The exact contents of citations.db main_text.text for this paper — one flattened LaTeX string, title through conclusion, appendix excluded, unmodified except for removing email addresses. This is what our citation measures are computed over.
99,842 characters
Measuring the Euro Area Output Gap$^$
\maketitle
\vspace{-20pt}
\begin{abstract}
\noindent We measure the Euro Area (EA) output gap and potential output using a non-stationary dynamic factor model estimated on a large dataset of macroeconomic and financial variables. Our results indicate that, between 2012 and 2024, the EA economy was consistently tighter than suggested by institutional estimates, implying that its weak growth reflects a potential output problem rather than a business-cycle one. Moreover, we find that the decline in trend inflation---rather than economic slack---kept core inflation below 2\% before the pandemic, while demand forces explain at least 30\% of the post-pandemic rise in core inflation.
\end{abstract}
\thispagestyle{empty}
\footnotetext{We would like to thank for helpful comment Travis Berge, Danilo Cascaldi Garcia, Antonio Conti, Thiago Ferreira, Domenico Giannone, Manuel Gonzalez-Astudillo, Michele Lenza, Giovanni Pellegrino, and Riccardo Trezzi. This paper has benefited also from discussions with seminar participants at the Federal Reserve Board and with participants of several conferences. M. Barigozzi and C. Lissona gratefully acknowledge financial support from MIUR (PRIN2020, Grant 2020N9YFFE). Of course, any error is our responsibility.\smallskip
\noindent \textsc{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.}
\section{Introduction}
The decomposition of GDP in potential output---the level of output consistent with current technologies and ``normal'' use of capital and labor---and the output gap---the percentage deviation of GDP from its potential---is a fundamental task for policymakers. Potential output tells us how fast an economy can grow in the long run; the output gap helps assess the cyclical position of the economy and, thus, potential inflationary pressures \citep[e.g.,][]{jarocinski2018inflation, banbura_does_2023}. Both measures are central to the common Euro Area (EA) monetary policy and the fiscal policy of individual countries---they are among the main pillars of the EA fiscal surveillance framework, ultimately affecting the fiscal capacity of each member country \citep{EC_Stability}. However, since both quantities are unobserved, policymakers need a model to 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 macroeconomic and financial variables. Compared to the prevailing literature, which focuses on theoretical structural models with few variables of interest, we adopt a 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 \citep{cette2016pre}.
We conduct our analysis on a new large-dimensional dataset comprising 118 EA economic indicators from 2001:Q1 to 2025:Q1. Four 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) 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), and as of 2025:Q1, 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 both implement structural reforms and promote productivity-enhancing investments, and support aggregate demand more forcefully during downturns to mitigate recession-induced output losses.
Second, we find that, on average, the Okun's law relationship and the Phillips Curve are satisfied in our model, even though we do not impose either of them; hence, cyclical movements in real activity, unemployment, and inflation are interconnected \citep[see, e.g.,][]{BianchiNicoloSong}.
Third, we find that core inflation remained below 2\% after the GFC, not because there was slack in the economy, but rather because trend inflation decreased by one percentage point---in line with the idea that inflation expectations de-anchored on the downside after the GFC \citep{CiccarelliOsbat, CorselloNeriTagliabracci}. Moreover, we show that the output gap contributed to at least 30\% of the post-pandemic increase in core inflation, thus supporting existing literature that suggests demand forces played a substantial role in the rise of post-pandemic inflation \citep{AscariTrezzi,GiannonePrimicieri,Canovaetal}. Finally, our output gap measure yields better inflation forecasts than those derived from commonly used methods. These results confirm that our data-driven measure is economically meaningful and valuable for policy analysis.
Fourth, we find that growth financed through household debt is not sustainable in the long run; hence, policies aimed at boosting debt-financed household consumption or residential investment would deliver only short-term gains.
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 jointly with the Kalman smoother \citep{doz2012quasi,barigozzi_quasi_2022}. Then, we extract a common trend from the estimated common factors and compute the cyclical component by subtracting the common trend from the common factors. 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. Our model belongs to the class of unobserved component models, which is among the most indicated ones for extracting the transitory component of GDP \citep{canova2022faq}.
This model is an enhanced version of the model \citet{barigozzi_measuring_2021} used to measure the US output gap, as it does not require a long sample to identify the trend and deals with the Covid pandemic. Specifically, we use a three-step estimation procedure to account for the latter. First, we estimate the model using only pre-Covid data. Second, we estimate the effects induced by the Covid shock (level-shift and increased volatility). Third, we re-estimate the model on the full dataset after purging the data from Covid-induced dynamics.
Our model is both large-dimensional and non-stationary. A large-dimensional model enables us to capture well-established co-movements in macroeconomic variables. It is now widely recognized that cross-sectional aggregation of a large number of series allows to consistently disentangle the co-movements in the data from idiosyncratic dynamics \citep{stock_chapter_2016}, and that a rich information set is necessary to obtain meaningful estimates of potential output and the output gap \citep{buncic_discovering_2022}. To the best of our knowledge, this is the first paper that estimates the EA output gap with such a rich information set.
Allowing for non-stationarity enables us to go beyond the common practice of pre-transforming data into stationary, mean-centered variables. This approach preserves features that are crucial for identifying the common trend, which would otherwise be lost if the data were differenced to achieve stationarity \citep{ng2018comments}. This is important not only for estimating the output gap but also because accurately accounting for low-frequency movements helps eliminate any potential confounding effect when estimating the relationship between real activity and inflation over the business cycle \citep{BianchiNicoloSong}.
\paragraph{Related literature.} How to estimate potential output and the output gap has been a hotly debated topic for several decades \citep[see, e.g.,][]{canova2022faq}. 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 \citep{havik2014production} and the IMF \citep{demasi1997} or New-Keynesian DSGE models \citep{JPT,burlon2020reliable,furlanetto2021output}.
The statistical approach uses (univariate or multivariate) statistical models, sometimes paired with some macroeconomic relationships of interest, e.g., the Phillips Curve. For example, many papers rely on univariate models \citep[e.g.,][]{morley2003beveridge,kamber2018intuitive,hamilton2018you,phillips2021business,phillips2021boosting,hartl2022fractional}, while few others employ multivariate non-stationary methods, which are either low-dimensional models \citep{jarocinski2018inflation,ManuelGA,toth_multivariate_2021, hasenzagl2022model}, or medium-size, but stationary, models \citep{aastveit2014estimating,morley2020estimating,morley_estimating_2023,OgapBilkent}. Most of these works focus on the US, while only a few focus on the EA, the most recent being \citet{morley_estimating_2023}.
\paragraph{Structure of the paper.} The rest of the paper is organized as follows. In Section \ref{sec::data}, we present the data used in our analysis, and in Section \ref{sec::method}, we present the model and the estimation strategy. Then, in Section \ref{sec::poutgap}, we present our estimate of potential output and the output gap, and in Section \ref{sec::OL&PC}, we dive deep into the economic content of these estimates, focusing on the Okun's law and the Phillips correlation. Next, we focus on inflation. Section \ref{sec::InflationDynamics} interprets inflation dynamics after the GFC through the lens of our model, and Section \ref{sec::InflationForecasting} assesses the ability of our output gap estimate to forecast inflation. Finally, Section \ref{sec::Credit} looks into what signals our model takes from financial indicators to estimate the output gap, and Section \ref{sec::conclude} concludes.
A Supplemental Appendix contains additional details on the model, showing various robustness analyses, and reporting additional results. Specifically, Appendix \ref{datades} provides full details on the dataset, while Appendixes \ref{app:ass}, \ref{sec::estdetail}, and \ref{app::confbands} provide additional details about the model. Next, Appendixes \ref{sec::covid} and \ref{app::notvpars} provide robustness analysis. Appendix \ref{sec::altTC} compares our output gap estimate with that obtained with alternative statistical methodologies. Lastly, Appendix \ref{sec::altTCest} compares our output gap estimate with that obtained with alternative estimation strategies for the common trend, and Appendix \ref{sec::realtime} assesses the reliability of our output gap estimate.
\section{A large Euro Area dataset} \label{sec::data}
We construct a large macroeconomic dataset of $n=118$ EA series, observed from 2001:Q1 to 2025:Q1 ($T=97$). 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, 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 \citep[see, for example,][]{mccracken_fred-md_2016,mccracken_fred-qd_2020}.
In terms of which and how many series to include for each category, we use a mix of economic and 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 \citep{boivin2006more,lucioADFM}. 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 $I(2)$ behavior in price indexes \citep{stock_chapter_2016,mccracken_fred-md_2016,mccracken_fred-qd_2020}. Appendix \ref{datades} provides the complete list of variables included in the dataset, along with their sources and treatment.
\section{Methodology}\label{sec::method}
In Section \ref{sec::mod}, we outline the model and its main features---we discuss all the details, formal assumptions, and further comments in Appendix \ref{app:ass}. In Section \ref{sec::fattoni}, we sketch how we estimate the model while referring the reader to Appendix \ref{sec::estdetail} for a step-by-step guide on how estimation is carried out in practice and Appendix \ref{app::confbands} for the bootstrap procedure used to measure uncertainty around our estimates.
\subsection{The model}\label{sec::mod}
We denote the observed $i$-th time series at a given quarter $t$ as $y_{it}$, with $1\le i\le 118$ and $\text{2001:Q1} \le t\le \text{2025:Q1}$. In our non-stationary dynamic factor model, each variable is the sum of (i) a secular component $\mathrm{D}_{it}$, which is treated either as deterministic or stochastic, (ii) $q$ common factors $\mathbf{f}_t=(f_{1t}\cdots f_{qt})'$, which capture the macroeconomic long- and short-run co-movements and have a dynamics governed by a VAR, and (iii) an idiosyncratic component $\xi_{it}$, which captures local dynamics 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 $\mathrm{D}_{it}$ is a stochastic or a deterministic process. Second, according to the nature of the idiosyncratic component, that is, whether $\xi_{it}$ has a stochastic trend or it is stationary.
In particular, we model $\mathrm{D}_{it}$ as a local linear trend for GDP to account for the well-documented slowdown in productivity \citep{cette2016pre} and for households' financial liabilities ({\small HHLB}) and households' long-term loans ({\small HHLB.LLN}), which make up more than 80\% of total household's liabilities, whose average growth rate has slowed down consistently since the GFC. {For these variables, we say that $i\in \mathcal{L}_1$}. Moreover, we model $\mathrm D_{it}$ as a local level model for the unemployment rate ({\small UNETOT}) to account for relevant labor market features \citep{cette2016pre}---this includes the reallocation of employees across sectors, which contributed to the slowdown in the EA productivity growth---and for all consumer price inflation indexes ({\small HICPOV, HICPNEF, HICPG, HICPSV, HICPNG, HICPFD}) as well as oil and natural gas prices ({\small POIL, PNGAS}) to account for the slowdown in inflation occurred after the GFC---accounting for low-frequency dynamics in inflation is crucial to avoid potential confounding effects which may alter the relation between inflation and real activity over the business cycle \citep{BianchiNicoloSong}. For these variables, we say that $i\in \mathcal{L}_0$.\footnote{Appendix \ref{app::notvpars} shows results for the estimate of the output gap when removing the time variation in the secular trends.} For all other series, $\mathrm{D}_{it}$ is either a linear trend with a constant slope, in which case we say that $i\in\mathcal I_b$, or $\mathrm D_{it}$ is just a constant equal to $\mathrm D_{i0}$. To determine $\mathcal I_b$, we test the significance of the sample mean of $\Delta{y}_{it}$ (see Appendix \ref{datades}).
As for the idiosyncratic components, if $\xi_{it}\sim I(1)$, then we say that $i\in\mathcal I_1$ and we model $\xi_{it}$ as a random walk, while if $\xi_{it}\sim I(0)$, we say that $i\notin\mathcal I_1$ and we leave its dynamics unspecified to avoid over-parametrization of the model. To determine $\mathcal I_1$, we employ the test proposed by \cite{bai2004panic} for the null hypothesis of an idiosyncratic unit root (see Appendix \ref{datades}).
Furthermore, we capture the effect of the Covid shock, which generated a large shift both in the levels \citep{ng2021modeling,stockcomovement}, and in the volatility \citep{lenza_how_2022,carriero_addressing_2022} of most macroeconomic EA series, through an additional common factor $g_t$ \citep{stockcomovement}, and a scalar $s_t$ scaling the conditional volatility of the latent factors \citep{lenza_how_2022}. While we model the former to have an impact on all series only in 2020 and 2021, we allow the latter to have an effect that persists even after the recovery from the pandemic. These choices reflect the fact that mobility restrictions and lockdowns in the EA have been on and off until early 2022, while in the US, they were enforced only at the beginning of the pandemic.
Formally, the model reads as follows:
\begin{align}
y_{it}\ &=\ \mathrm{D}_{it} + \bm\lambda_{i}^\prime\mathbf {f}_{t} + \gamma_i\hspace{1pt} g_t \mathbb I_{\text{\tiny 2020:Q1$\le\! t\!\le$2021:Q4}} + \xi_{it}, \hspace{10pt}&& 1\le i\le 118,\;\;\text{\footnotesize 2001:Q1}\le t \le \text{\footnotesize 2025:Q1},
\label{eq::obseq} \\[3pt]
\mathrm{D}_{it}\ &=\ \mathrm{D}_{it-1} + b_{i,t-1}\mathbb I_{i\in\mathcal I_b} + \epsilon_{it}\hspace{1pt}, && \epsilon_{it}\sim (0,\sigma_{\epsilon_i}^2\mathbb I_{i\in \mathcal{L}_0}),\label{eq::seccomp}\\[3pt]
b_{it}\ &=\ b_{it-1} +\eta_{it}\hspace{1pt}, && \eta_{it}\sim(0,\sigma_{\eta_i}^2\mathbb I_{i\in \mathcal{L}_1}), \label{eq::betat} \\[3pt]
\mathbf{f}_t\ &=\sum_{j=1}^{p}\mathbf A_j\mathbf{f}_{t-j} + \{s_t \mathbb I_{\,\text{\tiny $t\!\ge\!$ 2020:Q1}} + \mathbb I_{\,\text{\tiny $t\!<\!$ 2020:Q1} })\}\mathbf{u}_t\hspace{1pt}, &&\mathbf{u}_t\stackrel{{i.i.d.}}{\sim}(\mathbf{0},\boldsymbol{\Sigma}_u), \label{eq::faceq}\\[3pt]
\xi_{it}\ &=\xi_{it-1} \mathbb I_{i\in\mathcal I_1}+ e_{it}, && e_{it}\stackrel{}{\sim} ({0},\sigma_{e_i}^2), \label{eq::idioeq}
\end{align}
where $\mathbb I_A=1$, if $A$ is true, and $\mathbb I_A=0$, otherwise, $\bm\lambda_i$, $\mathbf f_t$, $\mathbf u_t$ are $q$-dimensional vectors, and $\mathbf A_j$ and $\bm\Sigma_u$ are $q\times q$. We set $p=2$ based on the BIC criterion for a VAR on the estimated factors, and $q=4$ based on standard criteria implemented on differenced data \citep{bai2002determining,hallin_determining_2007,ABC}. Furthermore, following the criteria by \citet{barigozzi_large-dimensional_2021} and \citet{ACFZ} applied on the zero-frequency spectral density matrix of the differenced data, we impose the presence of one common trend, ${\tau}_t$, driving the non-stationarity in the factors $\mathbf{f}_t$. This finding is consistent with many theoretical models in which a common trend is the sole driver of long-run dynamics \citep[e.g.,][]{del2007fit}.
Given the above results, if follows that $\mathbf{f}_t$ is a cointegrated vector with cointegration rank $q-1=3$. Consequently, the VAR characteristic polynomial in \eqref{eq::faceq} has one root in $z=1$, while the remaining roots lie inside the unit circle. Hence, $\mathbf{f}_t$ also admits the following decomposition \citep{escribano1994cointegration}:
\begin{align}
\mathbf{f}_t\ &=\ \bm\psi \tau_t+\boldsymbol{\omega}_t, \qquad \boldsymbol{\omega}_t \sim (\mathbf{0}, \bm\Sigma_\omega), \label{sbeq::obsTR}
\end{align}
where $\bm\psi$ and $\bm\omega_t$ are $q$-dimensional vectors and $\bm\Sigma_\omega$ is $q\times q$.
In this paper, we assume that the common trend evolves as:
\begin{align}
\tau_t\ &=\ \tau_{t-1}+\nu_t, \qquad \nu_t \sim ( 0, \sigma_\nu^2).\label{sbeq::st1TR}
\end{align}
It is important to emphasize that, under our estimation approach, we do not need to impose a specific parametric model for the dynamic evolution of $\nu_t$; in fact, $\nu_t$ can still be an autocorrelated process. In this respect, our specification differs from the pure random walk assumption \citep[e.g.,][]{stockwatson88JASA} and is instead compatible with the ARIMA specification \citep[e.g.,][]{lippi_diffusion_1994,barigozzi_measuring_2021,morley2023simple}. Finally, we remain agnostic about the law of motion of the residual stationary component, $\boldsymbol{\omega}_t$.\footnote{Modeling the cycle as an AR(2) poses identification problems because the same state-space representation can equivalently be obtained with an ARMA(2,1) specification \citep{kim2022trend}.}
The model we just described is a modified version of the model \citet{barigozzi_measuring_2021} (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 between the factors to retrieve the common trends, which require longer 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 \eqref{sbeq::obsTR}-\eqref{sbeq::st1TR}, 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 \eqref{eq::obseq} and \eqref{sbeq::obsTR}, we obtain the decomposition of each observed variable:
\begin{equation}
y_{it}\ =\ \mathrm{D}_{it} + \boldsymbol{\lambda}_i'\boldsymbol{\psi} {\tau}_t + \boldsymbol{\lambda}_i' \boldsymbol{\omega}_t + \gamma_{i}g_t \mathbb{I}_{\text{\tiny 2020:Q1$\le\! t\!\le$2021:Q4}} + \xi_{it}.\label{eq::TrendCycleCovidIdioDec}
\end{equation}
Focusing on GDP, we define potential output, $\text{PO}_t$, and the output gap, $\text{OG}_t$, as:
\begin{align}
\text{PO}_t\ &=\ \mathrm{D}_{\text{\tiny GDP},t} + \boldsymbol{\lambda}'_{\text{\tiny GDP} }\boldsymbol{\psi}{\tau}_t,\label{POT}\\
\text{OG}_t\ &=\ \boldsymbol{\lambda}'_{\text{\tiny GDP}}\boldsymbol{\omega}_t.\label{OGT}
\end{align}
Hence, in our framework, potential output is the sum of the time-varying secular trend of GDP ($\mathrm{D}_{\text{\tiny GDP},t}$), which captures the long-run decline in EA output growth, and the part of GDP driven by the common trend component (${\tau}_t$); the output gap is the part of GDP driven by the stationary cyclical component ($\boldsymbol{\omega}_t$).
From the definition of potential output \eqref{POT} and output gap \eqref{OGT}, we left out the idiosyncratic component, $\xi_{\text{\tiny GDP},t}$, and the Covid component, $\gamma_{\text{\tiny GDP}}g_t$. While the idiosyncratic component is likely to be just a measurement error \citep{aruoba2016improving}, 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 \enquote{froze} due to the lockdowns. While this view is commonly accepted by European institutions \citep{thum2022ii}, 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.
\subsection{Estimating the model}\label{sec::fattoni}
To estimate potential output and the output gap, we first estimate the DFM \eqref{eq::obseq}-\eqref{eq::idioeq} using a three-step estimation procedure, and then we estimate the model for the common trend \eqref{sbeq::obsTR}-\eqref{sbeq::st1TR} on the estimated factors.
\paragraph{Estimating the dynamic factor model.}
To estimate the model in \eqref{eq::obseq}-\eqref{eq::idioeq}, we need to extract the latent states $\mathbf f_t$, $g_t$, $\mathrm D_{\text{\tiny i},t}$ (if $i\in\mathcal{L}_1$ or $i\in \mathcal{L}_0$), and $\xi_{it}$ (if $i\in\mathcal I_1$), and estimate the parameters $\bm\lambda_{i}$, $\gamma_i$, $\mathbf A_j$, $s_t$, $\bm\Sigma_u$, $\sigma_{e_i}^2$, $a_i$, and $b_i$ (if $i\in\mathcal I_b$), while we calibrate $\sigma^2_{\epsilon_i}$ (if $i\in\mathcal L_0$) and $\sigma^2_{\eta_i}$ (if $i\in\mathcal L_1$) following \cite{del2017safety}.\footnote{We calibrate the variances of the stochastic secular components so that, for $i\in \mathcal{L}_1$, the standard deviation of the secular trend is approximately $1\%$ over 100 years, while for $i\in \mathcal{L}_0$, itis approximately $1\%$ over 50 years. This calibration, \textit{de facto}, defines the secular trends by allowing the average growth rate of the variables in $\mathcal{L}_1$ and the average level of the variables in $\mathcal{L}_0$ to drift slowly over time. In principle, these variances could be estimated; however, given the short sample, we consider it more reliable to define the trends upfront as slow-moving processes, an approach also adopted by \citet{AhnLuciani2024}, which avoids the overfitting ``pile-up'' problem highlighted by \citet{kim2022trend}. Appendix \ref{app::notvpars} shows that the results are robust to reasonable changes to this calibration.} To do so, we use a three-step estimation procedure that we summarize below.
\medskip
\leftskip 1em
\parindent -1em
\textsc{Step 1: Estimate the model up to 2019:Q4 (pre-Covid step).} We obtain a preliminary estimate of the parameters using PCA for $I(1)$ data \citep{bai2004panic,barigozzi_large-dimensional_2021,onatski_spurious_2021}. Then, we run the EM algorithm, jointly with the Kalman smoother, as described in \citet{barigozzi_quasi_2022} in the high-dimensional case.\medskip
\textsc{Step 2: Estimate the Covid factor and volatility (Covid step).}
\leftskip 1em
\parindent 0em
\textsc{Covid factor} \citep{stockcomovement}. Using the parameter estimated over the pre-Covid period, we extract the latent states using data up to the end of the sample by running the Kalman smoother separately for the pre- and post-pandemic periods to prevent the pandemic observations from changing the pre-pandemic inference, following the approach of \citet{AhnLuciani2024}. This yields an estimate of the factors during Covid as if they had continued along their pre-pandemic dynamics, while all the Covid-specific co-movements are absorbed by the idiosyncratic component $\xi_{it}$. Since Covid represented a common shock affecting most (if not all) series in the dataset, we then estimate the Covid factor $\widehat g_t$ and its loadings $\widehat \gamma_i$ by PCA on the variance-covariance matrix of the idiosyncratic component for the period 2020:Q1-2021:Q4. \smallskip
\textsc{Covid volatilities} \citep{lenza_how_2022}. Lastly, we estimate the Covid volatilities $\widehat s_t$ for the period 2020:Q1-2025:Q1 by maximum likelihood and by using the factors extracted using pre-Covid parameters. We find that $\widehat s_t$ jumps 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 \ref{sec::covid}, we show how our measures would change if we do not explicitly model the effect of Covid, or if we use the exponential decay parametrization proposed by \citet{lenza_how_2022}. \smallskip
\leftskip 1em
\parindent 1em
The procedure just described is general and could be applied again should another large, non-economic event occur. Its purpose is twofold: first, to prevent an exceptional, non-economic event from contaminating the model's estimates going forward; and second, to provide a way to assess the impact of such a non-economic event on the economy. If the researcher's sole objective is to insulate the model estimates from the effects of the event, a simpler alternative is to append post Covid observations to pre-Covid ones.\footnote{It is common practice in dynamic factor models---whether stationary or non-stationary---to adjust for outliers on a series-by-series basis. However, applying this procedure to Covid data would be ill-advised, since much of the additional volatility during this period is pandemic induced rather than the result of measurement error \citep{ng2021modeling}. A univariate outlier adjustment method cannot distinguish between the two and would therefore risk removing economically relevant information. Indeed, when we apply such an adjustment, the estimated Covid factor is essentially zero, and the output gap appears flat, as if nothing had happened.}\medskip
\leftskip 1em
\parindent -1em
\textsc{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 $y_{it} - \widehat \gamma_i \widehat g_t$. Specifically, by using the factors estimated over the whole sample in Step 2 rescaled by $\widehat s_t$ in the last part of the sample, we estimate the parameters, $\widehat{\bm\lambda}_{i}$, $\widehat{\mathbf A}_j$, $\widehat{\bm\Sigma}_u$, $\widehat\sigma_{e_i}^2$, $\widehat a_i$, and $\widehat b_i$, by maximizing the expected likelihood. Finally, with the estimated parameters in hand, we obtain a final estimate of the states, $\widehat{\mathbf f}_t$, $\widehat {\mathrm D}_{i,t}$, and $\widehat{\xi}_{it}$, through the Kalman smoother again truncated in 2020:Q1 and reinitialized before iterating backward, as explained in Step 2.
\leftskip 0em
\parindent 1.5em
\paragraph{Estimating the common trend.}
Having estimated the model parameters and unobserved states, we can now estimate the common trend. To this end, we estimate the state-space model in \eqref{sbeq::obsTR}-\eqref{sbeq::st1TR} using the EM algorithm by replacing the true factors with the estimated ones. Since we are extracting one trend from four factors, we can identify the trend by properly initializing the loadings $\bm\psi$. Specifically, we set all entries of $\bm\psi$ to zero except for the one corresponding to the factor with the largest share of variance at frequencies below eight years, which is set equal to one. As for the initialization of $\sigma^2_\nu$, our experimentation has shown that it does not matter.
At convergence of the EM algorithm, we obtain a final estimate of the parameters, $\widehat{\bm\psi}$, $\widehat{\bm\Sigma}_\omega$, and $\widehat{\sigma}^2_\nu$, and using these estimates, we have a final estimate of the trend $\widehat \tau_t$ and of the cyclical component $\widehat{\bm\omega}_t=\widehat{\mathbf f}_t-\widehat{\bm\psi}\widehat \tau_t$, 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 \eqref{POT} and \eqref{OGT}, respectively.\bigskip
As shown in \citet{barigozzi_quasi_2022}, the estimation procedure we just outlined delivers consistent estimators of all parameters and of the factors, provided that $n$ and $T$ grow to infinity and the EM algorithm is initialized with the estimator of the loadings and factors introduced by \citet{barigozzi_large-dimensional_2021}. Furthermore, to prove this result, we neither have to impose the Gaussianity assumption nor have to require uncorrelatedness of the idiosyncratic components ($\xi_{it}$ if $i\notin \mathcal I_1$ or $e_{it}$ if $i\in\mathcal I_1$) across $i$ or $t$. Rather, we just have to impose standard moment conditions, such as existence and summability of 4th-order cumulants.
We conclude by comparing our approach with two alternative estimation strategies. First, our model implies that, in principle, the dynamics of the factors could be modeled directly using a VECM, thereby estimating the cointegration space in the first step. However, this approach appears to be quite unstable possibly due to the short sample (see Appendix \ref{sbsec::VECM} for a comparison). Second, in a recent paper, \citet{morley2023simple} suggest to smooth a preliminary estimate of the trend by appropriately rescaling the parameters of an ARMA fitted on $\nu_t$. While their univariate approach is prone to overfitting, our multivariate approach isolates low-frequency dynamics without the need for ex-post smoothing (see Appendix \ref{app:morley} for a comparison).
\section{Potential output and output gap of the Euro Area}\label{sec::poutgap}
Figure \ref{fig::pout} presents our potential output estimate (black line), both in $100\times\log$ levels (left plot) and in year-on-year (YoY) growth rates (right plot). The right plot also includes potential output growth estimates from the EC (red line) and IMF (blue line).
Four main results emerge from Figure \ref{fig::pout}. First, the GFC and the SDR had a permanent negative effect on the level of potential output (a hysteresis effect), consistent with \citet{schmoller2021deep}. Second, potential output growth decelerated after the GFC and slowed further following the SDR---average potential output growth was 2.1\% in 2008, while the subsequent peak was 1.6\% on the eve of the Covid pandemic. Third, although both the EC and IMF estimate a slowdown in potential growth after the GFC, neither estimate a slowdown after the SDR. Fourth, it is still too soon to determine whether the Covid recession had a long-term effect on potential output growth. The prevailing institutional view is that it did not \citep{thum2022ii}; however, while potential growth rebounded quickly after the Covid shock---peaking at 2\% in 2022:Q4---it subsequently declined to 0.95\% in 2025:Q1, below its pre-Covid pace. Overall, these results indicate a gradual slowdown of potential growth over the past two decades, punctuated by persistent losses following major crises.
\begin{figure}[ht] \caption{Potential output} \label{fig::pout}
\centering \footnotesize \sc \smallskip
\setlength{\tabcolsep}{.01\textwidth}
\begin{tabular}{cc}
$100\times\log$ levels & Year-on-year growth rates\\
\includegraphics[trim={2cm 9.1cm 2.2cm 9.5cm},clip,width = 0.475\textwidth]{Figures_0925/POl.pdf}&
\includegraphics[trim={2cm 9.1cm 2.2cm 9.5cm},clip,width = 0.475\textwidth]{Figures_0925/POg_EC.pdf} \\
\end{tabular}
\begin{tabular}{p{.98\textwidth}} \scriptsize Notes: \rm 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 right chart, the blue and red lines are the potential output estimates published by the European Commission and the IMF, respectively. The IMF estimate of YoY potential output growth reported in the right chart is the result of our own calculation. Indeed, the IMF publishes only an estimate of the output gap from which we backed out potential output. Thus, the blue line in the right chart does not account for any adjustment for Covid that the IMF might have done.
\end{tabular}
\end{figure}
Figure \ref{fig::ogapEC} shows our estimated output gap alongside the the EC and IMF estimates. The three measures display similar dynamics, with turning points occurring at the same dates. They also align closely up to the SDR, all indicating substantial overheating before the GFC and a persistently negative output gap during and between the two recessions. However, our estimate rises after the SDR, remaining around 2\% from 2017 until the Covid pandemic, implying a considerably tighter economy than suggested by the EC or IMF estimates. Similarly, our measure points to a much tighter post-Covid economy than those from the EC and the IMFs.
\begin{figure}[ht] \caption{Output gap}\label{fig::ogapEC}
\centering \footnotesize \sc \smallskip
\setlength{\tabcolsep}{.01\textwidth}
\begin{tabular}{cc}
Levels & Year-on-year growth rates\\
\includegraphics[trim={2cm 9.1cm 2.2cm 9.5cm},clip,width = 0.475\textwidth]{Figures_0925/OGl_EC.pdf}&
\includegraphics[trim={2cm 9.1cm 2.2cm 9.5cm},clip,width = 0.475\textwidth]{Figures_0925/OGg_EC.pdf}
\end{tabular}
\begin{tabular}{p{.98\textwidth}} \scriptsize Notes: \rm The black line is our estimate of the output gap (OG) in levels (left plot) and YoY growth rates (right plot)---the level of the output gap is the percentage deviation from potential, the YoY growth rates is $\text{OG}_t-\text{OG}_{t-4}$. Each black marker denotes one year (four quarters), starting from $2001$:Q1. The grey shaded areas are the 68$\%$ and 84$\%$ confidence bands. The red and blue lines are the output gap estimates published by the European Commission and the IMF, respectively.
\end{tabular}
\end{figure}
To conclude, Figure \ref{fig::decompgdp} shows GDP growth decomposed into the contribution of potential output, the output gap, the Covid factor, and the idiosyncratic component. During the Covid pandemic, the output gap subtracted 27.2 percentage points (p.p.) from quarter-on-quarter (QoQ) annualized GDP growth in 2020:Q2 (inset box, right chart). Given the -47\% QoQ annualized GDP decline, such a contribution may appear implausible. However, as explained in Section \ref{sec::fattoni}, our output gap captures only business-as-usual co-movements, while the Covid factor isolates the extraordinary co-movements induced by the pandemic. As shown in the inset box in the right plot in Figure \ref{fig::decompgdp}, the Covid factor accounts for an additional -15 p.p. of the 2020:Q2 contraction. Since then it contributed about +23 p.p. in 2020:Q3, before alternating between negative and positive values over the next six quarters, reflecting the tightening and easing of mobility restrictions.
\begin{figure}[h!] \caption{Decomposition of GDP growth} \label{fig::decompgdp}
\centering \footnotesize \smallskip
\setlength{\tabcolsep}{.01\textwidth}
\begin{tabular}{cc}
\textsc{Year-on-year growth rate} & \textsc{Quarter-on-quarter annualized growth rates}\\
\includegraphics[trim={1cm 9cm 1.4cm 9cm},clip,width = 0.475\textwidth]{Figures_0925/dec_GDP_EA.pdf}&
\begin{tikzpicture}
\node[anchor=south west,inner sep=0] (image) at (0,0) {\includegraphics[trim={1cm 9cm 1.4cm 9cm},clip,width = 0.475\textwidth]{Figures_0925/dec_QoQ_GDP_EA_1824.pdf}};
\begin{scope}[x={(image.south east)},y={(image.north west)}]
\node[anchor=south west,inner sep=0] (image) at (0.41,0.08)
{\setlength{\tabcolsep}{.0\textwidth} \tiny
\begin{tabular}{|C{.06\textwidth}C{.05\textwidth}C{.05\textwidth}C{.05\textwidth}C{.05\textwidth}|}\hline
& trend & cycle & idio & COV \\\hline
2020:Q2 & 0.8 &-27.2 &-5.5 &-15.0 \\
2020:Q3 & 1.0 &~22.0 &-2.4 &~23.3 \\\hline
\end{tabular}};
\end{scope}
\end{tikzpicture}
\end{tabular}
\begin{tabular}{p{.98\textwidth}} \scriptsize Notes: \rm The black line with dot markers is GDP growth. The bars represent the contribution of each component to GDP growth. The left plot shows YoY growth rates, while the right plot shows QoQ growth at an annual rate. Growth rates are computed using the log approximation.
\end{tabular}
\end{figure}
In summary, Figures \ref{fig::pout}--\ref{fig::decompgdp} show that the EA has a potential output issue, not a business cycle issue. Since the seminal work of \citet{BQ89}, the prevailing view has been that supply shocks have permanent effects on output, whereas demand shocks have only transitory effects. Empirical studies mostly support this perspective. For instance, \citet{ForniAmerican} show that the impulse response functions to a supply (demand) shock are almost identical to those of a permanent (transitory shock), and \citet{BenatiLubik} find that it is essentially impossible to detect aggregate demand shocks that permanently affect GDP. Nevertheless, some recent studies challenge this view. For example, \citet{FurlanettoetaAEJmacro} provide evidence of hysteresis effects, showing that demand-driven recessions can lead to lasting output losses.
Based on this evidence, we assume that growth in the common trend $\tau_t$---and thus potential output growth---is primarily driven by supply forces, except during recessions, while the cyclical common component $\bm \omega_t$---and thus the output gap---is mainly driven by demand forces. This distinction implies that, to foster long-run growth, European countries should implement structural reforms and promote productivity-enhancing investments, whereas to mitigate recession-induced output losses, they should support aggregate demand more forcefully during downturns. In contrast, policies aiming at stimulating household consumption and residential investments in normal times will have only short-term effects at best because, as we will show in Section \ref{sec::Credit}, growth financed through household debt is not sustainable in the long run.
\section{What about the Okun’s law and the Phillips curve?}\label{sec::OL&PC}
Our estimate of the output gap has a different meaning than that of the EC and IMF, which derive the output gap and potential output according to the so-called \enquote{production function approach} \citep{kiley2013output}. In production-function-based models, the output and unemployment gaps are related through the Okun's law, and the output gap is related to inflation through the Phillips curve. Thus, in these models, the unemployment gap decreases whenever the output gap increases, and vice-versa, and low inflation suggests a negative output gap, while high inflation indicates a positive gap.
In our model (like any statistical model), we do not impose any Okun's law or Phillips curve. Thus, we must be careful when we compare our estimate with that of the EC or the IMF because their output gap measures are designed to indicate potential inflation pressure, whereas ours is not. Nonetheless, in Sections \ref{sec::OL} and \ref{sec::PC}, we show that, on average, our model satisfies the Okun's law relationship and exhibits Phillips correlation.
\subsection{The Okun's law}\label{sec::OL}
The left plot in Figure \ref{fig::OkunLaw} 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. Over the whole sample, on average, for every percentage point increase in the output gap, the unemployment gap decreases 0.6 p.p. (grey dotted line). Moreover, this correlation has decreased after Covid from -0.56 (blue dash-dotted line) to -0.23 (red solid line), suggesting a (temporary) disconnect between the labor and goods and services market post-Covid in the EA as also noted by \citet{berson2025explaining}. The right plot in Figure \ref{fig::OkunLaw} shows the expanding window estimate of the Okun’s law coefficient $\beta$ in the regression $(\text{UR}_t - D_{\text{UR},t}) = \alpha + \beta \text{OG}_t + \varepsilon_{\text{UR},t}$, where $D_{\text{UR},t}$ is the time-varying mean of the unemployment rate defined in \eqref{eq::seccomp}. On average, the Okun's law coefficient varies between -0.55 to -0.60.\footnote{We also estimated the relationship between the output gap and hours worked gap and find a positive correlation in line with the results of \citet{morley_estimating_2023}.}
\begin{figure}[ht!]\caption{Okun's Law}\label{fig::OkunLaw}
\centering \footnotesize\sc \smallskip
\setlength{\tabcolsep}{.01\textwidth}
\begin{tabular}{cc}
Unemployment rate gap vs. output gap & Okun's law coefficient (expanding window)\\
\includegraphics[trim={1.8cm 9.1cm 2.2cm 9.5cm},clip,width = 0.475\textwidth]{Figures_0925/scatter_OG_UG.pdf} &
\includegraphics[trim={2cm 9.1cm 2cm 9.5cm},clip,width = 0.475\textwidth]{Figures_0925/coeffplot_OL.pdf}\\
\end{tabular}
\begin{tabular}{p{.98\textwidth}} \scriptsize Notes: \rm
The left chart shows the Okun’s law relationship with the output gap on the horizontal axis and the unemployment rate gap on the vertical axis. Each circle corresponds to an output gap - unemployment gap pair at a given time $t$. The grey dotted, blue dashed-dotted, and red solid lines are the least squares fit lines for the full, pre-Covid, and post-Covid samples, respectively, obtained by omitting the observation for 2020:Q2 (orange dot).
~~~The right chart shows the least squares estimate, based on an expanding window starting from 2015:Q1, of the Okun’s law slope $\beta$ given by the regression $(\text{UR}_t - D_{\text{UR},t}) = \alpha + \beta \text{OG}_t + \varepsilon_{\text{UR},t}$, where $D_{\text{UR},t}$ is the time-varying mean of the unemployment rate defined in \eqref{eq::seccomp}. Each dot is an estimate of $\beta$ while the whiskers are $\pm$ one HAC standard errors.
\end{tabular}
\end{figure}
To further corroborate the intuition that there is a tight relationship between our output gap estimate and labor market indicators, Figure \ref{fig::urshock_g} 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 to the common component of the unemployment rate.\footnote{The lag-$h$ GIRF of all variables is obtained by computing the differences between the $h$-step ahead forecast of their 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 \citep{banbura2015conditional,crump2021large}.} Results confirm that our model, on average, associates an unemployment rate increase with an output gap decrease. After the shock, 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.
\begin{figure}[ht!]\caption{Generalized Impulse Response Functions to a shock to the unemployment rate}\label{fig::urshock_g}
\centering \scriptsize \sc \smallskip
\setlength{\tabcolsep}{0\textwidth}
\begin{tabular}{ccc}
Common component: UR & Common component: GDP & Potential output and output gap \\
\includegraphics[trim={2cm 9.1cm 2.2cm 9.5cm},clip,width = 0.33\textwidth]{Figures_0925/UR_UR_GIRF_chi.pdf} &
\includegraphics[trim={2cm 9.1cm 2.2cm 9.5cm},clip,width = 0.33\textwidth]{Figures_0925/GDP_UR_GIRF_chi.pdf} &
\includegraphics[trim={2cm 9.1cm 2.2cm 9.5cm},clip,width = 0.33\textwidth]{Figures_0925/OG_PO_UR_GIRF.pdf} \\
\end{tabular}
\begin{tabular}{p{.98\textwidth}} \scriptsize Notes: \rm
The black solid/dashed lines are the GIRFs to a 1 p.p. shock to the common component of the unemployment rate. The major ticks in the x-axis represent quarters after the shock.
\end{tabular}
\end{figure}
\subsection{The Phillips curve}\label{sec::PC}
The left plot in Figure \ref{fig::PhillipsCurve} 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. Over the whole sample, on average, for every percentage point increase in the output gap, the core inflation gap increases 4 basis points (dotted grey line). Moreover, this correlation has increased significantly after Covid from 0.022 (dashed-dotted blue line) to 0.079 (solid red line). The right plot in Figure \ref{fig::PhillipsCurve} shows the expanding window estimate of the slope of the Phillips curve $\alpha$ in the following expectation-augmented specification \citep[e.g.,][]{conti2021resurrecting}: $\pi_t = c + \alpha \text{OG}_t + \beta \pi_{t-1} + \gamma \pi^e_t + \varepsilon_{\pi,t}$, where $\pi_t $ is core inflation and $\pi^e_t$ are the long-run (5-year ahead) inflation expectations in the Survey of Professional Forecasters. The results suggest that the relationship between inflation and the output gap has strengthened after Covid, a point also made by \citet{Lane2024}.
\begin{figure}[ht!]\caption{Phillips Curve}\label{fig::PhillipsCurve}
\centering \sc \smallskip
\setlength{\tabcolsep}{0\textwidth}
\begin{tabular}{cp{.02\textwidth}c}
\footnotesize Core Inflation gap vs Output gap && \footnotesize Phillips Curve slope (expanding window)\\
\includegraphics[trim={1.8cm 9.1cm 2.2cm 9.5cm},clip,width = 0.49\textwidth]{Figures_0925/scatter_OG_IOC.pdf} & &
\includegraphics[trim={2cm 9.1cm 2cm 9.5cm},clip,width = 0.49\textwidth]{Figures_0925/coeffplot_PC.pdf}\\
\end{tabular}
\begin{tabular}{p{\textwidth}} \scriptsize Notes: \rm
The left chart shows the Phillips curve relationship with the output gap on the horizontal axis and the core inflation gap on the vertical axis. Each circle corresponds to an output gap - core inflation gap pair at time $t$. The grey dotted, blue dashed-dotted, and red solid lines are the least squares fit lines for the full, pre-Covid, and post-Covid samples, respectively, obtained by omitting the observation for 2020:Q2 (orange dot).
~~~The right chart shows the least squares estimate, based on an expanding window starting from 2015:Q1, of the slope of the Phillips curve given by the following expectation-augmented specification: $\pi_t = c + \alpha \text{OG}_t + \beta \pi_{t-1} + \gamma \pi^e_t + \varepsilon_{\pi,t}$, where $\pi_t $ is core inflation and $\pi^e_t$ are the long-run (5-year ahead) inflation expectations in the Survey of Professional Forecasters. Each dot is an estimate of $\alpha$ while the whiskers are $\pm$ one HAC standard errors.
\end{tabular}
\end{figure}
To further corroborate the intuition that there is a relationship between our output gap estimate and inflation indicators, Figure \ref{fig::coreshock_g} 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 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.
\begin{figure}[ht!]\caption{Generalized Impulse Response Functions to a shock to core inflation}\label{fig::coreshock_g}
\centering \scriptsize \sc \smallskip
\setlength{\tabcolsep}{0\textwidth}
\begin{tabular}{ccc}
Common component: core HICP & Common component: GDP & Potential output and output gap \\
\includegraphics[trim={2cm 9.1cm 2.2cm 9.5cm},clip,width = 0.33\textwidth]{Figures_0925/CoreHICP_CoreHICP_GIRF_chi.pdf} &
\includegraphics[trim={2cm 9.1cm 2.2cm 9.5cm},clip,width = 0.33\textwidth]{Figures_0925/GDP_CoreHICP_GIRF_chi.pdf} &
\includegraphics[trim={2cm 9.1cm 2.2cm 9.5cm},clip,width = 0.33\textwidth]{Figures_0925/OG_PO_CoreHICP_GIRF.pdf}
\end{tabular}
\begin{tabular}{p{.98\textwidth}} \scriptsize Notes: \rm
The black solid/dashed lines are the GIRF to a 0.5 p.p. shock to the common component of core inflation. The major ticks in the x-axis represent quarters after the shock.
\end{tabular}
\end{figure}
\section{Inflation dynamics through the lens of our model}\label{sec::InflationDynamics}
In Section \ref{sec::PC}, we show that, although we did not create it specifically to signal inflationary pressures, our output gap measure does provide insights related to inflation dynamics. In light of these results, further inspection of Figure \ref{fig::ogapEC} raises two important questions: How can we reconcile an output gap of 2\% between 2017 and 2019 when inflation was just 1\%, well below the 2\% ECB target? How does our model interpret post-pandemic inflation dynamics?
To answer these questions, in Figure \ref{fig::CoreDecomp}, we plot the decomposition of core inflation (\small{HICPNEF}) implied by our model:
\begin{align}
y_{\text{\tiny HICPNEF},t}\ &=\ \bar{\pi}^c_t + \tilde{\pi}^c_t + \gamma_{\text{\tiny HICPNEF}}g_t \mathbb{I}_{\text{\tiny 2020:Q1$\le\! t\!\le$2021:Q4}} + \xi_{\text{\tiny HICPNEF},t},\label{eq::CoreDecomp}\\
\bar{\pi}^c_t\ &=\ \mathrm{D}_{\text{\tiny HICPNEF},t} + \boldsymbol{\lambda}'_{\text{\tiny HICPNEF} }\boldsymbol{\psi}{\tau}_t,\label{TrendInf}\\
\tilde{\pi}^c_t\ &=\ \boldsymbol{\lambda}'_{\text{\tiny HICPNEF}}\boldsymbol{\omega}_t,\label{InfGap}
\end{align}
where $\bar{\pi}^c_t$ denotes trend core inflation---defined as the sum of the secular component of core inflation and the portion of core inflation driven by the common trend---and $\tilde{\pi}^c_t$ denotes the inflation gap---defined as the portion of core inflation driven by the common cyclical component.
Consistent with \citet{LucreziaInflation}, we estimate that, following the GFC, trend inflation decreased from 2\% to about 1\% in 2016, where it remained stable until the Covid pandemic. This finding is in line with \citet{CiccarelliOsbat} and \citet{CorselloNeriTagliabracci}, who show that inflation expectations de-anchored on the downside after the SDR. We therefore conclude that core inflation remained persistently below 2\% after the GFC primarily because trend core inflation decreased, not because there was slack in the economy, which explains why we estimate an output gap above 2\% when core inflation was 1\%.
\begin{figure}[ht!]\caption{Decomposition of core inflation }\label{fig::CoreDecomp}
\centering \footnotesize \smallskip
\setlength{\tabcolsep}{.01\textwidth}
\begin{tabular}{cc}
\textsc{Year-on-year inflation} & \textsc{Quarter-on-quarter annualized inflation}\\
\begin{tikzpicture}
\node[anchor=south west,inner sep=0] (image) at (0,0) {\includegraphics[trim={1.3cm 9cm 1cm 9cm},clip,width = 0.485\textwidth]{Figures_0925/dec_HICPNEF_EA.pdf}};
\begin{scope}[x={(image.south east)},y={(image.north west)}]
\node[anchor=south west,inner sep=0] (image) at (0.19,0.74)
{\setlength{\tabcolsep}{.0\textwidth} \tiny
\begin{tabular}{|C{.075\textwidth}C{.05\textwidth}C{.05\textwidth}C{.05\textwidth}C{.05\textwidth}|}\hline
& data & trend & cycle & idio \\\hline
2019:Q4 & 1.2 & 0.9 & 0.3 & ~0.0 \\
2023:Q1 & 5.4 & 2.3 & 1.6 & ~1.5 \\
2025:Q1 & 2.7 & 2.5 & 0.4 & -0.2 \\\hline
\end{tabular}};
\end{scope}
\end{tikzpicture} &
\includegraphics[trim={1.3cm 9cm 1cm 9cm},clip,width = 0.485\textwidth]{Figures_0925/dec_QoQ_HICPNEF_EA_1824.pdf}
\end{tabular}
\begin{tabular}{p{.98\textwidth}} \scriptsize Notes: \rm The black line with dot markers is core inflation. The bars represent the contribution of each component to core inflation. The left plot shows YoY inflation, while the right plot shows QoQ inflation at an annual rate.
\end{tabular}
\end{figure}
Next, we turn our attention to post-pandemic inflation dynamics. As shown in the inset box of the left chart in Figure \ref{fig::CoreDecomp}, YoY core inflation increased about 4.2 p.p. between 2019:Q4 and 2023:Q1. Trend inflation and the core inflation gap jointly account for about 65\% of this 4.2 p.p. increase, while idiosyncratic factors drive the remaining portion. Since its 5.4\% peak in 2023:Q1, YoY core inflation decreased to 2.7\% in 2025:Q1. The core inflation gap accounts 40 basis points of this decline, and the idiosyncratic components for 250 basis points, while trend inflation increased 20 basis points.
As we explained at the end of Section \ref{sec::poutgap}, the cyclical common component---therefore, the core inflation gap---primarily reflects demand forces. Under this assumption, the results of the decomposition in Figure \ref{fig::CoreDecomp} show that demand forces accounted for at least 30\% of the post-pandemic increase in core inflation, thus supporting existing literature that indicates that demand dynamics played a significant role in the inflation surge following the pandemic \citep{AscariTrezzi,GiannonePrimicieri,Canovaetal}.
Figure \ref{fig::CoreDecomp} clearly shows that other significant factors shaping post-pandemic inflation dynamics are idiosyncratic. For example, lingering Covid-specific effects (e.g., supply chain bottlenecks) that are not captured by the Covid factor. Another possibility is that the concurrent sharp rise in oil and natural gas prices following the onset of the Russia-Ukraine war induced second-round effects beyond those experienced in the pre-Covid sample. Finally, labor market tightness might have induced a non-linear response of prices that our linear model fails to capture. These factors might have a more or less persistent effect. Thus, our estimate of the role of demand forces should be considered a lower bound.
\section{Has our output gap measure predictive power for inflation?}\label{sec::InflationForecasting}
Any output gap measure \textit{must} be good at predicting inflation to be considered a credible indicator of current/future inflationary pressure. We therefore evaluate the ability of our output gap estimate to forecast year-on-year inflation one year ahead and compare its performance with that of other output gap estimates. In practice, we replicate the analysis conducted in \citet{banbura_does_2023}, and we employ the following model:
\begin{equation} \label{eq::ADL}
\pi_{t+4} = \alpha \pi_{t} + \beta OG_{t} + v_{t+4},
\end{equation}
where $\pi_{t} = 100\log({P_{t}}/{P_{t-1}})$ is the quarter-on-quarter inflation rate in quarter $t$, $P_t$ is the harmonized consumer price index (either headline or core), $\pi_{t+4} = \sum_{i=1}^4 \pi_{t+i}$ is year-on-year inflation in quarter $t+4$, and $OG_{t}$ is the output gap.\footnote{\citet{banbura_does_2023} labeled model \eqref{eq::ADL} the ``benchmark model,'' and they show that, despite being very simple, it delivers decent forecasts compared to more complex alternative models.} Alongside our output gap estimate, we consider estimates from different univariate and multivariate statistical models: the HP filter, the filter by \citet{hamilton2018you}, the boosted HP filter by \citet{phillips2021boosting}, the \cite{christianofitzgerald}filter, the Butterworth filter as recommended by \citet{canova2022faq}, and the large Bayesian VAR approach by \citet{morley_estimating_2023}---Appendix \ref{sec::altTC} describes these alternative models.
We look at the forecasting performance of the different output gap measures over three distinct samples: the full sample (2015:Q4--2025:Q1), a pre-Covid sample, 2015:Q4--2019:Q4, and a post-Covid sample, 2022:Q1--2025:Q1. The forecasting exercise is an expanding window exercise, where the first window is a 60-quarter window. All models are estimated up to time $t$ and used to forecast inflation at $t+4$.
Table \ref{tab::s1519} compares the forecasting performance of the different output gap measures in terms of relative Root Mean Squared Error (RMSE), where values below one indicate superior forecasting performance when using our output gap estimate. As shown in rows (1)-(8), over the full sample our measure performs about as well as other statistical output gap estimates. However, this average masks important differences across subperiods. Excluding the Covid pandemic, in the pre-Covid sample---when inflation was low and stable---our output gap measure outperforms all alternatives in forecasting headline inflation and most alternatives in forecasting core inflation, often significantly so according to the \citet{DieboldMariano} test of equal predictive accuracy. In the post-Covid period, when inflation surged and then declined, our model continues to outperform all other measures, sometimes by a substantial margin, though not always significantly. It is worth noting, however, that the post-Covid sample includes only 13 observations, which limits the power of the \citet{DieboldMariano} test.
\begin{table}[ht!]\caption{4-quarter ahead year-on-year inflation forecasting} \label{tab::s1519}
\centering
\small \textit{Relative Root Mean Squared Errors}\\\smallskip
\setlength{\tabcolsep}{0pt}
\begin{tabular}{C{.05\textwidth}L{.35\textwidth}|C{.1\textwidth}C{.1\textwidth}|C{.1\textwidth}C{.1\textwidth}|C{.1\textwidth}C{.1\textwidth}} \hline\hline
& & \multicolumn{2}{c|}{2015:Q4-2025:Q1} & \multicolumn{2}{c}{2015:Q4-2019:Q4} \vline & \multicolumn{2}{c}{2022:Q1-2025:Q1} \\ \hline
& Output Gap Measure & Headline & Core & Headline & Core & Headline & Core\textcolor{white}{$^*$} \\ \hline
\scriptsize (1) & \small HP Filter ($\lambda = 1600$) & 1.03 & 0.97 & 0.90 & 1.00\textcolor{white}{$^*$} & 1.02 & 0.93 \\
\scriptsize (2) & \small HP Filter ($\lambda = 51200$) & 1.02 & 0.96 & 0.91 & 1.01\textcolor{white}{$^*$} & 0.99 & 0.94 \\
\scriptsize (3) & \small Hamilton Filter & 1.00 & 0.97 & 0.98 & 0.97$^*$ & 1.01 & 0.94 \\
\scriptsize (4) & \small Boosted HP Filter ($\lambda = 1600$) & 1.01 & 0.97 & 0.89 & 0.96\textcolor{white}{$^*$} & 0.93 & 0.93 \\
\scriptsize (5) & \small Boosted HP Filter ($\lambda = 51200$) & 1.01 & 0.97 & 0.89 & 0.99\textcolor{white}{$^*$} & 0.96 & 0.93 \\
\scriptsize (6) & \small Christiano-Fitzgerald Filter & 1.04 & 0.97 & 0.89 & 0.82$^*$ & 1.01 & 0.92 \\
\scriptsize (7) & \small Butterworth Filter & 1.02 & 0.97 & 0.94 & 0.95$^*$ & 1.00 & 0.93 \\
\scriptsize (8) & \small Multivariate Beveridge-Nelson & - & - & 0.91 & 0.98$^*$ & - & -
\\ \hline
\end{tabular}
\begin{tabular}{p{\textwidth}} \scriptsize {\sc Notes}: \rm The table shows the relative RMSE of forecasting year-on-year inflation using \eqref{eq::ADL}, where $\text{OG}_t$ is either our output gap estimate, or an alternative output gap estimate. Our benchmark output gap estimate is always the numerator of the RMSE; therefore, values below 1 indicate a better forecasting performance when using our benchmark output gap estimate. Asterisks denote statistical significance at the 10\% confidence level according to the \citet{DieboldMariano} test of equal predictive accuracy. Rows (1)--(8) compare our benchmark estimate with alternative models. In row (8) forecasts are obtained with the output gap measure by \cite{morley_estimating_2023} which is available only until 2021:Q3; accordingly, we report results only for the pre-Covid forecasting exercise.
\end{tabular}
\end{table}
In conclusion, the results in this section demonstrate that our output gap measure is not only a measure of the cyclical position of the economy but also a reliable inflation gauge.
\section{The role of credit indicators}\label{sec::Credit}
As discussed in Section \ref{sec::OL&PC}, in production-function-based models, low inflation signals a negative output gap, whereas high inflation indicates a positive one. \citet{borio_rethinking_2017} challenge this view, arguing that credit expansions---especially since the late 1990s---have often resulted in unsustainable growth episodes without any corresponding rise in inflation. Hence, they argue that financial indicators are essential for a meaningful assessment of the business cycle. Supporting this perspective, \citet{berger2022unified} show that much of the U.S. economy's overheating before the GFC originated from credit and housing market imbalances. Similarly, \citet{claessens_how_2012}, \citet{runstler_business_2018}, and \citet{winter2022joint} find that business and financial cycles are correlated and tend to co-move over the medium run.
To explore the role of financial conditions in our framework, we examine which signals the model extracts from credit indicators when estimating the output gap. We focus on household liabilities, which have become a key driver of the business cycle in many advanced economies since the early 2000s \citep{mian2017household}. While in the 1990s non-financial corporations dominated the financial cycle, households drove both the pre-GFC leverage boom, which boosted demand, and the subsequent deleveraging, which curtailed it \citep{mian2018finance,reichlin_financial_2020,reichlin_financial_2020wp}. For the EA, \citet{gambetti2017loan} find that loan supply shocks significantly affect business cycle fluctuations.
Figure \ref{fig::scenarioHH} shows the impact of a scenario in which household liabilities increase faster than projected in the baseline for about 3\sfrac{1}{2} years. In this scenario, household liabilities reach a level about 6\sfrac{1}{4} p.p. higher than in the baseline before returning to baseline after about 8 years.\footnote{We calibrate this scenario by comparing the actual path of household liabilities between 2003:Q1 and 2011:Q4 with a counterfactual linear trend (red line, top-left panel of Figure \ref{fig::scenarioHH}). The smoothed difference (using a fifth-degree polynomial) is then added to the unconditional forecast (red line, top-right panel), generating the conditional scenario (blue line) used to simulate the dynamic responses of all variables in the model.}
\begin{figure}[ht!]\caption{Scenario analysis} \label{fig::scenarioHH} \centering
\centering \footnotesize \sc \smallskip
\setlength{\tabcolsep}{.005\textwidth}
\begin{tabular}{cc}
\hspace{12pt}Household liabilities &\hspace{10pt} Household liabilities \\[-1pt]
\scriptsize \hspace{12pt}(Data and counterfactual) & \hspace{12pt}\scriptsize (Conditional and unconditional forecast)\\
\includegraphics[trim={2.05cm 9.5cm 2.2cm 9.5cm},clip,width = 0.475\textwidth]{Figures_0925/HHLB_scenario.pdf} &
\includegraphics[trim={2.05cm 9.5cm 2.2cm 9.5cm},clip,width = 0.475\textwidth]{Figures_0925/HHLB_forecast.pdf}\\[5pt]
\hspace{12pt}GDP &\hspace{10pt} Potential output and output gap\\[-1pt]
\scriptsize \hspace{12pt}(Scenario dynamic effects) & \scriptsize \hspace{12pt}(Scenario dynamic effects)\\
\includegraphics[trim={2.05cm 9.5cm 2.2cm 9.5cm},clip,width = 0.475\textwidth]{Figures_0925/GDP_HHLB_GIRF_chi.pdf} &
\includegraphics[trim={2.05cm 9.5cm 2.2cm 9.5cm},clip,width = 0.475\textwidth]{Figures_0925/OG_PO_HHLB_GIRF.pdf}\\
\end{tabular}
\begin{tabular}{p{.975\textwidth}} \scriptsize Notes: \rm In the upper-left chart, the black line is the data (in 100$\times$log-levels), and the red line is a linear path starting from the value of household liabilities in 2003:Q1 and ending in 2011:Q4. In the upper-right chart, the black line are the data,the blue line is the scenario we simulate, and the red line is the forecast of household liabilities when no alternative scenario is imposed. In the lower charts, the black solid/dashed lines are the dynamic effects of the simulated scenario.
\end{tabular}
\end{figure}
The lower charts in Figure \ref{fig::scenarioHH} show the dynamic effects of this scenario on the log level of GDP and on the output gap and potential output.\footnote{This is equivalent to computing the effect of a sequence of shocks. Hence, it is estimated the same way we estimated the GIRFs in Section \ref{sec::OL&PC}.} Specifically, GDP increases for a little over three years, reaching a peak at 3 p.p. above the baseline. Then, it declines 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 that debt-driven expansions deliver only temporary gains, as growth financed through household debt is not sustainable in the long run.
\section{Conclusions} \label{sec::conclude}
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 output gap estimate is in line with those published by the EC and the IMF in most of the sample. However, it diverges notably after the SDR, indicating that the EA economy was considerably 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, to achieve stronger and more resilient growth, European countries should prioritize structural reforms and promote productivity-enhancing investments, while supporting aggregate demand more decisively during downturns to prevent recession-induced output losses. In contrast, policies aiming at boosting debt-financed household consumption or residential investment would deliver only short-lived gains, as our findings indicate that growth financed through household debt is not sustainable in the long run.
Moreover, although we did not create our output gap measure specifically to signal inflationary pressures, it does provide insights related to inflation dynamics because, on average, the Phillips Curve is satisfied. In particular, we find that core inflation remained below 2\% after the GFC, not because there was slack in the economy, but rather because trend inflation decreased by one percentage point---in line with the idea that inflation expectations de-anchored on the downside after the GFC \citep{CiccarelliOsbat, CorselloNeriTagliabracci}. Finally, we show that the output gap contributed to at least 30\% of the post-pandemic increase in core inflation, thus supporting existing literature that suggests demand forces played a substantial role in the rise of post-pandemic inflation \citep{AscariTrezzi,GiannonePrimicieri,Canovaetal}.
\begin{thebibliography}{}
\bibitem[\protect\citeauthoryear{Aastveit and Trovik}{Aastveit and
Trovik}{2014}]{aastveit2014estimating}
Aastveit, K.~A. and T.~Trovik (2014).
\newblock Estimating the output gap in real time: {A} factor model approach.
\newblock {\em The Quarterly Review of Economics and Finance\/}~{\em 54},
180--193.
\bibitem[\protect\citeauthoryear{Ahn and Luciani}{Ahn and
Luciani}{2025}]{AhnLuciani2024}
Ahn, H.~J. and M.~Luciani (2025).
\newblock Common and idiosyncratic inflation.
\newblock {\em Journal of Applied Econometrics\/}.
\newblock forthcoming.
\bibitem[\protect\citeauthoryear{Alessi, Barigozzi, and Capasso}{Alessi
et~al.}{2010}]{ABC}
Alessi, L., M.~Barigozzi, and M.~Capasso (2010).
\newblock Improved penalization for determining the number of factors in
approximate factor models.
\newblock {\em Statistics \& Probability Letters\/}~{\em 80\/}(23-24),
1806--1813.
\bibitem[\protect\citeauthoryear{Aruoba, Diebold, Nalewaik, Schorfheide, and
Song}{Aruoba et~al.}{2016}]{aruoba2016improving}
Aruoba, S.~B., F.~X. Diebold, J.~Nalewaik, F.~Schorfheide, and D.~Song (2016).
\newblock Improving {GDP} measurement: {A} measurement-error perspective.
\newblock {\em Journal of Econometrics\/}~{\em 191}, 384--397.
\bibitem[\protect\citeauthoryear{Ascari, Bonomolo, Hoeberichts, and
Trezzi}{Ascari et~al.}{2023}]{AscariTrezzi}
Ascari, G., P.~Bonomolo, M.~Hoeberichts, and R.~Trezzi (2023).
\newblock The euro area great inflation surge.
\newblock Policy Brief 548, SUERF.
\bibitem[\protect\citeauthoryear{Avarucci, Cavicchioli, Forni, and
Zaffaroni}{Avarucci et~al.}{2022}]{ACFZ}
Avarucci, M., M.~Cavicchioli, M.~Forni, and P.~Zaffaroni (2022).
\newblock The main business cycle shock(s): Frequency-band estimation of the
number of dynamic factors.
\newblock Cepr discussion paper no. dp17281.
\bibitem[\protect\citeauthoryear{Bai}{Bai}{2004}]{bai2004estimating}
Bai, J. (2004).
\newblock Estimating cross-section common stochastic trends in nonstationary
panel data.
\newblock {\em Journal of Econometrics\/}~{\em 122}, 137--183.
\bibitem[\protect\citeauthoryear{Bai and Ng}{Bai and
Ng}{2002}]{bai2002determining}
Bai, J. and S.~Ng (2002).
\newblock Determining the number of factors in approximate factor models.
\newblock {\em Econometrica\/}~{\em 70}, 191--221.
\bibitem[\protect\citeauthoryear{Bai and Ng}{Bai and Ng}{2004}]{bai2004panic}
Bai, J. and S.~Ng (2004).
\newblock A {PANIC} attack on unit roots and cointegration.
\newblock {\em Econometrica\/}~{\em 72}, 1127--1177.
\bibitem[\protect\citeauthoryear{Ba{\'n}bura and Bobeica}{Ba{\'n}bura and
Bobeica}{2023}]{banbura_does_2023}
Ba{\'n}bura, M. and E.~Bobeica (2023).
\newblock Does the {Phillips} curve help to forecast euro area inflation?
\newblock {\em International Journal of Forecasting\/}~{\em 39}, 364--390.
\bibitem[\protect\citeauthoryear{Ba{\'n}bura, Giannone, and Lenza}{Ba{\'n}bura
et~al.}{2015}]{banbura2015conditional}
Ba{\'n}bura, M., D.~Giannone, and M.~Lenza (2015).
\newblock Conditional forecasts and scenario analysis with vector
autoregressions for large cross-sections.
\newblock {\em International Journal of Forecasting\/}~{\em 31\/}(3), 739--756.
\bibitem[\protect\citeauthoryear{Barigozzi, Lippi, and Luciani}{Barigozzi
et~al.}{2021}]{barigozzi_large-dimensional_2021}
Barigozzi, M., M.~Lippi, and M.~Luciani (2021).
\newblock Large-dimensional {dynamic} {factor} {models}: {Estimation} of
{impulse}-{response} {functions} with {$I(1)$} cointegrated factors.
\newblock {\em Journal of Econometrics\/}~{\em 221}, 455--482.
\bibitem[\protect\citeauthoryear{Barigozzi and Luciani}{Barigozzi and
Luciani}{2023}]{barigozzi_measuring_2021}
Barigozzi, M. and M.~Luciani (2023).
\newblock Measuring the {Output} {Gap} using {Large} {Datasets}.
\newblock {\em The Review of Economics and Statistics\/}~{\em 105}, 1500--1514.
\bibitem[\protect\citeauthoryear{Barigozzi and Luciani}{Barigozzi and
Luciani}{2024}]{barigozzi_quasi_2022}
Barigozzi, M. and M.~Luciani (2024).
\newblock Quasi {maximum} {likelihood} {estimation} and {inference} of {large}
{approximate} {dynamic} {factor} {models} via the {EM} algorithm.
\newblock Finance and Economics Discussion Series 2024-086, Board of Governors
of the Federal Reserve System.
\bibitem[\protect\citeauthoryear{Benati and Lubik}{Benati and
Lubik}{2021}]{BenatiLubik}
Benati, L. and T.~A. Lubik (2021).
\newblock Searching for hysteresis.
\newblock Working Paper 21-03, Federal Reserve Bank of Richmond.
\bibitem[\protect\citeauthoryear{Berger, Richter, and Wong}{Berger
et~al.}{2022}]{berger2022unified}
Berger, T., J.~Richter, and B.~Wong (2022).
\newblock A unified approach for jointly estimating the business and financial
cycle, and the role of financial factors.
\newblock {\em Journal of Economic Dynamics and Control\/}~{\em 136}, 104315.
\bibitem[\protect\citeauthoryear{Bergholt, Canova, Furlanetto, Maffei-Faccioli,
and Ulvedal}{Bergholt et~al.}{2025}]{Canovaetal}
Bergholt, D., F.~Canova, F.~Furlanetto, N.~Maffei-Faccioli, and P.~Ulvedal
(2025).
\newblock What drives the recent surge in inflation? the historical
decomposition roller coaster.
\newblock {\em American Economic Journal: Macroeconomics\/}.
\newblock forthcoming.
\bibitem[\protect\citeauthoryear{Berson, Botelho, Da~Silva, Foroni, Mohr,
Schroeder, and Wei{\ss}ler}{Berson et~al.}{2025}]{berson2025explaining}
Berson, C., V.~Botelho, A.~D. Da~Silva, C.~Foroni, M.~Mohr, C.~Schroeder, and
M.~Wei{\ss}ler (2025).
\newblock Explaining the resilience of the euro area labour market between 2022
and 2024.
\newblock {\em Economic Bulletin Articles\/}~{\em 8}.
\bibitem[\protect\citeauthoryear{Bianchi, Nicol\'o, and Song}{Bianchi
et~al.}{2023}]{BianchiNicoloSong}
Bianchi, F., G.~Nicol\'o, and D.~Song (2023).
\newblock Inflation and real activity over the business cycle.
\newblock Finance and Economics Discussion Series 2023-038, Board of Governors
of the Federal Reserve System.
\bibitem[\protect\citeauthoryear{Blanchard and Quah}{Blanchard and
Quah}{1989}]{BQ89}
Blanchard, O.~J. and D.~Quah (1989).
\newblock The dynamic effects of aggregate demand and supply disturbance.
\newblock {\em The American Economic Reivew\/}~{\em 79}, 655--673.
\bibitem[\protect\citeauthoryear{Boivin and Ng}{Boivin and
Ng}{2006}]{boivin2006more}
Boivin, J. and S.~Ng (2006).
\newblock Are more data always better for factor analysis?
\newblock {\em Journal of Econometrics\/}~{\em 132}, 169--194.
\bibitem[\protect\citeauthoryear{Borio}{Borio}{2014}]{borio2014financial}
Borio, C. (2014).
\newblock The financial cycle and macroeconomics: {W}hat have we learnt?
\newblock {\em Journal of Banking \& Finance\/}~{\em 45}, 182--198.
\bibitem[\protect\citeauthoryear{Borio, Disyatat, and Juselius}{Borio
et~al.}{2017}]{borio_rethinking_2017}
Borio, C., P.~Disyatat, and M.~Juselius (2017).
\newblock Rethinking potential output: embedding information about the
financial cycle.
\newblock {\em Oxford Economic Papers\/}~{\em 69}, 655--677.
\bibitem[\protect\citeauthoryear{Buncic, Pagan, and Robinson}{Buncic
et~al.}{2023}]{buncic_discovering_2022}
Buncic, D., A.~Pagan, and T.~Robinson (2023).
\newblock Recovering stars in macroeconomics.
\newblock {\em Available at SSRN 4562801\/}.
\bibitem[\protect\citeauthoryear{Burlon and D’Imperio}{Burlon and
D’Imperio}{2020}]{burlon2020reliable}
Burlon, L. and P.~D’Imperio (2020).
\newblock Reliable real-time estimates of the euro-area output gap.
\newblock {\em Journal of Macroeconomics\/}~{\em 64}, 103191.
\bibitem[\protect\citeauthoryear{Canova}{Canova}{2025}]{canova2022faq}
Canova, F. (2025).
\newblock {FAQ: How do I estimate the output gap?}
\newblock {\em The Economic Journal\/}~{\em 135}, 59--80.
\bibitem[\protect\citeauthoryear{Carriero, Clark, Marcellino, and
Mertens}{Carriero et~al.}{2024}]{carriero_addressing_2022}
Carriero, A., T.~E. Clark, M.~Marcellino, and E.~Mertens (2024).
\newblock Addressing {COVID-19} outliers in {BVARs} with stochastic volatility.
\newblock {\em The Review of Economics and Statistics\/}~{\em 106}, 1403--1417.
\bibitem[\protect\citeauthoryear{Cette, Fernald, and Mojon}{Cette
et~al.}{2016}]{cette2016pre}
Cette, G., J.~Fernald, and B.~Mojon (2016).
\newblock {The pre-Great Recession slowdown in productivity}.
\newblock {\em European Economic Review\/}~{\em 88}, 3--20.
\bibitem[\protect\citeauthoryear{Christiano and Fitzgerald}{Christiano and
Fitzgerald}{2003}]{christianofitzgerald}
Christiano, L.~J. and T.~J. Fitzgerald (2003).
\newblock The bandpass filter.
\newblock {\em International Economic Review\/}~{\em 44}, 435--465.
\bibitem[\protect\citeauthoryear{Ciccarelli and Osbat}{Ciccarelli and
Osbat}{2017}]{CiccarelliOsbat}
Ciccarelli, M. and C.~Osbat (2017).
\newblock Low inflation in the euro area: Causes and consequences.
\newblock Occasional Paper Series 181, European Central Bank.
\bibitem[\protect\citeauthoryear{Claessens, Kose, and Terrones}{Claessens
et~al.}{2012}]{claessens_how_2012}
Claessens, S., M.~A. Kose, and M.~E. Terrones (2012).
\newblock How do business and financial cycles interact?
\newblock {\em Journal of International Economics\/}~{\em 87}, 178--190.
\bibitem[\protect\citeauthoryear{Conti}{Conti}{2021}]{conti2021resurrecting}
Conti, A.~M. (2021).
\newblock Resurrecting the {P}hillips curve in low-inflation times.
\newblock {\em Economic Modelling\/}~{\em 96}, 172--195.
\bibitem[\protect\citeauthoryear{Corsello, Neri, and Tagliabracci}{Corsello
et~al.}{2021}]{CorselloNeriTagliabracci}
Corsello, F., S.~Neri, and A.~Tagliabracci (2021).
\newblock Anchored or de-anchored? that is the question.
\newblock {\em European Journal of Political Economy\/}~{\em 69}, 102031.
\bibitem[\protect\citeauthoryear{Crump, Eusepi, Giannone, Qian, and
Sbordone}{Crump et~al.}{2021}]{crump2021large}
Crump, R.~K., S.~Eusepi, D.~Giannone, E.~Qian, and A.~M. Sbordone (2021).
\newblock A large bayesian var of the united states economy.
\newblock {\em FRB of New York Staff Report\/}~(976).
\bibitem[\protect\citeauthoryear{De~Masi}{De~Masi}{1997}]{demasi1997}
De~Masi, P. (1997).
\newblock {IMF} estimates of potential output: Theory and practice.
\newblock IMF Working Paper 177.
\bibitem[\protect\citeauthoryear{Del~Negro, Giannone, Giannoni, and
Tambalotti}{Del~Negro et~al.}{2017}]{del2017safety}
Del~Negro, M., D.~Giannone, M.~P. Giannoni, and A.~Tambalotti (2017).
\newblock Safety, liquidity, and the natural rate of interest.
\newblock {\em Brookings Papers on Economic Activity\/}~{\em Spring 2017},
235--316.
\bibitem[\protect\citeauthoryear{Del~Negro, Schorfheide, Smets, and
Wouters}{Del~Negro et~al.}{2007}]{del2007fit}
Del~Negro, M., F.~Schorfheide, F.~Smets, and R.~Wouters (2007).
\newblock On the fit of new {K}eynesian models.
\newblock {\em Journal of Business \& Economic Statistics\/}~{\em 25},
123--143.
\bibitem[\protect\citeauthoryear{Diebold and Mariano}{Diebold and
Mariano}{1995}]{DieboldMariano}
Diebold, F.~X. and R.~S. Mariano (1995).
\newblock Comparing predictive accuracy.
\newblock {\em Journal of Business and Economic Statistics\/}~{\em 13},
253--263.
\bibitem[\protect\citeauthoryear{Doz, Giannone, and Reichlin}{Doz
et~al.}{2012}]{doz2012quasi}
Doz, C., D.~Giannone, and L.~Reichlin (2012).
\newblock A quasi-maximum likelihood approach for large, approximate dynamic
factor models.
\newblock {\em The Review of Economics and Statistics\/}~{\em 94}, 1014--1024.
\bibitem[\protect\citeauthoryear{Elfsbacka~Schm{\"o}ller and
Spitzer}{Elfsbacka~Schm{\"o}ller and Spitzer}{2021}]{schmoller2021deep}
Elfsbacka~Schm{\"o}ller, M. and M.~Spitzer (2021).
\newblock Deep recessions, slowing productivity and missing {(dis-) inflation}
in the euro area.
\newblock {\em European Economic Review\/}~{\em 134}, 103708.
\bibitem[\protect\citeauthoryear{Escribano and Pe{\~n}a}{Escribano and
Pe{\~n}a}{1994}]{escribano1994cointegration}
Escribano, A. and D.~Pe{\~n}a (1994).
\newblock Cointegration and common factors.
\newblock {\em Journal of Time Series Analysis\/}~{\em 15}, 577--586.
\bibitem[\protect\citeauthoryear{{European Commission}}{{European
Commission}}{2018}]{EC_Stability}
{European Commission} (2018).
\newblock Staff working document on the review of the flexibility under the
{Stability and Growth Pact}.
\newblock COM(2018) 335.
\bibitem[\protect\citeauthoryear{Forni, Gambetti, Granese, Sala, and
Soccorsi}{Forni et~al.}{2025}]{ForniAmerican}
Forni, M., L.~Gambetti, A.~Granese, L.~Sala, and S.~Soccorsi (2025).
\newblock An {American} macroeconomic picture: Supply and demand shocks in the
frequency domain.
\newblock {\em American Economic Journal: Macroeconomics\/}.
\newblock forthcoming.
\bibitem[\protect\citeauthoryear{Furlanetto, Gelain, and Sanjani}{Furlanetto
et~al.}{2021}]{furlanetto2021output}
Furlanetto, F., P.~Gelain, and M.~T. Sanjani (2021).
\newblock Output gap, monetary policy trade-offs, and financial frictions.
\newblock {\em Review of Economic Dynamics\/}~{\em 41}, 52--70.
\bibitem[\protect\citeauthoryear{Furlanetto, Lepetit, Robstad, Rubio-Ram\'irez,
and Ulvedal}{Furlanetto et~al.}{2025}]{FurlanettoetaAEJmacro}
Furlanetto, F., A.~Lepetit, O.~Robstad, J.~Rubio-Ram\'irez, and P.~Ulvedal
(2025).
\newblock Estimating hysteresis effects.
\newblock {\em American Economic Journal: Macroeconomics\/}~{\em 17}, 25--70.
\bibitem[\protect\citeauthoryear{Gambetti and Musso}{Gambetti and
Musso}{2017}]{gambetti2017loan}
Gambetti, L. and A.~Musso (2017).
\newblock Loan supply shocks and the business cycle.
\newblock {\em Journal of Applied Econometrics\/}~{\em 32}, 764--782.
\bibitem[\protect\citeauthoryear{Giannone and Primiceri}{Giannone and
Primiceri}{2024}]{GiannonePrimicieri}
Giannone, D. and G.~Primiceri (2024).
\newblock The drivers of post-pandemic inflation.
\newblock Working Paper 32859, National Bureau of Economic Research.
\bibitem[\protect\citeauthoryear{Gonz\'alez-Astudillo}{Gonz\'alez-Astudillo}{2019}]{ManuelGA}
Gonz\'alez-Astudillo, M. (2019).
\newblock An output gap measure for the euro area: Exploiting country-level and
cross-sectional data heterogeneity.
\newblock {\em European Economic Review\/}~{\em 120}, 103301.
\bibitem[\protect\citeauthoryear{Hallin and Liška}{Hallin and
Liška}{2007}]{hallin_determining_2007}
Hallin, M. and R.~Liška (2007).
\newblock Determining the {number} of {factors} in the {general} {dynamic}
{factor} {model}.
\newblock {\em Journal of the American Statistical Association\/}~{\em 102},
603--617.
\bibitem[\protect\citeauthoryear{Hamilton}{Hamilton}{2018}]{hamilton2018you}
Hamilton, J.~D. (2018).
\newblock Why you should never use the {Hodrick-Prescott} filter.
\newblock {\em The Review of Economics and Statistics\/}~{\em 100}, 831--843.
\bibitem[\protect\citeauthoryear{Hartl, Tschernig, and Weber}{Hartl
et~al.}{2022}]{hartl2022fractional}
Hartl, T., R.~Tschernig, and E.~Weber (2022).
\newblock Solving the unobserved components puzzle: {A} fractional approach to
measuring the business cycle.
\newblock mimeo.
\bibitem[\protect\citeauthoryear{Hasenzagl, Pellegrino, Reichlin, and
Ricco}{Hasenzagl et~al.}{2019}]{LucreziaInflation}
Hasenzagl, T., F.~Pellegrino, L.~Reichlin, and G.~Ricco (2019).
\newblock The inflation puzzle in the euro area.
\newblock VoxEU 16 October 2019, CEPR.
\bibitem[\protect\citeauthoryear{Hasenzagl, Pellegrino, Reichlin, and
Ricco}{Hasenzagl et~al.}{2022}]{hasenzagl2022model}
Hasenzagl, T., F.~Pellegrino, L.~Reichlin, and G.~Ricco (2022).
\newblock {A Model of the Fed's View on Inflation}.
\newblock {\em The Review of Economics and Statistics\/}~{\em 104}, 686--704.
\bibitem[\protect\citeauthoryear{Havik, Mc~Morrow, Orlandi, Planas, Raciborski,
R{\"o}ger, Rossi, Thum-Thysen, Vandermeulen, et~al.}{Havik
et~al.}{2014}]{havik2014production}
Havik, K., K.~Mc~Morrow, F.~Orlandi, C.~Planas, R.~Raciborski, W.~R{\"o}ger,
A.~Rossi, A.~Thum-Thysen, V.~Vandermeulen, et~al. (2014).
\newblock The production function methodology for calculating potential growth
rates \& output gaps.
\newblock Economic Papers 535, Directorate General Economic and Financial
Affairs (DG ECFIN), European Commission.
\bibitem[\protect\citeauthoryear{\.{I}pek and Kısacıko\u{g}lu}{\.{I}pek and
Kısacıko\u{g}lu}{2025}]{OgapBilkent}
\.{I}pek, M.~S. and B.~Kısacıko\u{g}lu (2025).
\newblock Estimating euro area output gap dynamics: Evidence from the updated
{Area-Wide Model Database}.
\newblock {\em European Economic Review\/}.
\newblock forthcoming.
\bibitem[\protect\citeauthoryear{Jaroci{\'n}ski and Lenza}{Jaroci{\'n}ski and
Lenza}{2018}]{jarocinski2018inflation}
Jaroci{\'n}ski, M. and M.~Lenza (2018).
\newblock An inflation-predicting measure of the output gap in the euro area.
\newblock {\em Journal of Money, Credit and Banking\/}~{\em 50}, 1189--1224.
\bibitem[\protect\citeauthoryear{Justiniano, Primiceri, and
Tambalotti}{Justiniano et~al.}{2013}]{JPT}
Justiniano, A., G.~Primiceri, and A.~Tambalotti (2013).
\newblock Is there a trade-off between inflation and output stabilization?
\newblock {\em American Economic Journal: Macroeconomics\/}~{\em 5}, 1--31.
\bibitem[\protect\citeauthoryear{Kamber, Morley, and Wong}{Kamber
et~al.}{2018}]{kamber2018intuitive}
Kamber, G., J.~Morley, and B.~Wong (2018).
\newblock Intuitive and reliable estimates of the output gap from a
{B}everidge-{N}elson filter.
\newblock {\em The Review of Economics and Statistics\/}~{\em 100}, 550--566.
\bibitem[\protect\citeauthoryear{Kiley}{Kiley}{2013}]{kiley2013output}
Kiley, M.~T. (2013).
\newblock Output gaps.
\newblock {\em Journal of Macroeconomics\/}~{\em 37}, 1--18.
\bibitem[\protect\citeauthoryear{Kim and Kim}{Kim and Kim}{2022}]{kim2022trend}
Kim, C.-J. and J.~Kim (2022).
\newblock Trend-cycle decompositions of real {GDP} revisited: {C}lassical and
{B}ayesian perspectives on an unsolved puzzle.
\newblock {\em Macroeconomic Dynamics\/}~{\em 26}, 394--418.
\bibitem[\protect\citeauthoryear{Lane}{Lane}{2024}]{Lane2024}
Lane, P.~R. (2024).
\newblock The 2021-2022 inflation surges and the monetary policy response
through the lens of macroeconomic models.
\newblock Speech by Philip R. Lane, Member of the Executive Board of the ECB,
at the SUERF Marjolin Lecture hosted by the Banca d’Italia Rome, 18
November 2024.
\bibitem[\protect\citeauthoryear{Lenza and Primiceri}{Lenza and
Primiceri}{2022}]{lenza_how_2022}
Lenza, M. and G.~E. Primiceri (2022).
\newblock How to estimate a vector autoregression after {March} 2020.
\newblock {\em Journal of Applied Econometrics\/}~{\em 37}, 688--699.
\bibitem[\protect\citeauthoryear{Lippi and Reichlin}{Lippi and
Reichlin}{1994}]{lippi_diffusion_1994}
Lippi, M. and L.~Reichlin (1994).
\newblock Diffusion of {technical} {change} and the {decomposition} of {output}
into {trend} and {cycle}.
\newblock {\em The Review of Economic Studies\/}~{\em 61}, 19--30.
\bibitem[\protect\citeauthoryear{Luciani}{Luciani}{2014}]{lucioADFM}
Luciani, M. (2014).
\newblock Forecasting with approximate dynamic factor models: {T}he role of
non-pervasive shocks.
\newblock {\em International Journal of Forecasting\/}~{\em 30}, 20--29.
\bibitem[\protect\citeauthoryear{McCracken and Ng}{McCracken and
Ng}{2020}]{mccracken_fred-qd_2020}
McCracken, M. and S.~Ng (2020).
\newblock {FRED}-{QD}: {A} {quarterly} {database} for {macroeconomic}
{research}.
\newblock NBER working paper 26872.
\bibitem[\protect\citeauthoryear{McCracken and Ng}{McCracken and
Ng}{2016}]{mccracken_fred-md_2016}
McCracken, M.~W. and S.~Ng (2016).
\newblock {FRED}-{MD}: {A} {monthly} {database} for {macroeconomic} {research}.
\newblock {\em Journal of Business \& Economic Statistics\/}~{\em 34},
574--589.
\bibitem[\protect\citeauthoryear{Mian and Sufi}{Mian and
Sufi}{2018}]{mian2018finance}
Mian, A. and A.~Sufi (2018).
\newblock Finance and business cycles: {T}he credit-driven household demand
channel.
\newblock {\em Journal of Economic Perspectives\/}~{\em 32}, 31--58.
\bibitem[\protect\citeauthoryear{Mian, Sufi, and Verner}{Mian
et~al.}{2017}]{mian2017household}
Mian, A., A.~Sufi, and E.~Verner (2017).
\newblock Household debt and business cycles worldwide.
\newblock {\em The Quarterly Journal of Economics\/}~{\em 132}, 1755--1817.
\bibitem[\protect\citeauthoryear{Morley, Nelson, and Zivot}{Morley
et~al.}{2003}]{morley2003beveridge}
Morley, J., C.~R. Nelson, and E.~Zivot (2003).
\newblock Why are the {Beveridge-Nelson} and unobserved-components
decompositions of {GDP} so different?
\newblock {\em The Review of Economics and Statistics\/}~{\em 85}, 235--243.
\bibitem[\protect\citeauthoryear{Morley, Rodríguez-Palenzuela, Sun, and
Wong}{Morley et~al.}{2023}]{morley_estimating_2023}
Morley, J., D.~Rodríguez-Palenzuela, Y.~Sun, and B.~Wong (2023).
\newblock Estimating the euro area output gap using multivariate information
and addressing the {COVID}-19 pandemic.
\newblock {\em European Economic Review\/}~{\em 153}, 104385.
\bibitem[\protect\citeauthoryear{Morley, Tran, and Wong}{Morley
et~al.}{2024}]{morley2023simple}
Morley, J., T.~D. Tran, and B.~Wong (2024).
\newblock A simple correction for misspecification in trend-cycle
decompositions with an application to estimating $r$.
\newblock {\em Journal of Business \& Economic Statistics\/}~{\em 42},
665--680.
\bibitem[\protect\citeauthoryear{Morley and Wong}{Morley and
Wong}{2020}]{morley2020estimating}
Morley, J. and B.~Wong (2020).
\newblock Estimating and accounting for the output gap with large {B}ayesian
vector autoregressions.
\newblock {\em Journal of Applied Econometrics\/}~{\em 35}, 1--18.
\bibitem[\protect\citeauthoryear{Ng}{Ng}{2018}]{ng2018comments}
Ng, S. (2018).
\newblock Comments on the cyclical sensitivity in estimates of potential
output.
\newblock {\em Brookings Papers on Economic Activity\/}~{\em 49}, 412--423.
\bibitem[\protect\citeauthoryear{Ng}{Ng}{2021}]{ng2021modeling}
Ng, S. (2021).
\newblock Modeling macroeconomic variations after {COVID}-19.
\newblock NBER working paper 29060.
\bibitem[\protect\citeauthoryear{Onatski and Wang}{Onatski and
Wang}{2021}]{onatski_spurious_2021}
Onatski, A. and C.~Wang (2021).
\newblock Spurious {factor} {analysis}.
\newblock {\em Econometrica\/}~{\em 89}, 591--614.
\bibitem[\protect\citeauthoryear{Phillips and Jin}{Phillips and
Jin}{2021}]{phillips2021business}
Phillips, P.~C. and S.~Jin (2021).
\newblock Business cycles, trend elimination, and the {HP} filter.
\newblock {\em International Economic Review\/}~{\em 62}, 469--520.
\bibitem[\protect\citeauthoryear{Phillips and Shi}{Phillips and
Shi}{2021}]{phillips2021boosting}
Phillips, P.~C. and Z.~Shi (2021).
\newblock Boosting: {W}hy you can use the {HP} filter.
\newblock {\em International Economic Review\/}~{\em 62}, 521--570.
\bibitem[\protect\citeauthoryear{Plagborg-M{\o}ller, Reichlin, Ricco, and
Hasenzagl}{Plagborg-M{\o}ller et~al.}{2020}]{reichlin_financial_2020}
Plagborg-M{\o}ller, M., L.~Reichlin, G.~Ricco, and T.~Hasenzagl (2020).
\newblock When is growth at risk?
\newblock {\em Brookings Papers on Economic Activity\/}~{\em Spring 2020},
167--229.
\bibitem[\protect\citeauthoryear{Reichlin, Ricco, and Hasenzagl}{Reichlin
et~al.}{2020}]{reichlin_financial_2020wp}
Reichlin, L., G.~Ricco, and T.~Hasenzagl (2020).
\newblock Financial variables as predictors of real growth vulnerability.
\newblock Deutsche Bundesbank Discussion Papers 05/2020.
\bibitem[\protect\citeauthoryear{Rünstler and Vlekke}{Rünstler and
Vlekke}{2018}]{runstler_business_2018}
Rünstler, G. and M.~Vlekke (2018).
\newblock Business, housing, and credit cycles.
\newblock {\em Journal of Applied Econometrics\/}~{\em 33}, 212--226.
\bibitem[\protect\citeauthoryear{Stock and Watson}{Stock and
Watson}{1988}]{stockwatson88JASA}
Stock, J.~H. and M.~W. Watson (1988).
\newblock Testing for common trends.
\newblock {\em Journal of the American Statistical Association\/}~{\em 83},
1097--1107.
\bibitem[\protect\citeauthoryear{Stock and Watson}{Stock and
Watson}{2016}]{stock_chapter_2016}
Stock, J.~H. and M.~W. Watson (2016, January).
\newblock {Dynamic} {factor} {models}, {factor}-{augmented} {vector}
{autoregressions}, and {structural} {vector} {autoregressions} in
{macroeconomics}.
\newblock In J.~B. Taylor and H.~Uhlig (Eds.), {\em Handbook of
{Macroeconomics}}, Volume~2, pp.\ 415--525. Elsevier.
\bibitem[\protect\citeauthoryear{Stock and Watson}{Stock and
Watson}{2025}]{stockcomovement}
Stock, J.~H. and M.~W. Watson (2025).
\newblock Recovering from {COVID}.
\newblock Brookings Paper on Economic Activity Conference Draft, March 27-28,
2025.
\href{https://www.brookings.edu/wp-content/uploads/2025/03/5_Stock-Watson.pdf}{link}.
\bibitem[\protect\citeauthoryear{Thum-Thysen, Blondeau, d’Auria, D{\"o}hring,
Hristov, and Mc~Morrow}{Thum-Thysen et~al.}{2022}]{thum2022ii}
Thum-Thysen, A., F.~Blondeau, F.~d’Auria, B.~D{\"o}hring, A.~Hristov, and
K.~Mc~Morrow (2022).
\newblock Potential output and output gaps against the backdrop of the
{COVID}-19 pandemic.
\newblock {\em Quarterly Report on the Euro Area\/}~{\em 21}, 21--30.
\bibitem[\protect\citeauthoryear{Tóth}{Tóth}{2021}]{toth_multivariate_2021}
Tóth, M. (2021).
\newblock A {multivariate} {unobserved} {components} {model} to {estimate}
{potential} {output} in the {Euro} {Area}: {A} {production} {function}
{based} {approach}.
\newblock {ECB} working paper 2523.
\bibitem[\protect\citeauthoryear{Winter, Koopman, and Hindrayanto}{Winter
et~al.}{2022}]{winter2022joint}
Winter, J.~d., S.~J. Koopman, and I.~Hindrayanto (2022).
\newblock Joint decomposition of business and financial cycles: {E}vidence from
eight advanced economies.
\newblock {\em Oxford Bulletin of Economics and Statistics\/}~{\em 84}, 57--79.
\end{thebibliography}
\clearpage
\titleformat{\section}
{\titlerule[0.75pt]\normalfont\large\bfseries}{\thesection}{1em}{}[{\titlerule[0.75pt]}]
\titleformat{\subsection}
{\normalfont\normalsize\bfseries}{\thesubsection}{1em}{}[{\titlerule[0.2pt]}]
\titleformat{\subsubsection}
{\normalfont\normalsize\it}{\thesubsubsection}{1em}{}[{\titlerule[0.1pt]}]
\titlespacing*{\section}{0pt}{15pt}{6pt}
\titlespacing*{\subsection}{0pt}{12pt}{6pt}
\titlespacing*{\subsubsection}{0pt}{9pt}{6pt}
\pagestyle{fancy}
\fancyhf{}
\chead{\sc Supplementary material for the paper: Measuring the Euro Area Output Gap\\[-10pt]}
\cfoot{\sc Page \thepage\ of \pageref{LastPage}}
\gdef\thesection{\Alph{section}}
\gdef\thesubsection{\Alph{section}.\arabic{subsection}}
\gdef\thefigure{\Alph{section}\arabic{figure}}
\gdef\theequation{\Alph{section}\arabic{equation}}
\gdef\thetable{\Alph{section}\arabic{table}}
\setcounter{table}{0}
\setcounter{figure}{0}
\setcounter{equation}{0}
\setcounter{page}{1}
\begin{center}
\rule{\textwidth}{1pt}\\
\textit{Supplementary material for the paper:} \\
\Large{\bf Measuring the Euro Area Output Gap} \\[-10pt]
\rule{\textwidth}{1pt}\\[12pt]
\begin{tabular}{C{.3\textwidth}C{.3\textwidth}C{.3\textwidth}}
\normalsize Matteo Barigozzi & \normalsize Claudio Lissona & \normalsize Matteo Luciani \\[-6pt]
\small University of Bologna & \small University of Bologna & \small Federal Reserve Board \\[-6pt]
\footnotesize [email removed] & \footnotesize [email removed] & \footnotesize [email removed] \\
\end{tabular}
\end{center}
\footnotetext{\noindent M. Barigozzi and C. Lissona gratefully acknowledges financial support from MIUR (PRIN2020, Grant 2020N9YFFE).\smallskip
\noindent \textsc{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.}
\gdef {(\roman{footnote})}