if i ∈ I , R = z 1 (D1e) i σ2 if i ∈/ I , ei 1 where I denotes the set of series with only a constant intercept, while I denotes the set of series a b with a deterministic linear trend. Similarly, I denotes the set of series with a I(1) idiosyncratic 1 component. Initialization In order to apply the Kalman Filter and Smoother, we need initial estimates of all the quantities described in Equations (D1a)-(D1e). We denote with the superscript “19” all quantities computed with data up to 2019. Let y˘19 = (y19−a˘19− ˘ b19·t)/σ2 , where σ2 is the sample variance of ∆y19, it t i i ∆y19 ∆y19 it and a˘19 and ˘ b19 are estimated by regressing a constant i and a time i trend on y19, whenever i ∈ I or i i it b i = GDP, HHLB, HHLB.LLN . If i ∈ I a or i = UNETOT, HICPOV, HICPNEF we let y˘ it = y i 1 t 9−a˘1 i 9, where a˘1 i 9 isthesampleaverageofy it . Thestandardizedslopesaredenotedas bb1 i 9 =˘ b1 i 9/σ ∆ 2 y19 . Weinitialize i the loadings using the estimator of Barigozzi et al. (2021): the n×q matrix of estimated loadings Λb19 = (λb19,...,λb19)′ is obtained by principal components on the standardized first differences of 1 q the data, i.e. (∆y19 −∆y19)/σ2 , where ∆y19 is the sample mean of ∆y19. Given the loadings, it i ∆y19 i it i we also obtain a first estimate of the q common factors, bf19 = n−1Λb19′y˘19 and of the idiosyncratic t t components, ξb19 = y˘19 −λb19′ bf19. Furthermore, we obtain Ab19, j = 1,...,p, by fitting a VAR(p) on it it i t j bf t 19. Given the residuals u bt 19 = bf t 19 −Jb19bf t 1 − 9 1 , where Jb19 is the companion form representation of the autoregressive matrices Ab1 1 9,...,Ab1 p 9, an estimate of the latent variance is given by Σb1 u 9 = C d ov(u bt 19), where Cov is the sample covariance matrix. d We are left with the initialization of the variances of the time-varying parameters of the model and of the idiosyncratic components. We calibrate the variances of the time-varying parameters following Del Negro et al. (2019). In this way, we assume these quantities to slowly drift over time, with a standard deviation of approximately 1% over a century. This results in σ2 ≈ 4×10−3 for bηi i = GDP , σ bη 2 i ≈ 10−3 for i = HHLB, HHLB.LLN and σ bϵ 2 ≈ 10−2 for i = UNETOT, HICPOV, HICPNEF . For the idiosyncratic variances, if i ∈/ I 1 we set Rb i 19 = V d ar(ξb i 1 t 9). On the other hand, if i ∈ I 1 , we set Rb19 = 10−2, a value sufficiently high to guarantee more accurate estimates (Opschoor and van Dijk, i 2023). Table D1 provides an overview of the initial values of the states for the Kalman Filter. We denote as f19 ,D19 ,b19 and ζ19 the Kalman Filter estimates (up to 2019:Q4) of the states at time t|t t|t t|t t|t t, estimated using all the information up to time t. Similarly, we denote as f19 ,D19 ,b19 and ζ19 t|T t|T t|t t|T the Kalman Smoother estimates (up to 2019:Q4) of the states at time t, estimated using all the Page 9 of 26

Supplementary material for the paper: Measuring the Euro Area Output Gap Table D1: Initialization of states for the Kalman Filter f19 = bf19 0|0 0 (cid:16) (cid:17) (cid:16) (cid:17)−1 (cid:16) (cid:17) vec P1 0 9 |0 = I pq2 −Ab19⊗Ab19 vec Σb1 u 9 D19 = 0 if i ∈ I i,0|0 a D1 i, 9 0|0 = bb1 i 9 if i ∈ I b , i = GDP, HHLB, HHLB.LLN D19 = 0 if i = UNETOT,HICPOV,HICPNEF i,0|0 b1 i, 9 0|0 = bb1 i 9 if i ∈ I b , i = GDP, HHLB, HHLB.LLN PD(19) = 0 if i ∈ {I ,I } i,0|0 a b P i D ,0 (1 | 9 0 ) = (1−0 1 .99)2 σ bη 2 i if i = GDP, HHLB, HHLB.LLN PD(19) = σ2 if i = UNETOT,HICPOV,HICPNEF i,0|0 bϵi Pb(19) = 1 σ2 if i = GDP, HHLB, HHLB.LLN i,0|0 (1−0.99)2bηi ζ19 = ξbζ(19) if i ∈ I i,0|0 i1 1 (cid:16) (cid:17) P i ζ , ( 0 19 | ) 0 = (1−0 1 .99)2 V d ar ∆ξbit if i ∈ I 1 information up to time T. Estimation is carried on using a standardized version of the data in levels, that is:   y σ it 2 −aˇi if i ∈ {I b ,GDP, HHLB, HHLB.LLN} y˜ it =  y σ it ∆ 2 − y y¯ i i if i ∈ {I a ,UNETOT,HICPOV,HICPNEF} ∆yi with y˜ i 1 t 9 being the standardized data up to 2019. When computing σ ∆ 2 yi , a˘ i and˘ b i bb i ) for the entire sample, we treat Covid outliers as missing values. Step 1: Estimate the model up to 2019:Q4 (pre-Covid step) Given the initial values of the parameters and the states, we run the Kalman Filter and Smoother using standardized data up to 2019, y˜19 = (y˜ ,...,y˜ )′, to obtain a new estimate of the states, t 1,t n,t namely the factors f19 , the time-varying secular components D19 and slopes b19 , and the nont|T t|T t|t stationary idiosyncratic components ζ19 , along with the corresponding conditional covariances. t|T Given the smoothed states, we estimate all the parameters as follows: - Factor loadings: T ! T !−1 λb19′ = X(cid:16) y˜19−D19 −ζ19 (cid:17) f19′ X f19 f19′ +P19 i it i,t|T i,t|T t|T t|T t|T t|T t=1 t=1 Page 10 of 26

Supplementary material for the paper: Measuring the Euro Area Output Gap - Parameters of the law of motion of the common factors: T ! T ! Ab19 = X f19 f19′ +P19 X f19 f19′ +P19 t|T t−1|T t,t−1|T t−1|T t−1|T t−1|T t=2 t=2 1 T T ! Σb19 = X(cid:16) f19 f19′ +P19 (cid:17) −Ab19 X(cid:16) f19 f19′ +P19 (cid:17) u T t|T t|T t|T t|T t−1|T t,t−1|T t=2 t=2 - Slopes of secular trend: T ! T !−1 bb19 = X(cid:16) y˜19−λb19′ f19 −ζ19 (cid:17) t X t2 i it i t|T i,t|T t=1 t=1 - Variance of I(1) idiosyncratic components: 1 T 1 T σ2,19 = X(cid:16) ζ19 ζ19′ +Pζ(19) (cid:17) + X(cid:16) ζ19 ζ19′ +Pζ(19) (cid:17) − bei T i,t|T i,t|T i,t|T T it−1|T it−1|T i,t−1|T t=2 t=2 2 T − X(cid:16) ζ19 ζ19′ +Pζ(19) (cid:17) T it|T it−1|T i,t,t−1|T t=2 - Covariance prediction error: Rb i 19 = T 1 X T (cid:26)(cid:16) y˜ i 1 , 9 t −λb1 i 9′ f t 1 | 9 T −I i∈I b D1 i, 9 t|T −I i∈I1 ζ i 1 , 9 t|T (cid:17)2 +λb1 i 9′ P1 t, 9 T|T λb1 i 9+ t=1 o + I PD(19) +I Pζ(19) i∈I b i,t|T i∈I1 i,t|T Given the estimated factors, f19 we obtain an estimate of the trend and cyclical component t|T for the pre-Covid period by means of the EM algorithm. To run the algorithm, we need an initial estimate of the parameters ψ, Σ , and σ2. ω ν (a) We compute an initial estimate of ψ19, denoted as ψb(0),19 by PCA on the long-run variancecovariance matrix of the factors (Zhang et al., 2019). (b) Wecomputeaninitialestimateofthecommontrend, denotedasτ(0),19, byprojectingf19 onto bt t|T ψb(0),19. This also yields an initial estimate of the cyclical component ω bt (0) = f t 1 | 9 T −ψb(0),19τ bt (0),19. (c) Weinitializeσ2,19 followingDelNegroetal.(2017,2019),sothatσ2(0),19 = (400σ2 )−1,where ν bν b∆τ19 σ2 is the sample variance of ∆τ(0),19. We chose a very small value for the variance of σ2(0),19 b∆τ19 bt bν to incorporate our prior assumption of a slow-moving trend. (d) Lastly, since by construction ω(0),19 has a sample covariance matrix of reduced rank (q−1), in bt ordertoruntheEMalgorithmweinitializethiscovarianceasΣb( ω 0),19 = T−1PT t=1 ω bt (0),19ω bt (0),19′+ κI , where we set κ = 10−2. This choice is consistent with the recommendations by Opschoor q and van Dijk (2023) who show that smaller values of κ might be detrimental for the performance of the algorithm. Once the initial estimates for the algorithm have been computed, in the E-step we run the Kalman Filter and Smoother to obtain a new estimate of the trend, namely τ(1),19, along with an estimate t|T of its conditional variance and covariance, Pτ(1),19 and Pτ(1),19 , respectively. The smoothed trend is t|T t,t−1|T Page 11 of 26

Supplementary material for the paper: Measuring the Euro Area Output Gap then used to estimate the parameters in the M-step, until convergence of the algorithm is achieved. For a generic iteration k of the algorithm, the parameters are estimated as follow: - Trend loadings: T ! T !−1 ψb(k),19 = X f19 τ(k),19 X τ2(k),19+Pτ(k),19 t|T t|T t|T t|T t=1 t=1 - Variance of common trend: 1 T 1 T 2 T σ2(k),19 = X(cid:16) τ2(k),19+Pτ(k),19 (cid:17) + X(cid:16) τ2(k),19+Pτ(k),19 (cid:17) − X(cid:16) τ(k),19τ(k),19 +Pτ(k),19 (cid:17) bν T t|T t|T T t−1|T t−1|T T t|T t−1|T t,t−1|T t=2 t=2 t=2 - Covariance of transitory component Σb(k),19 = 1 X T (cid:26)(cid:16) f19 −ψb(k),19τ(k),19 (cid:17)(cid:16) f19 −ψb(k),19τ(k),19 (cid:17)′ +ψb(k),19Pτ(k),19ψb(k),19′ (cid:27) ω T t|T t|T t|T t|T t|T t=1 Thealgorithmisstoppedusingthelikelihood-basedcriterionofDozetal.(2012),withathresholdof 10−3. At convergence, we obtain an estimate of the trend and transitory component up to 2019:Q4, τ t 1 | 9 T and ω t 19 |T , respectively, along with the estimated parameters ψb19, σ bν 2,19 and Σb1 ω 9. Step 2: Estimate the Covid factor and volatility (Covid step) Given the estimated parameters up to 2019:Q4, we run the Kalman Filter and Smoother using all the (standardized) data, y˜ = (y˜ ,...,y˜ )′, to obtain the estimated states given the pre-Covid t 1,t n,t parameters. In doing so, we truncate the Kalman smoother in correspondence of 2020:Q1, to avoid spurious backward effects from the presence of Covid outliers. The estimated states are denoted as f(0) , D(0) , b(0) and ζ(0) . t|T t|T t|T t|T Given the smoothed states, let: ξbt = y˜ t −Λb19f t ( | 0 T ) −D( t 0 |T ) anddenoteasΞb = (ξ 1 ,...,ξ t )′ theT×nmatrixofidiosyncraticcomponents. Thenweestimatethe Covid factor by estimating the first principal component using the n×n variance-covariance matrix of the estimated idiosyncratic components from 2020:Q1 to 2021:Q4, denoted as Σb ΞC . This is the procedure proposed by Maroz et al. (2021) that we modify to account for non-stationarity in the idiosyncratic component. This done by partitioning the matrix of idiosyncratic components during the Covid period as ΞbC = (ΞbC,1|ΞbC,0 ), where ΞbC,1 and ΞbC,1 are the matrices of estimated idiosyncratic components in the period 2020:Q1 to 2021:Q4 for i ∈ I and i ∈ I , respectively. Then, we estimate 1 0 Σb ΞC (Hamilton, 2020, Chapter 17; Bai, 2004):   1 ΞbC,1 ′ΞbC,1 1 ΞbC,1 ′ΞbC,0 Σb ΞC =   (T 1 C)2 ΞbC,0 ′ΞbC,1 (TC 1 )3/ Ξb 2 C,0 ′ΞbC,0   (TC)3/2 TC Page 12 of 26

Supplementary material for the paper: Measuring the Euro Area Output Gap where Tc = 8 denotes the time-periods between 2020:Q1 and 2021:Q4. Given Σb ΞC , we obtain the Covid factor and the corresponding loadings as: √ γ b = n·Vb Ξ¯C 1 g b = √ n ·(Ξb CVb Ξ¯C ) where g b is the TC ×1 vector with entries g bt and Vb ΞC is the n×1 eigenvector corresponding to the largest eigenvalue of Σb ΞbC . Given g b , the associated loadings are γ b = (γ b1 ,...,γ bn )′. Next, give the estimate of states and the Covid factor, we account for the presence of changes in the volatility after the Covid shock by modifying the Lenza and Primiceri (2022) procedure to accommodate for quarterly data. Let s∗ = s I +(1−I ), the likelihood writes as: t t t≥t20Q1 t≥t20Q1 ( ) T T L(f(0)|A,Σ u ,s 1 ∗,...,s∗ T ) ∝ Y(cid:12) (cid:12)(s∗ t )2Σ u (cid:12) (cid:12) − 2 1 ·exp − 2 1X(cid:0) f t ( | 0 T ) −Af t ( − 0) 1|T (cid:1)′(cid:0) s2 t Σ u (cid:1)−1(cid:0) f t ( | 0 T ) −Af t ( − 0) 1|T (cid:1) t=2 t=2 ! ( ) T T ∝ Y (s∗ t )−n |Σ u |−T− 2 1 ·exp − 1 2 X(cid:0) f t ∗ | ( T 0)−Af t ∗ − (0 1 ) |T (cid:1)′ (Σ u )−1(cid:0) f t ∗ | ( T 0)−Af t ∗ − (0 1 ) |T (cid:1) t=2 t=2 where f∗(0) = f(0) /s∗, with corresponding variance-covariance matrix P∗(0). t|T t|T t t|T The maximum-likelihood estimators of A and Σ given the factors are: u T !−1 T ! A ˇ = X f∗(0)f∗(0)′ +P∗(0) X f∗(0) f∗(0)′ +P∗(0) t|T t−1|T t,t−1|T t−1|T t−1|T t−1|T t=2 t=1 1 T T ! Σ ˇ = X(cid:16) f∗(0)f∗(0)′ +P∗(0) (cid:17) −A ˇ X(cid:16) f∗(0)f∗(0)′ +P∗(0) (cid:17) u T t|T t|T t|T t|T t−1|T t,t−1|T t=2 t=2 Substituting A ˇ and Σ ˇ in the likelihood, we obtain the concentrated likelihood: u L(f(0)|A ˇ ,Σ ˇ ,s∗,...,s∗) = Y T (s∗)−n· (cid:12) (cid:12)Σ ˇ (cid:12) (cid:12) −T 2 u 1 T t (cid:12) u(cid:12) t=2 By numerically maximizing the concentrated likelihood we obtain the volatility parameters s. bt Step 3: Full sample estimation Given the Covid factor estimated in Step 2, we obtain all the other parameters in the model: - Factor loadings: T ! T !−1 λb ′ i = X(cid:16) y˜ it −D( i 0 ,t ) |T −ζ i ( , 0 t ) |T (cid:17) f t (0 | ) T ′ −γ bi g bt X f t (0 | ) T f t (0 | ) T ′ +P( t 0 | ) T t=1 t=1 - Slopes of secular trend: T ! T !−1 bb i = X(cid:16) y˜ it −λb ′ i f t ( | 0 T ) −ζ i ( , 0 t ) |T −γ bi g bt (cid:17) t X t2 t=1 t=1 Page 13 of 26

Supplementary material for the paper: Measuring the Euro Area Output Gap - Variance of I(1) idiosyncratic components: 1 T 1 T σ2 = X(cid:16) ζ(0) ζ(0)′ +Pζ(0) (cid:17) + X(cid:16) ζ(0) ζ(0)′ +Pζ(0) (cid:17) − bei T i,t|T i,t|T i,t|T T it−1|T it−1|T i,t−1|T t=2 t=2 2 T − X(cid:16) ζ(0) ζ(0)′ +Pζ(0) (cid:17) T it|T it−1|T i,t,t−1|T t=2 - Covariance prediction error: 1 ( T Rbi = T X(cid:16) y˜ i (0 ,t )−λb i (0)′ f t ( | 0 T ) −I i∈I b D( i 0 ,t ) |T −I i∈I1 ζ i ( , 0 t ) |T −γ bi g bt (cid:17)2 +λb ′ i P( t 0 ,T ) |T λbi + t=1 o + I PD(0) +I Pζ(0) i∈I b i,t|T i∈I1 i,t|T Given the estimated parameters, using data net of the Covid component, i.e. y˜ −γg , we obtain t bbt a final estimates of all the states, f , D , b , ζ and their conditional covariances with the t|T t|T t|T t|T Kalman Filter and Smoother. Note that, for this final run, the Kalman Filter and Smoother do not need to be truncated in 2019:Q4 because we have already controlled for the Covid pandemic. Withthefinalestimatesofthefactors, f weobtainafinalestimateofthetrendandtransitory t|T component by means of the EM algorithm. We initialize the EM algorithm with the estimated trend loadings up to 2019:Q4, normalized so that ψb19′ψb19 = 1. We denote this initial estimate as ψb(0). An initial estimate of the trend component is obtained by projecting f onto ψb(0), i.e. t|T τ(0) = (ψb(0)′ψb(0))−1ψb(0)′f . This also yields an initial estimate of the transitory component ω(0) = t t|T t f t|T −ψb(0)τ t (0). Finally, σ ν 2 is initialized as (400σ b∆ 2 τ )−1, where σ b∆ 2 τ is the sample variance of τ bt (0). After having computed the initial estimates for the algorithm, we estimate the trend via the usual EM algorithm. For a generic iteration k of the algorithm, we estimate the parameters as follow: - Trend loadings: T ! T !−1 ψb(k) = X f τ(k) X τ2(k)+Pτ(k) t|T t|T t|T t|T t=1 t=1 - Variance of common trend: 1 T 1 T σ2(k) = X(cid:16) τ2(k)+Pτ(k) (cid:17) + X(cid:16) τ2(k) +Pτ(k) (cid:17) − bν T t|T t|T T t−1|T t−1|T t=2 t=2 2 T − X(cid:16) τ(k)τ(k) +Pτ(k) (cid:17) T t|T t−1|T t,t−1|T t=2 - Covariance of transitory component Σb( ω k) = T 1 X T (cid:26)(cid:16) f t|T −ψb(k)τ t (k | ) T (cid:17)(cid:16) f t|T −ψb(k)τ t (k | ) T (cid:17)′ +ψb(k)P t τ | ( T k)ψb(k)′ (cid:27) t=1 Page 14 of 26

Supplementary material for the paper: Measuring the Euro Area Output Gap The algorithm is stopped using the likelihood-based criterion of Doz et al. (2012), with a threshold of 10−3. At convergence, we obtain an estimate of the trend and transitory components, τ and t|T ω t|T , respectively, along with the estimated parameters ψb , σ bν 2 and Σbω . Ultimately, given the estimated trend and transitory components, the estimated output gap and potential output are defined as: P d O t = D GDP,t|T +λb ′ GDP ψbτ t|T , O d G t = λb ′ GDP ω t|T E Confidence bands To obtain confidence bands for our quantities of interest, we follow the procedure outlined in Barigozzi and Luciani (2023). In particular, we simulate all the states in the model using the simulation smoother of Durbin and Koopman (2002), and we generate all the stationary residuals of the model using a stationary block bootstrap procedure (Politis and Romano, 1994). In practice, we have an estimate of all the states, namely f , D , b and ζ , an estimate of the Covid t|T t|T t|T t|T factor g and the estimated volatility parameters s , t≥t . Then, the algorithm is structured as bt bt 20Q1 follows: 1. Simulate the states by the simulation smoother (Durbin and Koopman, 2002) (a) Common factors: i. simulate ˜ f (b) ∼ N (cid:16) f ,P (cid:17) ; 1 1|T 1|T ii. simulate u˜(b) ∼ N (cid:16) 0 ,Σ ˇ (cid:17) ; t q u iii. for t = 2,...,T generate ˜ f (b) = Pp A ˇ ˜ f (b) +{s I +(1−I )}u˜(b). t k=1 k t−k t t≥t20Q1 t≥t20Q1 t (b) I(1) idiosyncratic components. For each i ∈ I : 1 i. simulate ζ ˜(b) ∼ N (cid:16) ζ ,Pζ (cid:17) ; i1 i,1|T i,1|T (cid:16) (cid:17) ii. simulate e˜(b) ∼ N 0,σ2 ; it bei iii. for t = 2,...,T generate ζ ˜(b) = ζ ˜(b) +e˜(b). it it−1 it (c) Time-varying secular components: • i = GDP, HHLB, HHLB.LLN : i. simulate ˜ b (b) ∼ N (cid:16) b ,Pb (cid:17) , and set D˜(b) =˜ b (b); i,1 i,1|T i,1|T i,1 i,1 (cid:16) (cid:17) ii. simulate η˜(b) ∼ N 0,σ2 ; t bηi iii. for t = 2,...,T generate ˜ b (b) =˜ b (b) +η˜(b); i,t i,t−1 t iv. for t = 2,...,T generate D˜(b) = D˜(b) +˜ b (b). i,t i,t−1 i,t • i = UNETOT, HICPOV, HICPNEF : i. simulate D˜(b) ∼ N (cid:16) b ,PD (cid:17) i,1 i,1|T i,1|T (cid:16) (cid:17) ii. simulate η˜(b) ∼ N 0,σ2 ; t bϵi iii. for t = 2,...,T generate D˜(b) = D˜ +η˜(b). i,t i,t−1 t Page 15 of 26

Supplementary material for the paper: Measuring the Euro Area Output Gap 2. Simulate the stationary residuals of the model, z = (z ,...,z )′, using a stationary blockt 1,t n,t bootstrap (Politis and Romano, 1994) with an average block length of four quarters. Denote the resulting simulated residuals as z˜(b) = (z˜(b) ,...,z˜(b))′. This step, along with step 2(b), t 1,t n,t gives us the simulated idiosyncratic component, i.e. ξ ˜(b) = ζ ˜(b) +z˜(b) if i ∈ I and ξ ˜(b) = z˜(b) i,t i,t i,t 1 i,t i,t if i ∈ I . 0 3. Generate the data. For t = 1,...,T, generate: (a) y˜ i ( t b) = D˜( it b)+λb ′ i ˜ f ( t b)+γ bi g bt +ξ ˜ i ( t b), for i = GDP,HHLB,HHLB.LLN,UNETOT,HICPOV,HICPNEF (b) y˜ i ( t b) = D i,t|T +λb ′ i ˜ f ( t b)+γ bi g bt +ξ ˜ i ( t b), for all other variables. 4. Using y˜(b) = (y˜(b) ,...,y˜(b))′, estimate the model as described in Appendix D to get a new t 1,t n,t estimate of the loadings, Λb (b) = (λb (b)′ ,...,λb (b)′)′, and all the other parameters in the model, 1 N as well as a new estimate of the states f (b) ,D (b) ,b (b) and ζ (b) . t|T t|T t|T t|T 5. Center the estimated states: ¯ f (b) = f − ˜ f (b), D ¯(b) = D −D ˜(b) +D (b) , ¯ b (b) = b − t|T t|T t t|T t|T t t|T t|T t|T ˜ b (b)+b (b) and ζ ¯(b) = ζ −ζ ˜(b)+ζ (b) . t t|T t t|T t t|T 6. Run the trend cycle decomposition on the estimated factors ¯ f (b) to get a new estimate of the t common trend τ (b), the transitory component ω (b) , and the parameter ψb (b). t|T t|T 7. Estimate potential output as P d O (b) = D¯(b) + λb (b)′ ψb (b)τ (b), and the output gap as t GDP,t|T GDP t|T O d G (b) = λb (b)′ ω (b) . t GDP t|T (b) RepeatingthisprocedureB times,weobtainadistributionoftheoutputgap: {O d G ,b = 1,...,B}. t Then, we construct the (1−α) confidence interval as [O d G t +z α/2 σ bt OG,O d G t +z 1−α/2 σ bt OG], where σ bt OG (b) is the sample standard deviation of {O d G t −O d G t } and z α/2 = −z 1−α/2 is the α/2-th quantile of a standard normal distribution. F Identification `a la Morley et al. (2023) Morley et al. (2023), henceforth MTW, propose an alternative trend smoothing approach to correct the estimated trend whenever it displays some serial correlation in first differences despite being assumed to be a random walk. We can apply the MTW approach in our setting by estimating an ARMA(1,1) model on the first difference of the estimated common trend ∆τ (0). The trend estimate bt corrected as in MTW is given by 1+θb ! ∆τ = ε , (F1) et 1−ϕb bt whereϕb andθb aretheestimatedARMAparameters,andε bt aretheARMAresiduals. Bycumulating ∆τ˜, we obtain the corrected estimate of the common trend, τ˜. t t In practice, smoothing as in (F1) seems to be less efficient than our proposal of using the Kalman smoother due to the presence of some residual Covid volatility in the estimated factors that affects the ARMA estimation. In particular, V d ar(∆τ bt ) = σ bν 2 = 0.092 (computed by excluding observation in 2020 and 2021 from the calculation) when estimating the model with our method, Page 16 of 26

Supplementary material for the paper: Measuring the Euro Area Output Gap and V d ar(∆τ et ) = (1+θb )(1−ϕb )−1σ bε 2 = 0.17, when estimating the model as in (F1), using MTW’s method, whereσ2 isthesamplevarianceofε . Althoughthesevaluesarequitesimilar, theybecome bε bt very different if we compute them using the whole sample; indeed, they increase to 0.095 and 0.52, respectively. This difference is essentially due to anomalous fluctuations in the ARMA residuals ε bt during 2020-2021. Figure F1 compares our output gap estimate with the one obtained by applying the correction proposed by Morley et al. (2023). As expected, the two approaches yield very similar estimates, but in 2021 and 2022, when the MTW corrections reduce the estimate of the output gap from +4% to +3%. Figure F1: Output gap estimate with identification `a la Morley et al. (2023) 6 4 2 0 -2 -4 -6 BLL MTW 2001 2003 2005 2007 2009 2011 2013 2015 2017 2019 2021 2023 Notes: The black bold line is our estimate of the output gap. The grey shaded areas are the 68% and 84% confidence bands. Thelightredlineistheestimatedoutputgapobtainedapplying the correction of the preliminary estimated trend proposed by Morleyetal.(2023)(MTW). G Comparison with alternative measures Inthissection,wecompareourestimatesofpotentialoutputandtheoutputgapwiththoseobtained using four different univariate filters and one multivariate approach: 1) The Hodrick and Prescott (1997) filter (HP), with two different values for the smoothing parameter λ: (i) λ = 1600, commonly used for quarterly data, (ii) λ = 51200, as proposed by Borio (2014) to capture variability at lower frequencies. 2) The Hamilton (2018) filter (Ham), where the trend is the 8−step ahead forecast of quarterly GDP growth, obtained using the 4 most recent values of quarterly GDP for each time t, and the cycle is the residual obtained from this regression. 3) The boosted HP filter (bHP) of Phillips and Shi (2021), which improves on the standard HP filter by applying the filter recursively on the residuals extracted from previous iterations. The number of iterations m is a tuning parameter that controls the intensity of the updating, and it is chosen to minimize the information criterion proposed by the authors. 4) The Christiano and Fitzgerald (2003) filter (CF), with cutoff frequencies for the transitory component between 8 and 32 quarters. 5) A Butterworth filter (BT) for the transitory component, as proposed by Canova (2022). This Page 17 of 26

Supplementary material for the paper: Measuring the Euro Area Output Gap filter can be cast in state-space form, and its squared-gain function defines the frequencies attributed to the cycles. Here, we employ a first-order polynomial n = 1, with a cutoff point for the frequency set at ω = 0.04 and scale G = 1. 0 6) The multivariate Beveridge-Nelson (BN) decomposition based on a large Bayesian VAR, as proposed by Morley et al. (2023). The authors estimate the Euro Area output gap from 1999:Q1 to 2021:Q3. Here, due to the lack of availability of their data, we keep their original estimates, truncating the figure in correspondence with our starting point, i.e. 2001:Q1. Figure G1 presents the results of this exercise. Overall, the output gap obtained with our methodology aligns with those estimated with univariate models in terms of peaks and troughs. However, there are several differences in terms of shape and amplitude. Figure G1: Output gap estimates with alternative methods 0 HP filter 0 Hamilton filter 000000 Boosted HP filter 6 6 4 5 4 2 2 0 0 0 -2 -2 -4 -4 -6 -5 -6 -8 -8 -10 BLL -10 -10 BLL - - 1 1 4 2 H H P P : : 6 6 = = 1 5 6 2 0 0 0 00 -15 B H L A L M - - 1 1 4 2 b b H H P P : : 6 6 = = 1 5 6 2 0 0 0 00 2001 2003 2005 2007 2009 2011 2013 2015 2017 2019 2021 2023 2001 2003 2005 2007 2009 2011 2013 2015 2017 2019 2021 2023 2001 2003 2005 2007 2009 2011 2013 2015 2017 2019 2021 2023 Christiano-Fitzgerald filter 0 Butterworth filter 000000 Multivariate BN 6 6 6 4 4 4 2 2 2 0 0 0 - - 4 2 - - 4 2 -2 -6 -6 -8 -4 -8 -10 -6 B C L F L -12 B B L T L - - 1 1 2 0 B B L N L -14 2001 2003 2005 2007 2009 2011 2013 2015 2017 2019 2021 2023 2001 2003 2005 2007 2009 2011 2013 2015 2017 2019 2021 2023 2001 2003 2005 2007 2009 2011 2013 2015 2017 2019 2021 2023 Notes: Theblacklineisourbenchmarkestimateandthegreyshadedareasarethe68%and84%confidencebands,thered andbluelinesarealternativeestimates. H Accounting for Covid H.1 Why adjusting for the Covid shock matters Section 3.2 explained how we accounted for the Covid shock when estimating the model. The upper plots in Figure H1 compare our benchmark estimates with the one we would have obtained if we had estimated the model over the full sample without applying any adjustment for the Covid shock (“no adj.”). As can be seen, ignoring the Covid shock affects the estimates of potential output and the output gap throughout the sample, which is undesirable; moreover, ignoring the Covid shock distorts the estimate of common dynamics in 2020 and 2021. HavingshownthataccountingfortheCovidshockisnecessary,thequestioniswhetherastrategy differentfromtheoneweadoptedwouldhavebeendesirable. Forexample,whatifwehadestimated all parameters up to 2019:Q4 and then extracted the states by simply truncating the Kalman Smoother? Despite being effective, this strategy is sub-optimal because estimating the parameters Page 18 of 26

Supplementary material for the paper: Measuring the Euro Area Output Gap Figure H1: Output gap when using pre-Covid parameters or without Covid adjustment Potential output growth Output gap tnemtsujda divoC oN 8 6 4 4 2 2 0 0 -2 -4 -2 -6 -8 -4 GDP -10 BLL BLL noadj. -12 noadj. -6 2002 2004 2006 2008 2010 2012 2014 2016 2018 2020 2022 2001 2003 2005 2007 2009 2011 2013 2015 2017 2019 2021 2023 sretemarap divoC-erP 8 6 4 4 2 2 0 0 -2 -4 -2 -6 -8 -4 GDP -10 BLL BLL pars2019 -12 pars2019 -6 2002 2004 2006 2008 2010 2012 2014 2016 2018 2020 2022 2001 2003 2005 2007 2009 2011 2013 2015 2017 2019 2021 2023 Notes: Theblacksolidlineisourbenchmarkestimateandthegreyshadedareasarethe68%and84%confidencebands,the blackdashedlineisGDPYoYgrowthrate,theredlinesaretheestimatesobtainedwithtwoalternativeestimationstrategies: 1. (left)fixingparametersestimatedupto2019:Q4(pars2019);2. (right)estimatingthemodelwithnoadjustmentforCovid (noadj.). up to 2019 becomes less and less justifiable as new data come in. Moreover, if the increase in volatility induced by the Covid shock turns out to be very persistent, confidence intervals would be underestimated because they only account for the pre-Covid volatility regime. By accounting for the Covid shock, we avoid both these issues. The lower plots in Figure H1 compare our benchmark estimate with the one obtained by estimating the parameters up to the last quarter of 2019 (“pars 2019”). The two estimates are very similar up to the pandemic, after which the estimate using the up-to-2019 parameters points towards much larger fluctuations in potential output growth and a much larger output gap, which, if taken at face value, signals a very tight economy. H.2 Covid volatiliy Section 3.2 explained how we accounted for the effect of the Covid shock on the volatility of the common factors. Specifically, we follow Lenza and Primiceri (2022) and introduce a factor s that t scales the volatility of the common factors from the beginning of the pandemic onward. Lenza and Primiceri (2022) analyzed monthly US data and impose an exponential decay for s starting in June t 2020. In contrast, we estimate one parameter for each period starting in 2020:Q1. This choice is motivated by both the different impact and policy response of the Covid pandemic in Europe and by the fact that, as Morley et al. (2023) pointed out, quarterly data do not allow for a sharp Page 19 of 26

Supplementary material for the paper: Measuring the Euro Area Output Gap identification of the decay parameter. The left plot in Figure H2 shows the estimated scaling factor s obtained under both our t parametrization (black line) and under the exponential decay parametrization proposed by Lenza and Primiceri (2022) (red line). Our estimate of the volatility closely tracks the evolution of the pandemic, as it spikes in the first two quarters of 2020 when mobility restrictions were most stringent in Europe, and pick-up again in 2021:Q1 when the spread of the Delta variant reached its peak. Moreover, we estimate that the volatility is very persistent—the decay we get is very close to 1 (≈ 0.98)—much more persistent than estimated by Lenza and Primiceri (2022), who estimated their model on monthly US data. This difference is likely due to the evolution of the pandemic in Europe, where mobility restriction measures were much more restrictive than in the US, lasted for longer, and were also implemented in 2021. Moreover, the Russia-Ukraine war had a much larger impact on Europe by pushing natural gas (and gasoline prices to a lesser extent) to the roof and creating a lot of macro-financial uncertainty. This result motivates the need to allow for time variation in the factor volatility until the end of the sample. Figure H2: Output gap estimate with alternative Covid volatility Estimated Covid volatility Output gap 5.5 BLL LP 6 5 4 4.5 4 2 3.5 0 3 -2 2.5 2 -4 1.5 -6 BLL LP 1 2019Q4 2020Q2 2020Q4 2021Q2 2021Q4 2022Q2 2022Q4 2023Q2 2001 2003 2005 2007 2009 2011 2013 2015 2017 2019 2021 2023 Notes: Intheleftplot,theblacklineisourestimateofCovidvolatility(st),theredlineistheestimateobtainedbyassuming the exponential decay parametrization of the Covid volatility as in Lenza and Primiceri (2022). In the right plot, the black lineisourbenchmarkestimateandthegreyshadedareasarethe68%and84%confidencebands,theredlineistheestimate obtainedwiththeexponentialdecayparametrizationoftheCovidvolatility. As shown in the right plot in Figure H2, the two parametrizations lead to virtually identical results. H.3 Alternative estimator for the Covid factor The approach we described in Appendix D to estimate the Covid factor has the benefit of retaining all the information in ξbt , but it has the problem of relying only on eight data points. That said, since we are only interested in the first eigenvector of Σb , and given the extent to which the series ΞbC co-moved during the Covid period, even a few data points should be informative. However, the estimates could be imprecise, thereby motivating an alternative estimation strategy. As an alternative approach, we estimate the Covid factor by estimating the first principal component using the TC ×TC variance-covariance matrix of the estimated idiosyncratic components from 2020:Q1 to 2021:Q4, denoted as Σ ˜ ΞC . In order to estimate Σ ˜ ΞC , we consider ξbit if i ∈ I 0 and Page 20 of 26

Supplementary material for the paper: Measuring the Euro Area Output Gap ∆ξbit if i ∈ I 1 , i.e. we take first-differences of all non-stationary idiosyncratic components.(xi) We obtain the Covid factor and the corresponding loadings as: √ g˜ = TC ·V ˜ Ξ¯C 1 γ˜ = √ ·(Ξb CV ˜ Ξ¯C ) TC where g˜ is the TC ×1 vector with entries g˜ and V˜ is the TC ×1 eigenvector corresponding to t ΞC the largest eigenvalue of Σ ˜ . Given g˜, the associated loadings are γ˜ = (γ˜ ,...,γ˜ )′. Thissecondstrategyall Ξ o b w C sustoestimateΣ ˜ withN datapoints, ther 1 ebyyiel n dingamorepre- ΞbC cise estimate. However, this comes at the cost of missing important information due to differencing of the non-stationary idiosyncratic component. Which one of the two approaches is better? As a robustness exercise, in this Appendix we look at what would have been the output gap estimate, had we adopted the second strategy to estimate the Covid factor that we just laid out. As shown in Figure H3, the results obtained with the alternative Covid factor are almost identical to those obtained in the benchmark specification. This is not surprising, since the co-movoments observed in most of the series during the Covid period are so large to be easily identified even with a limited range of observations. Figure H3: Output gap estimate with the alternative Covid factor Potential Output: YoY Growth Rates Output Gap: Levels 6 4 4 2 2 0 0 -2 -2 -4 -4 GDP BLL -6 BLL BLL BLL -6 2002 2004g2006 2008 2010 2012 2014 2016 2018 2020 2022 2001 2003g2005 2007 2009 2011 2013 2015 2017 2019 2021 2023 Notes: The black solid line is our benchmark estimate and the grey shaded areas are the 68% and 84% confidence bands. The red solid lines are the estimates obtained with the alternative Covid factor (BgLL). The level of the output gap is the percentagedeviationfrompotential. I No time-varying parameters Figure I1 compares the benchmark estimates of the output gap (right plot) and YoY potential output growth (left plot) with those obtained without allowing for a local linear trend for GDP, household liabilities, and long-term loans and no time-varying mean for the unemployment rate. Removing the time variation in the secular trends leads to a flatter estimate of potential output growth in the post-pandemic periods. This result shows that allowing for a time-varying trend (xi)In this case ξbi will be a T −1 vector denoting at time t the level of the idiosyncratic component for i∈I 0 and the growth rate of the idiosyncratic component for i∈I . 1 Page 21 of 26

Supplementary material for the paper: Measuring the Euro Area Output Gap for GDP is crucial to properly capture the slowdown in potential output in the latter part of the sample. Figure I1: Output gap estimate when imposing no time-varying parameters Potential Output: YoY Growth Rates Output Gap: Levels 6 4 4 2 2 0 0 -2 -2 -4 -4 GDP BLL -6 BLL noTVpars noTVpars -6 2002 2004 2006 2008 2010 2012 2014 2016 2018 2020 2022 2001 2003 2005 2007 2009 2011 2013 2015 2017 2019 2021 2023 Notes: The black solid line is our benchmark estimate, the grey shaded areas are the 68% and 84% confidence bands, the black dashed line is GDP YoY growth rate, the red line is the estimate obtained without time-varying parameters (no TV pars). Theleveloftheoutputgapisthepercentagedeviationfrompotential. J Real-time reliability There is skepticism in the literature on the reliability of model-based output gap estimates in real time because of the size of end-of-sample revisions (Orphanides and van Norden, 2002). Modelbased estimates of the output gap are subject to revisions in real time because new information leadstochangesinboththeestimatesofthemodelparametersandthelatentstates. Inthissection, we assess the reliability of our output gap estimate through a quasi-real-time exercise on expanding windows, where the first window begins in 2000:Q1 ends in 2015:Q1 (T = 57). Looking at the upper charts in Figure J1, it is clear that the model is slow in recognizing how deepthe2012Sovereigndebtcrisisis, onlydoingsooncedatafor2016becomesavailable. Inasmuch asthisresultisdisappointing,wecannothelpbutnoticethat60observationsareprobablytoofewto pin down the output gap accurately. As more information becomes available, the model’s estimate of the output gap stabilizes, so much so that from 2017 onward, the quasi-real-time estimate is very close to the final estimate. Moving to the Covid pandemic and its aftermath, it is evident that our strategy for adapting to the COVID shock was viable only starting from the second half of 2020 or even 2021. Moreover, it is reasonable to assume that anybody would have understood that doing nothing and allowing the Covid shock to affect the estimates was a huge mistake. Thus, we consider the quasi-real-time performance of the simple strategy of freezing the parameters to pre-Covid data. As shown by the green line, if we had followed this approach, we would have had very reliable output gap estimates, as opposed to extreme estimates, had we chosen to do nothing. Finally, by comparing the red and the green lines, we can appreciate the impact of the adjustment we implemented to account for the Covid shock. Page 22 of 26

Supplementary material for the paper: Measuring the Euro Area Output Gap Figure J1: Output Gap: quasi-real time, expanding window 2015:Q4 2016:Q4 2017:Q4 8 8 8 6 6 6 4 4 4 2 2 2 0 0 0 -2 -2 -2 -4 -4 -4 -6 -6 -6 -8 -8 -8 -10 -nal -10 -nal -10 -nal BLL BLL BLL -12 -12 -12 2001 2003 2005 2007 2009 2011 2013 2015 2017 2019 2021 2023 2001 2003 2005 2007 2009 2011 2013 2015 2017 2019 2021 2023 2001 2003 2005 2007 2009 2011 2013 2015 2017 2019 2021 2023 2018:Q4 2019:Q4 2020:Q4 8 8 8 6 6 6 4 4 4 2 2 2 0 0 0 -2 -2 -2 -4 -4 -4 -6 -6 -6 -8 -8 -8 -nal -10 -nal -10 -nal -10 pars.19 BLL BLL noadj. -12 -12 -12 2001 2003 2005 2007 2009 2011 2013 2015 2017 2019 2021 2023 2001 2003 2005 2007 2009 2011 2013 2015 2017 2019 2021 2023 2001 2003 2005 2007 2009 2011 2013 2015 2017 2019 2021 2023 2021:Q4 2022:Q4 2023:Q4 8 8 8 6 6 6 4 4 4 2 2 2 0 0 0 -2 -2 -2 -4 -4 -4 -6 -6 -6 -8 - B n L a L l -8 - B n L a L l -8 - B n L a L l -10 pars19 -10 pars19 -10 pars19 noadj. noadj. noadj. -12 -12 -12 2001 2003 2005 2007 2009 2011 2013 2015 2017 2019 2021 2023 2001 2003 2005 2007 2009 2011 2013 2015 2017 2019 2021 2023 2001 2003 2005 2007 2009 2011 2013 2015 2017 2019 2021 2023 Notes: The black line is the estimate of the output gap obtained using the final sample of data, up to 2023:Q4. The red line inFigureJ1displaystheevolutionofourquasi-real-timeestimateoftheoutputgapobtainedusingtheprocedureoutlinedin Section 3. The blue line shows the estimate we would have obtained without any adjustment for the Covid shock. Lastly, the greenlineistheestimatewewouldhaveobtainedifwehadfrozentheparameterestimateatthevaluein2019:Q4. Figure J2: Output gap: quasi-real time results Adjusted for Covid HP CF 8 8 8 6 6 6 4 4 4 2 2 2 0 0 0 -2 -2 -2 -4 -4 -4 -6 -6 -6 -8 -8 -8 -10 -10 -10 -12 -12 -12 2001 2003 2005 2007 2009 2011 2013 2015 2017 2019 2021 2023 2001 2003 2005 2007 2009 2011 2013 2015 2017 2019 2021 2023 2001 2003 2005 2007 2009 2011 2013 2015 2017 2019 2021 2023 Notes: theblacklineistheestimateoftheoutputgapobtainedoverthefullsample,andthethingreylinesaretheestimate obtained on all the other subsamples. Each black dots represent the estimate of the output gap for quarter Q and year Y obtainedonthesampleendingatquarterQ andyearY. Figure J2 compares the quasi-real-time estimate of the output gap from our model with those obtained from an HP filter and a Christiano-Fitzgerald band-pass filter. As mentioned earlier, our estimate’s weakness is that the one obtained on the sample ending in 2015 is quite different from Page 23 of 26

Supplementary material for the paper: Measuring the Euro Area Output Gap the final estimate. However, its strength is that from 2017 onward, the quasi-real-time estimates converge to the final and are very robust. In contrast, the HP filter and the Christiano-Fitzgerald filter yield quasi-real-time estimates that are never to far from the final estimate. However, they converge to the final estimate very late, making them unreliable for real-time analysis. References Ahn, S. C. and A. R. Horenstein (2013). Eigenvalue ratio test for the number of factors. Econometrica 81, 1203–1227. Alessi, L., M.Barigozzi, andM.Capasso(2010). Improvedpenalizationfordeterminingthenumber of factors in approximate factor models. Statistics & Probability Letters 80(23-24), 1806–1813. Antolin-Diaz, J., T. Drechsel, and I. Petrella (2017). Tracking the slowdown in long-run GDP growth. The Review of Economics and Statistics 99, 343–356. Bai, J. (2004). Estimating cross-section common stochastic trends in nonstationary panel data. Journal of Econometrics 122, 137–183. Bai, J. and S. Ng (2002). Determining the number of factors in approximate factor models. Econometrica 70, 191–221. Bai, J. and S. Ng (2004). A PANIC attack on unit roots and cointegration. Econometrica 72, 1127–1177. Barigozzi, M., M.Lippi, andM.Luciani (2021). Large-dimensionaldynamic factormodels: Estimation of impulse-response functions with I(1) cointegrated factors. Journal of Econometrics 221, 455–482. Barigozzi, M.andM.Luciani(2023). MeasuringtheOutputGapusingLargeDatasets. The Review of Economics and Statistics 105, 1500–1514. Borio, C. (2014). The financial cycle and macroeconomics: What have we learnt? Journal of Banking & Finance 45, 182–198. Canova, F. (2022). FAQ: How do I estimate the output gap? mimeo. Casalis, A. and G. Krustev (2022). Cyclical drivers of euro area consumption: What can we learn from durable goods? Journal of International Money and Finance 120, 102241. Cette, G., J. Fernald, and B. Mojon (2016). The pre-Great Recession slowdown in productivity. European Economic Review 88, 3–20. Christiano, L. J. and T. J. Fitzgerald (2003). The bandpass filter. International Economic Review 44, 435–465. Del Negro, M., D. Giannone, M. P. Giannoni, and A. Tambalotti (2017). Safety, liquidity, and the natural rate of interest. Brookings Papers on Economic Activity Spring 2017, 235–316. Del Negro, M., D. Giannone, M. P. Giannoni, and A. Tambalotti (2019). Global trends in interest rates. Journal of International Economics 118, 248–262. Del Negro, M., F. Schorfheide, F. Smets, and R. Wouters (2007). On the fit of new Keynesian models. Journal of Business & Economic Statistics 25, 123–143. Page 24 of 26

Supplementary material for the paper: Measuring the Euro Area Output Gap Doz, C., D. Giannone, and L. Reichlin (2012). A quasi-maximum likelihood approach for large, approximate dynamic factor models. The Review of Economics and Statistics 94, 1014–1024. Durbin, J. and S. J. Koopman (2002). A simple and efficient simulation smoother for state space time series analysis. Biometrika 89, 603–616. Gordon,R.J.(2018). DecliningAmericaneconomicgrowthdespiteongoinginnovation.Explorations in Economic History 69, 1–12. Hallin, M. and R. Liˇska (2007). Determining the number of factors in the general dynamic factor model. Journal of the American Statistical Association 102, 603–617. Hamilton, J. D. (2018). Why you should never use the Hodrick-Prescott filter. The Review of Economics and Statistics 100, 831–843. Hamilton, J. D. (2020). Time series analysis. Princeton university press. Hodrick, R. J. and E. C. Prescott (1997). Postwar US business cycles: an empirical investigation. Journal of Money, Credit, and Banking, 1–16. Jarocin´ski, M. and M. Lenza (2018). An inflation-predicting measure of the output gap in the euro area. Journal of Money, Credit and Banking 50, 1189–1224. Lenza, M. and G. E. Primiceri (2022). How to estimate a vector autoregression after March 2020. Journal of Applied Econometrics 37, 688–699. Lippi, M. and L. Reichlin (1994). Diffusion of technical change and the decomposition of output into trend and cycle. The Review of Economic Studies 61, 19–30. Maroz, D., J. H. Stock, and M. W. Watson (2021). Comovement of economic activity during the Covid recession. mimeo. Morley, J., D. Rodr´ıguez-Palenzuela, Y. Sun, and B. Wong (2023). Estimating the euro area output gap using multivariate information and addressing the COVID-19 pandemic. European Economic Review 153. Morley, J., T. D. Tran, and B. Wong (2023). A simple correction for misspecification in trend-cycle decompositions with an application to estimating r. Journal of Business & Economic Statistics. available online. Ng, S. (2018). Comments on the cyclical sensitivity in estimates of potential output. Brookings Papers on Economic Activity 49, 412–423. Ng, S. (2021). Modeling macroeconomic variations after COVID-19. NBER working paper 29060. Onatski, A. (2009). Testing hypotheses about the number of factors in large factor models. Econometrica 77, 1447–1479. Opschoor, D. and D. J. van Dijk (2023). Slow Expectation-Maximization convergence in low-noise dynamic factor models. SSRN working paper 4408065. Orphanides, A. and S. van Norden (2002). The unreliability of output-gap estimates in real time. The Review of Economics and Statistics 84, 569–583. Phillips, P. C. and Z. Shi (2021). Boosting: Why you can use the HP filter. International Economic Review 62, 521–570. Page 25 of 26

Supplementary material for the paper: Measuring the Euro Area Output Gap Politis, D. N. and J. P. Romano (1994). The stationary bootstrap. Journal of the American Statistical Association 89, 1303–1313. T´oth, M. (2021). A multivariate unobserved components model to estimate potential output in the Euro Area: A production function based approach. ECB working paper 2523. Zhang, R., P. Robinson, and Q. Yao (2019). Identifying cointegration by eigenanalysis. Journal of the American Statistical Association 114, 916–927. Page 26 of 26