B Candelon
email: b.candelon@maastrichtuniversity.nl

S Tokpavi
email: sessi.tokpavi@u-paris10.fr

A Nonparametric Test for Granger-causality in Distribution with Application to Financial Contagion $

Keywords: Granger-causality, Distribution, Tails, Kernel-based test, Financial Spill-over

) and hence shares its main advantage, by checking a large number of lags with higher order lags discounted. Besides, our test is highly flexible as it can be used to check for Granger-causality in specific regions on the distribution supports, like the center or the tails. We prove that it converges asymptotically to a standard Gaussian distribution under the null hypothesis and thus it is free of parameter estimation uncertainty. Monte Carlo simulations illustrate the excellent small sample size and power properties of the test. This new test is applied for a set of European stock markets in order to analyse the spill-overs during the recent European crisis and to distinguish contagion from interdependence effects.

Introduction

Analysis of causal relationships holds an important part of the theoretical and empirical contributions in quantitative economics (See the special issues of the Journal of Econometrics in 1988 and2006). Although the concept of causality as defined by [START_REF] Granger | Investigating Causal Relations by econometric Models and Cross-Spectral Methods[END_REF] is broad and consists in testing transmission effects between the whole distribution of random variables, recent literature has proposed some weak versions of this concept, as the causality in the frequency domain or for specific distribution moments. For instance, Granger-causality in mean [START_REF] Granger | Testing for Causality: A Personal Viewpoint[END_REF][START_REF] Granger | Some Recent Developments in a Concept of Causality[END_REF] is widely used in macroeconomics. 1 Granger et al. (1986) also introduce the concept of Granger-causality in variance to test for causal effects in the second order moment between financial series.2 A unified treatment of Granger-causality in the mean and the variance is formalized by [START_REF] Comte | Second order noncausality in multivariate GARCH processes[END_REF].

More recently, some contributions have focused on the concept of Grangercausality in quantiles, an issue which is particularly important for non-Gaussian distributions that exhibit asymmetry, fat-tail characteristics and non-linearity [START_REF] Lee | Money Income Granger-Causality in Quantiles[END_REF][START_REF] Jeong | A Consistent Nonparametric Test for Causality in Quantile[END_REF]. Indeed, given these distributions, the dynamic in the tails can be rather different from the one in the center of the distribution. In this case, the information content of quantiles gives more insights on the distribution than the mean. [START_REF] Lee | Money Income Granger-Causality in Quantiles[END_REF] developed a parametric methodology for Granger-causality in quantiles which is based on the conditional predictive ability (CPA) framework of [START_REF] Giacomini | Tests of Conditional Predictive Ability[END_REF]. [START_REF] Jeong | A Consistent Nonparametric Test for Causality in Quantile[END_REF] introduce a non-parametric approach to test for causality in quantiles and apply it to the detection of causal relations between the crude oil price, the USD/GBP exchange rate, and the gold price. A closely related but different concept is the Granger-causality in tail-event by [START_REF] Hong | Granger Causality in Risk and Detection of Extreme Risk Spill-over between Financial Markets[END_REF], a tail-event being identified as a situation where the value of a time series is lower than its Value-at-Risk at a specified risk level. Hence the test checks whether an extreme downside movement for a given time series has a predictive content for an extreme downside movement for another time series, and has many potential applications in risk management.

All the tests of causality in quantiles and tail-events share the same limit that statistical inference is exclusively performed at a particular fixed level of the quantile. At this given level, the null hypothesis should not be rejected, while the opposite conclusion should hold for another quantile level. Indeed as emphasized by [START_REF] Granger | Time Series Concepts for Conditional Distributions[END_REF] and [START_REF] Engle | CAViaR: Conditional autoregressive Value at Risk by regression quantile[END_REF], time series behavior of quantiles can vary considerably across the distribution because of long memory or non-stationarity. Hence, a Granger-causality test in quantiles or tail-events which does not consider a large number of quantiles simultaneously over the distribution support would be restrictive. Given that the predictive distribution of a time series is entirely determined by its quantiles, testing for Granger-causality for the range of quantiles over the distribution support is equivalent to testing for Granger-causality in distribution.

Very few papers developed testing procedures for Granger-causality in the whole distribution in a time series context. The only exceptions to our knowledge include [START_REF] Su | A Consistent Characteristic Function-based test for Conditional Independence[END_REF],2008,2012,2013), [START_REF] Bouezmarni | Nonparametric Copula-Based Test for Conditional Independence with Applications to Granger Causality[END_REF] and [START_REF] Taamouti | Nonparametric estimation and inference for Granger causality measures[END_REF]. For example, [START_REF] Su | Conditional Independence Specification Testing for Dependent Processes with Local Polynomial Quantile Regression[END_REF] introduce a conditional independence specification test which can be used to test for Grangercausality in quantiles for a continuum values of quantile levels between (0, 1). [START_REF] Bouezmarni | Nonparametric Copula-Based Test for Conditional Independence with Applications to Granger Causality[END_REF] construct a nonparametric Granger-causality test in distribution based on conditional independence in the framework of copulas. 3 Our paper adds to this literature proposing a new methodology to test for Granger-causality in the whole distribution between two time series. Our testing procedure consists in dividing the distribution support of each series into a multivariate process of dynamic inter-quantile event variables, and by checking whether there is a spill-over effect between the two multivariate pro-cesses, analyzing their cross-correlations structure. The test draws from the generalized portmanteau test for independence between multivariate processes in [START_REF] Bouhaddioui | A generalized portmanteau test for independence of two infinite order vector autoregressive series[END_REF].

It is worth mentioning that although our approach checks for the strong version of the Granger-causality concept [START_REF] Granger | Investigating Causal Relations by econometric Models and Cross-Spectral Methods[END_REF], it is highly flexible as it can be used to test for causality in specific regions on the distribution supports, like the center or the tails (left or right). 4 For example, the test can be used to test for causality in the left-tail distribution for two time series. In this case the multivariate process of inter-quantile event variables should be defined so as to focus the analysis exclusively on this part of the distribution. This flexibility is one of the great advantage of our methodology compared to those based on copulas theory [START_REF] Bouezmarni | Nonparametric Copula-Based Test for Conditional Independence with Applications to Granger Causality[END_REF][START_REF] Taamouti | Nonparametric estimation and inference for Granger causality measures[END_REF]. It allows us to go beyond the simple rejection of the null hypothesis of Granger-causality for the whole distribution, as it provides us with the specific regions for which Grangercausality is rejected. Besides, our test statistic is a multivariate extension of the kernel-based nonparametric Granger-causality test in tail-event by [START_REF] Hong | Granger Causality in Risk and Detection of Extreme Risk Spill-over between Financial Markets[END_REF], and hence shares its main advantage: it checks for a large number of lags by discounting higher order lags. This characteristic is consistent with the stylized fact in empirical finance that recent events have much more influence in the current market trends than those older. In this line, our Granger-causality test in distribution is different from those available in the literature which check for causality uniformly for a limited number of lags.

Technically, we show that the test has a standard Gaussian distribution under the null hypothesis which is free of parameter estimation uncertainty.

Monte Carlo simulations reveal indeed that the Gaussian distribution provides a good approximation of the distribution of our test statistic, even in small samples. Moreover, the test has power to reject the null hypothesis of causality in distribution stemming from different sources including linear and non-linear causality in mean and causality in variance.

To illustrate the importance of this test for the empirical literature, we use it to better understand the spill-overs that have taken place within European stock markets during the recent crisis. Our Granger-causality test in distribution allows to consider asymmetry between markets (which is not possible using correlation), to take into account for break in volatility (as suggested by [START_REF] Forbes | No contagion, only interdependence: measuring stock market comovements[END_REF] and to distinguish between contagion and interdependence. Indeed, interdependence is a long run path and taking place in "normal periods" concerning hence the center of the distribution. On the contrary, contagion is detected by a short-run abrupt increase in the causal linkages taking place exclusively during crisis' period, i.e., in the tails of the distribution. As our test is designed to check for causality in specific regions of the distribution, it can be used to check for interdependence or contagion. Anticipating on our results, we find weak (resp. strong) support for interdependence (resp. contagion) during the recent crisis. Interestingly, we observe a strong asymmetry between causal tests in the right and left tails: Whereas spill-overs are important in crisis periods, they are only weakly present in upswing times. Such a result constitutes an important feature for the European stock markets.

The paper is sketched as follows: the second Section presents the Grangercausality test in distribution. The properties of this test are analysed in Section 3 via a Monte-Carlo simulation experiment. Section 4 proposes the empirical application whereas Section 5 concludes.

Nonparametric test for Granger-causality in distribution

This Section presents our kernel-based test for Granger-causality in distribution between two time series. As this test is a multivariate extension of the Granger-causality test in tail-event introduced by [START_REF] Hong | Granger Causality in Risk and Detection of Extreme Risk Spill-over between Financial Markets[END_REF], we begin with the presentation of [START_REF] Hong | Granger Causality in Risk and Detection of Extreme Risk Spill-over between Financial Markets[END_REF] test and then introduce the new approach.

Granger-causality in tail-event

For two time series X t and Y t , the Granger-causality test in tail-event developed by [START_REF] Hong | Granger Causality in Risk and Detection of Extreme Risk Spill-over between Financial Markets[END_REF] checks whether an extreme downside risk from Y t can be considered as a lagged indicator for an extreme downside risk for X t . [START_REF] Hong | Granger Causality in Risk and Detection of Extreme Risk Spill-over between Financial Markets[END_REF] identify an extreme downside risk as a situation where X t and Y t are lower than their respective Value-at-Risk (VaR) at a prespecified level α. Recall that VaR is a risk measure often used by financial analysts and risk managers to measure and monitor the risk of loss for a trading or investment portfolio. The VaR of an instrument or portfolio of instruments is the maximum dollar loss within the α%-confidence interval [START_REF] Jorion | Value at Risk: The New Benchmark for Managing Financial Risk[END_REF]. For the two time series X t and Y t , we have

Pr X t < V aR X t θ 0 X F X t-1 = α, (1)
Pr Y t < V aR Y t θ 0 Y F Y t-1 = α, (2) with V aR X t θ 0 X and V aR Y t θ 0 Y
the VaR of X t and Y t respectively at time t, θ 0 X and θ 0 Y the true unknown finite-dimensional parameters related to the specification of the VaR model for each variable. The information sets F X t-1 and F Y t-1 are defined as

F X t-1 = {X l , l ≤ t -1} , (3)
F Y t-1 = {Y l , l ≤ t -1} . (4)
In the framework of [START_REF] Hong | Granger Causality in Risk and Detection of Extreme Risk Spill-over between Financial Markets[END_REF], an extreme downside risk occurs at time t for X t , if the tail-event variable Z X t θ 0 X is equal to one, with

Z X t θ 0 X =    1 if X t < V aR X t θ 0 X 0 else. (5)
Similarly, an extreme downside risk for

Y t corresponds to Z Y t θ 0 Y taking value one, with Z Y t θ 0 Y =    1 if Y t < V aR Y t θ 0 Y 0 else. (6)
Hence, the time series Y t does not Granger-cause (in downside risk or tailevent at level α) the time series X t , if the following hypothesis holds

H 0 : E Z X t θ 0 X F X&Y t-1 = E Z X t θ 0 X F X t-1 , (7)
with

F X&Y t-1 = {(X l , Y l ) , l ≤ t -1} . ( 8
)
Under the null hypothesis and at the risk level α, it means that spill-overs of extreme downside movements from Y t to X t do not exist. [START_REF] Hong | Granger Causality in Risk and Detection of Extreme Risk Spill-over between Financial Markets[END_REF] propose a nonparametric approach for testing for the null hypothesis in (7) based on the cross-spectrum of the estimated bivariate process of tail-event variables

Z X t , Z Y t , with components Z X t ≡ Z X t θ X , Z Y t ≡ Z Y t θ Y , (9)
where θ X and θ Y are consistent estimators of the true unknown parameters θ 0 X and θ 0 Y , respectively. To present their test statistic, let us define the sample cross-covariance function between the estimated tail-event variables as

C (j) =            T -1 T t=1+j Z X t -α X Z Y t-j -α Y , 0 ≤ j ≤ T -1 T -1 T t=1-j Z X t+j -α X Z Y t -α Y , 1 -T ≤ j ≤ 0, (10)
with T the sample length, α X and α Y the sample mean of Z X t and Z Y t , respectively. The sample cross-correlation function ρ (j) is then equivalent to

ρ (j) = C (j) S X S Y , (11)
where S 2 X and S 2 Y are the sample variances of Z X t and Z Y t , respectively. Using the cross-correlation function, the kernel estimator for the cross-spectral density of the bivariate process of tail-event variables corresponds to

f (ω) = 1 2π T -1 j=1-T κ (j /M ) ρ (j) e -ijω , (12)
with κ (.) a given kernel function and M the truncation parameter. The truncation parameter M is function of the sample size T such that M → ∞ and M/T → 0 as T → ∞. The kernel is a symmetric function defined on the real line and taking value in [-1, 1], such that

κ (0) = 1, (13) ∞ -∞ κ 2 (z) dz < ∞. ( 14
)
Under the null hypothesis of non Granger-causality in tail-event from Y t to X t , the kernel estimator for the cross-spectral density is equal to

f 0 1 (ω) = 1 2π 0 j=1-T κ (j /M ) ρ (j) e -ijω . (15)
This suggests using the distance between the two estimators f (ω) and f 0 1 (ω) to test for the null hypothesis. [START_REF] Hong | Granger Causality in Risk and Detection of Extreme Risk Spill-over between Financial Markets[END_REF] consider the following quadratic form

L 2 f , f 0 1 = 2π π -π f (ω) -f 0 1 (ω) 2 dω, (16)
which is equivalent to

L 2 f , f 0 1 = T -1 j=1 κ 2 (j /M ) ρ 2 (j) . (17)
The test statistic is a standardized version of the quadratic form given by

U Y →X =   T T -1 j=1 κ 2 (j /M ) ρ 2 (j) -C T (M )   D T (M ) 1 2 , ( 18
)
and follows under the null hypothesis a standard Gaussian distribution, with C T (M ) and D T (M ) the location and scale parameters

C T (M ) = T -1 j=1 (1 -j /T ) κ 2 (j /M ) , (19)
D T (M ) = 2 T -1 j=1 (1 -j /T ) (1 -(j + 1) /T ) κ 4 (j /M ) . (20)

Granger-causality in distribution

In this section, we present our multivariate extension of the test of [START_REF] Hong | Granger Causality in Risk and Detection of Extreme Risk Spill-over between Financial Markets[END_REF] which helps checking for Granger-causality in the whole distribution between two time series.

Notations and the null hypothesis

The setting of our testing procedure is as follows. We consider a set A = {α 1 , ..., α m+1 } of m + 1 VaR risk levels which covers the distribution support of both variables X t and Y t , with 0% <= α 1 < ... < α m+1 <= 100%. For the first time series X t , the corresponding VaRs at time t are V aR X t,s θ 0 X , α s , s = 1, ..., m + 1, with

V aR X t,1 θ 0 X , α 1 < .... < V aR X t,m+1 θ 0 X , α m+1 , (21)
where the vector θ 0 X is once again the true unknown finite-dimensional parameter related to the specification of the VaR model for X t . We adopt the convention that V aR X t,s θ 0 X , α s = -∞ for α s = 0% and V aR X t,s θ 0 X , α s = ∞ for α s = 100%.

We divide the distribution support of X t into m disjoint regions, each related to the indicator or event variable

Z X t,s θ 0 X =    1 if X t ≥ V aR X t,s θ 0 X , α s and X t < V aR X t,s+1 θ 0 X , α s+1 0 else, (22)
for s = 1, ..., m. For illustration, let m + 1 = 5 and suppose that the set A = {α 1 , α 2 , α 3 , α 4 , α 5 } = {0%, 20%, 40%, 60%, 80%}. Now, let H X t θ 0 X be the vector of dimension (m, 1) with components the m event variables

H X t θ 0 X = Z X t,1 θ 0 X , Z X t,2 θ 0 X , ..., Z X t,m θ 0 X . ( 23
)
We similarly define for the second time series Y t these event variables collected in the vector

H Y t θ 0 Y , with H Y t θ 0 Y = Z Y t,1 θ 0 Y , Z Y t,2 θ 0 Y , ..., Z Y t,m θ 0 Y . ( 24
)
The time series Y t does not Granger-cause the time series X t in distribution if the following hypothesis holds

H 0 : E H X t θ 0 X F X&Y t-1 = E H X t θ 0 X F X t-1 . (25)
Therefore, Granger-causality in distribution from Y t to X t corresponds to Granger-causality in mean from

H Y t θ 0 Y to H X t θ 0 X .
When the null hypothesis of non causality in distribution holds, this means that the event variables defined for the variable Y t along its distribution support, do not have any predictive content for the dynamics of the same event variables over the distribution support of X t .

Remark that our null hypothesis is flexible enough as it can be used to check for Granger-causality in specific regions on the distribution supports, like the center or the tails (left or right). This can be done by restricting the set A = {α 1 , ..., α m+1 } of VaR levels to some selected values. For instance, we can check for Granger-causality in the left-tail distribution by setting A to A = {0%, 1%, 5%, 10%}. In this case, the rejection of the null hypothesis is of great importance in financial risk management, as it suggests the existence of spill-over effects from Y t to X t that take place in the lower tail. Similarly Granger-causality in the center of the distribution can be checked by setting for example A to A = {20%, 40%, 60%, 80%}. In the next subsection, we construct a nonparametric kernel-based test statistic to test for our general null hypothesis in (25), and analyze its asymptotic distribution.

Test statistic and asymptotic distribution

Bouhaddioui and Roy (2006) introduce a generalized portmanteau test for the independence between two infinite order vector auto-regressive (VAR) series.

Our test statistic relies for (25) on their work. However, the asymptotic analysis differs because (i) we are not in a VAR framework, (ii) and the event variables Z X t,s θ 0 X and Z Y t,s θ 0 Y are indicator variables which are not differentiable with respect to the unknown parameters θ 0

X and θ 0 Y , respectively. The latter challenge is solved relying on some asymptotic results in [START_REF] Hong | Granger Causality in Risk and Detection of Extreme Risk Spill-over between Financial Markets[END_REF].

To present the test statistic, let H

X t ≡ H X t θ X and H Y t ≡ H Y t θ Y
be the estimated counterparts of the multivariate processes of event variables

H X t θ 0 X and H Y t θ 0 Y , respectively, with θ X and θ Y √ T consistent estimators
of the true unknown parameter vectors θ 0 X and θ 0 Y . Denote Λ (j) the sample cross-covariance matrix between H X t and H Y t , with

Λ (j) ≡            T -1 T t=1+j H X t -A X H Y t-j -A Y T 0 ≤ j ≤ T -1 T -1 T t=1-j H X t+j -A X H Y t -A Y T 1 -T ≤ j ≤ 0, (26)
where the vector A

X (resp. A Y ) of length m is the sample mean of H X t (resp. H Y t ). The corresponding sample cross-correlation matrix R (j) equals R (j) = D Σ X -1/2 Λ (j) D Σ Y -1/2 , ( 27
)
where D (.) stands for the diagonal form of a matrix, Σ X and Σ Y the sample covariance matrices of H X t and H Y t , respectively. We consider the following weighted quadratic form that accounts for the dependence between the current value of H X t and lagged values of

H Y t T = T -1 j=1 κ 2 (j /M ) Q (j) , (28)
where κ (.) is a kernel function, M the truncation parameter and Q (j) equal to

Q (j) = T vec R (j) T Γ -1 X ⊗ Γ -1 Y vec R (j) , (29)
with Γ X and Γ Y the sample correlation matrix of H X t and H Y t , respectively. Following [START_REF] Bouhaddioui | A generalized portmanteau test for independence of two infinite order vector autoregressive series[END_REF], our test statistic is a centered and scaled version of the quadratic form in (28), i.e.,

V Y →X = T -m 2 C T (M ) (m 2 D T (M )) 1/2 , (30)
with C T (M ) and D T (M ) as defined in ( 19) and ( 20) respectively. The above test statistic generalizes in a multivariate setting the one in [START_REF] Hong | Granger Causality in Risk and Detection of Extreme Risk Spill-over between Financial Markets[END_REF].

Indeed when m is equal to one, which corresponds to the univariate case where each of the vectors H X t and H Y t has only one event variable, the test statistic V Y →X in ( 30) is exactly equal to the [START_REF] Hong | Granger Causality in Risk and Detection of Extreme Risk Spill-over between Financial Markets[END_REF] test statistic in (18).

The following proposition gives the asymptotic distribution of our test statistic.

Proposition 1. Suppose that Assumptions of Theorem 1 in [START_REF] Hong | Granger Causality in Risk and Detection of Extreme Risk Spill-over between Financial Markets[END_REF] hold. Then under the null hypothesis of no Granger-causality in distribution as stated in (25), we have

V Y →X = T -m 2 C T (M ) (m 2 D T (M )) 1/2 -→ d N (0, 1) .
Assumptions of Theorem 1 in [START_REF] Hong | Granger Causality in Risk and Detection of Extreme Risk Spill-over between Financial Markets[END_REF] impose some regulatory conditions on the time series X t and Y t , on the VaR models used including smoothness, moment conditions and adequacy, on the kernel function κ (.), and also on the truncation parameter M . The latter should be equal to

M = cT v with 0 < c < ∞, 0 < v < 1/2, v < min 2 d-2 , 3 d-1 if d ≡ max (d X , d Y ) > 2 and d X (resp. d Y )
is the dimension of the parameter θ X (resp. θ Y ). See Hong et al. (2009, pp. 275) for a complete discussion on these assumptions.

The proof of Proposition 1 proceeds as follows. Consider the following decomposition of our test statistic

V Y →X = T * -m 2 C T (M ) (m 2 D T (M )) 1/2 + T -T * (m 2 D T (M )) 1/2 , (31)
with T * the pseudo version of the weighted quadratic form in (28-29) computed using the true correlation matrices Γ X and Γ Y , i.e.,

T * = T -1 j=1 κ 2 (j /M ) Q * (j) , (32)
Q * (j) = T vec R (j) T Γ -1 X ⊗ Γ -1 Y vec R (j) . ( 33
)
Under the decomposition in (31), the proof of Proposition 1 is given by the following two lemmas:

Lemma 2. Under Assumptions of Theorem 1 in [START_REF] Hong | Granger Causality in Risk and Detection of Extreme Risk Spill-over between Financial Markets[END_REF], we have

T * -m 2 C T (M ) (m 2 D T (M )) 1/2 -→ d N (0, 1) . ( 34
)
Lemma 3. Under Assumptions of Theorem 1 in [START_REF] Hong | Granger Causality in Risk and Detection of Extreme Risk Spill-over between Financial Markets[END_REF], we have

T -T * (m 2 D T (M )) 1/2 -→ p 0. ( 35
)
The proofs of these two Lemmas are reported in Appendix A.

Small sample properties

In this section, we study the finite sample properties of our test via Monte

Carlo simulation experiments. We analyze the size in the first part of the section and the remaining one is devoted to the analysis of the power.

Empirical size analysis

We simulate the size of the nonparametric test of Granger-causality in distribution assuming the following data generating process (DGP) for the first time series

X t :          X t = σ t v t , σ 2 t = 0.1 + 0.5σ 2 t-1 + 0.2X 2 t-1 , v t ∼ m.d.s. (0, 1) ,
which corresponds to a GARCH(1,1) model. We make the assumption that the second time series Y t follows the same process. Because the two processes are generated independently, there is no Granger causality in distribution between them. For a given value of the sample size T ∈ {500, 1.000, 2.000}, and for each simulation, we compute our test statistic in (30) and make inference using the asymptotic Gaussian distribution. For the computation of the test statistic, we need to specify a model to estimate the VaRs (at the risk level α 1 , ..., α m+1 ) and the m event variables for each variable X t and Y t . The m + 1 VaRs are computed using a GARCH(1,1) model estimated by quasi-maximum likelihood.

The estimated values of the m + 1 VaRs at time t are

V aR X t,s = σ t,X q ( v t , α s ) , s = 1, ..., m + 1, (36)
where σ t,X is the fitted volatility at time t, and q ( v t , α s ) the empirical quantile of order α s of the estimated standardized innovations. We proceed similarly to compute the m + 1 VaRs and the corresponding m event variables for the second time series Y t . Note that we set the parameter m + 1 to 14 and the set

A to A = {α 1 , α 2 , .
.., α 14 } = {0%, 1%, 5%, 10%, 20%, ..., 90%, 95%, 99%}, which covers regions in the tails and the center of the distribution support of each time series. 6 We also need to make a choice about the kernel function in order to compute our test statistic. We consider the four different usual kernels, i. Indeed, the rejection frequencies are close to the nominal risk levels. Hence, the standard Gaussian distribution provides asymptotically a good approximation of the distribution of our test statistic. This result seems to hold regardless of the kernel function used and the value of the truncation parameter M .

Empirical power analysis

We now simulate the empirical power of our test. Since causality in distribution springs from causality in moments such as mean or variance, we assume different DGPs which correspond to these cases. The first DGP assumes the existence of a linear Granger-causality in mean in order to generate data under the alternative hypothesis:

         Y t = 0.5Y t-1 + u t,Y , u t,Y = σ t,Y v t,Y , σ 2 t,Y = 0.1 + 0.5σ 2 t-1,Y + 0.2u 2 t-1,Y , (37)
         X t = 0.5X t-1 + 0.3Y t-1 + u t,X , u t,X = σ t,X v t,X , σ 2 t,X = 0.1 + 0.5σ 2 t-1,X + 0.2u 2 t-1,X , (38)
where both v t,Y and v t,X are martingale difference sequences with mean 0 and variance 1. The empirical powers of our test are computed over 500 simulations for T ∈ {500, 1.000, 2.000}. As in the analysis of the size, we consider three values of the truncation parameter M, and two nominal risk levels η = 5%, 10%.

The results are reported in Table B.2, only for the Daniell kernel to save space.7

For comparison we also display in Table B.2 results for the Granger-causality test in mean. In order to have a fair comparison, we do not use the usual parametric Granger-causality test in mean derived from a vector autoregressive model. We consider instead the kernel-based non-parametric Granger-causality test in mean introduced by [START_REF] Hong | Testing for Independence Between Two Covariance Stationary Time Series[END_REF]. Results in Table B.2 show that our kernel-based nonparametric test for Granger-causality in distribution has appealing power properties. For instance, the rejection frequencies of the null hypothesis for (T, M ) = (500, 6) are equal to 93.6% and 95.6% for η = 5% and 10%, respectively. For T = 1.000, 2.000 the powers are equal to one. The rejection frequencies of the Granger-causality test in mean are always equal to 100% and hence are slightly higher than the ones of our Granger-causality test in distribution for the smallest sample. This result is expected as the assumed causality in distribution springs from causality in mean.

To stress the relevance of our testing approach, we consider a second representation of the DGPs under the alternative hypothesis, assuming causality in distribution stemming from a non-linear form of causality in mean. Precisely, we generate data for the time series Y t using the specification in (37), and the second time series is generated as follows is equal to 75.2% for η = 5%, while it is only equal to 18.2% for the causality test in mean in the same configuration. Remark that for our test, the power drops as the truncation parameter M increases. Moreover, the power increases as the sample size increases and converges to 100%.

         X t = 0.5X t-1 + 0.3Y 2 t-1 + u t,X , u t,X = σ t,X v t,X , σ 2 t,X = 0.1 + 0.5σ 2 t-1,X + 0.2u 2 t-1,X . (39)
Lastly, we generate data under the alternative hypothesis, assuming Grangercausality in variance. Formally, we suppose once again that Y t has the specification in (37), and X t is generated as

         X t = 0.5X t-1 + u t,X , u t,X = σ t,X v t,X , σ 2 t,X = 0.1 + 0.5σ 2 t-1,X + 0.2u 2 t-1,X + 0.7Y 2 t-1 . (40)
Results displayed in capital markets to finance their domestic investments and boost their growth, they have been highly subject to financial instability.

It is of major importance for empirical studies to evaluate the importance of these spill-overs. Theoretically it relies on the crisis-contingent theories, which explain the increase in market cross-correlation after a shock issued in an origin country as resulting from multiple equilibria based on investor psychology; endogenous liquidity shocks causing a portfolio recomposition; and/or political disturbances affecting the exchange rate regime.8 9 The presence of spill-overs during a crisis can be thus tested empirically by a significant and transitory increase in cross-correlation between markets. (See inter alia [START_REF] King | Transmission of volatility between stock markets[END_REF], Calvo and Reinhart, 1995and Baig and Goldfajn, 1998). Nevertheless, this intuitive approach, which presents the advantage of simplicity as it avoids the identification of the transmission channels, presents many shortcomings:

First, [START_REF] Forbes | No contagion, only interdependence: measuring stock market comovements[END_REF] show that an increase in correlation can be exclusively driven by an higher volatility during crisis periods. In such a case, it could not be attributed to a stronger economic interdependence. To correct for this potential bias, they thus propose to use a modified version of the correlation 10 and test for its temporary increase during crisis period.

Second, correlation is a symmetrical measure: an increase in the correlation between markets i and j does not provide any information on the direction of the contagion (from i to j, from j to i, or both). For such a reason, [START_REF] Bodart | Evidence of interdependence and contagion using a frequency domain framework[END_REF] prefer to consider an indicator of causality to measure spillovers. It is thus possible to evaluate asymmetrical spill-overs, which can then move from i to j, j to i or in both directions. Besides, using Granger-causality approach requires the estimation of multivariate dynamic models which are less prone to potential misspecification issues.

It is, more or less, feasible to tackle both these shortcomings in a classical framework. Nevertheless, even if comparing causality between pre-and crisis periods allows to evaluate spill-overs, it does not permit to separate interdependence and contagion. Interdependence deals with the long run structural links between markets. It thus provides information on the extend to which markets are integrated. Therefore, interdependence should be analysed without considering extreme positive or negative events. On the contrary, contagion deals with short-run abrupt increases in the causal linkages and takes place exclusively during crisis' period. Thus, testing for contagion requires to exclusively focus on the extremal left tail of the distribution, as it is performed in extreme value theory (see Hartman et al., 2004). Our Granger-causality test in distribution allows to tackle all these issues. Indeed, it offers an asymmetric measure of spill-overs, based on a dynamic representation. Besides, it is possible to investigate if causality has increased for the whole distribution but also for specific percentiles of the distribution, in particular those located at the left tail or right tails, corresponding to extreme events.

As an illustration, we analyse the recent European crisis and consider a set of 12 European daily stock market indices (Austria, Belgium, Finland, France, Germany, Greece, Ireland, Italy, Luxemburg, the Netherlands, Portugal and Spain) downloaded from datastream ranging from January 1, 2007 to May 6, 10 In fact, they are using the unconditional correlation instead of the conditional one.

2011 (i.e. T = 1.134 observations). The first empirical illustration consists in testing for interdependence. It is performed implementing the pairwise Grangercausality test for the whole distribution but removing crisis's periods, i.e. the right and left tails. Then, in a second analysis, we repeat this analysis for the left tail in order to test for contagion during crisis. This part refers to the EVT approach of spill-overs and extend [START_REF] Hartmann | Asset market linkages in crisis periods[END_REF]. Similarly, the test is conducted for the right tail, i.e. upswing period. We can then compare the strength of contagion during crises vs boom periods and check in which periods contagion is the most significant.

The general design of the Granger-causality test in distribution to test for spill-over

To implement the Granger-causality test in distribution in our empirical illustration, we first need to compute for each index, m + 1 series of VaRs corresponding to m+1 risk levels α s s = 1, ..., m+1, which cover its distribution support. As for the Monte Carlo simulations, we consider the following set for the VaR levels A = {0%, 1%, 5%, 10%, ..., 90%, 95%, 99%} with m + 1 = 14. To compute the VaRs, we use a semi-parametric model. Formally, we suppose that each index returns series R i,t i = 1, ..., 12, follows an AR (m)-GARCH (p, q) model, with:

R i,t = m j=1 φ i,j R i,t-j + ε i,t , (41)
ε i,t = σ i,t v i,t , (42)
σ 2 i,t = κ i + q j=1 γ i,j ε 2 i,t-j + p j=1 β i,j σ 2 i,t-j , (43)
and v i,t an i.i.d. innovation with mean zero and unit variance. The choice for an AR (m)-GARCH (p, q) is in line with the [START_REF] Forbes | No contagion, only interdependence: measuring stock market comovements[END_REF] correction.

It accounts for volatility increase that biases the causality analysis. For each index, this model is estimated by quasi-maximum likelihood method. Hence, the m + 1 series of VaRs are obtained as:

V aR i t,s = m j=1 φ i,j R i,t-j + σ t,i q ( v i,t , α s ) , s = 1, ..., m + 1, ( 44
)
with σ t,i the fitted volatility at time t for the index number i, and q ( v i,t , α s ) the empirical quantile of order α s of the estimated standardized innovations v i,t .

Table B.5 displays the estimation results of the AR(m)-GARCH(p, q) models for the indices. As shown through the Ljung-Box test applied to the residuals and their squares, the retained specifications succesfully capture the dependence in the first two moments.

With the fitted series of VaRs at hand, we calculate for each index, the multivariate process of dynamic inter-quantiles event variables, and compute for each couple (i, j) of indices our kernel-based non parametric test statistic

V j→i as defined in (30). For the computation we use the Daniell kernel and set the truncation parameter M to 1.5T 0.3 which leads to the value of M = 12

for the whole sample of length T = 1.134.

Testing for interdependence

To test for interdependence, we follow the general design of the pairwise test of Granger-causality in distribution as described above except that we remove from the distribution the extreme events. The new set A of VaRs risk levels is equal to A = {20%, 30%, ..., 70%, 80%} with m + 1 = 7. European stock markets integration is strongly vulnerable to negative, and to a lesser extend positive, shocks. European policy makers should acknowledge it and set up structural measures to limit it.

Conclusion

A kernel-based non-parametric test for Granger-causality in distribution between two time series is proposed in this paper. The test checks for spill-overs between the multivariate processes of dynamics inter-quantile event variables associated to each variable. Beyond the existing approaches our testing methodology has two main advantages. First, it can be used to test for Granger-causality in specific regions of the distributions, like the center or the tails (left and right).

Second, it checks for a large number of lags discounting higher order lags, and hence is consistent against causality which carries over long distributional lags.

We show that the test has a standard Gaussian distribution under the null hypothesis which is free of parameter estimation uncertainty. We run a Monte Carlo simulations exercise which shows that the Gaussian distribution is valid in small samples. The test also has very appealing power properties in various settings including linear and non-linear causality in mean and causality in variance.

In an empirical part we implement our testing procedure to 12 European daily stock market indices to analyze spill-overs during the recent European crisis. As our test is designed to check for causality in specific regions of the distribution (center or tails), it can be used to test for both the presence of interdependence and contagion. Indeed interdependence can be checked through Granger-causality in the center of the distribution, as interdependence is a long run path that takes place in normal periods. On the contrary, contagion refers to a short-run abrupt increase in the causal linkages taking place exclusively during crisis period, and can be tested via Granger-causality in distribution's tails.

The empirical results indicate that European stock market integration is far from being achieved, because we observe very weak evidence supporting interdependence. On contrary, our results support the presence of contagion, with a strong asymmetry between contagion in the right and left tails. More precisely, contagion is important in crisis periods, whereas it is weak in upswing times.

Such a result constitutes an important feature for the European stock markets, and policy makers should acknowledge it in designing structural measures for financial stability.

the correlations between Z X t,k and Z Y t-j,p with k = 1, ...m and p = 1, ..., m. Let us denote ρ k,p (j) such correlation, i.e.,

ρ k,p (j) = corr Z X t,k , Z Y t-j,p . (A.6)
With the definition of the vec operator, it is easy to see that vec R (j) has m 2 components given by ρ k,p (j), k = 1, ...m and p = 1, ..., m. Consequently, the pseudo quadratic form T * in (A.5) becomes

T * = T T -1 j=1 κ 2 (j /M ) m k=1 m p=1 ρ 2 k,p (j) , = m k=1 m p=1   T T -1 j=1 κ 2 (j /M ) ρ 2 k,p (j)   . (A.7)
For a given value of the couple (k, p), the quadratic form in the bracket of (A.7) corresponds to the uncentered and unscaled test statistic in [START_REF] Hong | Granger Causality in Risk and Detection of Extreme Risk Spill-over between Financial Markets[END_REF] for the Granger-causality from Z Y t,p to Z X t,k , and these statistics are obviously independent. We deduce from this that under Assumptions of Theorem

1 in Hong et al. (2009) T T -1 j=1 κ 2 (j /M ) ρ 2 k,p (j) -→ d N (C T (M ) , D T (M )) . (A.8)
Remark that the event variables in [START_REF] Hong | Granger Causality in Risk and Detection of Extreme Risk Spill-over between Financial Markets[END_REF] are related to onesided regions, whereas we have to deal with two-sided regions in our framework.

However in their proof of Theorem 1, it is easy to see that all of the technical results remain valid even in the case of two-sided regions. For this, it suffices to replace Z 1t (θ 1 ) and Z 2t (θ 2 ) by Z X t,k (θ X ) and Z Y t,p (θ Y ), with θ X and θ Y any vector in the parameter spaces Θ X and Θ Y , respectively. Moreover, the martingale difference sequence W 1t (θ 1 ) and W 2t (θ 2 ) in the proof of Theorem 1 in [START_REF] Hong | Granger Causality in Risk and Detection of Extreme Risk Spill-over between Financial Markets[END_REF] must be replaced by W X t,k (θ X ) and W Y t,p (θ Y ) respectively in order to deal with two-sided regions, with

W X t,k (θ X ) = Z X t,k (θ X ) -Z X t,k θ 0 X -E Z X t,k (θ X ) F X t-1 + E Z X t,k θ 0 X F X t-1 = Z X t,k (θ X ) -Z X t,k θ 0 X -F X V aR X t,k+1 (θ X ) -F X V aR X t,k (θ X ) + + F X V aR X t,k+1 θ 0 X -F X V aR X t,k θ 0 X , W Y t,p (θ Y ) = Z Y t,p (θ Y ) -Z Y t,p θ 0 Y -E Z Y t,p (θ Y ) F Y t-1 + E Z Y t,p θ 0 Y F Y t-1 = Z Y t,p (θ Y ) -Z Y t,p θ 0 Y -F Y V aR Y t,p+1 (θ Y ) -F Y V aR Y t,p (θ Y ) + + F Y V aR Y t,p+1 θ 0 Y -F Y V aR Y t,p θ 0 Y ,
with F X (.) and F Y (.) the conditional cumulative distribution functions of X and Y respectively. Hence, we can conclude using (A.8) that the pseudo quadratic form T * in (A.7) has the limiting distribution .9) and this completes the proof of LEMMA 2. .14) with Λ (j) the sample cross-covariance matrix at lag-order j, Σ X and Σ Y the covariance matrices of the true multivariate processes of event variables H

T * -→ d N m 2 C T (M ) , m 2 D T (M ) . (A
T = T T -1 j=1 κ 2 j M vec Λ (j) T Σ -1 X ⊗ Σ -1 Y vec Λ (j) , (A.13) T * = T T -1 j=1 κ 2 j M vec Λ (j) T Σ -1 X ⊗ Σ -1 Y vec Λ (j) , ( A
T -T * = T T -1 j=1 κ 2 j M vec Λ (j) T Σ -1 X ⊗ Σ -1 Y -Σ -1 X ⊗ Σ -1 Y vec Λ (j) . (A.15)
Now, let us study the asymptotic behavior of Σ X . The components of this matrix are given by the covariance between the estimated event variables Z X t,k , k = 1, ..., m. Let C k,p be a typical element of Σ X with

C k,p = cov Z X t,k , Z X t,p . (A.16)
Let C 0 k,p be the true value of C k,p , i.e., the covariance between the true event variables Z X t,k θ 0 X and Z X t,p θ 0 X . Note that C 0 k,p is a typical element of Σ X . The difference between C k,p and C 0 k,p can be decomposed as follows

C k,p -C 0 k,p = M 1 θ X + M 2 θ X + M 3 θ X , (A.17) with M 1 θ X = T -1 T t=1 Z X t,k -Z X t,k θ 0 X Z X t,p θ 0 X -π X p (A.18) M 2 θ X = T -1 T t=1 Z X t,k θ 0 X -π X k Z X t,p -Z X t,p θ 0 X (A.19) M 3 θ X = T -1 T t=1 Z X t,k -Z X t,k θ 0 X Z X t,p -Z X t,p θ 0 X , (A.20)
where we replace the sample means π X k and π X p of Z t,k and Z t,p by their true respective values π

X k = E Z X t,k θ 0 X and π X p = E Z X t,p θ 0 X .
Using the following result in the proof of theorem A.3 in [START_REF] Hong | Granger Causality in Risk and Detection of Extreme Risk Spill-over between Financial Markets[END_REF] sup

θ X ∈Θ X M 1 (θ X ) = O p T -1/2 , (A.21)
with θ X any √ T -consistent estimator of θ 0 X in the space Θ X , we have for the first term A.33) where Λ (j) is the cross-covariance matrix at lag-order j of the true event variables H X t θ 0 X and H Y t θ 0 Y . Let us first consider B 1 (T ) in (A.32). Since Λ (j) is defined as the estimated cross-covariance matrix at lag-order j between

M 1 θ X = O p T -1/2 . (A.22) B 2 (T ) = T -1 j=1 κ 2 j M vec (Λ (j)) T vec (Λ (j)) , (
H X t = Z X t,1 , Z X t,2 , ..., Z X t,m , (A.34)
and A.35) its components are given by the covariances between Z X t,k and Z Y t-j,p , k = 1, ..., m, p = 1, ..., m. Denote C k,p (j) such covariances. Using the definition of the vec operator we thus have A.36) which can be rewritten as

H Y t = Z Y t,1 , Z Y t,2 , ..., Z Y t,m , (
B 1 (T ) = T -1 j=1 κ 2 j M m k=1 m p=1 C 2 k,p (j) -C 2 k,p (j) , (
B 1 (T ) = T -1 j=1 κ 2 j M m k=1 m p=1 C k,p (j) -C k,p (j) 2 + C k,p (j) -C k,p (j) C k,p (j) = m k=1 m p=1 Q 1 + Q 2 , (A.37)
with

Q 1 = T -1 j=1 κ 2 j M C k,p (j) -C k,p (j) 2 , (A.38) Q 2 = T -1 j=1 κ 2 j M C k,p (j) -C k,p (j) C k,p (j) . (A.39)
Using the results of Theorem A.1 in [START_REF] Hong | Granger Causality in Risk and Detection of Extreme Risk Spill-over between Financial Markets[END_REF], that is

Q 1 = O p M 1/2 /T , (A.40) Q 2 = O p M 1/2 /T , (A.41)
we have Notes: Each entry of the table gives the p-value (in %) of the test of causality in the center of the distribution from the index j in column towards the index i in row. Entries with the rejection of the null hypothesis at the 5% nominal risk level are emphasized in bold. The last column labelled "Sum" provides for a given index in row, the number of time it is Granger-caused by the others. Similarly, the last row labelled "Sum" provides for a given index in column, the number of time it Granger-causes others indices. The entry corresponding to the last row and the last column gives the total number of significant Granger-causality in the system. The tests are performed over the period ranging from January 1, 2007 to May 6, 2011, with a total of T = 1134 observations. Notes: Each entry of the table gives the p-value (in %) of the test of causality in left-tail distribution from the index j in column towards the index i in row. Entries with the rejection of the null hypothesis at the 5% nominal risk level are emphasized in bold. The last column labelled "Sum" provides for a given index in row, the number of time it is Granger-caused by the others. Similarly, the last row labelled "Sum" provides for a given index in column, the number of time it Granger-causes others indices. The entry corresponding to the last row and the last column gives the total number of significant Granger-causality in the system. The tests are performed over the period ranging from January 1, 2007 to May 6, 2011, with a total of T = 1134 observations. Notes: Each entry of the table gives the p-value (in %) of the test of causality in right-tail distribution from the index j in column towards the index i in row. Entries with the rejection of the null hypothesis at the 5% nominal risk level are emphasized in bold. The last column labelled "Sum" provides for a given index in row, the number of time it is Granger-caused by the others. Similarly, the last row labelled "Sum" provides for a given index in column, the number of time it Granger-causes others indices. The entry corresponding to the last row and the last column gives the total number of significant Granger-causality in the system. The tests are performed over the period ranging from January 1, 2007 to May 6, 2011, with a total of T = 1134 observations.

B 1 (T ) = O p M 1/2 /T . (A.42)

  Figure B.1 displays the support of X t , along with the VaRs and the event variables defining the m = 4 distinct regions. 5

  e. the Daniell (DAN), the Parzen (PAR), the Bartlett (BAR) and the Truncated uniform (TR) one. Lastly for the choice of the truncation parameter M , we use three different values: M = [ln (T )], M = 1.5T 0.3 and M = 2T 0.3 , with [.] the integer part of the argument. These rates lead to the values M = 6, 10, 13 for T = 500, M = 7, 12, 16 for T = 1.000, and M = 8, 15, 20 for T = 2.000. These values cover a range of lag orders for the sample sizes considered.

  proof of Lemma 3 proceeds by combining elements in the proofs of Proposition 3.2 in[START_REF] Bouhaddioui | A generalized portmanteau test for independence of two infinite order vector autoregressive series[END_REF] and Theorems A.1 and A.3 in[START_REF] Hong | Granger Causality in Risk and Detection of Extreme Risk Spill-over between Financial Markets[END_REF]. Formally, given thatD T (M ) = M ∞ 0 κ 4 (z) dz [1 + o (1)] , (A.11)as M → ∞, the proof of Lemma 3 can be established showing thatT -T * = O p M/T 1/2 . (A.12)Based on Lemma 4.1 in El Himdi and Roy (1997), the quadratic forms T and T * can be rewritten in term of cross-covariances as

Table B .

 B 3 reports the rejection frequencies over 500 simulations. The presentation is similar to TableB.2. We observe that while the Granger-cauality test in mean fails to reject the null hypothesis for most of the simulations, our test still exhibits good power in detecting this non-linear form of causality. For illustration the rejection frequency of the null hypothesis for (T, M ) =(500, 6)

  Table B.4 are qualitatively similar to the ones in TableB.3. Our causality test in distribution has good powers in rejecting the null hypothesis, while the causality test in mean exhibits low powers. Overall the reported values are lowers to the ones in Tables B.2 and B.3. This pattern can be explained by the fact that (i) causality in variance takes place mainly in the tails, (ii) and the dynamics of the tails are more difficult to fit due to the lack Recent financial crises have all been characterized by quick and large regional spill-overs of negative financial shocks. For example, consecutively to the Greek distress (2009), South European countries have been contaminated, facing skyrocketing refinancing rates. Besides it has impacted North European states in an opposite way. Considered as safe harbors for investors, they were able to refinance their debt on markets at lower rates. It is obvious that the degree of globalization within European Union as well as the low degree of fiscal federalism has fostered the speed as well as the amplitude of the transmission mechanism of such a shock. And as Southern European countries used foreign

	4. Empirical part
	of data.

Table B

 B Austrian and the French stock markets are the most integrated ones, as they are affected by 3 other European markets, respectively.On the contrary, Greece, Ireland, Italy, Luxemburg and Netherlands appear as independent from the other markets. It is interesting to notice that the causal matrix is not symmetric: France which is among the most caused markets does not affect any market. It hence supports our choice for causality as a measure of spill-over rather than correlation. The most causal markets are Netherlands, Greece and Portugal. The presence of these two last countries is interesting as they were among the main drivers of the European crisis. Their causal importance, which can be qualified as systemic for the rest of Europe, should have constituted a signal of alarm at the edge of the crisis. 95%, 99%, 100%}, i.e., m + 1 = 4 and results are reported in TableB.8. It appears that "positive" contagion is only supported in 7.5% of the cases and concerns mainly Spain as spill-overs driver, and Luxembourg, Germany and Belgium as spill-overs receivers. The asymmetry between causal tests in the right and left tail is striking. Whereas spill-overs are important in crisis periods, they are only weakly present in upswing times. Such a feature highlights that

	The last column labelled "Sum" provides for a given index in row, the number of
	time it is Granger-caused by the others. Similarly, the last row labelled "Sum"
	provides for a given stock market index the number of time it Granger-causes
	others stock market indices. Lastly, the entry corresponding to the last row
	and last column gives the total number of significant Granger-causality cases
	for our set of indices. It turns out that interdependence is supported in only
	9.8% of the cases (13 cases out of 132). This result indicates that European
	stock market integration is far from being achieved. Looking at the countries
	results, we observe that the

.6 displays the results of the test. The reported values are the p-values in percentage. Hence, for a nominal risk level of 5%, we reject the null hypothesis of no-causality from index j to index i when the reported value is lower than 5%. Test statistics corresponding to the rejection of the null hypothesis of no causality are put in bold.

4.3. Testing for contagion

Contagion is apprehended implementing our Granger-causality test in lefttail distribution. The set A of VaRs risk levels is now set as A = {0%, 1%, 5%, 10%} with m + 1 = 4. Table

B

.7 displays the outcomes of the tests. The number of causal pairs increases to 35.6% of the cases supporting hence the presence of contagion. We observe that the most causal markets are Portugal, Italy, Netherlands, Greece and Ireland, and except Netherlands, this group includes all the countries in turmoils (Portugal, Italy, Greece and Ireland), around which the crisis was build. On the other side, the most caused markets are Austria, Belgium, Italy, France, Luxembourg and Greece. Remark the predominant role of Italy and Greece in the system, which cause and are caused in many cases.

The Granger-causality test is now repeated for the left-tail distribution with A = {90%,

  Notes: The table displays the empirical sizes (in %) of the Granger-causality test in distribution. Rejection frequencies are reported over 500 simulations for two nominal risk levels η, with T the sample size and M the truncation parameter. DAN, BAR, PAR and TR refer to the Daniell, the Bartlett, the Parzen, and the truncated kernels, respectively.Notes: The table displays the empirical powers (in %) of the Granger-causality test in distribution. Rejection frequencies are reported over 500 simulations for two nominal risk levels η, with T the sample size and M the truncation parameter. For comparison, we also report the rejection frequencies of the kernel-based nonparametric test in mean. For both test, results are reported for the Daniell kernel. Data are generated under the alternative hypothesis assuming Granger-causality in mean.Notes: The table displays the empirical powers (in %) of the Granger-causality test in distribution. Rejection frequencies are reported over 500 simulations for two nominal risk levels η, with T the sample size and M the truncation parameter. For comparison, we also report the rejection frequencies of the kernel-based nonparametric test in mean. For both test, results are reported for the Daniell kernel. Data are generated under the alternative hypothesis assuming nonlinear Granger-causality in mean.Notes: The table displays the empirical powers (in %) of the Granger-causality test in distribution. Rejection frequencies are reported over 500 simulations for two nominal risk levels η, with T the sample size and M the truncation parameter. For comparison, we also report the rejection frequencies of the kernel-based nonparametric test in mean. For both test, results are reported for the Daniell kernel. Data are generated under the alternative hypothesis assuming Granger-causality in variance.

		Table B.2: Empirical powers of the Granger-causality test in distribution: Table B.3: Empirical powers of the Granger-causality test in distribution: Table B.4: Empirical powers of the Granger-causality test in distribution: SUM 3 2 1 3 1 0 0 0 0 0 2
		DGP1 DGP2 DGP3
	T 500 1000 2000	Table B.1: Empirical sizes of the Granger-causality test in distribution M η DAN BAR PAR 6 5% 5.80 5.80 5.60 10% 10.40 10.40 11.40 10 5% 5.20 5.60 5.80 10% 10.00 9.60 10.20 13 5% 5.60 4.80 5.20 10% 10.40 10.60 10.20 7 5% 5.00 4.20 4.40 10% 9.00 9.40 8.40 12 5% 5.00 5.20 5.00 10% 9.40 10.00 9.00 16 5% 5.40 5.60 6.20 10% 10.00 10.20 10.00 8 5% 6.60 7.00 6.60 10% 11.80 11.80 13.40 15 5% 5.60 6.20 6.40 10% 10.20 11.20 11.60 20 5% 5.00 5.00 6.60 10% 10.60 10.80 11.20 T M η Distribution Mean 6 5% 93.60 100.00 10% 95.60 100.00 500 10 5% 91.80 100.00 10% 93.20 100.00 13 5% 90.40 100.00 10% 92.40 100.00 7 5% 100.00 100.00 10% 100.00 100.00 12 5% 100.00 100.00 1000 10% 100.00 100.00 16 5% 100.00 100.00 10% 100.00 100.00 8 5% 100.00 100.00 10% 100.00 100.00 15 5% 100.00 100.00 2000 10% 100.00 100.00 20 5% 100.00 100.00 10% 100.00 100.00 T M η Distribution Mean 6 5% 75.20 18.20 10% 82.80 21.40 500 10 5% 68.20 16.20 10% 76.60 20.60 13 5% 60.20 15.80 10% 70.60 20.60 7 5% 97.80 20.60 10% 98.40 21.60 12 5% 95.80 19.20 1000 10% 97.00 23.20 16 5% 93.00 19.20 10% 95.80 22.60 8 5% 100.00 24.20 10% 100.00 28.60 15 5% 100.00 19.60 2000 10% 100.00 24.40 20 5% 100.00 17.60 10% 100.00 21.80 T M η Distribution Mean 6 5% 51.40 18.80 10% 61.40 23.00 500 10 5% 50.20 20.40 10% 60.40 25.80 13 5% 48.20 21.40 10% 57.80 25.80 7 5% 79.80 17.80 10% 86.80 21.60 12 5% 72.40 22.60 1000 10% 81.60 27.40 16 5% 67.00 21.80 10% 76.20 29.00 8 5% 99.20 15.40 10% 99.40 19.40 15 5% 96.60 18.80 2000 10% 97.60 23.40 20 5% 93.00 19.00 10% 95.80 24.40 BEL FI FR GER GRE 2.3 27.1 14.0 27.0 38.9 100.0 27.6 19.9 64.4 53.9 98.3 100.0 10.4 90.8 22.6 28.0 38.6 100.0 8.2 0.0 26.1 27.4 8.6 100.0 0.2 47.3 28.7 50.7 60.6 100.0 59.6 5.2 11.8 15.7 90.6 42.6 80.9 72.9 9.6 46.2 97.7 65.3 7.2 12.0 91.6 99.3 75.4 15.5 23.7 73.7 86.9 84.4 40.3 0.2 9.2 71.7 96.6 61.5 64.7 34.9 Table B.6: Results of bilateral tests of Granger causality in the center of the distribution IE IT LU NL PT ES 79.1 2.7 48.2 1.0 19.7 29.6 4.2 5.3 18.3 31.5 85.6 6.4 21.0 70.2 60.1 79.8 21.8 3.7 39.4 10.2 2.6 58.3 0.1 44.7 36.9 98.4 5.1 16.3 14.0 98.4 28.3 30.3 55.6 92.1 72.8 70.2 100.0 15.9 25.4 85.6 23.4 42.6 29.7 35.6 21.2 81.2 77.3 100.0 45.3 44.0 46.4 91.9 16.1 73.2 9.7 30.7 100.0 95.0 48.3 11.0 43.0 42.7 100.0 69.2 64.6 53.1 100.0 25.8 1.3 71.9	TR 4.40 9.80 5.60 11.60 5.40 11.80 5.20 10.80 6.00 11.80 6.20 11.40 5.20 8.80 6.20 11.60 4.80 9.60
		AT	100.0	0.5	80.9	42.0	95.9	54.1	42.4	65.0	37.6	94.9	99.1	58.2
			AT	BEL	FI	FR	GER	GRE	IE	IT	LU	NL	PT	ES

Table B .

 B 7: Results of bilateral tests of Granger causality in left-tail distribution

Table B .

 B 8: Results of bilateral tests of Granger causality in right-tail distribution

See inter alii[START_REF] Sims | Money, Income and Causality[END_REF][START_REF] Sims | Comparison of Interwar and Postwar Business Cycles: Monetarism Reconsidered[END_REF] who tests for Granger-causality in mean between money and income.

This concept is further explored by[START_REF] Cheung | A causality in variance test and its application to financial market prices[END_REF],[START_REF] Kanas | Mean and variance causality between official and parallel currency markets: Evidence from four Latin American Countries[END_REF],Hafner and Herwartz (2004), to cite but a few.

See also[START_REF] Taamouti | Nonparametric estimation and inference for Granger causality measures[END_REF] for another approach from the copulas theory.

Note that[START_REF] Candelon | Testing for Granger-Causality in Distribution Tails: An Application to Oil Markets Integration[END_REF] introduce a parametric test to check for Grangercausality in distribution tails, but the methodology does not apply for other regions of the distribution like the center.

Remark that we do not consider the event variable corresponding to the extreme m + 1 region identified by Xt ≥ V aR X t,m+1 θ 0 X , α m+1 . Indeed this variable is implicitly defined by the first m event variables.

Recall that for αs = 0%, the VaR corresponds to -∞.

Results for the other kernels are similar and available from the authors upon request. The only exception occurs for the uniform kernel which has a relatively low power, because of its uniform weighting which does not discount higher order lags.

see[START_REF] Rigobon | Identification Through Heteroskedasticity: Measuring 'Contagion' between Argentinian and Mexican Sovereign Bonds[END_REF] for a survey.

On contrary, according to the non-crisis-contingent theories, the propagation of shocks does not lead to a shift from a good to a bad equilibrium, but the increase in cross-correlation is the continuation of linkages (trade and/or financial) existing before the crisis.

Appendix A.1. Proof of LEMMA 2 Lemma 2: Under Assumptions of Theorem 1 in [START_REF] Hong | Granger Causality in Risk and Detection of Extreme Risk Spill-over between Financial Markets[END_REF], we have

1/2 -→ d N (0, 1) . (A.1)

Proof: Consider the pseudo version of the weighted quadratic form T * defined as

where Γ X (resp. Γ Y ) is the correlation matrix of the true unknown multivariate process of event variables H

where the event variables Z X t,s θ 0 X , s = 1, ..., m, are related to distinct regions on the distribution support of X t . Hence, they are mutually independent, and the associated correlation matrix Γ X is equal to the identity matrix. The same reasoning applies for H Y t θ 0 Y , with the consequence that Γ Y is also equal to the identity matrix. Hence, the pseudo weighted quadratic form T * defined in (A.2-A.3) takes the expression

Since R (j) is defined as the cross-correlation matrix at lag-order j between

, its components are given by Similar arguments apply for the last two terms, with the consequence that

We deduce that

Using the same reasoning for the elements of Σ Y we have that

The rest of the proof proceeds by showing that

We decompose B (T ) into two parts

For the second term B 2 (T ), using the Markov inequality, we have

We deduce from (A.42) and (A.43) that

and

This completes the proof of Lemma 3. (41)(42)(43). We report the parameters estimates followed in brackets by the student statistics. The two last columns give the results of the Ljung-Box test applied to the serie of the standardized innovations v i,t and its square, respectively, with 6 the number of lags. The critical value for the rejection of the null hypothesis at the 5% nominal risk level is equal to 12.59.

Appendix B. Tables and Figures

Economic Series, University Carlos III de Madrid.