Modeling the Economic Impact of the COVID-19 Pandemic Using Dynamic Panel Models and Seemingly Unrelated Regressions

Abstract

The importance of assessing and estimating the impact of the COVID-19 pandemic on financial markets and economic activity has attracted the interest of researchers and practitioners in recent years. The proposed study aims to explore the pandemic’s impact on the economic activity of six Euro area economies. A class of dynamic panel data models and their corresponding Seemingly Unrelated Regression (SUR) models are developed and applied to model the economic activity of six Eurozone countries. This class of models allows for common and country-specific covariates to affect the real growth, as well as for cross-sectional dependence in the error processes. Estimation and inference for this class of panel models are based on both Bayesian and classical techniques. Our findings reveal that significant heterogeneity exists among the different economies with respect to the explanatory/predictive factors. The impact of the COVID-19 pandemic varied across the Euro area economies under study. Nonetheless, the outbreak of the COVID-19 pandemic profoundly affected real economic activity across all regions and countries. As an exogenous shock of such magnitude, it caused a sharp increase in overall uncertainty that spread quickly across all sectors of the global economy.

1. Introduction

When the COVID-19 pandemic broke out in the Chinese city of Wuhan back in December 2019, very few expected that it would have such severe global repercussions. Apart from the obvious health-related issues and the eventual high mortality rates, the pandemic has brought about drastic changes in the way of living around the globe. Prior to the effective release of vaccines, governments had to implement unprecedented measures, such as severe lockdowns, distancing measures and even border closures, to control the spread of the pandemic that resulted in a vast deceleration in global economic activity. The outbreak of the COVID-19 pandemic profoundly affected real economic activity across all regions and countries, as the exogenous shock had a forceful impact on the global economy that was significantly higher than that of the 2008 Global Financial Crisis (GFC), the most recent severe disruption in the global economic and financial system. As a matter of fact, the global economy experienced a real-term contraction exceeding 3% in 2020, marking the worst downturn since the Second World War.

The pandemic shock was, more or less, unique, as it was reflected in a sizeable reduction both in aggregate demand and aggregate supply. Social distancing and lockdown measures, as well as a rise in economic uncertainty, resulted in the suspension of non-essential consumption for a protracted period of time, while the pandemic’s impact on production and supply chains (interrupted transports and logistics, backlogs) explains the decline in aggregate supply. Under this specific backdrop, businesses were forced to lay off employees in an effort not to go under water. As a consequence, the observed surge in unemployment rates was practically unavoidable. The specific mix led to a global economic recession that was not driven by a typical cause, such as the outbreak of a financial crisis, due to an economy overburdened by high levels of private or public debt, or a higher inflationary environment. With hindsight, the COVID-19 recession was short-lived, but its repercussions proved to be widespread, as they diffused to almost every segment of the global economy.

The unprecedented measures and the corresponding policies taken by the governments around the world to control the spread of the pandemic had a huge impact on economic activity. On the one hand, despite the creation of effective vaccines, the continuous global vaccination efforts, and the supportive macroeconomic policy measures to reduce the impact of the pandemic, the economic recovery has been asymmetric across countries, since some countries have controlled the virus due to high vaccine coverage, while others continued to encounter new waves of infections and deaths. On the other hand, the spread of the pandemic has highlighted the importance of interdependencies across different countries’ economic activity, which depend on business links and other existing relationships.

While many research studies have highlighted the financial and economic consequences of the COVID-19 pandemic, fewer studies have explored the impact of the pandemic on economic activity in Euro area countries. Assessing and estimating the potential impact of COVID-19 in European countries will enable governments, companies, civil society, and individuals to better cope with the consequences of COVID-19 or another pandemic in the future. The findings of the analysis could be of substantial importance for policymakers to obtain further insight into the economic benefits of globally coordinated policy responses to face a pandemic. In this context, the aim of the proposed study is to figure out the impact of the COVID-19 pandemic on economic activity in Euro area countries by considering the following research questions: (i) What is the impact of COVID-19 pandemic on the GDP growth of European countries as a result of the health-related issues and the corresponding measures taken by governments? (ii) Which variables/factors influenced GDP growth at the country level during the outbreak of COVID-19 pandemic? (iii) What are the interdependencies across different countries’ economic activities? To answer these research questions, it is critical to comprehend and synthesize data from different countries to reduce the risk of lacking relevant information. The economic effects of COVID-19 may vary depending on common or specific macroeconomic and financial covariates. A country-level analysis or comparison that takes into account several factors will yield a more reliable estimate of the pandemic’s economic impact.

The main goal of this study is to effectively measure the impact of a severe exogenous shock and, as such, of an unanticipated one, like the COVID-19 pandemic, on the aggregate European economic activity of a specific country, or that of a group of countries that exhibit some degree of economic homogeneity (e.g., a group of countries that are part of an economic and/or monetary union). More specifically, this study focuses on the aftermath of the pandemic on Greek economic activity, as well as on that of five other Euro area economies, namely those of Belgium, France, Germany, Italy, and Spain. The effect of the shock is measured through the use of dynamic panel data models, as the aim is to explore the ability of several common and country-specific covariates/factors to capture the aggregate shock. The advantages of the proposed panel model are the following: the model is able (i) to capture the impact across countries over time, particularly if there are cross-sectional and time series variations in the effect of the pandemic during its evolution; (ii) to estimate the impact of underlying determinants on GDP growth simultaneously for all the analysed countries, while controlling for individual heterogeneity; and (iii) to minimize problems such as omitted variables and heteroscedasticity, particularly for a short sample (see Baltagi 2005). Since GDP growth may have some persistence, a lagged dependent variable is usually included in the model to account for autoregressive dynamics that typically appear in economic series. In our analysis, we also consider the Bayesian approach to inference for dynamic panel data models. Adoption of the Bayesian framework can be advantageous on grounds of generality, accuracy, and flexibility. Moreover, the Markov chain Monte Carlo (MCMC) sampling-based approach provides an ideal way to extract any posterior summary of interest for the model parameters and cross-sectional parameters in the covariance structure of the errors and, in addition, to construct their posterior densities.

The present paper aims to provide new evidence of the impact of the COVID-19 pandemic on economic activity in European countries. It makes several contributions to the literature. First, a class of dynamic panel data models with common and specific covariates and cross-sectional dependence of the error processes is introduced and implemented in the empirical analysis. The proposed modeling framework accounts for individual country heterogeneity that allows the model coefficients to be different, i.e., country-specific. It reveals dynamic relationships among macroeconomic variables and yields consistent estimates because it includes lagged values of both the dependent and explanatory covariates that capture both short and long-run effects (Pesaran et al. 1999). It also allows for variation in cross-sectional dimensions of the data and infers interesting interrelationships/dependences among the analysed dependent variables. Second, the panel data modeling framework allows for constructing inferential methods based on both Bayesian and classical techniques. Specifically, a Markov chain Monte Carlo (MCMC) sampling scheme is adopted to simulate draws from the posterior distribution of model parameters, while Ordinary Least Squares (OLS) and feasible Generalized Least Squares (GLS) estimators are obtained by using the Seemingly Unrelated Regression (SUR) model parametrization of the dynamic panel model. Third, the proposed study has a European-wide focus. The proposed panel data modeling framework and inferential methods are employed for the analysis of real GDP growth series in order to investigate the impact of several macroeconomic and financial covariates and that of the COVID-19 pandemic on the real economic activity of six Euro area countries. The findings of the underlying analysis reveal that the exogenous shock caused by the pandemic has a significant negative effect on the real GDP growth of all the countries under study. The factors that proxy the state and spread of the pandemic appear in all estimated models.

The remainder of the paper is organized as follows. A literature review is presented in Section 2. The proposed dynamic panel data models and the corresponding Seemingly Unrelated Regression models are introduced in Section 3. The Bayesian approach to inference for the estimation of the model parameters is presented in Section 4. The dataset employed in the analysis is outlined in Section 5, while in Section 6, the empirical application using macroeconomic, financial and COVID-19-related data is presented. Finally, Section 7 concludes with a brief discussion.

2. Literature Review

The importance of assessing and estimating the impact of the COVID-19 pandemic on financial markets and economic activity has attracted the interest of researchers and practitioners in recent years. In a highly connected and integrated world, the repercussions of the pandemic go beyond just health-related concerns and mortality. Governments have been obliged to take unprecedented measures, such as severe lockdowns, to control the spread of the pandemic that are practically close to a halt to or shutdown of economic activity.

Several studies have attempted to assess the pandemic’s impact on financial markets and economic activity by using econometric modeling approaches and panel data models in particular. For instance, panel data models have been implemented to investigate the relationship between COVID-19 testing and COVID-19 cases (see, for example, Cirakli et al. 2022) and the effectiveness of a variety of non-pharmaceutical interventions and measures used by governments to control the evolution and spread of COVID-19 (see, for example, Ashraf 2020; Lopez-Mendoza et al. 2024). Due to the fact that the COVID-19 epidemic spread faster than previous diseases, a plethora of research studies investigated its impact on different aspects of financial and economic activity. Several studies analyse the impact of COVID-19 on financial markets (Ashraf 2020; Chowdhury et al. 2022; Bille and Caporin 2022; Ben-Ahmed et al. 2022; Padungsaksawasdi and Treepongkaruna 2023; Ullah 2023; Yiu and Tsang 2023; Pham and Chu 2024, among several others), European trade flows (Caporale et al. 2023), foreign investment flows (Kocak and Baris-Tuzemen 2022), international trade (Kim et al. 2023; Tudorache and Nicolescu 2023), shipping trade (Xu et al. 2021), the labour market (Crossley et al. 2021), and tourism (Scarlett 2021). Other studies examine the impact of the COVID-19 pandemic on economic uncertainty (Altig et al. 2020; Ng 2021; Abdelkafi et al. 2023) and investigate its economic effects (Erdogan et al. 2020; Chudik et al. 2021; Milani 2021; Chowdhury et al. 2022; Matei 2023; Vrontos et al. 2024). More recent studies (e.g., Liu and Chu 2024) investigate the impact of financial technology on economic growth during the pandemic using panel regression models.

Below, we briefly present and discuss some of the key research studies, which have employed panel data models in various macroeconomic and financial settings. Hoogstrate et al. (2000) employ dynamic panel data models, using pooling techniques, to forecast the real GDP growth rates of 18 Organization for Economic Cooperation and Development (OECD) countries for an extended period (1948–1990). In particular, they explore whether the use of pooling techniques improves the forecasting performance compared to individual country forecasts. Their findings reveal that OLS-based pooled forecasts produce smaller median Mean Squared Forecast Error (MSFE) metrics than the corresponding OLS-based individual country forecasts for small and/or moderate data samples. Kholodilin et al. (2008) forecast the annual growth rates of GDP for each of the sixteen German States using dynamic panel data models. Their main contribution is the simultaneous construction of GDP forecasts for all States, as well as the use of panel data models that allow both for temporal and spatial interdependence in the regional growth rates. Their main finding is that pooled models that account for spatial dependence produce the most accurate forecasts across multiple forecasting horizons. Simionescu et al. (2016) follow a dynamic panel data modeling approach to analyse real GDP growth rates in European Union (EU) countries. More specifically, they explore the relation between employment and real GDP growth, as well as the effect that lagged values of GDP growth have on the current level of real GDP growth. Their study covers the EU-28 countries for the period between 2004 and 2015. Their findings reveal that lagged values of real GDP affect the current level of real GDP growth. In addition, the results confirm that there is a relation between employment and real GDP growth.

Antunes et al. (2018) employ a dynamic panel probit specification to enhance the forecast accuracy of early warning models in the context of banking crises. In particular, they explore the information content and signaling power of exuberance indicators in predicting systemic banking crises that are rare events. Several macroeconomic, financial and bank-specific variables are used as signaling variables, along with a binary crisis indicator as the dependent variable. Their sample consists of 22 countries, including the United Kingdom. The main findings show that equity prices, house price growth, the credit-to-GDP gap and the debt service ratio are among the most useful indicators for signaling emerging banking crises. In addition, their study reveals that adding a dynamic component to the multivariate model substantially improves the accuracy of the models, both in and out of sample. Ho and Saadaoui (2022) explore the relation between bank credit and GDP growth in seven Asian countries by taking into account the level of the credit-to-GDP ratio in a dynamic threshold panel model context for the 1993 to 2019 period. More specifically, the authors detect whether threshold effects exist in the specific relationship; i.e., they explore whether the slopes of the relationship between bank credit and economic growth are different in a statistical sense before and after an estimated value of the threshold variable. Their primary empirical finding confirms the existence of threshold effects in the relationship between bank credit and economic growth. They conclude that their empirical findings could lead to further investigation of the existence of threshold effects in the relation between economic growth and various types of credit in various regions of the world. This specific investigation could support the formulation of sound policy recommendations regarding macroprudential supervision.

The literature that employs a dynamic panel data model framework and explores the effects of the COVID-19 pandemic on various macroeconomic variables of interest, such as economic growth, unemployment, inflation, etc., is still under development. Some illustrative examples are presented below. Apergis and Apergis (2021) explore the impact of the pandemic shock on the economies of a set of OECD countries for the period between March 2020 and January 2021 using a Bayesian panel vector autoregressive framework. This study estimates the potential economic cost of COVID-19 with respect to industrial production. It covers 35 countries that account for over 90% of the global purchasing power parity-adjusted income. They show that in terms of the number of incidences, the COVID-19 shock begins to explain industrial production in the second month after the initial shock, with 24.06% of its total variance explained, and this remains persistent, gradually increasing over the following 22 months, reaching 61.06%. Long et al. (2022) investigate the impact of the COVID-19 pandemic on the economy and whether central bank agility is able to ease the negative effect of the pandemic. Employing a panel model and data from 38 countries for the period between January 2020 and June 2021, this study explores the influence of the pandemic on employment and inflation. They find that the pandemic has had a profound effect on the economy, as it has exacerbated inflation and unemployment on a global scale. They go on to examine the central banks’ role in dealing with the effects of the pandemic. Their study reveals that the more active a central bank is, the more effective it is at alleviating inflation. Larson and Sinclair (2022) find that variation in state-level COVID-19-related state of emergency declarations is useful for predicting the aggregate nationwide volume of initial unemployment insurance claims. The specific employment-related indicator is an important barometer of current economic conditions in the U.S. Their dataset includes the 50 U.S. states plus the District of Columbia and Puerto Rico. They present nowcasts of the advance estimates of initial claims employing different information sets and data. They show that a panel model performs well shortly after a structural break, like the one realized after the outbreak of the pandemic, if relevant information is used. In their study, relevant information is proxied by the variation in emergency declaration dates across U.S. states. The primary model is a panel model with dummy variables representing the distance between a given week t and state i’s emergency declaration date. The panel regression is weighted based on covered employment for each state for a designated week. Their proposed panel model outperforms models that include Google Trends data, as well as autoregressive models. All models miss the initial shock, but in the subsequent early periods, the panel model leads to substantially better forecasts. Eventually, time-series models catch up. More specifically, the panel model exhibits the lowest mean absolute error (MAE) for the full sample, and is statistically significantly better versus the model that included Google Trends data.

In summary, dynamic panel data models have been implemented in the literature to forecast economic activity and GDP growth, particularly before the arrival of the COVID-19 pandemic, in order to assess the forecasting ability of the proposed models, indicating that panel models increase the forecasting accuracy. During the appearance and the spread of COVID-19, several panel data models were proposed in the literature to assess the impact of the pandemic on financial markets and on economic activity around the globe. The proposed study aims to explore the pandemic’s impact on economic activity for six Euro area economies by implementing dynamic panel data models that allow us to investigate the effect of several common and country-specific covariates/factors on GDP growth. The specific modeling approach also allows for heterogeneity in the panel model parameters across different countries and for cross-sectional dependence/covariance in the error processes. In addition, with regard to the panel model parameter estimation, a classical as well as a Bayesian approach to inference is implemented. One other important topic for future research is that of variable/factor selection in the panel data model, which allows for different cross-sectional units, i.e., countries, to be affected by different covariates. The advantage of the proposed inferential framework is that it can be extended to accommodate model uncertainty regarding the set of covariates to be included in the model by constructing a stochastic Markov chain Monte Carlo (MCMC) algorithm and thus visits/jumps between different panel models and provides posterior model probabilities.

3. Dynamic Panel Model Specification

Dynamic panel data models have been used extensively in financial and macroeconomic studies. The specific class of models has several interesting characteristics due to its dynamic structure that allows for lagged dependent variables in the right-hand side of the model specification, as well as due to the panel structure that facilitates the modeling of both cross-sectional and time-series dynamics and relations. This section presents the dynamic panel model used to assess the impact of the COVID-19 pandemic on aggregate real economic activity and on real GDP growth in particular. The following general dynamic panel data model is considered:

$$y_{i,t}={\delta }_i+\sum _{p=1}^P{\alpha }_{p,i}{f}_{p,t}+\sum _{q=1}^Q{\gamma }_{q,i}{f}_{q,i,t}+{ϵ}_{i,t},\mathrm{where}{ϵ}_{i,t}=\varphi {ϵ}_{i,t-1}+u_{i,t},$$

(1)

where $i=1,2,\dots ,N$ denotes the cross-sectional units of the panel (e.g., countries); $t=1,2,\dots ,T$ denotes the time period, $y_{i,t}$ denotes the economic dependent variable, i.e., real GDP growth for six Euro area countries; $f_{p,t}$, $p=1,\dots ,P$ is a set of P exogenous covariates or predictors (e.g., a global volatility/uncertainty index, or a World Pandemic index) that are common for all cross-sectional units i of the panel, $i=1,\dots ,N$, and affect the dependent variables through separate/different coefficients ${\alpha }_{p,i}$; $f_{q,i,t}$, $q=1,\dots ,Q$ is a set of Q cross-sectional specific explanatory variables (e.g., real productivity growth, rate of unemployment, consumer inflation, etc.) that are different for each cross-sectional unit i with separate coefficients ${\gamma }_{q,i}$; and ${ϵ}_{i,t}$ is a zero-mean first-order autoregressive process, AR(1), which captures the dynamics of the panel data model. The error term $u_{i,t}$ is assumed to be serially uncorrelated, i.e., $E\left(u_{i,s}{u}_{i,t}\right)=0$, for all $t\ne s$ and for all i, but it is heterogeneous and correlated across i, i.e., $E\left(u_{i,t}{u}_{j,t}\right)\ne 0,$ for all i and j, and, therefore, allows for cross-sectional dependence across units. The specific model has the following nonlinear representation:

$$y_{i,t}={\delta }_i+\varphi \left(y_{i,t-1}-{\delta }_i\right)+\sum _{p=1}^P{\alpha }_{p,i}(f_{p,t}-\varphi f_{p,t-1})+\sum _{q=1}^Q{\gamma }_{q,i}(f_{q,i,t}-\varphi f_{q,i,t-1})+u_{i,t},$$

or

$$y_{i,t}=\varphi y_{i,t-1}+{\delta }_i(1-\varphi )+\sum _{p=1}^P{\alpha }_{p,i}(f_{p,t}-\varphi f_{p,t-1})+\sum _{q=1}^Q{\gamma }_{q,i}(f_{q,i,t}-\varphi f_{q,i,t-1})+u_{i,t}.$$

Let us denote $y_{i,t}^{\ast }=y_{i,t}-\varphi y_{i,t-1}$, $f_{p,t}^{\ast }=f_{p,t}-\varphi f_{p,t-1}$, and $f_{q,i,t}^{\ast }=f_{q,i,t}-\varphi f_{q,i,t-1}$. Then, the model can be written as:

$$y_{i,t}^{\ast }={\delta }_i(1-\varphi )+\sum _{p=1}^P{\alpha }_{p,i}{f}_{p,t}^{\ast }+\sum _{q=1}^Q{\gamma }_{q,i}{f}_{q,i,t}^{\ast }+u_{i,t}.$$

The proposed dynamic panel model is general, in the sense that it allows common covariates and specific cross-sectional unit factors to affect the dependent variables. In addition, it allows for cross-sectional dependence in the innovation process, and, in this way, it may capture unobserved random effects (e.g., pandemic effects). In addition, it provides a natural link/connection to the system of dependent variables; it nests several panel model specifications and models with common factors only (if ${\gamma }_{q,i}=0$), as in the work of Meligkotsidou et al. (2012, 2014) that used dynamic panel models with common factors and cross-sectional dependence in the error process, as well as simple panel models without autoregressive dynamics (i.e., $\varphi =0$) that arise from the autoregressive modeling of the ${ϵ}_{i,t}$ process. Note also that the saturated model, which assumes that all $y_{i,t}^{\ast }$, for $i=1,\dots ,N$ depend on all available covariates, emerges if the cross-sectional specific covariates are treated as common factors as well and impact all the dependent variables of the underlying system.

A useful parameterisation of the specific dynamic panel model that provides a link with multivariate multiple regression type models, and can be used for inferential purposes, can be derived as follows. Let $y_t^{\ast }={\left(y_{1,t}^{\ast },\dots ,y_{N,t}^{\ast }\right)}^{\prime }$ be an $N\times 1$ vector of observations of the economic dependent variables (conditional on $\varphi$) on N cross-section units at time t, $t=2,\dots ,T$. Suppose also that there are $1+P+NQ$ available covariates that can be used to model each of the N cross-sectional dependent variables, and denote by $z_t={\left(z_{1,t},z_{2,t},\dots ,z_{1+P+NQ,t}\right)}^{\prime }=(1-\varphi ,f_{1,t}^{\ast },\dots ,f_{P,t}^{\ast },f_{1,1,t}^{\ast },\dots ,f_{Q,1,t}^{\ast },f_{1,2,t}^{\ast },\dots ,f_{Q,2,t}^{\ast },\dots ,f_{1,N,t}^{\ast },\dots ,$

$f_{Q,N,t}^{\ast }{)}^{\prime }$ the $(1+P+NQ)\times 1$ vector of covariates at time t, $t=2,\dots ,T$. Then, $y_t^{\ast }$ can be modeled as follows:

$$y_t^{\ast }=\left(I_N\otimes z_t^{\prime }\right)S\beta +u_t,u_t\sim N_N\left(0,\Sigma \right),$$

where $\left(I_N\otimes z_t^{\prime }\right)$ denotes the Kronecker product of the $N\times N$ identity matrix with the covariate vector $z_t^{\prime }$ (of dimension $1\times (1+P+NQ)$), S is an $N(1+P+NQ)\times N(1+P+Q)$ selection matrix with elements equal to 0 or 1, and $\beta$ is an $N(1+P+Q)\times 1$ vector of the dynamic panel model parameters, i.e., $\beta ={\left({\theta }_1^{\prime },{\theta }_2^{\prime },\dots ,{\theta }_N^{\prime }\right)}^{\prime }$, with ${\theta }_1={\left({\delta }_1,{\alpha }_{1,1},\dots ,{\alpha }_{P,1},{\gamma }_{1,1},\dots ,{\gamma }_{Q,1}\right)}^{\prime }$, …, ${\theta }_N={\left({\delta }_N,{\alpha }_{1,N},\dots ,{\alpha }_{P,N},{\gamma }_{1,N},\dots ,{\gamma }_{Q,N}\right)}^{\prime }$. The innovation vector $u_t={\left(u_{1,t},u_{2,t},\dots ,u_{N,t}\right)}^{\prime }$, $t=2,\dots ,T$, is assumed to be independently and normally distributed with zero mean and covariance matrix $\Sigma$. That is, it is assumed that the N cross-sectional units are dependent through the covariance/correlation structure of the error terms (cross-sectional dependence). Now, if the $N(1+P+Q)\times N$ matrix $x_t$ is defined as $x_t^{\prime }=\left(I_N\otimes z_t^{\prime }\right)S$, the model is written in the following form:

$$y_t^{\ast }=x_t^{\prime }\beta +u_t,u_t\sim N_N\left(0,\Sigma \right).$$

(2)

In this parameterization of the dynamic panel model, the selection matrix, S, plays an important role. It specifies the set of covariates that appear in each cross-sectional equation and allows each cross-sectional dependent variable to be modeled using a different subset of the available covariates (common as well as cross-sectional specific factors). For example, if each cross-sectional dependent variable depends on P common factors and Q cross-sectional specific factors, then the selection matrix S is defined as above, i.e., it is an $N(1+P+NQ)\times N(1+P+Q)$ matrix, where its column has $N(1+P+NQ)-1$ zeros and unity. If S is the $N(1+P+NQ)\times N(1+P+NQ)$ identity matrix, then the specified panel model is the saturated model that allows that each cross-sectional dependent variable $y_{i,t}^{\ast }$, $i=1,\dots ,N$ to depend on all available covariates (i.e., in this case, the cross-sectional specific factors are treated as factors that affect any cross-sectional dependent variable of the system).

As a result, the dynamic panel model can be written in a matrix form of a Seemingly Unrelated Regression (SUR) model as follows: Consider having observed $T-1$ realizations of the dependent economic series $y_t^{\ast }$, $t=2,\dots ,T$. Denote by $Y^{\ast }={\left({y_2^{\ast }}^{\prime },\dots ,{y_T^{\ast }}^{\prime }\right)}^{\prime }$ the $N(T-1)\times 1$ vector of realizations of the dependent variables, by $X={\left(x_2,\dots ,x_T\right)}^{\prime }$ the $N(T-1)\times N(1+P+Q)$ matrix with covariates, and by $U={\left(u_2^{\prime },\dots ,u_T^{\prime }\right)}^{\prime }$ the $N(T-1)\times 1$ vector of innovations. Then, the corresponding SUR model is given by:

$$Y^{\ast }=X\beta +U,U\sim N_{N(T-1)}\left(0,I_{T-1}\otimes \Sigma \right).$$

(3)

Next, the Bayesian approach to inference for the dynamic panel model parameters is presented, i.e., inference for $\beta$, $\Sigma$ and $\varphi$ for a given specification of the dynamic panel model. Note, however, that the classical approach to inference can also be implemented for the dynamic panel model parameters, in addition to several nested model specifications, by using the Seemingly Unrelated Regression model formulation of Equation (3), producing Ordinary Least Squares (OLS) and feasible Generalised Least Squares (GLS) estimators.

4. Bayesian Inference for a Given Model

This section presents the Bayesian approach for estimating the parameters of the dynamic panel model specification (corresponding to a specific selection matrix, S) for the N dependent cross-sectional unit equations with $1+P+NQ$ covariates, i.e., common and specific factor coefficients. The parameter vector $\theta$ of the dynamic panel model is $\theta ={\left(\beta ,\Sigma ,\varphi \right)}^{\prime }$, which includes the vector of coefficients $\beta$ of the covariates, the parameters in the covariance matrix $\Sigma$, as well as the autoregressive coefficients $\varphi$. Bayesian inferences regarding the parameter vector $\theta$ conditional on data Y are made via the posterior density $\pi \left(\theta \mid Y\right)$. Using Bayes’ theorem, this density takes the form $\pi \left(\theta \mid Y\right)=cL\left(Y\mid \theta \right)\pi \left(\theta \right)$ for some normalizing constant c, likelihood function $L\left(Y\mid \theta \right)$, and prior density $\pi \left(\theta \right)$. In this context, Markov chain Monte Carlo (MCMC) sampling strategies are adopted in order to simulate from the posterior distribution of the model parameters.

4.1. Likelihood Function

Under the assumption of the multivariate normal error distribution, the likelihood function for the dynamic panel model for a sample of T observations, $y_{i,t}$, $t=1,2,\dots ,T$, or $T-1$ observations for $y_{i,t}^{\ast }$, $t=2,\dots ,T$, can be written as:

$$\begin{array}{ccc}\hfill L\left(Y^{\ast }|X,\beta ,\Sigma ,\varphi \right)& =& {\left(2\pi \right)}^{-(T-1)N/2}{\left|\Sigma \right|}^{-(T-1)/2}exp\left\{-\frac{1}{2}\sum _{t=2}^T\left[{\left(y_t^{\ast }-x_t^{\prime }\beta \right)}^{\prime }{\Sigma }^{-1}\left(y_t^{\ast }-x_t^{\prime }\beta \right)\right]\right\}\hfill \\ & =& {\left(2\pi \right)}^{-(T-1)N/2}{\left|\Sigma \right|}^{-(T-1)/2}exp\left\{-\frac{1}{2}tr\left[{\Sigma }^{-1}\sum _{t=2}^T\left(y_t^{\ast }-x_t^{\prime }\beta \right){\left(y_t^{\ast }-x_t^{\prime }\beta \right)}^{\prime }\right]\right\}\hfill \\ & =& {\left(2\pi \right)}^{-(T-1)N/2}{\left|\Sigma \right|}^{-(T-1)/2}exp\left\{-\frac{1}{2}{\left(Y^{\ast }-X\beta \right)}^{\prime }\left(I_{T-1}\otimes {\Sigma }^{-1}\right)\left(Y^{\ast }-X\beta \right)\right\}.\hfill \end{array}$$

4.2. Prior Specification

The Bayesian approach to inference requires specifying prior distributions for the model parameters $\theta ={\left(\beta ,\Sigma ,\varphi \right)}^{\prime }$. The prior specification considered is the following: independent prior distributions on $\beta$, $\Sigma$, and $\varphi$, i.e., $\pi (\beta ,\Sigma ,\varphi )=\pi \left(\beta \right)\pi (\Sigma )\pi \left(\varphi \right)$ are assumed. Proper prior distributions for the panel model parameters are adopted. Specifically, a multivariate normal prior distribution for the parameter vector $\beta$ is assumed, denoted as $N_{N(1+P+Q)}\left(B_0,A_0\right)$, where $B_0$ is the $N(1+P+Q)\times 1$ vector of prior means and $A_0$ is an $N(1+P+Q)\times N(1+P+Q)$ covariance matrix of the prior. The probability density function of this multivariate normal prior distribution for $\beta$ is given by:

$$\pi \left(\beta \right)={\left(2\pi \right)}^{-\frac{N(1+P+Q)}{2}}{\left|A_0\right|}^{-\frac{1}{2}}exp\left\{-\frac{1}{2}{\left(\beta -B_0\right)}^{\prime }{A}_0^{-1}\left(\beta -B_0\right)\right\}.$$

For $\Sigma$, an inverted Wishart prior distribution is adopted, denoted by $IW(v,Q)$, where Q is an $N\times N$ hyperparameter matrix and v denotes the degrees of freedom. The probability density function of the inverted Wishart prior distribution for $\Sigma$ is given by:

$$\pi (\Sigma )={\left[2^{vN/2}{\pi }^{N(N-1)/4}\prod _{i=1}^N\Gamma \left(\frac{v+1-i}{2}\right){\left|Q\right|}^{-v/2}\right]}^{-1}{|\Sigma |}^{-(v+N+1)/2}exp\left[-\frac{1}{2}tr\left({\Sigma }^{-1}Q\right)\right],$$

where $v\ge N$. Finally, let $\pi \left(\varphi \right)$ be an appropriate prior distribution on the autoregressive coefficient with support in the stationary region. A uniform prior, i.e., $\varphi \sim U(-1,1)$, is considered.

4.3. MCMC Sampling Scheme

The MCMC sampling scheme is constructed by iteratively and successively sampling $\Sigma$, $\beta$, and $\varphi$ from their full conditional posterior distributions via two Gibbs and one Metropolis–Hastings step. The conditional posterior distributions of the model parameters are the following:

$$\pi \left(\Sigma \mid Y,X,\beta ,\varphi \right)\equiv IW\left((T-1)+v,\sum _{t=2}^T\left(y_t^{\ast }-x_t^{\prime }\beta \right){\left(y_t^{\ast }-x_t^{\prime }\beta \right)}^{\prime }+Q\right),$$

$$\pi \left(\beta \mid Y,X,\Sigma ,\varphi \right)\equiv N_{N(1+p+q)}\left({\beta }_1,A_1\right),$$

$$\mathrm{where}{A}_1={\left(A_0^{-1}+X^{\prime }\left(I_{(T-1)}\otimes {\Sigma }^{-1}\right)X\right)}^{-1},\mathrm{and}{\beta }_1=\left({\beta }_0^{\prime }{A}_0^{-1}+Y^{\ast \prime }\left(I_{(T-1)}\otimes {\Sigma }^{-1}\right)X\right)A_1,$$

$$\pi (\varphi \mid Y,X,\Sigma ,\beta )\propto exp\left\{-\frac{1}{2}\sum _{t=2}^T{\left(y_t^{\ast }-x_t^{\prime }\beta \right)}^{\prime }{\Sigma }^{-1}\left(y_t^{\ast }-x_t^{\prime }\beta \right)\right\}\pi \left(\varphi \right).$$

The proposed sampling scheme is known as the Metropolis-within-Gibbs scheme, where the components of the parameter vector, whose full conditional distributions are of a known form, are updated directly by using Gibbs sampling (i.e., for $\beta$ and $\Sigma$), while those with non-standard conditional distributions (i.e., for $\varphi$) are updated by using Metropolis–Hastings steps. In the Metropolis–Hastings step, a random walk chain is adopted, where candidate (proposal) values are drawn from candidate-generating densities with means, i.e., the current value of $\varphi$ plus noise. In the empirical application, a Gaussian random-walk-generating density is used, i.e., ${\varphi }^{can}\sim N({\varphi }^{current},V)$.

The proposed MCMC sampling scheme was implemented in the empirical application to produce draws from the posterior distribution of the panel data model parameters. The output sample of every MCMC run was constructed as follows. First, a large sample was taken and an initial (burn-in) part of it was discarded (we considered a burn-in period of 1000 iterations) after a visual inspection of the mixing performance of the simulated chains through MCMC for each model parameter. Then, the resulting samples were checked for convergence by using the test proposed by Geweke (1992). The corresponding z-scores indicate that the convergence of the Markov chain has been achieved. Estimated posterior means and standard deviations for the model parameters and convergence diagrams and histograms of the posterior sample of the parameters that indicate the good mixing performance of the algorithm are illustrated in the corresponding Tables and Figures.

5. Data

This study examines the effects of the COVID-19 pandemic on economic activity in six Eurozone countries; Germany, France, Italy, Spain, Greece, and Belgium. This particular group of countries was selected with the aim of including economies of varying sizes and importance for the analysis that still share several key characteristics. It is expected that these countries will display some uniformity and consistency due to their high level of economic integration and interdependence as members of the same economic and monetary union. Consequently, both Germany, the largest economy in the Eurozone, and Belgium, significantly smaller by comparison, were added to this group. Additionally, the selection process also considered another factor: the severity and the impact of the pandemic on these economies.

Greece is a rather interesting case; one needs to bear in mind that on the aggregate demand side, the pandemic caused a significant decrease in net (difference in demand between non-essential and essential goods and services) global demand for goods and services. This was primarily driven by a decline in non-essential demand, as well as a reduced demand for day-to-day services, due to social distancing and lockdown measures. Thus, the repercussions of COVID-19 were not equally allocated across a country’s economy, let alone across different countries with dissimilar macroeconomic dynamics. The tourism- and hospitality-related sectors, for example, were hard hit due to social distancing, lockdowns, and border closures. Airlines were forced to ground their fleets and fire staff, while cruises were cancelled and hotels ceased operations. In countries with stricter lockdown measures, bars, cafes, and restaurants were out of operation for a protracted period of time. Moreover, there were spillover effects to other related industries that rely on tourism and entertainment. Having said this, countries in which tourism makes a sizeable contribution to GDP should have been impacted more, at least theoretically; countries like Greece, Portugal, Mexico, and Spain that are heavily reliant on tourism (more than 15% of GDP) should have been severely and asymmetrically hit relative to other countries with less sizeable tourism and related sectors.

A country’s economic activity is measured by how much its real GDP (RGDP) changes every quarter. RGDP growth is calculated using differences in the natural logarithm of the corresponding index in each country. We use the seasonally and calendar-adjusted chain-linked index (adjusted over time from 2001—Q1 to 2021—Q3). This transformation (difference in the natural logarithm) achieves stationarity of the dependent variables. This study considers seven data points in time from when the COVID-19 pandemic started in Europe to when it was still ongoing, that is, from 2020—Q1 to 2021—Q3.

The initial group of covariates in the panel data models is considered universal for all countries, intended to capture global effects due to the widespread impact of the COVID-19 pandemic. The group of common covariates includes three variables: the West Texas Intermediate oil price index, the World Uncertainty index, and the World Pandemic Uncertainty index. These factors are chosen to reflect the economic consequences of an exogenous shock like the COVID-19 outbreak. Such variables have also been utilized in previous studies, for example, by Chudik et al. (2021) and Vrontos et al. (2024). The data related to the set of common factors are publicly available and have been obtained from the Federal Reserve Bank of the St. Louis’ FRED database.

Apart from the set of common factors, several country-specific factors are employed in the analysis. The specific set includes the lagged values of the real GDP growth series, as it is well recognized in the literature that these RGDP growth series often show significant autocorrelation. To deal with this autocorrelation and to use its informational value for projected estimates, four autoregressive terms (${\mathrm{RGDP}}_{t-1}$, ${\mathrm{RGDP}}_{t-2}$, ${\mathrm{RGDP}}_{t-3}$, and ${\mathrm{RGDP}}_{t-4}$) are used in the models.

In addition, other nine country-specific explanatory variables are taken into account, i.e., real productivity, car registrations, the consumer inflation rate, the producer inflation rate, the construction production index, long interest rates, equity market returns, and the Organization for Economic Cooperation and Development (OECD) leading indicator. Thus, the set of country-specific covariates contains macroeconomic and financial-market-related indicators, the majority of which are widely followed by both policymakers and practitioners, and have been used in the existing literature, see, for example, Morley and Wong (2020), Baumeister and Guérin (2021), Chudik et al. (2021), Cimadomo et al. (2022), Berger et al. (2023), and Vrontos et al. (2024). The list of predictor variables/factors and their transformations to achieve stationarity are presented in

Table 1. Set of predictor variables—overview.

Code	Predictor Variables	Transformation
1	Real GDP Growth—lag one (${\mathrm{RGDP}}_{t-1}$)	$\Delta ln$, quarter-on-quarter % change
2	Real GDP Growth—lag two (${\mathrm{RGDP}}_{t-2}$)	$\Delta ln$, quarter-on-quarter % change
3	Real GDP Growth—lag three (${\mathrm{RGDP}}_{t-3}$)	$\Delta ln$, quarter-on-quarter % change
4	Real GDP Growth—lag four (${\mathrm{RGDP}}_{t-4}$)	$\Delta ln$, quarter-on-quarter % change
5	Oil WTI (OIL)	$\Delta ln$, quarter-on-quarter % change
6	World Uncertainty Index (WUI)	$\Delta lv$
7	World Pandemic Index (WPI)	$\Delta lv$
8	Real Productivity (RPROD)	$\Delta ln$, quarter-on-quarter % change
9	Car Registrations (CREG)	$\Delta ln$, quarter-on-quarter % change
10	Rate of Unemployment (UNEM)	$\Delta lv$
11	Consumer Price Index (CPI)	$\Delta lv$
12	Producer Price Index (PPI)	$\Delta lv$
13	Construction Volume Index of Production	$\Delta ln$, quarter-on-quarter % change
	(CONPROD)
14	Long Run Interest Rates (LONGRs)	$\Delta lv$
15	Stock Index return (STOCK)	$\Delta ln$, quarter-on-quarter % change
16	OECD Leading Indicator (LEAD)	$\Delta ln$, quarter-on-quarter % change

This table reports detailed information about the set of predictor variables and the corresponding transformation used in the analysis; $\Delta ln$ denotes first differences of logarithms and $\Delta lv$ denotes first differences.

. The data for the six Eurozone countries analysed, i.e., Belgium, France, Germany, Greece, Italy, and Spain, are publicly available on the Eurostat, European Central Bank (ECB) and OECD databases. Please note that the original set of explanatory and/or predictor variables was considerably larger. Given, however, the possible multicollinearity issues due to high correlations among the predictor variables, the set of variables employed in the empirical analysis is reduced to those outlined above.

Table 2. Descriptive statistics.

Dependent Variable	Mean	StDev	Median	Q1	Q3	Skewness	Kurtosis	Min	Max
RGDP—Germany	0.26	1.80	0.44	−0.12	0.80	−1.73	23.43	−10.53	8.66
RGDP—France	0.28	2.65	0.35	0.03	0.66	0.87	32.69	−14.46	17.04
RGDP—Italy	0.01	2.47	0.21	−0.23	0.41	0.47	28.32	−13.50	14.85
RGDP—Spain	0.27	2.95	0.58	−0.01	0.90	−2.12	34.53	−19.42	15.53
RGDP—Greece	−0.09	2.48	0.14	−0.78	1.05	−2.51	15.60	−14.44	5.22
RGDP—Belgium	0.38	1.99	0.43	0.21	0.69	−1.35	32.55	−12.37	11.23
Predictors—Common Factors	Mean	StDev	Median	Q1	Q3	Skewness	Kurtosis	Min	Max
OIL (5)	1.03	17.30	3.92	−5.36	10.99	−1.22	6.19	−70.10	38.01
WUI (6)	−0.26	35.38	−4.24	−21.57	29.09	−0.06	2.52	−80.60	68.15
WPI (7)	0.16	2.12	0.00	−0.00	0.00	2.85	25.64	−8.68	13.37
Predictors—Germany-Specific Factors	Mean	StDev	Median	Q1	Q3	Skewness	Kurtosis	Min	Max
RPROD (8)	0.21	0.79	0.26	−0.05	0.57	−1.38	10.19	−3.62	2.65
CREG (9)	−0.27	11.11	0.08	−2.26	2.64	0.54	15.40	−46.45	58.52
UNEM (10)	−0.05	0.23	−0.10	−0.20	0.10	0.32	2.51	−0.50	0.50
CPI (11)	0.01	0.47	−0.04	−0.25	0.27	0.26	4.52	−1.47	1.610
PPI (12)	0.03	0.73	0.06	−0.28	0.42	−1.08	8.16	−3.45	1.80
CONPROD (13)	3.50	25.50	4.89	−14.72	17.70	0.53	3.04	−37.85	80.00
LONGR (14)	−0.06	0.26	−0.10	−0.25	0.16	0.15	2.36	−0.58	0.53
STOCK (15)	0.63	8.67	1.64	−2.93	6.46	−1.18	4.98	−31.83	15.04
LEAD (16)	−0.06	2.09	−0.07	−1.06	1.06	0.29	4.76	−5.79	6.62
Predictors—France-Specific Factors	Mean	StDev	Median	Q1	Q3	Skewness	Kurtosis	Min	Max
RPROD (8)	0.21	0.91	0.17	−0.08	0.49	2.34	25.92	−3.63	6.09
CREG (9)	−1.18	18.25	1.05	−7.57	6.20	0.30	9.72	−70.42	82.62
UNEM (10)	−0.01	0.29	−0.03	−0.13	0.13	1.80	12.99	−0.83	1.57
CPI (11)	0.00	0.44	0.01	−0.26	0.26	−0.30	4.12	−1.49	1.08
PPI (12)	0.04	1.24	0.00	−0.68	0.82	−0.03	5.61	−4.61	4.09
CONPROD (13)	0.30	9.07	1.98	−10.11	7.65	−0.25	2.08	−17.00	22.25
LONGR (14)	−0.05	0.38	−0.10	−0.25	0.19	−0.02	4.00	−1.20	1.12
STOCK (15)	0.40	7.85	1.99	−3.00	6.25	−1.27	4.64	−27.01	11.83
LEAD (16)	−0.15	1.75	0.15	−1.41	1.01	−0.27	3.68	−4.81	4.84
Predictors—Italy-Specific Factors	Mean	StDev	Median	Q1	Q3	Skewness	Kurtosis	Min	Max
RPROD (8)	0.02	0.74	0.02	−0.40	0.41	0.15	3.55	−2.01	2.10
CREG (9)	−0.61	13.13	0.15	−3.21	2.90	1.83	22.76	−51.22	80.66
UNEM (10)	0.00	0.44	0.00	−0.20	0.20	0.05	9.07	−1.90	1.80
CPI (11)	−0.02	0.42	−0.04	−0.25	0.17	−0.15	4.14	−1.31	1.21
PPI (12)	0.04	1.26	0.03	−0.66	0.73	−1.36	10.96	−6.58	3.21
CONPROD (13)	0.55	11.29	−0.71	−9.86	10.92	0.29	2.23	−18.77	35.11
LONGR (14)	−0.06	0.33	−0.05	−0.29	0.19	0.07	3.04	−0.87	0.94
STOCK (15)	−0.26	8.60	1.52	−5.05	5.92	−1.12	4.12	−30.39	11.44
LEAD (16)	−0.06	1.57	−0.01	−0.97	0.88	0.16	5.03	−4.48	5.84
Predictors—Spain-Specific Factors	Mean	StDev	Median	Q1	Q3	Skewness	Kurtosis	Min	Max
RPROD (8)	0.14	1.26	0.21	−0.30	0.66	−1.99	20.91	−7.45	5.36
CREG (9)	−0.38	10.19	−0.20	−2.90	3.28	1.56	18.80	−38.37	59.61
UNEM (10)	0.06	0.69	−0.10	−0.40	0.43	1.47	6.70	−1.10	2.80
CPI (11)	−0.02	0.78	−0.07	−0.48	0.38	0.02	3.92	−2.45	1.97
PPI (12)	0.05	1.71	0.00	−0.99	0.97	−0.21	5.57	−6.53	5.39
CONPROD (13)	−0.06	10.47	1.80	−7.24	7.93	−0.60	3.27	−33.28	21.47
LONGR (14)	−0.05	1.54	−0.09	−0.44	0.29	−0.89	11.86	−7.53	5.70
STOCK (15)	−0.04	8.54	1.06	−2.85	5.62	−0.85	3.64	−23.44	17.43
LEAD (16)	−0.09	1.57	0.04	−1.12	0.85	−0.41	3.51	−4.41	3.91
Predictors—Greece-Specific Factors	Mean	StDev	Median	Q1	Q3	Skewness	Kurtosis	Min	Max
RPROD (8)	−0.12	3.07	−0.21	−1.30	1.33	−0.70	12.63	−15.75	11.99
CREG (9)	−0.55	18.75	0.76	−4.18	4.04	0.67	22.46	−93.57	107.91
UNEM (10)	0.05	0.81	−0.10	−0.43	0.30	0.66	3.70	−1.90	2.40
CPI (11)	−0.04	0.75	−0.06	−0.44	0.34	0.36	3.85	−1.83	2.13
PPI (12)	0.04	3.54	−0.05	−2.16	2.12	0.12	4.84	−11.94	11.24
CONPROD (13)	2.79	30.05	10.81	−26.16	24.05	−0.35	2.22	−58.38	74.73
LONGR (14)	−0.06	0.26	−0.07	−0.23	0.15	−0.23	3.31	−0.84	0.54
STOCK (15)	−1.55	13.71	−0.70	−9.19	8.87	−0.59	3.57	−47.93	24.40
LEAD (16)	0.01	1.18	0.08	−1.00	1.10	−0.19	1.90	−2.38	2.27
Predictors—Belgium-Specific Factors	Mean	StDev	Median	Q1	Q3	Skewness	Kurtosis	Min	Max
RPROD (8)	0.21	0.92	0.20	−0.09	0.66	−0.29	12.49	−4.19	4.33
CREG (9)	−0.18	10.37	0.09	−3.15	3.94	0.48	11.50	−36.16	50.38
UNEM (10)	0.00	0.41	0.00	−0.30	0.23	0.16	2.91	−0.80	1.00
CPI (11)	−0.01	0.68	−0.02	−0.34	0.37	−0.59	4.23	−2.10	1.43
PPI (12)	0.11	2.63	0.12	−1.38	1.18	0.67	4.93	−6.17	9.05
CONPROD (13)	0.62	10.13	3.45	−6.26	8.35	−0.55	2.01	−17.83	17.20
LONGR (14)	−0.06	0.28	−0.11	−0.23	0.15	0.07	2.59	−0.72	0.63
STOCK (15)	0.49	8.17	1.84	−1.85	5.49	−1.73	7.77	−37.02	12.77
LEAD (16)	−0.05	2.10	−0.07	−1.10	0.63	0.47	4.68	−5.64	7.13

This table reports summary statistics for the dependent variables (individual country’s real GDP growth rates), as well as the common and country-specific predictor variables.

reports summary statistics for both the dependent and explanatory/predictor variables employed in the panel model specifications. More specifically, various descriptive statistics are estimated for the dependent, common, and country-specific factors, including the mean, median, standard deviation, and 0.25 and 0.75 quantiles, as well as kurtosis and skewness. Generally, these statistics reveal significant variability in the real GDP growth series, showing both skewness and excess kurtosis (fat tails).

6. Empirical Design and Analysis

In this section, the empirical analysis is presented. The aim is to estimate different panel model specifications in order to detect the most critical factors that determine the real GDP growth of different Euro area countries and to assess the impact of these covariates (for the World Pandemic Index (WPI) in particular) on the economic activity of the analysed countries during the COVID-19 pandemic. Below, the panel data models used in this study are outlined in detail, the estimated panel model parameters are presented, and the main findings of the analysis are discussed.

6.1. Panel Model Specifications

The following panel model specifications are employed:

$M_1$: 

A panel model with common covariates, without autoregressive dynamics:

$y_{i,t}={\delta }_i+{\sum }_{p=1}^P{\alpha }_{p,i}{f}_{p,t}+u_{i,t}$,

$y_t=x_t^{\prime }\beta +u_t=\left(I_N\otimes z_t^{\prime }\right)S\beta +u_t,u_t\sim N_N\left(0,\Sigma \right)$,

$Y=X\beta +U,U\sim N_{NT}\left(0,I_T\otimes \Sigma \right)$,

where $y_{i,t}$ denotes the real GDP growth of country i at time t, $i=1,2,\dots ,N$, $t=1,2,\dots ,T$; $f_{p,t}$ denotes the p-th exogenous common covariate/predictor at time t, $p=1,\dots ,P$; $u_{i,t}$ is the error process; $y_t$ is the $N\times 1$ vector of real GDP growth for the analysed countries at time t; $z_t={\left(1,f_{1,t},\dots ,f_{P,t}\right)}^{\prime }$ is the $(1+P)\times 1$ vector of common covariates at time t; $\left(I_N\otimes z_t^{\prime }\right)$ of dimension $N\times N(1+P)$ is the Kronecker product of the $N\times N$ identity matrix with the covariate vector $z_t^{\prime }$; S is an $N(1+P)\times N(1+P)$ selection matrix with elements equal to 0 or 1; $\beta$ is an $N(1+P)\times 1$ vector of the panel model parameters, i.e., $\beta ={\left({\delta }_1,{\alpha }_{1,1},\dots ,{\alpha }_{P,1}\dots ,{\delta }_N,{\alpha }_{1,N},\dots ,{\alpha }_{P,N}\right)}^{\prime }$; $x_t^{\prime }=\left(I_N\otimes z_t^{\prime }\right)S$ is an $N\times N(1+P)$ matrix; $Y={\left({y_1}^{\prime },{y_2}^{\prime },\dots ,{y_T}^{\prime }\right)}^{\prime }$ is the $NT\times 1$ vector of real GDP growth values; $X={\left(x_1,x_2,\dots ,x_T\right)}^{\prime }$ is the $NT\times N(1+P)$ matrix of covariates; and $U={\left(u_1^{\prime },\dots ,u_T^{\prime }\right)}^{\prime }$ is the $NT\times 1$ vector of innovations.

$M_2$: 

A panel model with common covariates, with autoregressive dynamics:

$y_{i,t}={\delta }_i+{\sum }_{p=1}^P{\alpha }_{p,i}{f}_{p,t}+{ϵ}_{i,t}$, $\mathrm{where}{ϵ}_{i,t}=\varphi {ϵ}_{i,t-1}+u_{i,t}$,

$y_{i,t}^{\ast }={\delta }_i(1-\varphi )+{\sum }_{p=1}^P{\alpha }_{p,i}{f}_{p,t}^{\ast }+u_{i,t}$,

$y_t^{\ast }=x_t^{\prime }\beta +u_t=\left(I_N\otimes z_t^{\prime }\right)S\beta +u_t$, $u_t\sim N_N\left(0,\Sigma \right)$,

$Y^{\ast }=X\beta +U,U\sim N_{N(T-1)}\left(0,I_{T-1}\otimes \Sigma \right)$,

where $y_{i,t}$ denotes the real GDP growth of country i at time t, $i=1,2,\dots ,N$, $t=1,2,\dots ,T$, $f_{p,t}$, denotes the p-th exogenous common covariate/predictor at time t, $p=1,\dots ,P$, ${ϵ}_{i,t}$ follows an autoregressive model of order one, AR(1), $u_{i,t}$ is the error process, $y_{i,t}^{\ast }=y_{i,t}-\varphi y_{i,t-1}$ is an $N\times 1$ vector at time t, $t=2,\dots ,T$, $z_t={\left(1-\varphi ,f_{1,t}^{\ast },\dots ,f_{P,t}^{\ast }\right)}^{\prime }$ is the $(1+P)\times 1$ vector with $f_{p,t}^{\ast }=f_{p,t}-\varphi f_{p,t-1}$ of common covariates at time t, $\left(I_N\otimes z_t^{\prime }\right)$ of dimension $N\times N(1+P)$ is the Kronecker product of the $N\times N$ identity matrix with the covariate vector $z_t^{\prime }$, S is an $N(1+P)\times N(1+P)$ selection matrix with elements equal to 0 or 1, $\beta$ is an $N(1+P)\times 1$ vector of the panel model parameters, i.e., $\beta ={\left({\delta }_1,{\alpha }_{1,1},\dots ,{\alpha }_{P,1}\dots ,{\delta }_N,{\alpha }_{1,N},\dots ,{\alpha }_{P,N}\right)}^{\prime }$, $x_t^{\prime }=\left(I_N\otimes z_t^{\prime }\right)S$ is an $N\times N(1+P)$ matrix, $Y^{\ast }={\left({y_2^{\ast }}^{\prime },\dots ,{y_T^{\ast }}^{\prime }\right)}^{\prime }$ is the $N(T-1)\times 1$ vector, $X={\left(x_2,\dots ,x_T\right)}^{\prime }$ is the $N(T-1)\times N(1+P)$ matrix of covariates, and $U={\left(u_2^{\prime },\dots ,u_T^{\prime }\right)}^{\prime }$ is the $N(T-1)\times 1$ vector of innovations.

$M_3$: 

A panel model with common and country-specific covariates, without autoregressive dynamics:

$y_{i,t}={\delta }_i+{\sum }_{p=1}^P{\alpha }_{p,i}{f}_{p,t}+{\sum }_{q=1}^Q{\gamma }_{q,i}{f}_{q,i,t}+u_{i,t}$,

$y_t=x_t^{\prime }\beta +u_t=\left(I_N\otimes z_t^{\prime }\right)S\beta +u_t,u_t\sim N_N\left(0,\Sigma \right)$,

$Y=X\beta +U,U\sim N_{NT}\left(0,I_T\otimes \Sigma \right)$,

where $y_{i,t}$ denotes the real GDP growth of country i at time t, $i=1,2,\dots ,N$, $t=1,2,\dots ,T$, $f_{p,t}$ denotes the p-th exogenous common covariate/predictor at time t, $p=1,\dots ,P$, with coefficients ${\alpha }_{p,i}$, $f_{q,i,t}$ denotes the q-th exogenous country-specific covariate, with coefficient ${\gamma }_{q,i}$, $u_{i,t}$ is the error process, $y_t$ is the $N\times 1$ vector of real GDP growth values for the analysed countries at time t, $z_t={\left(1,f_{1,t},\dots ,f_{P,t},f_{1,1,t},\dots ,f_{Q,1,t},f_{1,2,t},\dots ,f_{Q,2,t},\dots ,f_{Q,N,t}\right)}^{\prime }$ is the $(1+P+NQ)\times 1$ vector of covariates at time t, $t=1,\dots ,T$, $\left(I_N\otimes z_t^{\prime }\right)$ of dimension $N\times N(1+P+NQ)$ is the Kronecker product of the $N\times N$ identity matrix with the covariate vector $z_t^{\prime }$, S is an $N(1+P+NQ)\times N(1+P+Q)$ selection matrix with elements equal to 0 or 1, $\beta$ is an $N(1+P+Q)\times 1$ vector of the panel model parameters, i.e., $\beta ={\left({\delta }_1,{\alpha }_{1,1},\dots ,{\alpha }_{P,1},{\gamma }_{1,1},\dots ,{\gamma }_{Q,1},\dots ,{\delta }_N,{\alpha }_{1,N},\dots ,{\alpha }_{P,N},{\gamma }_{1,N},\dots ,{\gamma }_{Q,N}\right)}^{\prime }$, $x_t^{\prime }=\left(I_N\otimes z_t^{\prime }\right)S$ is an $N\times N(1+P+Q)$ matrix, $Y={\left({y_1}^{\prime },{y_2}^{\prime },\dots ,{y_T}^{\prime }\right)}^{\prime }$ is the $NT\times 1$ vector of real GDP growth values, $X={\left(x_1,x_2,\dots ,x_T\right)}^{\prime }$ is the $NT\times N(1+P+Q)$ matrix of covariates, and $U={\left(u_1^{\prime },\dots ,u_T^{\prime }\right)}^{\prime }$ is the $NT\times 1$ vector of innovations.

6.2. Empirical Results

This section presents the empirical findings. As noted earlier, the panel data models use several predictor variables/factors to investigate the impact of the COVID-19 pandemic on real GDP growth. Furthermore, there is an interest in examining the drivers of economic activity during turbulent times using the proposed panel models and their associated SUR representation. The modeling framework is implemented on the real economic activity of six Eurozone economies of different scale, namely Germany, France, Italy, Spain, Greece, and Belgium, using quarterly data for the period 2001:Q2–2021:Q3. The models have been estimated under parameter heterogeneity across countries without and with cross-sectional dependence of the error processes.

First, the results obtained from the panel data model $y_{i,t}={\delta }_i+{\sum }_{p=1}^P{\alpha }_{p,i}{f}_{p,t}+u_{i,t}$ or $Y=X\beta +U,U\sim N_{NT}\left(0,I_T\otimes \Sigma \right)$ (Model Specification M1), which uses three common covariates, i.e., the change in WTI oil price index (OIL), the World Uncertainty Index (WUI), and the World Pandemic Uncertainty Index (WPI), without employing autoregressive dynamics for the error process, are presented and discussed. Model parameter estimates are obtained by using OLS based on the SUR representation and by applying the Bayesian approach to inference, i.e., the MCMC sampling scheme described in Section 4, which was implemented with over 5000 iterations to produce a sample from the posterior distribution of the model parameters.

Table 3. Estimates and standard errors (in parentheses) of the parameters of panel model specification $M1$ with common covariates for the real GDP growth of the analysed countries.

Panel A: OLS
Predictor Variables	Germany	France	Italy	Spain	Greece	Belgium
$\delta$	0.295	0.376	0.093	0.382	−0.021	0.455
	(0.183)	(0.265)	(0.244)	(0.289)	(0.245)	(0.190)
OIL	0.010	−0.022	−0.014	−0.013	0.010	−0.011
	(0.011)	(0.016)	(0.014)	(0.017)	(0.014)	(0.011)
WUI	−0.008	−0.013	−0.014	−0.011	−0.006	−0.009
	(0.005)	(0.008)	(0.007)	(0.008)	(0.007)	(0.005)
WPI	−0.274	−0.476	−0.469	−0.616	−0.502	−0.445
	(0.087)	(0.126)	(0.116)	(0.137)	(0.116)	(0.091)
Panel B: Bayesian
Predictor Variables	Germany	France	Italy	Spain	Greece	Belgium
$\delta$	0.293	0.370	0.087	0.376	−0.025	0.450
	(0.183)	(0.263)	(0.242)	(0.286)	(0.247)	(0.188)
OIL	0.009	−0.021	−0.014	−0.013	0.010	−0.011
	(0.011)	(0.016)	(0.015)	(0.018)	(0.015)	(0.012)
WUI	−0.008	−0.013	−0.014	−0.011	−0.006	−0.009
	(0.005)	(0.008)	(0.007)	(0.009)	(0.007)	(0.006)
WPI	−0.272	−0.472	−0.465	−0.613	−0.501	−0.442
	(0.089)	(0.129)	(0.118)	(0.140)	(0.118)	(0.093)

This table reports parameter estimates and standard errors (in parentheses) of the common predictor variables in the panel data model $y_{i,t}={\delta }_i+{\sum }_{p=1}^P{\alpha }_{p,i}{f}_{p,t}+u_{i,t}$ or $Y=X\beta +U,U\sim N_{NT}\left(0,I_T\otimes \Sigma \right)$ (Model Specification M1) of the real GDP growth for the analysed countries based on the sample period from 2001:Q2 to 2021:Q3. OLS denotes the Least Squares-based estimates and Bayesian refers to the posterior means and standard deviations (in parentheses) of the model parameters obtained by the MCMC algorithm. The common predictor variables are the WTI oil price (OIL), the World Uncertainty Index (WUI), and the World Pandemic Uncertainty Index (WPI).

reports the corresponding OLS-based parameter estimates and standard errors (Panel A), as well as the posterior means and standard deviations (Panel B) of the common predictor factors of the panel data model. The findings reveal that all the estimated intercepts ${\delta }_i$, $i=1,\dots ,N$, are positive, apart from Greece (−0.021), ranging from 0.093 for Italy to 0.455 for Belgium; however, only the population intercept parameter of Belgium (estimated at 0.455) is statistically significant. We observe that the World Pandemic Uncertainty Index, WPI, appears in all countries with a statistically significant negative coefficient, ranging from −0.274 for Germany to −0.616 for Spain, indicating that, as expected, the COVID-19 pandemic has negatively influenced each country’s real economic activity. However, the impact varies across the countries studied. The estimates suggest that the pandemic’s effect was more severe in Spain (−0.616), Greece (−0.502), France (−0.476), Italy (−0.469), and Belgium (−0.445) relative to Germany (−0.274). Almost identical findings arise from the estimates based on the MCMC posterior means.

Table 4. Posterior means and standard deviations (in parentheses) of the covariance matrix $\Sigma$ of the residuals of the panel model specification $M1$ with common covariates for the six European countries’ real GDP growth.

Covariances
	Germany	France	Italy	Spain	Greece	Belgium
Germany	2.80
	(0.46)
France	3.65	5.88
	(0.63)	(0.96)
Italy	3.39	5.29	4.97
	(0.58)	(0.87)	(0.80)
Spain	3.99	6.18	5.62	6.99
	(0.69)	(1.03)	(0.93)	(1.14)
Greece	1.96	3.07	2.87	3.97	5.01
	(0.49)	(0.72)	(0.66)	(0.82)	(0.81)
Belgium	2.63	4.09	3.74	4.44	2.32	3.03
	(0.45)	(0.68)	(0.62)	(0.74)	(0.52)	(0.49)

This table reports posterior means and standard deviations (in parentheses) of the elements of the covariance matrix $\Sigma$ of the residuals of the panel data model $y_{i,t}={\delta }_i+{\sum }_{p=1}^P{\alpha }_{p,i}{f}_{p,t}+u_{i,t}$ or $Y=X\beta +U,U\sim N_{NT}\left(0,I_T\otimes \Sigma \right)$ (Model Specification M1) of the real GDP growth of the analysed countries based on the sample period from 2001:Q2 to 2021:Q3.

presents the posterior means and standard deviations for the elements of the covariance matrix $\Sigma$ of the residuals of the panel model. All the covariances take statistically significant positive values, with a minimum value of 1.96 between Germany and Greece and a maximum of 6.18 between France and Spain, indicating strong evidence for cross-sectional dependence/correlation among the error terms of the analysed countries.

Next, the focus is shifted to the results obtained from the dynamic panel model $y_{i,t}={\delta }_i+{\sum }_{p=1}^P{\alpha }_{p,i}{f}_{p,t}+{ϵ}_{i,t}$, where ${ϵ}_{i,t}=\varphi {ϵ}_{i,t-1}+u_{i,t}$ or $Y^{\ast }=X\beta +U,U\sim N_{N(T-1)}\left(0,I_{T-1}\otimes \Sigma \right)$ (Model Specification M2), that uses the three common covariates and allows for autoregressive dynamics for the error process. Model parameter estimates are obtained by implementing the MCMC sampling algorithm of Section 4 for 20,000 iterations to generate a sample from the posterior distribution of the model parameters. The algorithm converges to the posterior distribution of model parameters very fast, i.e., in a small fraction of iterations. This is clearly depicted in the convergence plots and the histograms of the posterior sample of parameters of the panel model presented in Figure 1, Figure 2, Figure 3 and Figure 4. The shapes of the posterior distribution of all ${\delta }_i$ and ${\alpha }_{p,i}$, $1,\dots ,P$ (Figure 2), as well as of the common $\varphi$ autoregressive parameter (Figure 3), indicate a symmetric normal-like distribution, while the posterior distribution of the variances ${\sigma }_i^2$ and covariances ${\sigma }_{ij}$ (Figure 4) exhibits positive skewness, indicating deviation from normality.

In

Table 5. Posterior means and standard deviations (in parentheses) of the parameters of dynamic panel model specification $M2$ with common covariates and autoregressive dynamics for the real GDP growth of the analysed countries.

Predictor Variables	Germany	France	Italy	Spain	Greece	Belgium
$\delta$	0.317	0.408	0.127	0.397	−0.033	0.485
	(0.232)	(0.335)	(0.306)	(0.366)	(0.298)	(0.239)
OIL	0.007	−0.027	−0.019	−0.017	0.009	−0.015
	(0.012)	(0.017)	(0.015)	(0.018)	(0.015)	(0.012)
WUI	−0.008	−0.012	−0.013	−0.010	−0.003	−0.009
	(0.005)	(0.008)	(0.007)	(0.008)	(0.007)	(0.005)
WPI	−0.316	−0.556	−0.536	−0.681	−0.524	−0.495
	(0.091)	(0.132)	(0.120)	(0.143)	(0.115)	(0.094)
Common $\varphi$	0.155
	(0.052)

This table reports posterior means and standard deviations (in parentheses) of the parameters of the common predictor variables in the dynamic panel data model $y_{i,t}={\delta }_i+{\sum }_{p=1}^P{\alpha }_{p,i}{f}_{p,t}+{ϵ}_{i,t}$, where ${ϵ}_{i,t}=\varphi {ϵ}_{i,t-1}+u_{i,t}$ or $Y^{\ast }=X\beta +U,U\sim N_{N(T-1)}\left(0,I_{T-1}\otimes \Sigma \right)$ (Model Specification M2) of the real GDP growth for the analysed countries based on the sample period from 2001:Q2 to 2021:Q3. The common predictor variables are the WTI oil price (OIL), the World Uncertainty Index (WUI), and the World Pandemic Uncertainty Index (WPI).

, the posterior means and standard deviations of the parameters of the common predictor variables, as well as of the common autoregressive coefficient $\varphi$ in the dynamic panel data model, are presented. The results are similar in spirit with those of the model specification $M1$ and show that all the estimated intercepts ${\delta }_i$, $i=1,\dots ,N$, are positive, apart from Greece (−0.033), ranging from 0.127 for Italy to 0.485 for Belgium. Again, only the population intercept parameter of Belgium (estimated at 0.485) is statistically significant. The World Pandemic Uncertainty Index, WPI, has significant negative exposure for all the analysed countries, ranging from −0.316 for Germany to −0.681 for Spain, which indicates considerable variability in the impact of the World Pandemic Uncertainty Index on the economic activity of the analysed countries. Based on the $a_{3,i}$ model estimates, the effect of the pandemic seems to be significant for Spain (−0.681), France (−0.556), Italy (−0.536), Greece (−0.524), and Belgium (−0.495) relative to the effect on Germany (−0.316). Therefore, both panel model specifications, $M1$ and $M2$, identify the World Pandemic Uncertainty Index as an important covariate that influences the economic activity in the analysed countries. Another important finding is that the common autoregressive coefficient $\varphi$ is statistically significant and positive, estimated at 0.155, indicating positive autocorrelation dynamics in real GDP.

The posterior means and standard deviations of the elements of the covariance matrix $\Sigma$ of the residuals of the dynamic panel model specification $M2$ with common covariates and autoregressive dynamics for the six European countries’ real GDP growth are reported in

Table 6. Posterior means and standard deviations (in parentheses) of the covariance matrix $\Sigma$ of the residuals of the dynamic panel model specification $M2$ with common covariates and autoregressive dynamics for the six European countries’ real GDP growth.

Covariances
	Germany	France	Italy	Spain	Greece	Belgium
Germany	3.01
	(0.50)
France	3.91	6.23
	(0.68)	(1.03)
Italy	3.60	5.58	5.19
	(0.62)	(0.93)	(0.86)
Spain	4.31	6.58	5.94	7.40
	(0.75)	(1.10)	(1.00)	(1.22)
Greece	2.18	3.20	2.96	4.04	4.89
	(0.51)	(0.73)	(0.67)	(0.83)	(0.80)
Belgium	2.81	4.33	3.92	4.72	2.42	3.19
	(0.49)	(0.72)	(0.66)	(0.79)	(0.53)	(0.52)

This table reports posterior means and standard deviations (in parentheses) of the elements of the covariance matrix $\Sigma$ of the residuals of the dynamic panel data model $y_{i,t}={\delta }_i+{\sum }_{p=1}^P{\alpha }_{p,i}{f}_{p,t}+{ϵ}_{i,t}$, where ${ϵ}_{i,t}=\varphi {ϵ}_{i,t-1}+u_{i,t}$ or $Y^{\ast }=X\beta +U,U\sim N_{N(T-1)}\left(0,I_{T-1}\otimes \Sigma \right)$ (Model Specification M2) of the real GDP growth of the analysed countries based on the sample period from 2001:Q2 to 2021:Q3.

. The estimated covariance structure is similar in spirit with those of the panel model specification $M1$, indicating some level of robustness in the estimated covariance matrix $\Sigma$. Specifically, all the estimated covariances are positive and statistically significant, with a minimum value of 2.18 between Germany and Greece and a maximum of 6.58 between France and Spain, indicating cross-sectional dependence/correlation among the error terms of the analysed countries and pointing out the need for allowing for cross-sectional dependence in the corresponding panel data models.

Next, the panel data model $y_{i,t}={\delta }_i+{\sum }_{p=1}^P{\alpha }_{p,i}{f}_{p,t}+{\sum }_{q=1}^Q{\gamma }_{q,i}{f}_{q,i,t}+u_{i,t}$ is considered, i.e., the equivalent Seemingly Unrelated Regression (SUR) representation $Y=X\beta +U,U\sim N_{NT}\left(0,I_T\otimes \Sigma \right)$ (Model Specification M3), which allows for common covariates, $f_{p,t}$, $p=1,\dots ,P$, and specific cross-sectional unit explanatory variables, $f_{q,i,t}$, $q=1,\dots ,Q$, to impact the panel model’s dependent variable series. The covariance structure of $\Sigma$ allows for cross-sectional dependence of the innovation process if $\Sigma$ is not diagonal, and in this way, it may capture unobserved random (pandemic, war, geopolitical, etc.) effects and provide a natural link/connection to the system of dependent variables. Note also that in this model specification, the set of country-specific covariates is augmented by the lagged values of the individual country’s real GDP series. More specifically, the model includes four autoregressive terms for each country/economy (${\mathrm{RGDP}}_{t-1}$, ${\mathrm{RGDP}}_{t-2}$, ${\mathrm{RGDP}}_{t-3}$, ${\mathrm{RGDP}}_{t-4}$) in the model to manage autocorrelation issues and to utilize their informational content for the projected estimates. In this sense, the proposed panel model specification, $M3$, has a dynamic structure that allows for lagged dependent variables to affect current and/or future GDP series.

First, the panel model specification $M3$ with common and country-specific covariates with cross-sectional independent errors (i.e., identical to estimating univariate regression models) is estimated for the analysed countries.

Table 7. Estimates and standard errors (in parentheses) of the parameters of panel model specification $M3$ with common and country-specific covariates with cross-sectional independent errors for the real GDP growth of the analysed countries.

Predictor Variables	Germany	France	Italy	Spain	Greece	Belgium
${\delta }_i$	0.532 *	0.441	0.141	0.892 *	0.110	0.931 *
	(0.170)	(0.230)	(0.178)	(0.208)	(0.250)	(0.151)
${\mathrm{RGDP}}_{t-1}$ (1)	−0.707 *	−0.260	−0.508 *	−0.591 *	−0.294	−0.401 *
	(0.212)	(0.147)	(0.182)	(0.135)	(0.158)	(0.116)
${\mathrm{RGDP}}_{t-2}$ (2)	−0.697 *	−0.888 *	−0.950 *	−0.796 *	0.006	−1.200 *
	(0.128)	(0.137)	(0.185)	(0.125)	(0.144)	(0.108)
${\mathrm{RGDP}}_{t-3}$ (3)	0.046	−0.182	−0.283 *	−0.047	0.261 *	−0.115
	(0.112)	(0.093)	(0.113)	(0.082)	(0.115)	(0.075)
${\mathrm{RGDP}}_{t-4}$ (4)	0.028	−0.019	−0.042	0.149	0.104	−0.127
	(0.094)	(0.078)	(0.100)	(0.075)	(0.127)	(0.076)
OIL (5)	0.019	0.015	−0.023	−0.004	0.005	0.001
	(0.013)	(0.016)	(0.017)	(0.018)	(0.024)	(0.010)
WUI (6)	−0.005	0.003	−0.006	−0.004	−0.007	−0.0001
	(0.004)	(0.005)	(0.005)	(0.006)	(0.007)	(0.003)
WPI (7)	−0.505 *	−0.876 *	−0.971 *	−1.193 *	−0.650 *	−0.936 *
	(0.088)	(0.129)	(0.161)	(0.136)	(0.138)	(0.070)
RPROD (8)	0.0007	1.607 *	0.274	0.186	0.119	1.030 *
	(0.321)	(0.337)	(0.280)	(0.266)	(0.119)	(0.228)
CREG (9)	0.006	0.012	−0.014	0.069	0.048 *	0.015
	(0.025)	(0.013)	(0.031)	(0.036)	(0.024)	(0.016)
UNEM (10)	−1.938 *	0.019	1.010	−1.486 *	−0.836	−0.699 *
	(0.853)	(0.881)	(0.662)	(0.381)	(0.463)	(0.317)
CPI (11)	−0.621	−0.171	1.103	−0.035	−0.797 *	0.313
	(0.473)	(0.514)	(0.559)	(0.300)	(0.383)	(0.197)
PPI (12)	−0.381	−0.340	0.128	−0.133	0.100	−0.125
	(0.326)	(0.218)	(0.209)	(0.169)	(0.096)	(0.066)
CONPROD (13)	0.007	0.012	0.024	0.011	0.0008	0.016
	(0.006)	(0.019)	(0.018)	(0.019)	(0.008)	(0.012)
LONGR (14)	−0.054	0.192	−0.112	0.071	0.581	−0.040
	(0.652)	(0.455)	(0.646)	(0.139)	(1.119)	(0.439)
STOCK (15)	0.044 *	0.015	0.029	−0.017	−0.005	0.026
	(0.020)	(0.027)	(0.027)	(0.030)	(0.024)	(0.019)
LEAD (16)	0.405 *	0.174	0.630 *	0.090	−0.296	0.156 *
	(0.099)	(0.134)	(0.173)	(0.178)	(0.248)	(0.077)

This table reports parameter estimates and standard errors (in parentheses) of the common and country-specific predictor variables in the panel data model $y_{i,t}={\delta }_i+{\sum }_{p=1}^P{\alpha }_{p,i}{f}_{p,t}+{\sum }_{q=1}^Q{\gamma }_{q,i}{f}_{q,i,t}+u_{i,t}$ or the equivalent Seemingly Unrelated Regression (SUR) representation $Y=X\beta +U,U\sim N_{NT}\left(0,I_T\otimes \Sigma \right)$ (Model Specification M3) with cross-sectional independent errors (i.e., identical to estimating univariate regression models) of the real GDP growth for the analysed countries based on the sample period from 2001:Q2 to 2021:Q3. The predictor variables are the real GDP growth lagged once (${\mathrm{RGDP}}_{t-1}$), the real GDP growth lagged twice (${\mathrm{RGDP}}_{t-2}$), the real GDP growth lagged thrice (${\mathrm{RGDP}}_{t-3}$), the real GDP growth lagged four times (${\mathrm{RGDP}}_{t-4}$), the WTI oil price (OIL), the World Uncertainty Index (WUI), the World Pandemic Uncertainty Index (WPI), the real productivity growth (RPROD), the change in car registrations (CREGs), the rate of unemployment (UNEM), the growth in the Consumer Price Index (CPI), the growth in the Producer Price Index (PPI), the Construction Volume Index of Production (CONPROD), the long-run interest rate (LONGR), the stock index return (STOCK), and the change in the OECD Leading Indicator (LEAD). Asterisks (*) are used to denote statistical significant model parameters.

presents the OLS estimates and standard errors (in parentheses) of the panel model parameters. The results show that the estimated constants ${\delta }_i$ are positive and range from 0.110 for Greece to 0.931 for Belgium, see, for example, the estimated ${\delta }_i$ parameters of Germany (0.532), Spain (0.892), and Belgium (0.931), which appear to be statistically significant. Turning to the autoregressive terms, there are important lagged real GDP predictors for all the analysed countries, something that was expected based on the nature of real GDP growth series. In particular, for Germany, Spain, and Belgium, the first two autoregressive terms are significant, while for Italy, the third lagged variable is also significant; for France, only the second autoregressive term is significant, and for Greece, only the third one is significant. The World Pandemic Uncertainty Index (WPI) appears to be statistically significant with a negative exposure in all analysed countries, varying from −0.505 for Germany to −1.193 for Spain. Therefore, each country’s real economic activity was negatively influenced by the COVID-19 pandemic. However, the magnitude of impact varies significantly across the countries, with a larger effect on Spain (−1.193), Italy (−0.971), Belgium (−0.936), and France (−0.876) relative to Greece, and especially to Germany. The rate of unemployment (UNEM) also had a significant negative effect on German (−1.938), Spanish (−1.486), and Belgian (−0.699) real GDP growth. In contrast, the OECD leading indicator (LEAD) has a significant positive effect on the real GDP of Germany (0.405), Italy (0.630), and Belgium (0.156). Other important predictors appear to be the real productivity growth (RPROD), which has a positive impact on French (1.607) and Belgian (1.030) real GDP growth; car registrations, CREG (0.048); the change in the consumer price index for Greece (−0.797); and the stock market return for Germany (0.044).

Next, the results of the panel model specification $M3$ with common and country-specific covariates and cross-sectional dependence for the error processes of real GDP growth are presented.

Table 8. Estimates and standard errors (in parentheses) of the parameters of panel model specification $M3$ with common and country-specific covariates with cross-sectional dependence of the errors for the real GDP growth of the analysed countries.

Predictor Variables	Germany	France	Italy	Spain	Greece	Belgium
${\delta }_i$	0.421 *	0.594 *	0.132	0.573 *	0.155	0.742 *
	(0.159)	(0.191)	(0.175)	(0.196)	(0.245)	(0.131)
${\mathrm{RGDP}}_{t-1}$ (1)	−0.415 *	−0.396 *	−0.510 *	−0.321 *	0.007	−0.323 *
	(0.176)	(0.106)	(0.124)	(0.094)	(0.134)	(0.083)
${\mathrm{RGDP}}_{t-2}$ (2)	−0.428 *	−0.671 *	−0.664 *	−0.457 *	0.069	−0.745 *
	(0.108)	(0.103)	(0.123)	(0.090)	(0.118)	(0.084)
${\mathrm{RGDP}}_{t-3}$ (3)	0.145	−0.108	−0.088	0.110	0.185	−0.007
	(0.095)	(0.074)	(0.084)	(0.067)	(0.100)	(0.062)
${\mathrm{RGDP}}_{t-4}$ (4)	0.091	0.084	0.076	0.220 *	0.146	0.036
	(0.082)	(0.064)	(0.076)	(0.062)	(0.107)	(0.062)
OIL (5)	0.011	0.007	−0.003	−0.012	−0.013	−0.001
	(0.012)	(0.014)	(0.015)	(0.015)	(0.022)	(0.009)
WUI (6)	−0.007	−0.006	−0.009	−0.009	−0.007	−0.004
	(0.004)	(0.005)	(0.005)	(0.006)	(0.007)	(0.003)
WPI (7)	−0.443 *	−0.877 *	−0.883 *	−1.013 *	−0.637 *	−0.770 *
	(0.082)	(0.108)	(0.121)	(0.117)	(0.132)	(0.065)
RPROD (8)	−0.110	0.544 *	0.114	0.174	−0.146	0.502 *
	(0.255)	(0.221)	(0.164)	(0.155)	(0.098)	(0.144)
CREG (9)	0.0006	0.015	0.00004	0.035	0.011	0.004
	(0.020)	(0.009)	(0.018)	(0.021)	(0.019)	(0.010)
UNEM (10)	−1.205	0.335	−0.252	−0.486 *	−0.473	−0.405 *
	(0.686)	(0.577)	(0.382)	(0.227)	(0.374)	(0.197)
CPI (11)	−0.500	−0.233	0.050	−0.060	−0.668 *	−0.076
	(0.380)	(0.355)	(0.348)	(0.193)	(0.316)	(0.133)
PPI (12)	−0.192	−0.207	0.009	0.003	0.065	−0.061
	(0.268)	(0.158)	(0.141)	(0.118)	(0.081)	(0.045)
CONPROD (13)	0.005	0.009	0.005	0.014	0.0003	0.012
	(0.005)	(0.014)	(0.012)	(0.013)	(0.007)	(0.008)
LONGR (14)	0.199	0.308	0.082	−0.032	1.145	0.059
	(0.535)	(0.310)	(0.388)	(0.083)	(0.924)	(0.290)
STOCK (15)	0.044 *	0.029	0.031	0.005	0.009	0.026
	(0.017)	(0.020)	(0.019)	(0.021)	(0.021)	(0.014)
LEAD (16)	0.276 *	0.169	0.416 *	0.088	−0.139	0.131 *
	(0.083)	(0.095)	(0.117)	(0.118)	(0.202)	(0.053)

This table reports parameter estimates and standard errors (in parentheses) of the common and country-specific predictor variables in the panel data model $y_{i,t}={\delta }_i+{\sum }_{p=1}^P{\alpha }_{p,i}{f}_{p,t}+{\sum }_{q=1}^Q{\gamma }_{q,i}{f}_{q,i,t}+u_{i,t}$ or the equivalent Seemingly Unrelated Regression (SUR) representation $Y=X\beta +U,U\sim N_{NT}\left(0,I_T\otimes \Sigma \right)$ (Model Specification M3) with cross-sectional dependence in the error processes of the real GDP growth for the analysed countries based on the sample period from 2001:Q2 to 2021:Q3. The predictor variables are the real GDP growth lagged once (${\mathrm{RGDP}}_{t-1}$), the real GDP growth lagged twice (${\mathrm{RGDP}}_{t-2}$), the real GDP growth lagged thrice (${\mathrm{RGDP}}_{t-3}$), the real GDP growth lagged four times (${\mathrm{RGDP}}_{t-4}$), the WTI oil price (OIL), the World Uncertainty Index (WUI), the World Pandemic Uncertainty Index (WPI), the real productivity growth (RPROD), the change in car registrations (CREGs), the rate of unemployment (UNEM), the growth in the Consumer Price Index (CPI), the growth in the Producer Price Index (PPI), the Construction Volume Index of Production (CONPROD), the long-run interest rate (LONGR), the stock index return (STOCK), and the change in the OECD Leading Indicator (LEAD). Asterisks (*) are used to denote statistical significant model parameters.

reports the feasible GLS estimates and corresponding standard errors (in parentheses) of the panel model parameters. The results indicate that the estimated intercepts ${\delta }_i$ are positive and, in most of the cases, statistically significant, apart from for Italy and Greece, ranging from 0.132 for Italy to 0.742 for Belgium. With respect to the autoregressive dynamics of the analysed series, it is observable that the autoregressive terms, mainly at lags one and two, are significant for the real GDP of Germany, France, Italy, and Belgium, while for Spain, the lagged third term is also significant. In contrast, for Greece, none of the autoregressive terms appear to be critical. Regarding the important common and country-specific macroeconomic and financial covariates, the analysis reveals that the World Pandemic Uncertainty Index (WPI) is universally significant. As was expected, it has a negative effect on all countries’ GDPs, with a higher impact mainly on Spain (−1.013), Italy (−0.883), France (−0.877), Belgium (−0.770), and Greece (−0.637). In Germany, the impact is relatively lower (−0.443). The critical predictors seem to be the OECD leading indicator, with a positive impact on the real GDP of Germany (0.276), Italy (0.416), and Belgium (0.131), and the rate of unemployment, which has a significant negative effect on Spanish (−0.486) and Belgian (−0.405) real GDP growth. Other important predictors appear to be the real productivity growth index, with a positive impact on French (0.544) and Belgian (0.502) real GDP growth; the change in the consumer price index for Greece (−0.668); and the equity market return for Germany (0.044). Therefore, based on this model and the respective GLS estimates, there are differences with respect to the findings about the intercepts, the autoregressive terms, and, in some cases, the magnitude of the effect of several predictors on the GDP growth of the analysed countries compared to previews modeling approaches.

Table 9. Estimates of the correlation coefficients and the elements of the covariance matrix $\Sigma$ of the residuals of the panel data model specification $M3$ with common and country-specific covariates with cross-sectional dependence for the error process of the six European countries’ real GDP growth.

Panel A: Correlations
	Germany	France	Italy	Spain	Greece	Belgium
Germany	1.00
France	0.76	1.00
Italy	0.83	0.94	1.00
Spain	0.78	0.93	0.94	1.00
Greece	0.68	0.72	0.75	0.74	1.00
Belgium	0.76	0.91	0.92	0.91	0.69	1.00
Panel B: Covariances
	Germany	France	Italy	Spain	Greece	Belgium
Germany	1.64
France	1.58	2.64
Italy	1.93	2.76	3.28
Spain	1.96	2.96	3.33	3.81
Greece	1.95	2.60	3.04	3.24	4.97
Belgium	1.12	1.71	1.91	2.04	1.76	1.32

This table reports estimates of the correlation coefficients and of the elements (variances and covariances) of the covariance matrix $\Sigma$ of the residuals of the panel data model $y_{i,t}={\delta }_i+{\sum }_{p=1}^P{\alpha }_{p,i}{f}_{p,t}+{\sum }_{q=1}^Q{\gamma }_{q,i}{f}_{q,i,t}+u_{i,t}$ or the equivalent Seemingly Unrelated Regression (SUR) representation $Y=X\beta +U,U\sim N_{NT}\left(0,I_T\otimes \Sigma \right)$ (Model Specification M3) with common and country-specific predictor variables, and cross-sectional dependence in the error processes of the real GDP growth for the analysed countries based on the sample period from 2001:Q2 to 2021:Q3.

presents the estimates of the correlation coefficients (Panel A) and of the variances/covariances (Panel B) of the residuals of the panel model. All the estimated correlations/covariances are positive, indicating strong evidence of cross-sectional dependence among the error terms of the analysed countries. Evidence of cross-sectional correlation seems reasonable, since the underlying effect of the COVID-19 pandemic can be considered as a latent unobserved factor that influences the economic activity of the analysed Eurozone countries in such a highly connected and integrated world.

7. Conclusions

The main objective of this analysis was to evaluate the impact of the COVID-19 pandemic on real economic activity in several European countries using dynamic panel data models and various common and country-specific factors. A class of dynamic panel data models and their corresponding Seemingly Unrelated Regression (SUR) models are developed and applied to model the economic activity and, in particular, the real GDP growth of the six Eurozone countries. This class of models allows for common and country-specific covariates to affect the real GDP, as well as for cross-sectional dependence in the error processes. We consider the problem of estimation and inference for this class of panel models by using Bayesian and classical techniques. Bayesian inference is implemented by applying a Markov chain Monte Carlo (MCMC) sampling scheme, while the classical approach to inference is performed by using OLS and feasible GLS estimates.

Another advantage of the proposed panel data modeling approach and the corresponding inferential framework is that it can be extended to account for model selection/uncertainty regarding the set of covariates to be included in the model. However, this interesting problem is not the purpose of the present analysis and remains open for future research in the context of dynamic panel data models. Under the Bayesian framework, model selection is based on the posterior distribution of the panel models or Bayes factors (Kass and Raftery 1995). Posterior dynamic panel model probabilities can be obtained by designing a Markov chain Monte Carlo (MCMC) stochastic search algorithm that visits (jumps between) a variety of panel models. It can be seen as a Bayesian-motivated stochastic search algorithm that produces panel models together with their posterior probabilities.

The policy implications of the study on assessing the impact of the COVID-19 pandemic are significant. Some of the most important implications are the following: (i) The impact of the pandemic was heterogeneous. In this sense, governments and central banks need to tailor their policy responses based on the specific economic conditions of individual economies and sectors. (ii) The interconnected character of the global economy demands that policymakers enhance international cooperation to address the challenges created by the pandemic effectively using collaborative efforts. (iii) This study highlights the importance of monitoring and analyzing numerous factors to assess the aggregate shock on real economies in order to inform fiscal, investment, trade, and healthcare policies. In this sense, it provides policymakers a flexible and adaptive framework to respond effectively to evolving economic conditions.

The findings of our analysis reveal that various explanatory and predictive factors are of crucial importance for different economies. In particular, the empirical results suggest that the outbreak and spread of the COVID-19 pandemic and the resulting increased uncertainty, depicted by the World Pandemic Uncertainty Index, had a significant negative effect on the economic activity of several European countries. There is evidence that different European economies were not uniformly affected by this exogenous shock, and that heterogeneous effects exist. Despite the underlying variation, the results seem to be robust to several panel model specifications.