Estimating uncertainty spillover effects across euro area using a regime dependent VAR model

: This paper investigates macroeconomic uncertainty spillover effects across countries and their impact on real economic activity in different economic periods, i

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Introduction
In recent years there has been a considerable debate on the effect of macroeconomic and financial uncertainty on economic fluctuations [see, for example, Angelini et al. 2019;Bachmann and Sims 2013;Bloom 2009;Bloom, Bond, and Van Reenen 2007;Caldara et al. 2016;Carriero, Clark, and Marcellino 2018;Cesa-Bianchi, Pesaran, and Rebucci 2020;Girardi and Reuter 2016;Jurado, Ludvigson, and Ng 2015;Ludvigson, Ma, and Ng 2021;Meinen and Roehe 2017;Mumtaz and Theodoridis 2015;Rossi and Sekhposyan 2015].A number of empirical studies have used a variety of measures for macroeconomic uncertainty, such as implied volatility of stock market or cross-sectional dispersion of stock returns.Jurado, Ludvigson, and Ng (2015) argue that, while some measures may be preferable to others because of their observability, their effectiveness largely relies on the degree of correlation with the latent processes.Consequently Jurado, Ludvigson, and Ng (2015), proposed a new measure of uncertainty that is related to the potential unpredictability of the state of the economy rather than its variability.In their empirical analysis, based on a macro VAR Jurado, Ludvigson, and Ng (2015), show that uncertainty is counter-cyclical and far more persistent than common uncertainty proxies.In a related paper Meinen and Roehe (2017), use different measures of uncertainty to investigate their impact on the dynamic of investment for four euro countries, namely France, Germany, Italy, and Spain.While they use a variety of uncertainty measures, the measure proposed by Jurado, Ludvigson, and Ng (2015) "generates remarkably robust investment dynamics across model specifications and countries" [Meinen and Roehe 2017, page 162].
The purpose of the present paper is to investigate spillover effects among macroeconomic uncertainty across four euro countries, notably Germany, France, Italy and Spain, which belong to the same currency area with strong economic links [see, for example Caporale and Girardi 2013;Costantini, Fragetta, and Melina 2014;Costantini and Sousa 2022;Potjagailo 2017].The empirical analysis is carried out over the period 1996:M7-2015:M12 using the macroeconomic uncertainty measure proposed by Jurado, Ludvigson, and Ng (2015),and compiled by Meinen and Roehe (2017) for these economies.
The approach by Bacchiocchi and Fanelli (2015), Bacchiocchi (2017) and Bacchiocchi, Castelnuovo, and Fanelli (2018) is used to account for the effects of the financial crisis, so to deal with volatility regimes and spillover effects. 1 Moreover, the different volatility regimes found in the data allow us to take into account reverse causality (or feedback) between core and periphery countries during the financial crisis period by identifying the structural shocks through a full on-impact matrix in the second regime.This implies that in the first regime spillover effects propagate from core to periphery countries, while in the second regime (financial crisis) spillover effects move also from periphery to core countries.
The paper also looks at the impact that uncertainty may have on real economic activity, namely industrial production [see also Jurado, Ludvigson, and Ng 2015;Ludvigson, Ma, and Ng 2021;Meinen and Roehe 2017;Moore 2017] and unemployment rate (Caggiano, Castelnuovo, and Figueres 2017;Caggiano, Castelnuovo, and Groshenny 2014) taking into account spillover effects.The literature has largely debated on whether uncertainty is a source of business cycles fluctuations or a consequence of them.Several approaches based on a VAR model have been used to investigate the impact of uncertainty on the real economy [for a review, see Ludvigson, Ma, and Ng 2021].This paper examines the impact of (macro and financial) uncertainty on the real activity while dealing with spillover effects and related identification scheme by assuming exogeneity of uncertainty shocks with respect to business cycle Angelini and Fanelli (2019).
Along with the empirical analysis, this paper offers a Monte Carlo study and a set of robustness checks for the empirical application.The Monte Carlo study considers three different data generating processes.For the robustness checks, we first use an alternative measure of uncertainty, namely the stock market volatility [see Bloom 2009;Meinen and Roehe 2017] for the four euro countries within the SVAR framework by Bacchiocchi and Fanelli (2015), Bacchiocchi (2017) and Bacchiocchi, Castelnuovo, and Fanelli (2018).S e cond, with the same SVAR framework, we investigate the role of US uncertainty for the spillover effects using the macroeconomic and financial indicators by Jurado, Ludvigson, and Ng (2015) and Ludvigson, Ma, and Ng (2021), respectively.Third, we evaluate the impact of the US macroeconomic uncertainty on the real economic activity (industrial production) of the four euro countries.Lastly, a different approach for the identification of the SVAR based on the contribution by Lanne and Lütkepohl (2008) is applied to delve into spillover effects across the four euro countries with the macroeconomic uncertainty indicator by Jurado, Ludvigson, and Ng (2015).
The main results of the paper are as follows.First, the Monte Carlo exercise documents the importance of accounting for spillover effects when investigating the impact of uncertainty on economic variables.Second, the empirical results clearly indicate that there are macroeconomic spillovers between the four euro countries, with some feedback from periphery economies, notably Italy, to the core economies during the financial crisis 1 The empirical literature has used different models to deal with spillover effects.For example Rossi and Sekhposyan (2017), study spillover effects for both output growth-and inflation-based uncertainty in the euro Area using a VAR model [see Klößner and Wagner 2013], Caggiano, Castelnuovo, and Figueres (2020) use a nonlinear Smooth Transition VAR (STVAR) model to capture economic policy uncertainty spillovers from the US to Canada, and Cipollini and Mikaliunaite (2020) apply a global VAR, based on the approach by Greenwood-Nimmo, Nguyen, and Shin (2021), to macroeconomic and financial data in the euro Area Greenwood-Nimmo, Nguyen, and Shin (2021) extend the approach by (Diebold and Yilmaz 2012;Diebold and Yilmaz 2014).period.Third, the impact of uncertainty on real economic activity is dampened when the spillover effects are accounted for.Lastly, the additional findings in the robustness checks are qualitatively similar to those displayed in the main empirical analysis.In particular, spillover effects are observed even when US (macro and financial) uncertainty is taken into account.Further, US macroeconomic uncertainty impacts negatively on the real economic activity (industrial production) of the four euro countries.
The rest of the paper is organized as follows.Section 2 describes data and methodology.Section 3 presents the Monte Carlo exercise.Section 4 presents the empirical analysis.Section 5 concludes.

Data and methodology
This paper uses the measure of uncertainty proposed by Jurado, Ludvigson, and Ng (2015),a sc o m p u t e d by Meinen and Roehe (2017) in their analysis, as to investigate uncertainty in France, Germany, Italy, and Spain.It also uses the stock market volatility index by Meinen and Roehe (2017) (see Section 4.3.1)asa proxy for financial uncertainty and we use the US macroeconomic and financial uncertainty measures by Ludvigson, Ma, and Ng (2021) as control variables in the VAR (see Section 4.3.2).Jurado, Ludvigson, and Ng (2015) aim to construct a measure of uncertainty that is "as free as possible both from the structure of specific theoretical models, and from dependencies on any single (or small number) of observable economic indicators" [Jurado, Ludvigson, andNg 2015 page 1178].More formally, denote by  y jt (h)theh-period head uncertainty in the series y jt ∈ Y t = (y 1t , … , y N y t ) ′ .The uncertainty is the conditional volatility of the purely unforecastable component of the future values of the series considered.More specifically, In ( 1), if the expectation today of the squared error in forecasting y jt+h rises, then the uncertainty in the series will also rise.In order to construct a macroeconomic uncertainty measure Jurado, Ludvigson, and Ng (2015), propose to aggregate individual uncertainty at each date by weights  j . (2) In order to yield estimates of (2), Jurado, Ludvigson, and Ng (2015) propose the following three-step procedure.First, factors are extracted from a large set of economic and financial indicators.These indicators accurately represents the information set I t .These factors are then used to approximate E[y jt+h |I t ], the forecastable component, by using a diffusion index.Second, given the h-step-ahead forecast error, V y jt+h , Jurado, Ludvigson, and Ng (2015) proceed to estimate the conditional volatility of this error, E .
To this end, a parametric stochastic volatility model for both the one-step-ahead prediction errors in y jt and the analogous forecast errors for the factors is applied.The volatility estimates are then used to compute the for h > 1, recursively.In the last step, the macroeconomic uncertainty in (2),  y t (h), is constructed from the individual uncertainty measures,  y jt (h). 2  Meinen and Roehe (2017) follow the approach by Jurado, Ludvigson, and Ng (2015) to estimate (2) for four euro countries, France, Germany, Italy, and Spain.They collect a large data set comprising between 137 and 143 macroeconomic and financial time series.The uncertainty series are constructed in such a way that nine broad fields of macroeconomic time series data are taken into account.More specifically, the macroeconomic categories considered are: (i) real output and income; (ii) employment and compensation; (iii) housing; (iv) consumption, orders, and inventory; (v) money and credit; (vi) bond and exchange rates; (vii) price indexes; 2 Equally-weighted average of individual uncertainties is considered.
(viii) stock market indexes; and (ix) international trade.3For the financial data Meinen and Roehe (2017), follow Jurado, Ludvigson, and Ng (2015), and construct a financial data set based on data obtained from Kenneth French's website at Dartmouth College.In particular, they use aggregate series for Europe for each country under investigation,4 that is Fama and French risk factors for Europe, 25 portfolios formed on size and book-to-market (5 × 5) for Europe, and the series termed R15 − R11, which is a spread computed from these portfolios [for details, see Appendix in Meinen and Roehe 2017].The number of financial series used in their analysis amounts to 29.In our empirical analysis, we use one-step-ahead macroeconomic uncertainty,  y t (1).
Meinen and Roehe (2017) also consider the implied or realized volatility of stock market returns (SVOL) for the four euro countries as a measure of uncertainty.In particular Meinen and Roehe (2017),usetheVDAX, which measures the implied volatility of the DAX, for Germany, concatenate the series of the actual volatility of the CAC40 and of the implied volatility measured by the VCAC for France, and compute the index based on the volatility of actual returns for the FTSE MIB and IGBM for Italy and Spain, respectively.
In order to evaluate the spillover effect of uncertainty shocks across the four countries under investigation, and the interactions between the real economic activity and the uncertainty measures, we use a SVAR model.Let Y t be an n × 1 vector of time series of interest.The SVAR model has the following representation where T is the sample length, p is the system lag order, C is a n × 1 constant, A i , i = 1, … , p are n × n matrices of parameters,  t = Be t , e t is the vector of mean zero, unit variance and uncorrelated structural shocks, and  t is a n × 1 vector of reduced form innovations with covariance matrix Σ  = BB ′ .It is straightforward to derive the Vector Moving Average (VMA) representation from (3) where •is a polynomial in the lag operator L of infinite order.In this setup the Impulse Response Functions (IRF) can be easily computed as follows where Ψ h is the matrix associated with the h-lag of Ψ(L), J = (I n , 0 n×n , … , 0 n×n ) is a selection matrix, A is the companion matrix associated with (3),andb j is the jth column of the matrix B. A necessary condition for the calculation of the IRF is that b j (and the matrix B) is identified, that is it can not be expressed as a linear combination of the other columns of B. The identification of the matrix B in (3) can be achieved using a different technique.In this paper we follow the idea proposed by Bacchiocchi and Fanelli (2015),whomake the identification possible using potential heteroscedasticity in the data, as an additional information [see also Angelini et al. 2019].This approach has two merits.First, it allows us to compute regime dependent IRFs.Therefore, it makes possible to evaluate IRFs during the pre-crisis period and the crisis period (see Section 4), which is the focus of this paper.Second, there is no need to impose strong 0 restrictions on the structural matrix.
As for the reduced covariance matrix, the following is required: Σ ,pre ≠ Σ ,crisis [see Assumption 1 in Bacchiocchi and Fanelli 2015].In particular, we specify Σ ,pre = B pre B ′ pre and Σ ,crisis = (B pre + B crisis )(B pre + B crisis ) ′ .In our notation, the subscripts "pre" and "crisis" indicate the matrices related to the pre-crisis period and the period of crisis, respectively.The SVAR model in (3) can then be generalized as follows where W t contains lagged regressors and a constant.In this formulation, both the slope autoregressive matrix A(t) and the covariance matrix Σ  (t) are regime dependent The regime-dependent SVAR described in (6) has different IRFs for each regime.In particular, we have where b pre,j is the jth column of the matrix B pre and b pre,j + b crisis,j is the jth column of the matrix B pre + B crisis .
For details about the estimation of the SVAR described in ( 6)-( 8),s e eBacchiocchi and Fanelli ( 2015) and Angelini et al. (2019).For the analysis in the paper, we consider only exactly identified specifications in line with the su cient conditions proposed by Rubio-Ramirez, Waggoner, and Zha (2010).

Monte Carlo analysis
In this section we conduct a set of Monte Carlo experiments to evaluate the consequences of the omission of spillover effects in the analysis.The Monte Carlo analysis looks at different scenarios, depending on the absence/presence of spillover effects.In particular, we proposed three different scenarios (different B pre and B crisis matrices).For simplicity, an economy with only two countries, say i and j, is considered.Further, the true Data Generating Process (DGP) is based on a trivariate SVAR with Y t = U i,t , U j,t , E j,t ,w h e r eU i,t and U j,t denote the measures of uncertainty for country i and j, respectively, and E j,t is a measure of real economic activity of country j, and a bivariate miss-specified SVAR with Ỹt = U j,t , E j,t is estimated for the evaluation of the omission of the spillover effects.In the Monte Carlo simulations, miss-specification is due to the assumption of uncorrelated uncertainty measures.The analysis is conducted for two different sample sizes, T 1 = T 2 = 100 and T 1 = T 2 = 500, where T 1 and T 2 denote the sample size of the trivariate and bivariate SVAR, respectively (see Tables 1-3), and for 10,000 replications.The identification scheme relies on the SVAR in ( 6)-( 8) and the matrices B pre and B crisis are defined as follows , where × indicates potential non-zero element.
For DGP 1 (see Table 1), we consider an economy where the uncertainty measure of country j has no effects on the real economic activity in country j (the 0 element in the third row, second column of the matrices B pre and B pre + B crisis in Table 1), the uncertainty measure of country i has a negative effect on real economic activity in country j, and the uncertainty measures of the two countries are positively related.
We estimate a true trivariate SVAR model for Y t , and a misspecified bivariate SVAR model with Ỹt (in the true DGP there are no direct effects from U j,t and E j,t ).The estimation results clearly show that the omission of spillover effects in the analysis point to misleading findings.More specifically, looking at the right part of Table 1, it emerges that, when the spillover effects between U i and U j are not accounted for, the impact of U j on E j tend to be different from 0 (even when the sample size T increases).
This result may imply that when spillover effects are omitted, a direct effect of U j on E j is observed.Therefore, potential spillover effects due to the characteristics of the economy should be taken into account.The second Monte Carlo exercise is based on the DGP illustrated in Table 2.It differs from DGP 1, as the uncertainty measure of country j has now a low effect on the real economic activity in country j, −0.05 in pre-crisis period (see second element in the third row of B pre ), and −0.06 during the period of the crisis (see second element in the third row of B pre + B pre ).
The results are similar to those obtained with DGP 1: the omission of spillover effects in the analysis point to misleading results.Indeed, the impact of U j on E j tends to be higher (in absolute value) than true one.In particular, the true on-impact effect of U j on E j is −0.05 and −0.06 for the pre-crisis period and during the period of the crisis, respectively, while the estimated values in the mis-specified model for T 1 = T 2 = 500 are −0.19 and −0.37.
The last simulation exercises (DGP 3) is based on an economy in which the uncertainty measures are not related across the two countries (no spillover effects).The matrices B pre and B crisis are described in Table 3.
Unlike previous DGPs, the impact of U j on E j in the mis-specified model is no longer statistically significant indicating that, when no spillover effects are present, the omission of one of the uncertainty measure in the system does not affect the relationship between the other uncertainty measure and the real economic activity.
The Monte Carlo simulations based on the three different DGP produce a clear set of results.When no spillovers are present or accounted for, the results are statistically insignificant.This implies that, when assessing the impact of macroeconomic uncertainty on real economic activity, the effects of uncertainty spillovers from other integrated economies must be taken into account.
See note in Table 1.

Empirical analysis
This section focuses on the empirical analysis and is divided into three parts.In Section 4.1,w e evaluate the spillover effects amongst one-step-ahead macroeconomic uncertainty (see Section 2)f o rt h e four euro countries using the SVAR framework by Bacchiocchi and Fanelli (2015), Bacchiocchi (2017) and Bacchiocchi, Castelnuovo, and Fanelli (2018).Section 4.2 investigates the impact uncertainty measures on real economic activity, namely industrial production and unemployment rate [see, for example, Caggiano, Castelnuovo, and

Macroeconomic uncertainty spillovers
Figure 1 illustrates the dynamics of the uncertainty measures across the four euro countries.Overall, uncertainty levels in Spain appear to be higher than the uncertainty in the other countries over the period under investigation.Furthermore, there are periods where uncertainties seem to co-move (for example during the financial crisis).Cross correlations among the uncertainty measures are also computed (see Table 4).The results indicate that the highest correlations are documented for the pair of Germany and France, and Italy and Spain, respectively.Further, there is clear-cut evidence of larger cross-correlations over the second sample period.
In order to evaluate the spillover effects between uncertainty measures across the four countries under investigation, we estimate the regime-dependent SVAR model in ( 6) with: where U j,t , j = Ger, Fra, Ita and Spa, denotes the uncertainty measures taken from Meinen and Roehe (2017) for Germany, France, Italy and Spain, respectively.Using the standard information criteria,5 the number of lags p in (3) is set to 2. The two volatility regimes that have been identified provide us with 20 moment conditions (the parameters in the matrices  pre and  crisis ,seeSection 2).6 Since 32 parameters in the matrices B pre and  B crisis should be estimated, at least 12 restrictions should be placed so to achieve identification.To do so, we impose a triangular (Cholesky) structure in the first sub-sample (pre-financial crisis period), which implies a more exogenous role of the core countries (Germany and France), but a complete non-recursive structure during the financial crisis period.On the other hand, during the crisis period, the impact matrix B pre + B crisis is full, and there are no restrictions on the causality direction between countries.Therefore, feedbacks can since the p-values are close to zero.Therefore, two different covariance matrices are set for the pre-crisis and the period of crisis, respectively.The results of the likelihood ratio test are not reported here and are available upon request from the authors.
be observed from core economies (Germany and France) to the periphery countries (Italy and Spain), and viceversa.The matrices B pre and B crisis are as follows The on-impact matrices B pre and B crisis capture the instantaneous (structural) relationships between uncertainties.As pointed out earlier (see Section 2), the matrix B pre contains the pre-crisis on-impact effects, while the matrix B pre + B crisis in ( 14) reports the instantaneous effects during and after the financial crisis.The estimates of the structural parameters are reported in Table 5.Interestingly, the structural parameters b crisis,12 , b crisis,13 and b crisis,14 are not statistically different from 0, indicating that Germany plays an exogenous role even during the financial crisis, that is uncertainty in Germany does not respond instantaneously to shocks to uncertainty in other countries.As such, Germany plays a dominant role among the four euro countries, in the sense that any external shock to the Eurozone, such as that resulting from the US sub-prime mortgage markets, affects the German economy directly and not via the other Eurozone countries, while the other euro countries may also be affected by the German economy.Moreover, looking at the elements on the main diagonal of the two matrices, the magnitude of the coe cients is higher during the financial crisis for all the countries except for Germany, denoting that, differently from the other countries, the variance of the shock related to the German uncertainty does not augment during the financial crisis.It would appear that Germany was not unduly affected by the financial crisis, that is effects on German uncertainty appear to be the same in the crisis period as they are in the pre-crisis period.
The IRFs are reported in Figure 2, where 90% confidence bands are computed following the bootstrap approach proposed in Kilian (1998).As previously highlighted, the analysis distinguishes between pre-financial crisis period (1996:M7-2007:M12) and crisis period (2008:M1-2015:M12) [see also    11) is estimated.90% confidence bands are computed following the bootstrap approach proposed by Kilian (1998).Angelini et al. 2019].The euro countries where affected by the credit crunch originating from the US and the resulting financial crisis.In addition, euro countries are blighted by the deficit crisis, as well as Italian banking crisis.Both have had a prolonged effect on these economies.
There are four noteworthy results.First, during the pre-financial crisis period (first regime), shocks to individual country's uncertainty seem to affect their own uncertainty ("self-effects") (see Figure 2).In particular, a rapid increase in uncertainty is observed across the four countries, with Germany showing the fastest decline thereafter.France, Italy and Spain display more persistence.
Second, also in the pre-crisis period, there is a clear evidence that shocks to core countries uncertainty propagate to periphery countries.7Notably, a shock on uncertainty in France propagates to Italy (the effect on Spain is substantially non-existent) and remains significant for more than a year.Similarly, spillover effects from Germany to Italy and Spain are observed, albeit for a shorter period.
Third, in the crisis period (second regime), the self-effects are still present, with a larger magnitude than that in the pre-crisis period.This finding is unsurprising in view of the financial crisis: Figure 1 points to a rise in uncertainty, especially over the period 2008-2012, in contrast to the level of uncertainty right before the period of the financial crisis.Fourth, the spillover effects from the core economies to the periphery countries seem to mimic those in the pre-financial crisis period, though Germany plays now a more relevant role in the transmissions of uncertainty over the two periphery countries.Interestingly, uncertainty in periphery economies now seems to exert some impact on the core countries.In particular, there is clear evidence of spillover effects from Italy to the core countries.The spillover effects last for a few horizons with a similar magnitude.On the other hand, in case of Spain, these effects are significant for a very short horizon. 8In the Italian case, the prolonged and individual-specific nature of the aftermath of the financial crisis on the Italian economy clearly had an effect on the core economies.Indeed, the ensuing banking crisis meant that the idiosyncratic shock has prolonged the recovery of the Italian economy and its effects were felt by other Eurozone economies.

Macroeconomic uncertainty shocks and their effects on the real economic activity
This section considers the effect of uncertainty shocks on the real economic activity, namely industrial production (IP) and unemployment rate (UR).Data for these variables are taken taken from Federal Reserve Economic Data (FRED).We consider the impact of uncertainty shocks on these macroeconomic variables one after another.We investigate the potential transmission channels of macro uncertainty on real economic activity.Specifically, we empirically assess the transmission in two scenarios: with and without spillover effects.First, we consider all the four uncertainties (with spillover effects) when studying the impact of these on the real economic activity of the country in question (see ( 15)), and then the macroeconomic uncertainty in this country only (without spillover effects) is considered for the impact on real economic activity (see ( 18)).
In the case with spillover effects, the SVAR model in ( 6) is expanded by adding one measure of economic activity where the variable E t is the measure of economic activity.For the analysis, the SVAR model in ( 15) is estimated 8 times, i.e. replacing E t with ΔIP j,t and ΔUR j,t , respectively, where j = Ger, Fra, Ita and Spa.To this end, let the structural matrices B pre and B crisis be as follows: This specification is consistent with the analysis reported in Section 4.1, where the reverse causality between uncertainties is only allowed for the financial crisis period.Also, in line with Bloom (2009), Ludvigson, Ma, and Ng (2021) and Angelini et al. (2019), the real economic activity has no instantaneous impact on uncertainty measures (see zeros in the last column of B pre and B crisis matrices).This specification is also consistent with the results in Caggiano, Castelnuovo, and Groshenny (2014), in which the effects of uncertainty measures on the real economic activity during normal times are negligible.Indeed, with this specification we can consider a full on-impact matrix (expect for the last column) during the financial crisis, the period in which the effects of uncertainty variables on the economy are stronger.Uncertainty shocks have a pronounced and significant effect on both real economic variables considered in the second period, while there are no effects in the pre-crisis period.The significant effects also take the forms of a feedback from periphery to core economies.
In the case without spillover effects, the SVAR model in ( 15) is now estimated considering one uncertainty measure and one measure of economic activity at a time The bivariate SVAR model in ( 18) is estimated 32 times, i.e. by replacing E t with ΔIP j,t and ΔUR j,t , respectively, where j = Ger, Fra, Ita and Spa and U j,t , with one uncertainty measures each time. 9The structural matrices B pre and B crisis are: The structure of B pre and B crisis implies that the real economic activity has no instantaneous impact on uncertainty measures.
Figures 3 and 4 illustrate the IRFs for the SVAR model in ( 15) with ΔIP and ΔUR, respectively during the financial crisis (the second regime). 10The red lines indicate the IRFs when spillover effects are taken into account while the dashed purple lines represent the IRFs when the spillover effects between uncertainties are omitted.In general, it emerges that uncertainty shocks have a negative (positive) effects on industrial production (unemployment) during the financial crisis.More specifically, macroeconomic uncertainty shocks in France and Italy exert a significant effect on the industrial production of the other economies, indicating that the macroeconomic uncertainties in France and in Italy during the financial crisis propagate across Europe.
When there are no spillover effects (dashed purple line in Figures 3 and 4), the results show that the impact of macroeconomic uncertainty real economic activity is higher in magnitude and more persistent.It is important to notice that these results are in line with those of the empirical literature for the euro area (Meinen and Roehe 2017; Popescu and Smets 2010). 11The results also show that there is a feedback from periphery to core countries during the crisis period.This may be due to the fact that both Italy and Spain have experienced prolonged downturns during the crisis period, as result of individual-specific aggregate shocks.In particular, in the case of Italy, the banking crisis was severe and prolonged, and this may have had adverse effects on trade and investment flows from the core economies to the periphery, thereby affecting the formers' recovery.However, this was not the case for unemployment.The likelihood is that while the prolonged crisis in the periphery countries has affected the industrial production in the core countries, this has not necessarily translated to the laying off workers by firms.There may be number of reasons for this.For instances, the firms may anticipate a recovery in the foreseeable future and laying off workers in the short-term could be unproductive and costly.
In brief, these findings seem to suggest that uncertainty plays a relevant role in affecting the real economic activity and that the omission of spillover effects in the specification leads to an overestimation, both in magnitude and in persistence, of these effects.

Robustness checks
In this section we report a set of robustness checks that complement the results in Sections 4.1 and 4.2.First, we repeat the same analysis reported in Sections 4.1 and 4.2 by considering a measure of financial uncertainty.Second, we investigate the spillover effects across countries while controlling for the US macroeconomic and financial uncertainty.Third, we examine the impact of the macroeconomic uncertainty of the four euro countries and of the US on the real economic activity, namely industrial production.Lastly, we use a different approach for the identification of the SVAR based on the contribution by Lanne and Lütkepohl (2008). 12

Financial uncertainty
This section deals with spillover effects across the four euro countries when we consider a measure of financial uncertainty.In particular, as a proxy of financial uncertainty, we consider the stock market volatility (SVOL) Bloom (2009), and we use the implied volatility of DAX, VCA, FTSE MIB and IGBM for Germany, France, Italy and Spain, respectively (see Section 2). 13Figure 5 reports the IRFs computed by estimating the SVAR model in ( 12)-( 14) with where SVOL i,t , i = Ger, Fra, Ita, Spa is the measure of stock market volatility for the ith country.The results confirm those reported in Section 4.1.Indeed, spillover effects are stronger in magnitude and persistence during the second regime.Moreover, during the financial crisis significant causal effects from periphery to core countries are observed.
In order to evaluate the effects of SVOL on real economic activity (we conduct the same analysis as in Section 4.2), Figure 6 reports the IRF related to the second volatility regime.The results are similar to those in Figure 3. Indeed, if we do not take into account the presence of spillover effects between SVOL measures, their impact on the real economic activity are overestimated.

The role of US uncertainty as a control variable for spillover effects
This section focuses on spillover effects across the four euro countries using the same structural model in Section 4.1, with the matrices B pre and B crisis described in ( 12)-( 14) and Y t in (11), and the additional macro and financial uncertainty measures as exogenous variables in the reduced form (6). In particular, for the reducedformwehave where U USM,t is the US macro uncertainty proposed by Jurado, Ludvigson, and Ng (2015) and US USF,t is the US financial uncertainty described in Ludvigson, Ma, and Ng (2021).The results illustrated in Figure 7 qualitatively replicate those in Figure 2, with a slightly less persistence of the spillover effects when controlling for the US uncertainty.14

The impact of US macroeconomic uncertainty on euro business cycle
This section focuses on the impact of US macro uncertainty on real economic activity (industrial production) of the four euro countries.In this case, the SVAR model in ( 15) is expanded by including U USM,t , that is the US macro uncertainty proposed by Jurado, Ludvigson, and Ng (2015) The structural matrices B pre and B crisis become Figure 8 illustrates the impulse responses.The results show that the effect of US macro uncertainty on the economic activity of the euro countries is statistically significant during the financial crisis (red lines).The largest negative impact on industrial seems to be exerted on Italy, Spain, and Germany.This effect is less pronounced in France.Before the financial crisis, this impact is almost nil.

Lanne and Lütkepohl's (2008) approach
This section provides a robustness check concerning the identification scheme of the SVAR.We use the approach by Lanne and Lütkepohl (2008), generalized by Lanne, Lütkepohl, and Maciejowska (2010).This approach assumes that the elements of the matrix B do not change across volatility regimes, but changes occur in the variances of structural shocks.Therefore, we have where Λ is a diagonal matrix containing the variances of structural shocks in the second regime.In the first regime, the variances are normalized to be the identity matrix I n .Note that in this case the matrix of on-impact casual effects could be completely full In order to be consistent with the analysis in Section 4.1, the autoregressive coe cients in the matrix A(t) are assumed to be regime dependent as in (6).2008)'s approach, slightly modified in our analysis in order to account for regime dependent autoregressive parameters, and the macroeconomic uncertainties measures used in Section 4.1.The IRFs replicates those in the main analysis, as spillover effects propagate from core to periphery countries.

Conclusions
Following the sub-prime financial crisis, there has been a heightened interest on the impact of aggregate uncertainty on economic fluctuations.The recent literature has proposed several measures of uncertainty so to analyze the impact of those on economic activity [see, for example, Angelini et al. 2019;Bloom 2009;Caldara et al. 2016;Jurado, Ludvigson, and Ng 2015;Ludvigson, Ma, and Ng 2021].The purpose of this paper is to investigate macroeconomic uncertainty spillovers across the main four euro countries, notably Germany, France, Italy and Spain.It uses the uncertainty measure proposed by Jurado, Ludvigson, and Ng (2015) and complied by Meinen and Roehe (2017) for the four economies.The paper also evaluates the impact of the uncertainty measures on economic activities, namely industrial production and unemployment rate.
The paper contributes to the literature in a number of ways.First, this paper shed valuable insight into the consequences of omitting potential spillover effects when investigating the impact of uncertainty on macroeconomic variables.This is first done by performing a preliminary Monte Carlos study on artificial data and then using data for the four euro countries.
Third, the paper uses a Structural Vector Autoregressive (SVAR) model recently proposed by Bacchiocchi and Fanelli (2015), Bacchiocchi (2017) and Bacchiocchi, Castelnuovo, and Fanelli (2018) that is suitable to deal with regime changes.
Lastly, the paper also performs a robustness check for the empirical analysis.First, it uses alternative measures of uncertainty within the SVAR framework by Bacchiocchi and Fanelli (2015), Bacchiocchi (2017) and Bacchiocchi, Castelnuovo, and Fanelli (2018), namely stock market volatility.Second, it examines the role of US uncertainty for the spillover effects using the macroeconomic and financial indicators by Jurado, Ludvigson, and Ng (2015) and Ludvigson, Ma, and Ng (2021) and the same SVAR framework.Third, it evaluates the impact of the US macroeconomic uncertainty on the real economic activity of the four euro countries.Lastly, it applies a different approach for the identification of the SVAR based on the contribution by Lanne and Lütkepohl (2008) to study spillover effects across the four euro countries while using the macroeconomic uncertainty indicator by Jurado, Ludvigson, and Ng (2015).
The main results of the paper show that spillover effects must be taken into account when investigating the impact of uncertainty on macro economic variables.Also, the empirical results clearly indicate that there are macroeconomic spillovers between the four euro countries and the impact of uncertainty on real economic activity is overestimated when the spillover effects are omitted.Lastly, the additional findings in the robustness checks are qualitatively similar to those displayed in the main empirical analysis.In particular, spillover effects are observed even when US (macro and financial) uncertainty is taken into account.Further, US macroeconomic uncertainty impacts negatively on the real economic activity of the four euro countries.
Pl e a s e n o t e: C h a n g e s m a d e a s a r e s ul t of p u blis hi n g p r o c e s s e s s u c h a s c o py-e di ti n g, fo r m a t ti n g a n d p a g e n u m b e r s m a y n o t b e r efl e c t e d in t his ve r sio n.Fo r t h e d efi nitiv e ve r sio n of t hi s p u blic a tio n, pl e a s e r ef e r t o t h e p u blis h e d s o u r c e.You a r e a d vis e d t o c o n s ul t t h e p u blis h e r's v e r sio n if yo u wi s h t o cit e t hi s p a p er. Thi s v e r sio n is b ei n g m a d e a v ail a bl e in a c c o r d a n c e wit h p u blis h e r p olici e s.
T 1 and T refer to the sample size of trivariate and bivariate SVARs, respectively; ( ) Standard errors are in brackets.

Figure 2 :
Figure 2: Macroeconomic uncertainty spillovers across Germany, France, Italy and Spain.U Ger , U Fra , U Ita ,andU Spa denote the uncertainty measures for Germany, France, Italy and Spain.Blue and red lines indicate the impulse response functions for the period 1996:M7-2007:M12 and 2008:M1-2015:M12, respectively.The VAR in (11) is estimated.90% confidence bands are computed following the bootstrap approach proposed byKilian (1998).

Figure 3 :
Figure 3: Macroeconomic uncertainty impacts on industrial production.ΔIP Ger , ΔIP Fra , ΔIP Ita ,andΔIP Spa denote the first difference of industrial production series for Germany, France, Italy, and Spain.Red lines indicate the impulse response functions for the second volatility regime (2008:M1-2015:M12) with 90% confidence bands computed following the bootstrap approach proposed by Kilian (1998) in case of spillover effects between uncertainty measures.Dashed purple lines indicate the impulse response functions for the second volatility regime without spillover.

Figure 4 :
Figure 4: Macroeconomic uncertainty impacts on unemployment rate.ΔUR Ger , ΔUR Fra , ΔUR Ita ,andΔUR Spa denote the first difference of unemployment rate series for Germany, France, Italy, and Spain.Red lines indicate the impulse response functions for the second volatility (2008:M1-2015:M12) with 90% confidence bands computed following the bootstrap approach proposed by Kilian (1998) in case of spillover effects between uncertainty measures.Dashed purple lines indicate the impulse response functions for the second volatility regime without spillover effects.

Figure 6 :
Figure 6: SVOL impacts on industrial production.ΔIP Ger , ΔIP Fra , ΔIP Ita ,andΔIP Spa denote the first difference of industrial production series for Germany, France, Italy, and Spain.Red lines indicate the impulse response functions for the second volatility regime (2008:M1-2015:M12) with 90% confidence bands computed following the bootstrap approach proposed by Kilian (1998) in case of spillover effects between uncertainty measures.Dashed purple lines indicate the impulse response functions for the second volatility regime without spillover effects.

Figure 7 :
Figure 7: Uncertainty spillovers across Germany, France, Italy and Spain when controlling for US macro and financial uncertainty.U Ger , U Fra , U Ita ,andU Spa denote the uncertainty measures for Germany, France, Italy and Spain.Blue and red lines indicate the impulse response functions for the period 1996:M7-2007:M12 and 2008:M1-2015:M12, respectively.The VAR in (20) is estimated.90% confidence bands are computed following the bootstrap approach proposed byKilian (1998).

Figure 8 :
Figure 8: U USM,t impacts on industrial production.ΔIP Ger , ΔIP Fra , ΔIP Ita ,andΔIP Spa denote the first difference of industrial production series for Germany, France, Italy, and Spain.Blue lines indicate the impulse response functions for the first regime (1996:M7-2007:M12) and red lines indicate the impulse response functions for the second regime (2008:M1-2015:M12).90% confidence bands are computed following the bootstrap approach proposed by Kilian (1998).

Figure 9 :
Figure 9: Uncertainty spillovers across Germany, France, Italy and Spain.U Ger , U Fra , U Ita ,andU Spa denote the uncertainty measures for Germany, France, Italy and Spain.Blue and red lines indicate the impulse response functions for the period 1996:M7-2007:M12 and 2008:M1-2015:M12, respectively.The VAR in (11) is estimated.90% confidence bands are computed following the bootstrap approach proposed byKilian (1998).The IRFs are estimated using theLanne and Lütkepohl's (2008) identification approach.

Figure 9
Figure9reports the IRFs computed using theLanne and Lütkepohl (2008)'s approach, slightly modified in our analysis in order to account for regime dependent autoregressive parameters, and the macroeconomic uncertainties measures used in Section 4.1.The IRFs replicates those in the main analysis, as spillover effects propagate from core to periphery countries.

Table 4 :
Cross-correlations of macroeconomic uncertainty measures across Germany, France, Italy and Spain.

Table 5 :
B pre and B crisis estimates for uncertainty measures.