Estimating the euro area output gap using multivariate information and addressing the COVID-19 pandemic

Abstract

We estimate the euro area output gap by applying the Beveridge–Nelson decomposition based on a large Bayesian vector autoregression. Our approach incorporates multivariate information through the inclusion of a wide range of variables in the analysis and addresses data issues associated with the COVID-19 pandemic. The estimated output gap lines up well with the CEPR chronology of the business cycle for the euro area and we find that hours worked, more than the unemployment rate, provides the key source of information about labor utilization in the economy, especially in pinning down the depth of the output gap during the COVID-19 recession when the unemployment rate rose only moderately. Our findings confirm that labor market adjustments to the business cycle in the euro area occur more through the intensive, rather than extensive, margin.

1. Introduction

The output gap, broadly defined as the deviation of real GDP from its trend, or potential output, is a crucial input into the formulation of monetary policy. It measures the degree of economic slack within the aggregate economy. While being a critical input for policy, the output gap is unobserved, and so needs to be estimated.

In this paper, we estimate the output gap for the euro area. Our key contributions in doing so are to utilize information from a relatively large set of 15 variables and addressing the occurrence of outlier observations associated with the COVID-19 pandemic. The latter issue should not be understated as real GDP growth in most of the world fell by over 20% quarter-on-quarter in annualized terms in 2020Q2 and then bounced back by similar magnitudes in the opposite direction in 2020Q3. To estimate the output gap, we apply the Beveridge and Nelson (1981) (BN) decomposition based on a large Bayesian vector autoregression (BVAR), as proposed in Morley and Wong (2020), but making three modifications for the application to the euro area data and accounting for COVID-19 outliers. First, we account for the fact that the euro area is an open economy and show that making the appropriate treatment of information in the foreign sector matters for estimating the output gap for the euro area, as opposed to simply following the typical closed-economy assumption for the United States. Secondly, we address the effects of the COVID-19 pandemic by following Lenza and Primiceri (2022) to account for the outlier observations during the pandemic. The approach by Lenza and Primiceri (2022) is appropriate for our setting given that, like them, we also work with a BVAR. However, we make some modifications in our application due to using quarterly data, as opposed to the monthly data considered in Lenza and Primiceri (2022). Third, relative to Morley and Wong (2020), we use a much shorter sample period as the euro area was only established in 1999Q1 and we consider the inclusion of data associated with the COVID-19 pandemic in our sample. The short sample makes the loss function based on out-of-sample forecasts that Morley and Wong (2020) adopt for calibrating the degree of shrinkage in the BVAR less suitable for the euro area. Instead, we adopt an alternative loss function to capture the idea that the shocks to the trend should not be excessively large, thus embedding the intuition of policy practitioners that the trend is “smooth”. The new loss function appears to work well within our setting, where, in particular, the procedure is able to recognize that the COVID-19 pandemic in 2020Q2 and 2020Q3 was largely a transitory drop in real GDP more than a downward shift in trend.

Our approach produces an estimated output gap that lines up well with the CEPR chronology of the business cycle for the euro area. While our estimates are comparable with other institutional estimates, such as by the IMF or OECD, we also document key differences, including finding a shallower output gap post-Global Financial Crisis to around 2016, implying trend output during that period, when the level of real GDP remained relatively flat, was not growing. In addition, we also find that hours worked is the key variable in terms of information about the euro area output gap. Notably, hours worked is extremely important in pinning down the depth of the output gap during the COVID-19 recession. This result in terms of the importance of hours worked stands in sharp contrast with the U.S. literature where various studies have shown that the unemployment rate is the main source of information for estimating the output gap (see, e.g., Morley and Wong, 2020, Berger et al., 2023, Fleischman and Roberts, 2011, Barbarino et al., 2020, González-Astudillo and Roberts, 2022). A likely reason for a different result is that euro area economies adjust more through the intensive, rather than extensive, margin in labor markets due to job retention policies and frictions such as high costs of firing and employment protection legislation.1 A key case in point is that the rise in the unemployment rate for the euro area during the COVID-19 recession was relatively modest when compared to changes in other economic indicators or to the rise the U.S. unemployment rate at the time.

The broader literature is replete with methods on how to estimate the output gap (see, e.g., Canova, 2020 and the references herein). Methods that rely on applying filters to real GDP, such as the HP filter or Bandpass filter, require neither parameter estimation nor explicit accounting for information other than real GDP. These methods can therefore never incorporate a direct role for multivariate information, such as from the unemployment rate or hours worked, to quantitatively pin down the level of output gap. Within the broader literature, research estimating the output gap for the euro area is also somewhat scant relative to that for the U.S. economy. This is perhaps not entirely surprising given a history of only slightly over 20 years, with such a short sample presenting obvious econometric challenges. The extant econometric research of estimating output gaps for the euro area or euro area economies using multivariate information has largely employed unobserved components models with a small set of variables.2 Seen from the perspective of extremely short samples, it is perhaps even less surprising that previous studies do not consider large multivariate models and that our challenge of estimating a BVAR aiming to utilize information from 15 variables to construct the euro area output gap should not be understated. Moreover, we also highlight that the outlier observations observed with the COVID-19 pandemic do not yet seem to have been satisfactorily addressed within the unobserved components framework and so our approach to dealing with these outliers should be of broader interest in its own right.3 Finally, we note that, given the BN decomposition is equivalent to an unrestricted unobserved components model (see Morley et al., 2003), the contrast to the extant research using unobserved components models is perhaps not as stark as it may at first appear.

The remainder of this paper proceeds as follows: Section 2 presents the multivariate BN decomposition approach and details on the modeling features that deviate from Morley and Wong (2020), as well as describing the data used in our analysis. Section 3 discusses the main estimation results, where we consider, amongst other issues, the informational content of the various variables, including the important roles of hours worked and the foreign sector. Section 4 concludes.

2. Methods and data

Our modeling framework builds on Morley and Wong (2020), who apply the BN decomposition based on a large BVAR to estimate the U.S. output gap. We first briefly present the approach proposed by Morley and Wong (2020), before motivating the modifications to the approach that we make to address issues with the data for the euro area. We then describe the data.

2.1. BN decomposition based on a BVAR

Beveridge and Nelson (1981) define the trend of a time series, $y_t$, as the long-horizon conditional forecast minus any deterministic drift. In particular, the BN trend, ${\tau }_t^{BN}$, is

$${\tau }_t^{BN}\equiv \underset{j\to \infty }{lim}{\mathbb{E}}_t[y_{t+j}-j\cdot \mu ],$$ (1)

where $\mu \equiv \mathbb{E}\left[\Delta y_t\right]$ is the deterministic drift, or unconditional mean of the growth rate, of $y_t$ and the BN trend will correspond to the conditional expectation of the trend in $y_t$ as long as the unconditional mean of the cyclical component is zero (see Morley et al., 2003). In our application $y_t$ is the natural logarithm of the euro area real GDP and $\Delta y_t$ is its first difference, which approximates the quarter-on-quarter real GDP growth rate. Next, consider a VAR(p) forecasting model in the companion form:

$${\mathit{X}}_t=\mathit{F}{\mathit{X}}_{t-1}+\mathit{H}{\mathit{e}}_t,$$ (2)

where ${\mathit{X}}_t\equiv {\left[{\tilde{\mathit{x}}}_t^{\prime },{\tilde{\mathit{x}}}_{t-1}^{\prime }\dots {\tilde{\mathit{x}}}_{t-p}^{\prime }\right]}^{\prime }$ and ${\tilde{\mathit{x}}}_t$ represents a $N\times 1$ vector of demeaned variables where ${\tilde{\mathit{x}}}_t\equiv {\mathit{x}}_t-\mathit{\mu }$ and $\mathit{\mu }$ is an $N\times 1$ vector of unconditional means for ${\mathit{x}}_t$.4 $\mathit{F}$ is a companion matrix, ${\mathit{e}}_t$ is a $N\times 1$ vector of forecast errors, and $\mathit{H}$ is a matrix that maps the forecast errors to the companion form. Let $\Delta {\tilde{y}}_t$, or demeaned output growth, be the $k$th element of ${\tilde{\mathit{x}}}_t$. Defining ${\mathit{\iota }}_k$ as a $1\times Np$ selector vector with 1 as its $k$th element and zero otherwise, Morley (2002) shows that the BN cycle of $y_t$, i.e., the output gap in our case, can be calculated as follows:

$$c_t^{BN}=-{\mathit{\iota }}_k\mathit{F}{\left[\mathit{I\; -\; F}\right]}^{-1}{\mathit{X}}_t.$$ (3)

At a high level, our approach involves specifying a BVAR, casting it into the form implied by Eq. (2), and then subsequently applying Eq. (3) to construct the estimated output gap. We can specify a standard BVAR for a demeaned vector of variables as

$${\tilde{\mathbf{x}}}_{\mathbf{t}}={\mathit{\Phi }}_{\mathbf{1}}{\tilde{\mathbf{x}}}_{\mathbf{t-1}}+\dots +{\mathit{\Phi }}_{\mathit{p}}{\tilde{\mathbf{x}}}_{\mathbf{t-p}}+{\mathit{e}}_{\mathit{t}}$$ (4)

where ${\mathit{\Phi }}_{\mathit{j}}$, $j\in \left\{1,\dots ,p\right\}$ are $N\times N$ matrices of VAR coefficients, $\mathbb{E}\left[{\mathit{e}}_{\mathit{t}}{{\mathit{e}}^{\prime }}_{\mathit{t}}\right]=\mathit{\Sigma }$ and $\mathbb{E}\left[{\mathit{e}}_{\mathit{t}}{{\mathit{e}}^{\prime }}_{\mathit{t-i}}\right]=\mathbf{0}$ $\forall i>0$. Sample averages are used to demean the variables, which is basically equivalent to setting a flat prior on the unconditional means. Shrinkage is then applied to the VAR coefficients using a Minnesota-type prior specification. Defining ${\varphi }_i^{jk}$ as the slope coefficient of the $i$th lag of variable $k$ in the $j$th equation of the VAR, and so elements of the ${\mathit{\Phi }}_{\mathit{j}}$ matrices in Eq. (4), the prior means and variances of the slope coefficients are given as follows:

$$\mathbb{E}\left[{\varphi }_i^{jk}\right]=0$$ (5)

$$Var\left[{\varphi }_i^{jk}\right]=\left\{\begin{array}{c}\frac{{\lambda }^2}{i^2},j=k\hfill \\ \frac{{\lambda }^2}{i^2}\frac{{\sigma }_j^2}{{\sigma }_k^2},\text{otherwise},\hfill \end{array}\right)$$ (6)

with the variances ${\sigma }_j^2$ and ${\sigma }_k^2$ set to the variances of residuals from AR(4) models estimated using least squares for the corresponding variables as per the usual practice (e.g., Banbura et al., 2010, Koop, 2013). Following the standard Minnesota prior structure, the factor $\frac{1}{i^2}$ shrinks coefficients at longer lags. Banbura et al. (2010) show how one can set a natural-conjugate prior with the prior implied by Eqs. (5), (6). The natural conjugate-prior is convenient as, conditional on $\lambda$, all the posterior moments of the model can be calculated analytically without requiring MCMC methods to approximate the posterior distribution. This provides important efficiency gains as it enables a quick evaluation of one-step ahead out-of-sample forecast errors for real GDP growth to select $\lambda$, as in Morley and Wong (2020). Evans and Reichlin (1994) demonstrate that the forecastability of real GDP growth is the key ingredient in obtaining the output gap, therefore parameter proliferation in the BVAR is a key source of potential overfitting, with overfitting leading to spurious forecastability. Thus, selecting $\lambda$ to minimize the overfitting of real GDP is a well-motivated approach.5

2.2. Allowing for a foreign sector

Because the euro area is a relatively open economy, we consider a standard two-block foreign and domestic structure, similar to some of the existing open-economy literature (see, e.g., Zha, 1999, Justiniano and Preston, 2010, Kamber and Wong, 2020), where we treat the foreign sector as being block-exogenous with respect to the euro area. Allowing the first $N^{\ast }$ variables in ${\tilde{\mathbf{x}}}_{\mathbf{t}}$ to be foreign variables, imposing block-exogeneity involves restricting the top-right $N^{\ast }\times N$ block of ${\mathit{\Phi }}_{\mathit{j}}$ matrices in Eq. (4) to be zero (i.e., the euro area variables do not have an effect on the foreign variables). On a more technical note, the zero restrictions with block-exogeneity mean that we lose the natural-conjugacy of the Normal-Inverse Wishart prior that Banbura et al. (2010) introduced for large BVARs. The loss of the natural-conjugate prior is nontrivial in our context because it was the analytical calculation of posterior moments that enabled Morley and Wong (2020) to quickly evaluate the out-of-sample forecasts across a variety of $\lambda$’s when choosing the optimal degree of shrinkage. In other words, the block-exogenous structure considered here imposes a very large computational cost.6 We return to this point later, as one would naturally ask whether it is worth paying the large computational cost for the sake of imposing block-exogeneity.

2.3. Treatment of outlier observations around the COVID-19 pandemic

A more recent issue is how to treat observations associated with the COVID-19 pandemic. As is now well-known, outlier observations during the pandemic period can induce nontrivial changes to the estimated parameters from a BVAR. We take a pragmatic approach in dealing with the COVID-19 data following the suggestions of Lenza and Primiceri (2022). We first rewrite Eq. (4) as

$${\tilde{\mathbf{x}}}_{\mathbf{t}}={\mathit{\Phi }}_{\mathbf{1}}{\tilde{\mathbf{x}}}_{\mathbf{t-1}}+\dots +{\mathit{\Phi }}_{\mathit{p}}{\tilde{\mathbf{x}}}_{\mathbf{t-p}}+s_t{\mathit{e}}_{\mathit{t}}.$$ (7)

The approach of Lenza and Primiceri (2022) treats $s_t$ as a parameter during COVID-19 pandemic, where $s_t$ allows the residual covariance matrix to scale up its regular value of $\mathit{\Sigma }$ by a factor of $s_t^2$ during the pandemic period. Once conditioning on $s_t$, we can specify the BVAR as follows:

$${\tilde{\mathbf{x}}}_{\mathbf{t}}/s_t={\mathit{\Phi }}_{\mathbf{1}}{\tilde{\mathbf{x}}}_{\mathbf{t-1}}/s_t+\dots +{\mathit{\Phi }}_{\mathit{p}}{\tilde{\mathbf{x}}}_{\mathbf{t-p}}/s_t+{\mathit{e}}_{\mathit{t}}.$$ (8)

As should be clear, if $s_t=1$ $\forall t$, the model collapses back into the standard BVAR in Eq. (4). Lenza and Primiceri (2022) essentially specify three parameters to model the effects of the COVID-19 pandemic: one for March 2020, one for April 2020, and one to govern decay back to normal after April 2020. Within the context of a six-variable monthly BVAR of the U.S. economy, their approach works relatively well. In our case, it is necessary to adjust their approach to the quarterly data we use. We similarly set three COVID-19 parameters; one each for 2020Q1, 2020Q2, and 2020Q3. That is, we set $s_t=1$ except for the three quarters, where we estimate a different $s_t$ for each quarter. We do not model a decay parameter because we find that the variation post 2020Q3 appears to be back to pre-pandemic levels in the sense that, when we estimated an additional fourth parameter for 2020Q4, we found the estimate was close to 1. However, we require one parameter each for 2020Q1, 2020Q2, and 2020Q3 because the data at the time were characterized by different magnitude changes in many variables in all three quarters, with much larger drops in many series in 2020Q2 than in 2020Q1 and large rebounds in 2020Q3.7

Lenza and Primiceri (2022) use a fully Bayesian approach by jointly estimate $s_t$ and $\lambda$. We take a more pragmatic approach because a fully Bayesian approach is considerably less computationally efficient. In particular, as our earlier discussion alluded to, we need to be able to quickly evaluate the model across different values of $\lambda$. Thus, we adopt a two-step approach, which is a hybrid of the Bayesian and maximum likelihood estimation (MLE) approaches discussed in Lenza and Primiceri (2022). Lenza and Primiceri (2022) show how to concentrate $s_t$ out of the likelihood function and so allow for estimation of the COVID-19 parameters using MLE. Following this, we first estimate our COVID-19 parameters using MLE and then we reweight the data with the MLE estimate of $s_t$ as per Eq. (8). It is straightforward, then, to estimate a standard BVAR with the reweighted data. One might naturally question moving between a frequentist and Bayesian approach from the first to second step. However, we verify this matters very little in our application and show that our baseline estimates of the output gap are essentially identical to taking a fully Bayesian approach to estimating $s_t$ when fixing $\lambda$ at its chosen value for the baseline. In terms of computational efficiency, our two-step approach is about a factor of 40 times faster than taking a fully Bayesian approach.8

2.4. A modified loss function for selecting the shrinkage hyperparameter

As previously noted, Morley and Wong (2020) selected $\lambda$ to minimize the one-step ahead out-of-sample forecast error of output growth. While this approach has proven to be suitable within the context of estimating the output gap for the U.S. economy (see, e.g., Berger et al., 2022, Berger et al., 2023), it poses two challenges within the context of our application. First, we have a much shorter sample relative to working with U.S. data, and, second, the occurrence of the COVID-19 pandemic in the sample requires addressing specific challenges with forecasting outlier observations. The sample we consider is short due to the relatively recent launch of the euro area, meaning we have only slightly more than 20 years of data, or around 90 quarterly observations, with which to fit a reasonably-sized VAR.9 Given a 15-variable BVAR with 4 lags, the available 90 or so quarterly observations does not allow for a sufficiently long out-of-sample sample period with which to calibrate $\lambda$. Moreover, given that observations related to the COVID-19 pandemic are reweighted by $s_t$, it is not clear how to implement an out-of-sample forecast, as the model is, strictly speaking, forecasting the reweighted observations. One possibility would be just to fix the value of the hyperparameter $\lambda$, as one often encounters in the BVAR literature, such as setting $\lambda =0.2$ (see, e.g., Carriero et al., 2015). While this approach proves viable for the pre-pandemic sample, we will show this is less suitable during the pandemic.

As an alternative, we adapt an approach proposed by Kamber et al. (2022) in a univariate context to our multivariate setting by selecting $\lambda$ to minimize the variance of the change in trend, i.e., we impose a relatively smooth trend. Kamber et al. (2022) apply this criterion with their univariate BN filter and find it is helpful in addressing COVID-19 outliers. In our setting, as our empirical results will show, minimizing the variance of the change in trend also appears to be a suitable way to deal with the COVID-19 recession, where real GDP fell by a lot, but the strength of the immediate rebound a priori suggests that the trend did not change by much. We will later present results that show that other approaches of setting $\lambda$ imply a larger change in trend with the large fall of GDP, which, while arguably plausible ex ante, appears unlikely ex post given what we now know about real GDP growth dynamics post-2020Q2.

2.5. Data

Our model includes 15 variables for the period from 1999Q1 to 2021Q4. The variables in the foreign block are U.S. real GDP and the real price of oil, while the variables in the domestic block are euro area real GDP, industrial production, employment, housing permits, CPI, the policy rate, hours worked, the term spread, capacity utilization, the unemployment rate, PMI, the risk spread, and the real effective exchange rate. When needed, the raw data are transformed to be stationary, with transformations generally consistent with the existing literature (see, e.g., Stock and Watson, 2012, McCracken and Ng, 2021), such as log differences for real activity variables and first differences for the policy rate, but keeping interest rate spreads in levels. Appendix A provides full details of data sources and transformations.

3. Results

3.1. Baseline estimates

The top panel of Fig. 1 presents the estimated output gap for our 15-variable BVAR with $\lambda =0.75$, which is selected based on minimizing the variance of the estimated trend shocks over the whole sample. Overall, our estimated output gap for the euro area accords well with all turning points of the business cycle as dated by CEPR ever since the monetary union came into existence. Our estimated output gap exhibits a large drop during the COVID-19 recession, reaching $-12\text{\%}$ in 2020Q2. Notably, the large drop is not persistent, as the estimated output gap suggests a large bounceback in 2020Q3, in line with the reversal of the large decrease in real GDP in 2020Q2 that can be related to the lockdowns to mitigate the initial spread of the virus and the subsequent opening up of the restricted sectors of the economy. Our output gap also lines up with the other two recessions in the sample in 2008–09, associated with the global financial crisis and sometimes referred to as the ‘Great Recession’, and 2010–12, associated with the European sovereign debt crisis.

Fig. 1.

Baseline results — Estimated output gap for the euro area. The shaded bars represent CEPR dating of recessions. The output gap is measured as a percentage deviation from trend. BN refers to our baseline estimate using Beveridge–Nelson decomposition; HP filter refers to Hodrick–Prescott filter; UCM refers to an unobserved components model by Tóth (2021).

The bottom panels of Fig. 1 compare our estimated output gap with external institutional estimates and other estimates. We note that the institutional estimates by the OECD, European Commission (EC), and the IMF are based on annual data, and so might not be fully comparable to our quarterly estimate. All three institutions base their estimates off a framework that assumes a production function for the total economy and defines the output gap as the difference between actual output and the estimated trend according to the production function. UCM refers to estimates for an unobserved components model by ECB staff based on quarterly data. We also compare our results against an output gap constructed using the HP filter with its typical smoothing parameter of 1600 for quarterly data. Our estimated output gap aligns reasonably closely with the external estimates, as well as the HP filter estimate. That said, a key departure is that the institutional estimates, including the UCM, consistently estimate a much lower output gap since the end of the Great Recession relative to our estimate and also to the HP filter estimate, which is remarkably similar to our estimate from 2009 on. In particular, the institutional estimates of the output gap at the trough of the euro area crisis are roughly similar in depth to the estimates of the output gap during the Great Recession, whereas ours and the HP filter estimate are higher. We note that real GDP in the euro area from the trough of the Great Recession to the trough of the European sovereign debt crisis only grew by 2.5% or roughly 0.6% per annum. From the external institutions’ estimates of the output gap, the implied trend growth is as much as 4% or roughly 1.2% per annum at a time when real GDP largely stagnated. In other words, our estimate implies relatively flat trend output after the Great Recession, whereas the institutional estimates largely assume trend output to be growing at a fast rate, even though actual output stagnated for around 7 years, not recovering to its previous peak at the onset of the Great Recession until early 2015. This is also why the HP filter, like our approach, estimates a much smaller output gap during the European sovereign debt crisis and thereafter. Because real output growth stagnated, implied trend growth for the HP filter also stagnated. Appendix B presents and discusses the implied trend based on our approach, as well as those from the other two quarterly measures of the output gap based on the UCM and HP filter.

We note that related work by Hasenzagl et al. (2019) and González-Astudillo (2019) suggest there was little slack in the euro area by 2017 and they estimate the output gap to have largely closed by 2017, possibly peaking in 2020Q1. While the timing of our estimated improvement in the output gap pre-dates theirs by about a year, our findings suggest a similar interpretation that the institutional measures may have been overly optimistic in terms of implied trend growth after the Great Recession. We consider a related issue in Appendix C of what a possible change in drift or trend growth, as well as the mean of any of the variables used in our BVAR, would imply for our estimation procedure. In short, allowing for a change in trend growth implies a slightly more negative output gap in the years leading up to the pandemic compared to our baseline (and thus closer to the institutional estimates), but this also implies a much larger drop in trend growth (and thus a shallower output gap) during the pandemic. In addition, when we compare the correlation of year-on-year euro area inflation relative to the three quarterly-based output gap estimates, our approach produces a correlation of 55% versus 41% and 45% for the UCM and HP filter respectively. While we caution that this correlation does not necessarily prove our approach provides a more accurate output gap measure, it does suggest that our approach produces a plausible estimate of the euro area output gap, at least when compared to other methods.

In terms of value-added of our approach compared to the HP filter given the similar output gap estimates after 2009, we highlight that it allows us to pinpoint which variables contain high informational content for the output gap, while the HP filter cannot do so given it is a univariate decomposition. We illustrate the benefits of an informational decomposition in Section 3.3. Also, as discussed in Section 3.6, our estimates are more reliable than the HP filter in a real-time setting. Thus, our approach seems to provide a more accurate prediction of revised HP filter estimates than the HP filter does itself, a result that is similar to what was found for U.S. data in Berger et al. (2023).

3.2. Effects of COVID-19 adjustments and shrinkage hyperparameter

Because most of our modifications compared to Morley and Wong (2020) are aimed at dealing with data associated with the COVID-19 pandemic, we next turn our attention to how these modifications affect the estimated output gap during the COVID-19 recession, as well as considering the robustness of pre-pandemic estimates. The top panel of Fig. 2 illustrates the importance of how we deal with the high volatility of the data around the onset of the pandemic. Our baseline involves the two-step approach of estimating different variance scale factors in the key quarters of 2020 and standardizing the data based on these estimates before estimating the BVAR. The blue dotted line presents estimates that do not account for changes in variance (i.e., estimating the BVAR without considering $s_t$ in Eq. (8) or, equivalently, setting $s_t=1$ even during the COVID-19 recession). We keep the shrinkage hyperparameter the same at $\lambda =0.75$ in this case to focus on how allowing for a change in variance deals with the COVID-19 data. Without an explicit treatment for the outlier observations associated with the COVID-19 pandemic, we can see a higher pre-pandemic output gap from about 2015 to 2019, indicating that the inclusion of data from during the COVID-19 pandemic without any adjustment would lead to nontrivial changes in the estimated BVAR parameters, a finding which mirrors what Lenza and Primiceri (2022) find with their six-variable BVAR for the U.S. economy. We also explore an alternative strategy suggested by Lenza and Primiceri (2022) of estimating the model parameters with just the pre-pandemic data to retain the pre-pandemic dynamics for forecasting and impulse response analysis. In this case, to be more in line with what would have been done with pre-pandemic estimation, we set $\lambda =0.2$ instead of 0.75. We find this strategy of just using pre-pandemic data for estimation may be a reasonable compromise in our setting if one does not want to grapple with how to model the COVID-19 data. The black line presents the estimates and there are only marginal differences in the estimated output gap around the pandemic, where estimating parameters with data up until 2019Q4 leads to a slightly lower output gap before the pandemic, while the estimated output gap in 2020Q2 is a bit shallower than with our two-step approach. We nonetheless note that the approach of estimating parameters only up to 2019Q4 will become increasingly indefensible over time the further we get away from 2020Q2. We therefore see our two-step approach as being more enduring going forward, especially as samples extend well beyond the COVID-19 pandemic. Moreover, we note that our sample is quite short relative to the application by Lenza and Primiceri (2022), so an explicit treatment of the outlier observations during the COVID-19 recession should be preferable to simply estimating parameters until 2019Q4 only.

Fig. 2.

Effects of COVID-19 adjustments. The shaded bars represent CEPR dating of recessions. The output gap is measured as a percentage deviation from trend.

A reasonable question is what would happen if we did not adopt the two-step approach, but rather did a full Bayesian estimation of the COVID-19 parameters, as done in Lenza and Primiceri (2022). In particular, our two-step approach is an approximation since our estimation conditions on MLE estimates of the COVID-19 parameters rather than treating them as random variables, as one would do in a fully Bayesian treatment. The bottom panel of Fig. 2 presents a comparison of results for our two-step approach and fully Bayesian estimation given a fixed $\lambda =0.75$. Essentially, the two are nearly indistinguishable. A key intuition of why this happens is because, with the pandemic, the increase in $s_t$ is so large during the COVID-19 recession that the COVID-19 observations are effectively reweighted to extremely small values even though the estimation methods yield somewhat different estimates for $s_t$.10 Because we scale by very large numbers whether we do the two-step or a fully Bayesian treatment, either strategy ends up with very similar estimated output gaps. We emphasize, though, that the two-step approach, while an approximation, is computationally about 40 times more efficient than a fully Bayesian treatment of the COVID-19 parameters. As both approaches turn out to produce results that are virtually indistinguishable, we thus view the efficiency gain as a good reason for us to adopt the two-step approach when needing repeated calculations, such as when determining the optimal $\lambda$ to minimize the variance of the change in trend.

Next, we explore the effects of changing the shrinkage hyperparameter on the estimated output gap during the COVID-19 pandemic, as well as on the pre-pandemic estimates. The top panel of Fig. 3 presents our baseline estimate with $\lambda =0.75$ against those for some other choices of the hyperparameter. The other choices for $\lambda$ are 0.075 corresponding to the value when detrending U.S. real GDP in Morley and Wong (2020),11 0.2 corresponding to the value suggested in Carriero et al. (2015), and 0.9 to determine what happens as shrinkage is relaxed further towards MLE. Consistent with the Minnesota-type prior structure, more shrinkage (i.e., smaller $\lambda$) reduces the amplitude of the output gap, as the prior shrinks towards real GDP being modeled as a random walk with no cyclical component. Relaxing the shrinkage more to $\lambda =0.9$ has barely any impact on the estimated output gap compared to our baseline case, suggesting our baseline shrinkage of $\lambda =0.75$ is already fairly loose. We note that pre-pandemic, the different values of $\lambda$ lead to very similar estimated output gaps. The most notable effect of $\lambda$ on the estimates is in terms of the COVID-19 recession, where the smaller values of 0.075 and 0.2 produce only a small negative estimated output gap in 2020Q2 when estimating the parameters with the full sample of data. The bottom panel of Fig. 3 reveals that a small negative output gap during the COVID-19 recession translates into a large drop in the estimated trend for $\lambda =0.2$, while our baseline case of $\lambda =0.75$ based on smoothing the trend has the much more intuitive result that the large fall in real GDP was largely transitory more than a downward shift in trend.

Fig. 3.

Effects of changing the shrinkage hyperparameter. The shaded bars represent CEPR dating of recessions. The output gap is measured as a percentage deviation from trend. Real GDP is reported in levels and normalized to 100 in 1999Q1.

3.3. Informational content of variables

A useful feature of our modeling approach is that the multivariate nature of the analysis conveniently offers an economic interpretation through the decomposition of the output gap into sources of forecast errors, or possibly identified structural shocks. This is a key benefit of our modeling approach compared to univariate estimates, even if the estimates end up being reasonably similar, such as we find with our estimates and the HP filter from 2009 on. Fig. 4 presents the standard deviations of the informational contributions for each of the 15 variables in the baseline model (full details of how we calculate the informational decompositions are provided in Appendix D). We observe that almost every variable contributes non-negligibly to the estimated output gap, perhaps excepting the term spread. We highlight two key insights from the informational decomposition. First, of the domestic variables, hours worked comes out as the most important in providing information about the estimated output gap, even in the presence of other labor market variables in the model such as the unemployment rate and employment. This result stands in sharp contrast to the analysis of Morley and Wong (2020), who show that the unemployment rate provides the bulk of the useful information needed to estimate the U.S. output gap, a result that is also reflected in Berger et al. (2023), and even in other modeling approaches used for estimating the U.S. output gap such as Barbarino et al., 2020, Fleischman and Roberts, 2011, and González-Astudillo and Roberts (2022).12 Second, the foreign block variables, even though there are only two of them, account for a reasonably large share of information used to estimate the euro area output gap, highlighting the relevance of considering an open-economy setting in our analysis.

Fig. 4.

Informational content of variables. Units are standard deviations. Contributions are calculated based on an informational decomposition of the estimated output gap into different types of forecast errors. IP is industrial production, CAPU is capacity utilization, PMI is the purchasing manager index for manufacturing output, and RER is the real exchange rate. The appendix provides full details of the data sources and transformations.

A natural question is whether one requires such a large set of variables in order to estimate the euro area output gap, especially given the similarity to the univariate estimate based on the HP filter. Also, in the context of the BN decomposition, the relatively short history of the euro area with a sample only starting in 1999Q1 raises questions about whether such a large model is appropriate given the proliferation of parameters to be estimated. From Evans and Reichlin (1994) and subsequently Morley and Wong (2020), we know that whether multivariate information matters for the estimation of the output gap is related to the extent a variables is relevant for forecasting or, more precisely, Granger-causing output growth.13 As Fig. 4 suggests, the forecast errors for almost all the variables have a non-negligible information content for the output gap, providing prima facie support for the idea that one needs a reasonably large model to account for all the relevant information to estimate the euro area output gap. Confirming this, we show in Appendix E that all the variables except the real exchange rate directly Granger-causes output growth in the full sample. However, if we restrict ourselves to a pre-pandemic sample, there is a bit more mixed evidence with Granger-causality tests for industrial production, the real exchange rate, the unemployment rate, and the PMI index not being significant.

As discussed by Morley and Wong (2020), if variables contain no extra relevant information for output growth, then one would estimate the same output gap in population when omitting these extra variables given that a reduced variable set would span the same information. By contrast, if omitted variables contain additional relevant forecasting information for output growth, then the estimated output gap will change, with the change reflecting the inclusion of relevant information to move the estimate closer to the true output gap. Therefore, the goal with a multivariate BN decomposition is to consider a model specification that spans all the relevant forecasting information for output growth, at which point the inclusion of any additional information would no longer change the output gap estimate, at least in population. We thus compared our baseline estimated output gap to output gaps estimated from (i) a univariate AR(4) model (ii) a 14-variable specification dropping the real exchange rate, (iii) an 11-variable specification dropping the real exchange rate, industrial production, unemployment, and the PMI, and (iv) a 14-variable specification dropping the term spread. The comparison to the univariate model explores whether one needs multivariate information at all in order estimate the euro area output gap using a BN decomposition. The 14-variable BVAR dropping the real exchange rate and the 11-variable BVAR are informed by the Granger-causality tests in Appendix E. The 14-variable BVAR dropping the term spread is motivated by the term spread having the smallest share in the informational decomposition in Fig. 4, and thus would be the variable to drop if one wanted to use the informational decomposition to pare down the model as suggested by Morley and Wong (2020).

Fig. 5 presents the estimated output gaps from the various specifications. As is clear, multivariate information is important to avoid an estimated gap that is very small in amplitude, which would imply that most movements in euro area log real GDP are effectively changes in trend. It should not be surprising that multivariate information matters for the estimation of the euro area output gap given that we can find evidence of Granger causality for at least of the 10 variables. Turning to the other multivariate models, dropping the real exchange rate has little effect on the estimated output gap. This is expected given the real exchange rate does not Granger-cause output growth. Therefore, if one is just concerned about estimating the output gap, and not understanding the possible role of variables, dropping the real exchange rate would be reasonable. However, the real exchange rate does have a higher informational contribution than five of the other variables in Fig. 4, including euro area real GDP. Thus, we include it in our baseline model. Meanwhile, dropping the term spread or considering the 11-variable specification changes the estimated output gap more substantially, but in different ways. While both these specifications imply a much larger positive output gap in the lead up to the pandemic, the 11-variable specification also estimates an even larger negative output gap in 2020Q2.

Fig. 5.

Estimated euro area output gap for various-sized models. The shaded bars represent CEPR dating of recessions. The output gap is measured as a percentage deviation from trend. Univariate AR(4) refers to estimated output gap from an AR(4) model. 11 variable refers to a model that drops the real exchange rate, industrial production, unemployment, and the PMI. Drop term spread and drop real exchange rate refer to models that drop noted variables.

Therefore, from this analysis of model size, we conclude that the minimal variable set to estimate the euro area output gap is relatively large, at least compared to what Morley and Wong (2020) found for the U.S. economy, where eight well-chosen variables from the 23-variable baseline BVAR were sufficient to span the relevant forecasting information. For the euro area, it appears the relevant variable set is very close to our baseline, where one would only consider dropping the real exchange rate if the interest was primarily in the estimated output gap and not the informational decomposition. Meanwhile, even if the informational contribution of the term spread to the output gap is quite small, it clearly contains unique relevant information for forecasting output growth and estimating the output gap. The importance of the term spread, despite the large multivariate information set, is consistent with a long tradition of viewing the term spread as important for forecasting future output growth and recessions in particular (see, e.g., Estrella and Mishkin, 1998). Together with our finding of a large role for the Gilchrist and Mojon (2018) risk spread in Fig. 4, there is a clear suggestion that it is important to include financial variables when estimating the euro area output gap.

3.4. The role of hours worked

While the standard deviations used for the informational decomposition in Fig. 4 provide useful clues as to which variables have been important on average, they may mask periods where certain variables have had particularly sizeable contributions to the estimated output gap. We thus also look at an historical informational decomposition. To maintain readability, the top panel of Fig. 6 presents the share of the forecast errors of hours worked (and the other 14 variables grouped together). Notably, hours worked is a particularly important variable for providing information about the depth of the output gap during the COVID-19 recession. We see that a large component of the negative output gap during the COVID-19 recession can be attributed to information about hours worked. Recall that this is in addition to other labor market variables such as unemployment and employment being in the model. We also note that hours worked no longer contributes much in 2020Q3, which suggests a bounceback of hours worked also provides information about output reverting to trend. In the bottom panel of Fig. 6, we consider the same historical informational decomposition, but now with $\lambda =0.2$. For most of the sample, we can see that the share of hours worked mirrors that for our baseline model with $\lambda =0.75$, re-emphasizing that, within a wide range, the specific value of the shrinkage hyperparameter has a meaningful impact on estimates only during the COVID-19 recession. Notably, the difference in the role of hours worked is by far most stark in the pandemic. Hours worked, while still important in the model with $\lambda =0.2$, has a smaller negative contribution than in the baseline case, which results in the model with $\lambda =0.2$ estimating a much smaller negative output gap in 2020Q2. Therefore, another way of understanding our results in terms of why we need to shrink less than typically done in other BVAR studies in order to produce an intuitive estimated output gap during the COVID-19 recession is that the shrinkage hyperparameter of $\lambda =0.75$ is allowing a greater degree of forecastability from hours worked to real GDP growth, which is important in pinning down the quantitative effects of the COVID-19 recession on the output gap.

Fig. 6.

The role of hours worked. The shaded bars represent CEPR dating of recessions. The output gap is measured as a percentage deviation from trend. Contributions are calculated based on an informational decomposition of the estimated output gap into different types of forecast errors.

Fig. 7 presents time series plots for some select variables from the model to provide further context for why hours worked matters so much in pinning down the depth of the output gap during the COVID-19 recession. The top left subplot shows the patterns in hours worked consistent with its contributions to the estimated output gap discussed earlier. The top right subplot suggests that, at least relative to what happened in the U.S. economy, the euro area unemployment rate rose very little during the COVID-19 recession, suggesting much of the labor utilization adjustment in response to the pandemic occurred on the intensive, rather than extensive, margin. Hours worked was not the only variable that experienced a large drop during the COVID-19 recession. Indeed, industrial production growth and capacity utilization, two variables that are also in our model, fell by a lot too. However, the magnitude of these falls was comparable to levels that they fell during the Great Recession in 2008–09. Because the fall in real GDP during the COVID-19 recession was much larger relative to the Great Recession and also during the recession associated with the European sovereign debt crisis in 2010–12, it is clearly necessary to also consider a variable that displays similar such dynamics when pinning down the depth of the output gap during the COVID-19 recession, which, visually inspecting the plots in Fig. 7, provides some intuition for why hours worked is such a crucial variable in our analysis.

Fig. 7.

Time series plots for selected variables. The shaded bars represent CEPR dating of recessions. Hours worked and capacity utilization are expressed in index terms. Industrial production growth is reported in percentage terms.

We stress that differences in labor market adjustments between the euro area and the U.S. economy pre-date the pandemic. Indeed, Burda and Hunt (2011) document the modest rise unemployment in Germany during the Great Recession owes much to adjustment on the intensive margin, or hours in particular, rather than the extensive margin.14 Two reasons for the adjustment via the intensive margin in the euro area are the high costs of firing and employment protection legislation, which Ohanian and Raffo (2012) point out large euro area countries such as Spain, France, Germany, and Italy rank very highly on internationally. That said, even if such mechanisms were already thought to play a crucial role pre-pandemic, they have arguably been even more relevant during the pandemic. The widespread availability of job retention schemes with a specific focus on preserving employer-employee matches to prevent job losses within the euro area provided very strong incentives for firms to utilize the intensive margin. By contrast, U.S. fiscal support was targeted more directly at supporting household income via pay checks and enhanced unemployment benefits, and so it was perhaps natural that U.S. firms responded during the COVID-19 downturn by adjusting the size of their labor forces (see ECB, 2020, ECB, 2021 for more details of the differences in labor markets and the role of hours worked in the euro area vis-a-vis the U.S.economy in particular). From this perspective, it is reassuring that these differences are also reflected in our results when compared to the analysis of the U.S. economy in Morley and Wong (2020) and Berger et al. (2023). The result also highlights a greater need for policymakers in the euro area to consider the output gap as a measure of the degree of economic slack rather than focus on other indicators such as the unemployment rate.

3.5. The role of the foreign sector

Another feature of our modeling approach is that we allow for two sectors, foreign and domestic, in the BVAR, with the foreign sector being treated as block-exogenous. We thus explore the importance of block-exogeneity for our results. Understanding its effects also has practical modeling implications. Because the block-exogeneity structure introduces a large degree of computational inefficiency due to moving away from the natural conjugate prior used by Morley and Wong (2020), it is natural to explore the benefits in return for this large inefficiency cost. The top panel of Fig. 8 compares our baseline model with a model that does not impose block-exogeneity.15 We can see that there are sizable differences in parts of the sample. In particular, not imposing block-exogeneity implies a more positive output gap just before the pandemic and a more negative output gap during the pandemic. While the euro area is not really a small economy per se, we take the perspective that it is small relative to the rest of the world given the size of the global economy is about five times larger than the euro area. In effect, then, block-exogeneity places a dogmatic prior that euro area variables do not Granger cause the U.S. real GDP and real oil prices, thus avoiding overfitting the data with our VAR given a relatively small sample and assuming block-exogeneity is a reasonable assumption.

Fig. 8.

The role of the foreign sector. The shaded bars represent CEPR dating of recessions. The output gap is measured as a percentage deviation from trend. The top panel presents the estimated output gap with and without block-exogeneity. The bottom panel decomposes the euro area output gap estimate into foreign versus domestic shocks.

While we show in Fig. 8 that imposing block-exogeneity leads to meaningful differences in the reduced-form VAR parameters and thus alters the estimated output gap, the block-exogenous structure also allows us to do structural analysis unlike with some other trend-cycle decompositions (e.g., HP filtering). In particular, block-exogeneity naturally leads to a structure where we can decompose the output gap into foreign and domestic shocks (again, see Appendix D for details). The bottom panel of Fig. 8 presents an historical decomposition of the baseline estimated output gap into foreign and domestic shocks. Overall, a substantial portion of the cyclical fluctuations for the euro area economic growth are driven by foreign shocks given its openness and against the background of an increased level of globalization over the sample period. During the COVID-19 recession in 2020Q2, for example, one-third of the large negative drop in the output gap is accounted for by foreign shocks, with the virus also spreading in the rest of the world, but two-thirds of the drop due to domestic conditions, mostly attributable to the enforcement of lockdown measures. In the years before the COVID-19 outbreak, the decomposition also indicates that the euro area output gap was bolstered by economic conditions abroad. Similarly, the decomposition suggests the negative output gap during the GFC recession was mainly driven by spillovers from outside the euro area, but the negative effects of the output gap during European sovereign debt recession were mostly driven by economic conditions within the euro area.

3.6. Reliability

As a final check on our results, we explore the real-time reliability of our modeling approach. We compare against the HP filter given it is a widely-used approach to estimating the output gap. For our procedure, $\lambda$ is re-optimized for each pseudo real-time sample by minimizing the variance of trend shocks to get a sense of how the criteria would have behaved in real-time with pre-pandemic data. A caveat to our analysis is that, because the sample is very short, we do not have a long evaluation sample to assess reliability, so these results should be interpreted as being suggestive at most.16 With such analysis, it is often a judgment call on what implies a “final” or ex post estimate since these are estimated models, and so parameters can continue to change in the future. The usual practice is to take the full-sample estimates as “final” and omit the last five years of the sample period for evaluation since the estimates near the end of the sample are often those that change the most. Therefore, while we present pseudo real-time estimates, there is less than 3 years since the COVID-19 recession in our sample, and so one should probably not yet evaluate the revision properties of the estimates during the pandemic period and beyond.

Table 1 reports the revision statistics of our estimated output gap relative to those for the HP filter. We also present plots of the real-time estimates in Appendix F, noting that revisions to both our approach and the HP filter can be nontrivial, especially during the pandemic. We assess reliability relative to the HP filter by drawing on Orphanides and van Norden (2002) to consider the correlation and size of revisions between the real-time and ex-post estimated output gaps, as well as whether the ex-post and real-time estimates have the same sign. Taking an evaluation sample up until 2019Q4, our multivariate approach outperforms the HP filter, whether in terms of correlation, size of the revisions as measured by root mean squared error (RMSE), or inferring the same sign. The 91% correlation between our real-time and ex-post estimates of the output gap even though we allow $\lambda$ to change in pseudo real-time is a strong indication that the multivariate information is helpful in inferring the final estimate of the output gap, even if the estimates are somewhat revised. This results also demonstrate why our multivariate approach is useful even if the final estimates are so similar for our approach and the HP filter since 2009. In particular, our method generally has smaller revisions relative to the HP filter, with our approach having an RMSE that is about 50% smaller. Notably, our method yields a sign of the real-time output gap that is the same as the ex-post output gap 96% of the time, while the HP filter infers the wrong sign more than half the time, possibly due to the known end-point issues.17

Table 1.

Revision statistics for 2014Q1–2019Q4. Correlation refers to the correlation between the pseudo real-time and ex-post estimates of the output gap. RMSE is the root mean square of the difference between the pseudo real-time and ex-post estimates of the output gap. Same sign is the proportion of times where the pseudo real-time and ex-post estimates of the output gap share the same sign.

	Multivariate BN	HP filter
Correlation	0.91	−0.65
RMSE	0.81	1.72
Same sign	0.96	0.42

Overall, our approach displays reasonable revision properties, at least when compared with the widely-used HP filter, although we again reiterate the caveat that the short evaluation period means we should only treat these results as suggestive at best. Ideally, one would have a much longer sample to do the real-time analysis, but the results from the short evaluation period appear supportive of our approach in comparison to the HP filter.

4. Conclusions

We have applied the Beveridge–Nelson decomposition based on a large Bayesian VAR in order to obtain an estimated output gap for the euro area. Our modeling approach extends the framework proposed by Morley and Wong (2020) by addressing the COVID-19 pandemic and also considering the short sample period given the relatively recent launch of the euro area. Our approach is successful in estimating an output gap for the euro area that lines up well with the CEPR chronology of the business cycle and is similar to other institutional estimates, but does not rely only on real GDP data or a production function approach with only a few variables.

With the caveat that any real-time evaluation in our setting is hamstrung by the short sample, our approach is promising in that it demonstrates reliability properties that appear better than those for the widely-used HP filter. Also, we show how to handle the unusual data dynamics associated with the COVID-19 pandemic, as well as the importance of including a foreign sector to account for the open economy features of the euro area.

From our empirical analysis, we find that data on hours worked contains important information in pinning down the depth of the euro area output gap during the COVID-19 recession. This result stands in sharp contrast to the U.S. economy, for which the unemployment rate appears to be the most important source of information in pinning down the output gap (see, e.g., Morley and Wong, 2020). Notably, this result supports a greater role of labor market adjustment along the intensive, rather than extensive, margin for the euro area.

Appendix A. Data plots and sources

See Fig. A.1, Fig. A.2 and Table A.1.

Fig. A.1.

Time series plots of the raw data. CAPU is capacity utilization, PMI is the purchasing manager index for manufacturing output, and RER is the real exchange rate. The CPI is measured using the harmonized index of consumer prices (HICP) for the euro area. Refer to Table A.1 for the relevant units.

Fig. A.2.

Time series plots of the transformed data. CAPU is capacity utilization, PMI is the purchasing manager index for manufacturing output, and RER is the real exchange rate. The CPI is measured using the harmonized index of consumer prices (HICP) for the euro area. Refer to Table A.1 for the relevant units and transformations.

Table A.1.

Data sources. Transformations. 1 - log level 2 - log differenced 3 - differenced.

Description	Units	Source	Haver or ECB SDW code	Transformation
U.S. real GDP	Billions of chained 2012 Dollars	Federal Reserve Economic Data (FRED)		2
Real brent oil price	US-dollar × Ratio (Euro/US-dollar)	World Bank	PEOBR@WBPRICES	1
Euro area real GDP	Millions of chained 2015 EUR	Eurostat	Q023YER@EUDATA, J025GDPT@EUDATA	2
Industrial production index (excl. construction)	Index 2010 = 100	OECD	C004IZ@OECDMEI, C023IZ@OECDMEI	2
Employment	Thousands of persons	AWM database and Eurostat	Q023LNN@EUDATA, J025TETE@EUDATA	2
Harmonized index of consumer prices (HICP)	Index 2010 = 100	Eurostat	Q023HI@EUDATA	2
Building permits: residential buildings	Index 2010 = 100	Statistical Office of the European Communities	S025QPRX@EUDATA	2
Nominal short-term interest rate	Percent	Eurostat	Q023STN@EUDATA	3
Hours worked	Index 2010 = 100	Eurostat	J025HETE@EUNA	1
Euro area 10-year government benchmark bond yield minus Euro 3-month Libor	Percent	ECB and Thomson Reuters
Capacity utilization in manufacturing	Percent	Eurostat	SUR.Q.I8.S.ECFIN.MAN013.TT
Unemployment rate	Percent	Eurostat	Q023URX@EUDATA
PMI Manufacturing output	50 = no change on previous month	Markit	SUR.M.I8.S.NTC.MANOUT.TT
Euro area credit risk spread	Index	Gilchrist and Mojon (2018)
Real effective exchange rate	Index	FRED		1

Appendix B. Estimates of trend output

Fig. B.1 plots the implied estimates of trend output from the three quarterly-based estimates of the output gap from the bottom right subplot of Fig. 1 in the main text; namely our approach, the UCM, and the HP filter. As discussed in Section 3.1, because real GDP was relatively flat between 2008 and 2012, the HP filter and our approach estimate a relatively flat path for trend output. By contrast, the trend for the UCM continued growing, implying a large negative gap. In the aftermath of the COVID-19 pandemic, the UCM trend has continued growing, somewhat linearly, which has resulted in an even larger negative gap opening up.

Fig. B.1.

Estimated trend output for the euro area. The shaded bars represent CEPR dating of recessions. Real GDP is reported in levels and normalized to 100 in 1999Q1. The estimated trends are measured as the level of output if the output gap were zero. BN refers to our baseline estimate using the Beveridge–Nelson decomposition; HP filter refers to Hodrick–Prescott filter; UCM refers to an unobserved components model by Tóth (2021). The one-sided HP filter is based on end-point estimates using only data up to each observation starting in 1999Q3.

In general, our approach leads to a more volatile trend estimate than the other two approaches. For example, the standard deviation of trend growth from the HP filter is 0.17% per quarter, compared with 0.36% and 0.50% for the UCM and our approach respectively. However, we make two points in this regard: First, our estimate is more volatile partly because trend output in our approach is not restricted by construction to be as smooth as the HP filter with its implied signal-to-noise ratio of 1/1600. While it is not certain whether trend output is actually more volatile given its unobservable nature, we do point out, as in the main text, that it is not obvious that our estimate is inherently less plausible given a (slightly) higher correlation with year-on-year inflation compared to the HP filter and UCM. Second, we note that both the UCM and HP filter estimates are two-sided or ‘smoothed’, and, therefore are less volatile by construction. By contrast, our approach is effectively a one-sided ‘filtered’ approach.18 Therefore, the comparison that the trend obtained via a BN decomposition relative to a smoothed measure is somewhat complicated by the fact that we are comparing a filtered estimate with a smoothed estimate. Nonetheless, it is possible to compare relative to a filtered estimate. As is well known, the HP filter can be viewed as a state-space model, and the one-sided HP filter can be seen as equivalent to a filtered estimate.19 The bottom panel of Fig. B.1 presents the estimated trend output from our approach relative to the one-sided HP filter. Here, it is less obvious that our approach is any less volatile than the one-sided HP filter. For this like-with-like comparison, the standard deviation of trend growth is very similar (0.50% for our approach relative to 0.52% for the one-sided HP filter). Therefore, the relatively more volatile trend from our approach is partly a function of comparing a filtered estimate to smoothed estimates. The large differences between the one-sided and two-sided HP filter estimates also relates to the relatively lack of real-time reliability of the HP filter estimates discussed in Section 3.6.

Appendix C. Allowing for changes in means and trend growth rates

We consider the possibility of changes in means for all the variables in our model. Note that a change in mean implies a change in the trend growth rate for a differenced variable. Failing to account for low frequency movement in means may contaminate the estimated output gap. First, in terms of the target variable, euro area output growth, as the BN trend is the long-horizon conditional expectation minus the deterministic drift (see Eq. (1) in the main text), a change in the mean growth rate implies a change in the drift, and so would have a direct effect on the estimated cycle. Second, even if the long-run growth rate of euro area real GDP is effectively constant within the sample period, because the means for the other variables may change, this could affect the estimate of the output gap through a change in the implied forecasting information embedded in these variables.

Because the sample period is short, it is very challenging to find any statistical evidence of breaks in means. We therefore allow for the possibility of low frequency movements in means by dynamically demeaning variables following an approach developed by Kamber et al. (2018) to deal with this possibility when applying the BN decomposition. That is, we demean variables relative to their last 40-quarter average levels. The intuition is that, if a change in the mean is large, dynamically demeaning observations would be able to at least partially account for this change. However, if the change is small, then the procedure would produce only a minimal amount of noise when estimating the mean compared to a full-sample estimate. For our robustness analysis, we apply dynamic demeaning to all the variables within the system.

Fig. C.1 presents the estimated output gap with and without dynamic demeaning. As presented, our estimated output gap is reasonably robust to allowing for the possibility of time-varying means across variables. That said, dynamically demeaning does lead to a somewhat lower estimated output gap between 2014 and 2019. Following the pandemic, whether one allows for time-varying means or not has very little effect on the estimated output gap. While dynamic demeaning does produce a slightly more negative output gap in the years leading up to the pandemic, and is therefore closer to the institutional estimates discussed earlier (see Fig. 1 in the main text), allowing for time-varying means also leads to a much larger implied drop in trend output during the pandemic. Because dynamic demeaning suggests a lower output gap before the start of the pandemic and both dynamic demeaning and the baseline estimate imply very similar output gaps at the depth of the pandemic, the estimated trend output when allowing for dynamic demeaning has to fall by more during the pandemic given that all fluctuations in output growth will be attributed to either changes in trend output or changes in the output gap. In other words, dynamic demeaning leads to a more volatile estimated trend than our baseline approach. While this does not definitively determine which estimate is more correct, we point out that estimating a dynamic mean when the true mean is effectively constant would inherently introduce additional estimation error compared to full-sample estimation and could show up in the form of a somewhat more volatile estimated trend.

Fig. C.1.

Allowing for breaks in means. The shaded bars represent CEPR dating of recessions. The output gap is measured as a percentage deviation from trend. Real GDP is reported in levels and normalized to 100 in 1999Q1. The estimated trends are measured as the level of output if the output gap were zero. Baseline refers to our baseline approach to estimate the output gap. Dynamic demeaning refers to demeaning all variables in the BVAR relative to their last 40-quarter average levels to allow for changes in means and drifts.

Appendix D. Informational decompositions

Following Morley and Wong (2020) and defining ${\mathit{\Gamma }}_i\equiv {\mathit{F}}^{\mathit{i}}{(\mathit{I}-\mathit{F})}^{-1}$, recursive substitution based on Eq. (3) in the main text yields the following expression:

$$c_t^{BN}=-{\mathit{\iota }}_k\left\{\sum _{i=0}^{t-1}{\mathit{\Gamma }}_{i+1}{\mathit{He}}_{t-i}\right\}.$$ (D.1)

This expression reveals that the estimated output gap via the multivariate BN decomposition is simply a linear function of the historical forecast errors contained in ${\mathit{e}}_{t-i}$. Consequently, we can back out the contribution of each variable’s forecast error to the BN cycle, which we term an ‘informational decomposition’. We note that the total sum of the variance shares of all variables do not sum up to the variance of the output gap because some of the forecast errors are correlated with each other.

However, given a structural identification scheme, the variances of the orthogonalized structural shocks would add up to the output gap variance. In Section 3.5, we block identify foreign shocks by using the small open economy structure where we assume domestic shocks have no effect on foreign variables. This effectively decomposes the ${\mathit{e}}_{t-i}$ into domestic and foreign shocks, which we present in the bottom panel of Fig. 8 in the main text. In this setting, we use a Cholesky decomposition of the reduced-form covariance matrix, where we group the first $N^{\ast }$ shocks as the foreign shocks. Note that our approach is widely used in small-open economy modeling (e.g., Zha, 1999), and only distinguishes foreign from domestic shocks, but does not distinguish between different types of foreign and domestic shocks (e.g., we do not separately identify and so cannot tell apart, say, a domestic productivity shock and a domestic monetary policy shock).

Meanwhile, given the large role of hours worked forecast errors in accounting for the output gap during the pandemic, as presented in Fig. 6 in the main text, the pandemic may have had an effect of distorting inferences about the more general informational content of hours worked. We thus explore how the informational decomposition presented in Fig. 4 changes if we restrict ourselves to a pre-pandemic sample. That is, we compare relative to calculating the contribution of each variable’s forecast error to the output gap using Eq. (D.1), but restricting ourselves to a sample from 1999Q1 to 2019Q4. The results are presented in Fig. D.1. For ease of comparison, we also present the informational shares in the baseline from Fig. 4 in the main text. The key role hours worked played in helping pin down the depth of the output gap during the pandemic does have an effect on the informational decomposition if we restrict ourselves to a pre-pandemic sample. Hours worked still remains important, although less so than a few other variables. However, hours worked is still more important than the unemployment rate and (marginally more so than) employment, supporting our conclusion that labor market adjustments have been more via the intensive, rather than extensive, margin in the euro area. Therefore, the important role of hours worked in providing information about economic slack in the euro area is robust even if we restrict ourselves to the pre-pandemic sample. As discussed in Section 3.4, this is an important distinction relative to the U.S. economy when comparing against the results in Berger et al. (2023) and Morley and Wong (2020), amongst others.

Fig. D.1.

Informational decomposition for full and pre-pandemic sample. Units are standard deviations. Contributions to the estimated output gap for each variable are calculated taking the standard deviation for the role of each forecast error from Eq. (D.1). Full sample indicates the period from 1999Q1 to 2021Q4. The pre-pandemic sample only accounts for the forecast errors from 1999Q1 to 2019Q4. IP is industrial production, CAPU is capacity utilization, PMI is the purchasing manager index for manufacturing output, and RER is the real exchange rate.

Appendix E. Granger causality

As discussed in Section 3.3, we consider Granger-causality tests as a starting point to understand which variables contain relevant forecasting information and are thus important for the estimation of the euro area output gap. All Granger-causality tests are done with 4 lags of the relevant variable and output growth. We consider Granger-causality tests on the pre-pandemic (i.e., 1999Q1–2019Q4) and full sample. For the latter, to maintain consistency with our model adjusting for the COVID-19 observations as per Lenza and Primiceri (2022), we divide by $s_t$ as in Eq. (8) in the main text before testing for Granger causality. Table E.1 presents the results of the Granger-causality tests. Only the real exchange does not appear to Granger-cause output growth for the full sample based on conventional levels of statistical significance. When we consider the pre-pandemic sample, the evidence of Granger causality for a number of variables is more mixed at conventional levels of statistical significance. In particular, we do not find evidence that industrial production, the real exchange rate, the unemployment rate, and the PMI index Granger-cause output growth. We thus use the Granger-causality tests to motivate our exploration in the main text of a 14-variable version of the model which omits the real exchange rate and an 11-variable version of the model which omits the four variables that do not Granger-cause euro area output growth in the pre-pandemic sample.

Table E.1.

Granger-causality test $p$-values. $p$-values for Granger-causality tests are based on hypothesis that the listed variable does not Granger-cause euro area output growth.

	1999Q1–2019Q4	1999Q1–2021Q4
U.S. real GDP	0.00	0.00
Real oil price	0.01	0.00
Industrial production	0.43	0.00
Employment	0.01	0.00
Housing permits	0.00	0.00
CPI	0.00	0.00
Policy rate	0.04	0.00
Hours worked	0.03	0.00
Term spread	0.00	0.00
Capacity utilization	0.01	0.00
Unemployment	0.63	0.03
PMI	0.72	0.01
Risk spread	0.00	0.00
Real exchange rate	0.27	0.58

Appendix F. Pseudo-real time estimates

The top two panels of Fig. F.1 present vintages of pseudo real-time estimates of the output gap for our approach and for the HP filter. That is, with each new observation of quarterly real GDP, we re-estimated the model. The plots in the top two panels display the pseudo real-time output gap estimates from 2014Q1. The bottom panel displays and compares the real-time and ex post estimates for both our method and the HP filter. In general, the revisions to the output gap estimates are nontrivial, especially during the pandemic.

Fig. F.1.

Pseudo real-time estimates of the euro area output gap. The shaded bars represent CEPR dating of recessions. The output gap is measured as a percentage deviation from trend. The solid line in the upper two panels are the estimated output gaps per the respective method with data up to 2021Q4. The dotted lines in the upper two panels represent the estimated pseudo real-time output gap per the respective methods using an expanding window of data.

Nonetheless, we uncover some interesting patterns in terms of revision properties. Focusing on the period from 2014 to just before the pandemic, each subsequent vintage estimate since the pandemic seems to revise the estimated output gap upwards. For our approach, this is partly because we increasingly loosen the shrinkage hyperparameter over time. For the HP filter, the revisions are likely due to the known end-point issues. During the COVID-19 recession, our approach takes about one quarter to learn the precise depth of the output gap. A possibility, then, is to consider the use of higher frequency data in conjunction with our approach to obtain more precise estimates in real time. For example, Berger et al. (2023) use a mixed-frequency BVAR in conjunction with the BN decomposition and pick up signals about the magnitude of the output gap in 2020Q2 within the quarter as monthly data for the U.S. economy became progressively available within the quarter and before the release of real GDP data for 2020Q2. However, we leave this for future research.

Appendix G. Supplementary data

The following is the Supplementary material related to this article.

MMC S1

.