Goodness-of-Fit Tests for Copulas of Multivariate Time Series

The asymptotic behaviour of the empirical copula constructed from residuals of stochastic volatility models is studied. It is shown that if the stochastic volatility matrix is diagonal, then the empirical copula process behaves like if the parameters were known, a remarkable property. However, that is not true if the stochastic volatility is genuinely non-diagonal. Applications for goodness-of-fit and structural change of the dependence between innovations are discussed.


Introduction
In many financial applications, e.g., pricing of options on multiple assets or exchange rates, multiname credit derivatives, portfolio management and risk management, it is necessary to model the dependence between different assets. That can be done simply by using copulas, which are distribution functions of multivariate uniform variables.
It has been shown, e.g., Embrechts et al. (2002) and Berrada et al. (2006), that the choice of the copula is of paramount importance since it can lead to significant differences in pricing. The same is true for measures of risk. From an economic point of view, it is also pertinent to try to find the kind of dependence linking several economic series. One could be interested for example in modeling the dependence between several exchange rates with respect to the US currency, or to show that there is a strong dependence between an exchange rate and the value of a commodity. Using the recent economic context in Greece, one could also be interested in modeling the dependence between exchange rates (Euro vs USD) and bond values. Actuaries also have to model dependence between pairs or multivariate vectors of data.
Motivated by actuarial, economic and financial applications, the problem of the choice of the copula to model dependence between data correctly is quite recent and has been tackled mainly for serially independent observations. See, e.g., Genest et al. (2009) for a comparison and review of the consistent goodness-of-fit tests that can be used in that context. However, in economic and financial applications, there is almost always serial dependence and when looking for the choice of a copula family, the serial dependence problem is either ignored, i.e., the data are not "filtered" to remove serial dependence, as in Schmid (2005, 2007) and Kole et al. (2007), or the data are "filtered" but the potential inferential problems of using these transformed data are not taken into account. For example, Panchenko (2005) uses a goodness-of-fit test on "filtered" data (residuals of GARCH models in his case), without proving that his proposed methodology works for residuals. However he mentioned in passing that working with residuals could destroy the asymptotic properties of his test. A similar situation appears in Breymann et al. (2003) where both the problem of working with residuals and the problem of the estimation of the copula parameters are ignored. The same criticisms can be addressed to van den Goorbergh et al. (2005) and Patton (2006). It seems that the first paper addressing rigorously the problems raised by the use of residuals in estimation and goodness-of-fit of copulas is Chen and Fan (2006). An unpublished document (Chen et al., 2005) also circulated some time ago proposing a kernel-based test using residuals of GARCH models. However the proof of their main result is missing and the technical report never got published. Using a multivariate GARCH-like model with diagonal innovation matrix, Chen and Fan (2006) showed the remarkable result that estimating the copula parameters using the rank-based pseudo-likelihood method of Genest et al. (1995) and Shih and Louis (1995) with the ranks of the residuals instead of the (non-observable) ranks of innovations, leads to the same asymptotic distribution. In particular, the limiting distribution of the estimation of the copula parameters does not depend on the unknown parameters used to estimate the conditional means and the conditional variances. That property is crucial if one wants to develop goodness-of-fit tests for the copula family of the innovations. In Chen and Fan (2006), the authors also propose ways of selecting copulas based on pseudo-likelihood ratio tests. However, their comparison test is not a goodness-of-fit test in the sense that one could select a model which is inadequate though better than the other proposed models.
In the present paper, goodness-of-fit tests are proposed, with the related statistics being functions of empirical processes, since tests based on empirical processes are generally consistent and more powerful than other classes of test statistics, including likelihood ratio tests. One extends the results of Chen and Fan (2006) by proving that under similar technical assumptions, the empirical copula process has the same limiting distribution as if one would have started with the innovations instead of the residuals. Other methods of estimation of the parameters of the copula families, not considered in Chen and Fan (2006), also share the same properties. For example, the asymptotic behaviour of Kendall's tau, Spearman's rho, van der Waerden and Blomqvist's coefficients are exactly the same as with serially independent observations. An immediate consequence is that all tools developed recently for the serially independent case remain valid for the residuals. In particular, one can use the 1-level and 2-level parametric bootstrap of Genest and Rémillard (2008) to estimate p-values of tests statistics, if the estimator is regular (Genest and Rémillard, 2008). That is the case for the usual estimators like pseudo likelihood estimators and moment estimators. Such properties are in sharp contrast with the ones using consecutive residuals of a single time series for testing serial independence (Ghoudi and Rémillard, 2010), where the limiting copula process does depend on parameters, even in simple ARMA models. It is also shown that when the volatility matrix is genuinely non-diagonal, then all these nice properties stop holding true. The estimation of the copula parameters and the limiting empirical copula depend on the conditional mean and conditional variance parameters. It is the case for general BEKK models, used in Patton (2006) and Dias and Embrechts (2009).
In what follows, one starts, in Section 2, by describing the model and discussing parameter estimation for copulas. Tests statistics based on the empirical copula process and the Rosenblatt's transform are then proposed in Sections 3 and 4 respectively, together with implementations of the parametric bootstrap. The main result for testing goodness-of-fit using the empirical copula process is given in Proposition 5, while its analog for using Rosenblatt's transform is given in Proposition 6. Change-point problems are discussed in Section 5, either for univariate series or copulas, while an example of application using some data of Chen and Fan (2006) is treated in Section 6. The main results on the convergence of the empirical processes are stated and proved in the Appendix.

Model and estimation
Following Chen and Fan (2006), one considers a stochastic volatility model for a multivariate time series X i , i.e., for i ≥ 1 and j = 1, . . . , d, di }, and where µ i , σ i are F i−i -measurable and independent of ε i . Here F i−1 contains information from the past and possible information from exogenous variables. That model, studied in Chen and Fan (2006), contains as a particular case, BEKK models (Engle and Kroner, 1995) with diagonal conditional volatility matrix. Note that in many applications, univariate stochastic volatility models are fitted separately to each time series (X ji ) n i=1 , j = 1, . . . , d. Given an estimation θ n of θ, compute the residuals e i,n = (e 1i,n , . . . , e di,n ) ⊤ , where Since the distribution function K is continuous, there exists a unique copula C (Sklar, 1959) so that for all x = (x 1 , . . . , x d ) ⊤ ∈ R d , where F 1 , . . . , F d are the marginal distribution functions of K, i.e., F j is the distribution function of ε ji . Setting U i = F(ε i ), one gets that U i has distribution C, denoted by U i ∼ C, i = 1, . . . , n.
Since the copula is independent of the margins, it is generally suggested 1 to remove their effect by replacing e i with the associated rank vector U i,n = (U 1i,n , . . . , U di,n ) ⊤ , U ji,n = Rank(e ji,n )/(n + 1), with Rank(e ji,n ) being the rank of e ji,n amongst e j1,n , . . . , e j1,n , j = 1, . . . , d. That can also be written as The main results on the paper are deduced from the asymptotic behaviour (see Appendix B) of the partial-sum empirical process The reason for introducing partial sums in (2) and (3) will become clear in Section 5 when one studies detection of change-points.
In the next two sections one will study tests of goodness-of-fit for parametric copula families, i.e., one wants to test the null hypothesis for some parametric family of copula C. Typical families are of the meta-elliptic type (Gaussian and Student copula) and Archimedean copulas (Clayton, Frank, Gumbel). As proposed in Dias and Embrechts (2004), Chen and Fan (2006) and Patton (2006), one could also consider mixtures of such copulas. These families are listed in Appendix D, together with their parameters.
It is assumed that K and F 1 , . . . , F d have continuous densities h, f 1 , . . . , f d respectively. As a result, the copula C has a density c which satisfies Under H 0 , each copula C φ is assumed to admit a density c φ satisfying hypotheses B1-B3 described in Appendix A, and depends on the parameter φ which must be estimated.
In what follows one lists some estimators of copula parameters and one also studies some of their asymptotic properties.
2.1.1. Pseudo likelihood estimators. In Chen and Fan (2006), it is shown that under smoothness conditions (conditions D, C, and N in their article), the pseudo maximum likelihood estimator is asymptotically Gaussian with covariance matrix depending only on c φ . Therefore, the asymptotic behaviour does not depend on the estimation of the parameter θ required for the evaluation of the residuals! In fact, it has the same representation as the estimator studied by Genest et al. (1995) in the serially independent case, i.e., if the parameter θ was known. More precisely, one has where W n = 1 √ n n i=1ċ and where J is the Fisher's information matrix (0,1) dċ Hereċ is the row vector given by the gradient of c φ with respect to φ. Note that W n Z n converges in law to W Z ∼ N(0, Σ), with Σ = J 0 0 J . It follows that Φ n converges in law to Φ ∼ N (0, J −1 + J −1 J J −1 ). Note also that E ΦW ⊤ = I, i.e., φ n is a regular estimator for φ in the sense of Genest and Rémillard (2008), where it is shown that regular estimators are essential for the validity of the parametric bootstrap procedure.
2.1.2. Two-stage estimators. In addition to pseudo likelihood estimators, one may consider a two-stage estimator. That is, suppose that φ = φ 1 φ 2 . Decompose also W n and Z n accordingly. Suppose that φ 1,n is an estimator of φ 1 that is regular in the sense that Φ 1,n = √ n(φ 1,n − φ 1 ) converges in law to Φ 1 ∼ N(0, Σ 1 ) and . Now define φ 2,n as the pseudo-likelihood estimator of the reduced log-likelihood viz It is then easy to check that W 2,n − Z 2,n = J 21 Φ 1,n + J 22 Φ 2,n , . As a result, E(Φ 2 W ⊤ 1 ) = 0 and E(Φ 2 W ⊤ 2 ) = I. This proves that φ n is a regular estimator of φ since (Φ, W) is a centered Gaussian vector with E(ΦW ⊤ ) = I. Two-stage estimation is often used for meta-elliptical copulas which depend on a correlation matrix ρ and possibly other parameters. It is known that ρ can be expressed in terms of functions of Kendall's tau, playing the role of φ 1 , while the remaining parameters are defined as φ 2 . In fact, τ jk = τ (U ji , U ki ) = 2 π arcsin(ρ jk ) (Fang et al., 2002). For example, in the Student copula case, φ 2 would be the degrees of freedom.
It follows from Proposition 1 in the next section that if τ jk,n is the empirical Kendall's tau for the pairs (U ji,n , U ki,n ), i = 1, . . . , n, then for all 1 ≤ j < k ≤ d, converge to centered Gaussian variables R K jk , where C (j,k) is the copula of (U ji , U ki ). Setting ρ jk,n = sin(πτ jk,n /2) + o P (1) 2 , it follows that As a result, if φ 1 is the vector of components ρ jk with 1 ≤ j < k ≤ d, and φ 2 does not depend on ρ, then E Φ 1 W ⊤ 1 = I and E Φ 1 W ⊤ 1 = 0. This shows that two-stage estimators are regular for meta-elliptic copulas families (defined in Appendix D.2).
Many copula families have parameters linked to rank-based measures of dependence. The most common Archimedean families (Clayton, Frank and Gumbel) can all be indexed by Kendall's tau (see Appendix D.1, Table 1), Gaussian copula has van der Waerden correlation matrix as parameters (see Appendix D.2) and the Plackett copula can be indexed by Spearman's rho (Nelsen, 2006). The estimation of these parameters when using the ranks of residuals is treated next.
2.2. Asymptotic behaviour of some rank-based dependence measures. In this section one investigates the asymptotic behaviour of four well-known rank-based dependence measures: Kendall's tau, Spearman's rho, van der Waerden and Blomqvist's coefficients. The main result is that these measures behave asymptotically like the ones computed from innovations, extending the results of Chen and Fan (2006). The proofs depend on the asymptotic behaviour of the empirical copula process and they are given in Appendix C.
2.2.1. Kendall's tau. τ jk,n , the empirical Kendall's coefficient for the pairs (e ji,n , e ki,n ), i = 1, . . . , n, is defined by where the pairs (e ji,n , e ki,n ) and (e jl,n , e kl,n ) are concordant if (e ji,n − e jl,n )(e ki,n − e kl,n ) > 0, i = l. Otherwise, they are said to be discordant. Its theoretical counterpart can be written as with values in [−1, 1] and with value 0 under independence. Letτ jk,n be Kendall's tau calculated with the pairs of innovations (ε ji , ε ki ), i = 1, . . . , n.
2.2.3. van der Waerden's coefficient. Let N and N −1 be respectively the distribution function and the quantile function of the standard Gaussian distribution. Then the van der Waerden's empirical coefficient ρ W jk,n is the correlation coefficient of the pairs (Z ji,n , Z ki,n ), i = 1, . . . , n, where Z ji,n = N −1 (U ji,n ). Its theoretical counterpart ρ W jk is defined by It has values in [−1, 1] and has value 0 under independence. Further letρ W jk,n be van der Waerden's coefficient calculated with the pairs (U ji , U ki ), i = 1, . . . , n.
3. Inference procedure using the empirical copula Tests of goodness-of-fit can be designed by computing some kind of distance between the empirical copula C n , defined by and the "best" representative C φ n of the parametric family C, since C n is a nonparametric estimator of the true copula C. Here, it is assumed that φ n is a rankbased estimator of φ, ı.e., φ n = T n (U 1,n , . . . , U n,n ), for some deterministic function T n (u 1 , . . . , u n ). For example, for testing H 0 , one could use the Cramér-von Mises type statistic based on the process According to Genest et al. (2009), S n is one of the best statistic constructed from A n for an omnibus test 3 , and is much more powerful and easier to compute than the Kolmogorov-Smirnov type statistic A n = sup u∈[0,1] d |A n (u)|. That is why the later is ignored in the present paper.
To be able to state the convergence result for S n , one needs to introduce auxiliary empirical processes. For any s ∈ [0, 1] and x ∈R d , set and β j,n (s, u j ) = α n (s, 1, . . . , 1, u j , 1, . . . , It is well known (Bickel and Wichura, 1971) that α n α 4 , where α is a C-Kiefer process, i.e., α is a continuous centered Gaussian process with Cov {α(s, u), Recall thatČ(1, ·) is the limit of the empirical copula process constructed from innovations; see, e.g., Gänßler and Stute (1987), Fermanian et al. (2004), Tsukahara (2005. The processČ, that could be called the Kiefer copula process, will be important in Section 5. It follows from Corollary 1 that the empirical copula process C n (1, u) = √ n(C n (u)− C(u)) converges toČ(1, u), which does not depend on the parameters of the conditional mean and conditional volatility. As a result, the limiting distribution of A n also shares that property, depending only on φ and Φ under H 0 . The basic result for testing goodness-of-fit using the empirical copula process is stated next. As in Genest and Rémillard (2008), assume, for identifiability purposes, that for every δ > 0, inf sup Before stating the main result of the section, one needs to extend the notion of regularity of φ n as defined in Genest and Rémillard (2008). One says that φ n is regular for φ if (α n , W n , Φ n ) (α, W, Φ) where the latter is centered Gaussian with E ΦW ⊤ = I. Note that the estimators described in Section 2.1 (under the additional assumptions of Chen and Fan (2006)) and Section 2.2 (under assumptions (A1)-(A6), stated in Appendix B) are all regular.
In fact, if ψ is a continuous function on the space C([0, 1]), then the statistic T n = ψ(A n ) converges in law to T = ψ(A). Moreover, the parametric bootstrap algorithm described next or the two-level parametric bootstrap proposed in Genest et al. (2009) can be used to estimate P -values of S n or T n .
3.1. Parametric bootstrap for S n . The following procedure leads to an approximate P -value for the test based on S n . The adaptations required for any other function of A n are obvious. It can be used only if there is an explicit expression for C φ . Otherwise, 2-level parametric bootstrap must be used.
1.-Compute C n as defined in (5) and estimate φ with φ n = T n (U 1,n , . . . , U n,n ). 2.-Compute the value of S n , as defined by (6). 3.-For some large integer N, repeat the following steps for every k ∈ {1, . . . , N}: (a) Generate a random sample Y 1,n , . . . , Y n,n from distribution C φ n and compute the pseudo-observations U An approximate P -value for the test is then given by Remark 1. Some authors, e.g., Kole et al. (2007), proposed tests statistics of the Anderson-Darling type, dividing A n (u) by C φ n (u){1 − C φ n (u)}, and then integrating or taking the supremum. As argued in Genest et al. (2009) and Ghoudi and Rémillard (2010), one should be very careful with these tests and in fact avoid them totally since the denominator only makes sense in the univariate case when parameters are not estimated. In the present context, the limiting distribution of such weighted processes has not been proven and in fact, Ghoudi and Rémillard (2010) gave an example where the limiting variance of the weighted process is infinite.

Inference procedure using Rosenblatt's transform
Based on recent results of Genest et al. (2009), one might also propose to use goodness-of-fit tests constructed from the Rosenblatt's transform (Rosenblatt, 1952). In their study, such tests were among the most powerful omnibus tests.
Recall that the Rosenblatt's mapping of a d-dimensional copula C is the mapping R from (0, 1) d → (0, 1) d so that u = (u 1 , . . . , u d ) → R(u) = (e 1 , . . . , e d ) with e 1 = u 1 and i = 2, . . . , d. Rosenblatt's transforms for Archimedean copulas and meta-elliptic copulas are quite easy to compute for any dimension; see, e.g., Rémillard et al. (2010). The usefulness of Rosenblatt's transform lies in the following properties (Rosenblatt, 1952): can be computed in a recursive way, this is particularly useful for simulation purposes. It follows that the null hypothesis H 0 : C ∈ C = {C φ ; φ ∈ O} can be written in terms of Rosenblatt's transforms viz.
Using an idea of Breymann et al. (2003), extending previous ideas of Durbin (1973) and Diebold et al. (1998), one can build tests of goodness-of-fit by comparing the and define where a ∨ b = max(a, b).
To define regular estimators in that setting, one needs to define It is easy to check that check that (B n , W n ) (B, W), where the joint law is Gaussian, and B is a C ⊥ -Brownian bridge. Now, when using Rosenblatt's transforms, one says that φ n is regular for φ if (B n , W n , Φ n ) (B, W, Φ) where the latter is centered Gaussian with E ΦW ⊤ = I. Again, all estimators described in Section 2.1 (under the additional assumptions of Chen and Fan (2006)) and Section 2.2 (under assumptions (A1)-(A6)) are all regular.
As in the case of copula processes studied in the previous section, in order to prove the following result, one must assume that R φ is Fréchet differentiable, i.e., One also has to assume that R is continuously differentiable with respect to u ∈ (0, 1). One can now state the main result of the section. The expression for D is given in Theorem 2 of Appendix B.
Remark 2. SetǓ i,n = R i /(n + 1), where R 1 , . . . , R n are the associated rank vectors of U 1 , . . . , U n , and letĚ i,n = Rφ n (Ǔ i ), whereφ n is the estimation of φ calculated withǓ i,n = R i /(n + 1), i = 1, . . . , n. Then, it follows from Theorem 2 thatĎ n D, whereĎ 4.1. A parametric bootstrap for S n . The following algorithm is described in terms of statistic S (B) n but can be applied easily to nay statistic of the form T n = ψ(D n ).
(2) For some large integer N, repeat the following steps for every k ∈ {1, . . . , N}: (a) Generate a random sample Y 1,n , . . . , Y n,n from distribution C φ n and compute the pseudo-observations U n,n , and compute and compute An approximate P -value for the test is then given by N k=1 1 S

Change-point problems
In this section, ones describes non-parametric tests for detecting change-points. First, inspired by Ghoudi and Rémillard (2010), detection of change-point for univariate series is tackled. Next, one proposes a new test for change-point detection for the copula, provided there is no change-point in the marginal distributions. 5.1. Detection of change-point for univariate series. Detection of change-point for the univariate series ε ji can be based on the process Under assumptions (A1)-(A6), A j,n A j , where The latter shows that the limiting distribution of the statistics are distribution free, converging respectively to That result extends the one obtained in Ghoudi and Rémillard (2010) for residuals of ARMA processes. Note that K j is a continuous centered Gaussian process with covariance given by Cov As remarked in Ghoudi and Rémillard (2010), that process appears as the limit of many other processes used in tests of change-point (Picard, 1985, Carlstein, 1988 and tests of independence (Blum et al., 1961, Ghoudi et al., 2001. Furthermore, tables for the limiting distribution of T  Table IV page 206 and Table I page 204 respectively. In case the sample size considered is not available in these tables, it is suggested in Ghoudi and Rémillard (2010) to use the simulations since K j also appear as the limit ofK j,n (s, u) = where R i is the rank of U i , amongst the i.i.d. uniform variables U 1 , . . . , U n .

5.2.
Detection of change-point for copulas. Suppose now that the null hypotheses that there is no change-point in the marginal distributions are all accepted.
Next, if one is interested in possible change-points in the dependence structure, one could do something similar to the previous section. That is the methodology proposed next. Previous work on structural change for copulas include the parametric change-point approach of Embrechts (2004, 2009), a filtering/nonparametric methodology proposed by Harvey (2010) and parametric/kernel-based approach proposed by Guégan and Zhang (2010). In the latter, to perform the test, the authors have to select a family for their so-called "static" copula. Their test is based on kernel estimates. Here, in contrast, one starts by performing a non-parametric change-point test. If the null hypothesis is accepted, then one may try to select a static copula. Furthermore, Guégan and Zhang (2010) work with residuals, without ever proving that the methodology is valid. In Harvey (2010), no residuals are used. The methodology is based on time-varying quantiles and some kind of filtering technique. It would be interesting to compare the approach proposed here to the one proposed in Harvey (2010).
Let's now describe the proposed methodology, which is closely related to the test of equality between two copulas proposed by Rémillard and Scaillet (2009). First, it is easy to check that if C Therefore change-point tests can be based on G n . For example, one could define and reject the null hypothesis for large values of T n . The limiting distribution of T n and G n is given next. Even if the law of G depends on the unknown copula C, it is easy to simulate independent copies, using a multiplier method adapted from Scaillet (2005) and Rémillard and Scaillet (2009). That technique is described next. 5.2.1. Multipliers method for T n . The following procedure leads to an approximate P -value for the test based on T n . The adaptations required for any other function of G n are obvious.
An approximate P -value for the test is then given by N k=1 1 S (k) n > S n /N.

Remark 3. Using Theorem 1, a non-parametric change-point for the innovations ε
Because of the form of the limiting distribution, one has to use multipliers technique to generate asymptotically independent copies . See Rémillard (2010) for details.

Example
In order to be able to make comparisons with Chen and Fan (2006), one of their data set is used, namely the Deutsche Mark/US and Japanese Yen/US exchanges rates, from April 28, 1988 to Dec 31, 1998. AR(3)-GARCH(1,1) and AR(1)-GARCH(1,1) models were fitted on the 2684 log-returns.
For such a large sample size, one must be sure that there is no structural changepoint. To that end, univariate change-point were performed first on the standardized residuals and the null hypothesis was accepted each time. Then, the copula changepoint test was performed, leading once again to the acceptance of the null hypothesis, since the p-value was estimated to be 33%, using N = 100 replications.
Next, the usual standard copula models (Gaussian, Student, Clayton, Frank, Gumbel) were checked for goodness-of-fit. In each case the null hypothesis was rejected since the p-value was estimated to be 0 (using N=100 replications). That shows the limitations of the model selection methodology proposed by Chen and Fan (2006). It can only be used to rank models, even if none is adequate, which is the case here.

Conclusion
The asymptotic behaviour of the empirical copula constructed from residuals of stochastic volatility models was studied. It was shown that if the stochastic volatility matrix is diagonal, then the empirical copula process behaves like if the parameters were known. That remarquable property makes it possible to construct consistent tests of goodness-of-fit for the copula of innovations. Tests of structural change in the dependence structure were also proposed.

Appendix A. Smoothness conditions for parametric bootstrap
Following Genest and Rémillard (2008), assume that the family of densities c φ satisfies (B1) The density c φ of C φ admits first and second order derivatives with respect to all components of φ. The gradient (row) vector with respect to φ is denoteḋ c φ , and the Hessian matrix is represented byc φ .
(B3) For every φ 0 ∈ O, there exist a neighborhood N of φ 0 and C θ 0 -integrable functions h 1 , h 2 : R d → R such that for every u ∈ (0, 1) d , Appendix B. Convergence of the partial-sum empirical processes In this section, one assumes a more general model than the one used in Section 2 where σ i is no longer a diagonal matrix, namely where the innovations ε i = (ε 1i , . . . , ε di ) ⊤ are i.i.d. with continuous distribution function K, and µ i , σ i are F i−i -measurable and independent of ε i . Here F i−1 contains information from the past and possible information from exogenous variables. Set Given an estimation θ n of θ, compute the residuals e i,n = (e 1i,n , . . . , e di,n ) ⊤ , where Further set Θ n = n 1/2 (θ n − θ). The goal is to study the asymptotic behaviour of the partial-sum empirical process K n defined by (3). The following assumptions are needed in order to prove the convergence of K n .
(A3) There exists a sequence of positive terms r i > 0 so that i≥1 r i < ∞ and such that the sequence max 1≤i≤n d i,n /r i is tight.
If σ is diagonal, (A7) is not needed for the convergence of K n . In that case, Remark 5. (Γ 1k θ) j = 0 for all θ and all j = k if and only if (Γ 1k ) jl = 0 for all l and all j = k. That can occur for example if In that case A i must be known since it is parameter free. This is true in particular if σ i is diagonal, in which case A i is the identity matrix. It then follows from (15) that Setting H i to be the diagonal matrix with (H i ) jj = (σ i ) jj , j = 1, . . . , d, then one can rewrite the model as Since A i is known, this model is a simple rescaling of a model with diagonal volatility, and as such has little interest. So if the model cannot be transformed into a diagonal one, the limiting empirical copula process is not parameter free.

Corollory 1. Under assumptions (A1)-(A6), if the volatility matrix is diagonal then
where β j (s, u j ) = α(s, . . . , 1, u j , 1, . . . , 1). If the volatility matrix is not diagonal, then Corollary 1 follows directly from Theorem 1, using Genest et al. (2007) for any A ⊂ S d . Using the multinomial formula, one has To prove the theorem, it suffices to show that for any 1 ≤ j ≤ d, uniformly in (s, x), µ j,n (s, x) converges in probability to s∂ x j K(x)(Γ 0 Θ) j + s d k=1 G jk (x)(Γ 1k Θ) j , and that for any |A| > 1, µ A,n (s, x) converges in probability to zero. These proofs will be done for j = 1 and A ⊃ {1, 2}, the other cases being similar. Also suppose that σ is diagonal. The general case is similar.
Let δ ∈ (0, 1) be given. From (A2), (A3) and (A5), one can find M > 0 such that if n is large enough, then P (B M,n ) > 1 − δ, where Because the closed ball of radius M is compact, it can be covered by finitely many balls of radius λ ∈ (0, 1).
Next, note that P ( ε 1i ≤ x 1 + η i,n (κ), ε 2i ≤ z 1 , . . . , ε di ≤ z d−1 | F i−1 ) is given by It follows from (A1), (A2) and (A6) that on B M,n , can be made arbitrarily small with large probability. The final step is to show thať µ 1,n (s, x; κ) = µ 1,n (s, x; κ) −Γ 1,n (s, x) can be made arbitrarily small by choosing κ 3 small enough. The proof is similar to the proof of Lemmas 7.1-7.2 in Ghoudi and Rémillard (2004). Suppose 1/2 < ν < 1 and set N n = ⌊n ν ⌋. Then, set y k = F −1 1 (k/N n ), 1 ≤ k < N n . Further set y 0 = −∞ and y Nn = +∞. Now, if y k ≤ x 1 < y k+1 , and z = (x 2 , . . . , x d ). First, note that one can coverR d by a finite number N n × J of intervals of the form [a, b) and set V i,n (x) = E{U i,n (x)|F i−1 }. One cannot directly with U i,n − V i,n . Better bounds are obtained by decomposing U i,n and V i,n as follows: Set and To complete the proof, it is enough to show thatμ ± 1,n can be made arbitrarily small. Only the proof for the + part is given, the other one being similar.
Finally, from (A1) and (A2), one may conclude that 1 √ n n i=1 V + i,n (y k+1 , v l ) − V + i,n (y k , u l ) is bounded, for some constants c 1 , . . . , c 4 depending on f 1 ∞ and g 1 ∞ , by That can be made as small as necessary, provided n is large, κ 3 is small and the mesh of the covering is small enough.
The following theorem give the result of the convergence of the empirical process based on Rosenblatt's transformation.
Theorem 2. Under Assumptions (A1)-(A6), if the volatility matrix is diagonal and if (φ n ) is regular for φ, then D n −Ď n 0 and D n D, where where B is a C ⊥ -Brownian bridge, E{B(u)W} = γ(u), E{κ(u)W} = 0, and , withŨ independent of all other observations.

Appendix C. Other proofs
Before starting the proofs, one states a lemma that is quite useful in some proofs.
By Corollary 1, C n =Č n +o P (1) Č , proving that R K jk,n converges to 8 C (j,k) dC (j,k) n . Next, it is easy to check that converges toČ. As a result, for any 1 ≤ j < k ≤ d, To compute the covariance between R K jk and W, note that E Č (u)W = C(u), and the latter is 0 if all but one u j is 1. As a result, using integration by parts, since τ jk = 4 C (j,k) dC (j,k) − 1. The proof of Propositions 2-4 is similar. It is sufficient to note that for the three estimators, one has √ n(ρ jk, for an appropriate distribution function J with left-continuous inverse L. 5 According to Genest and Rémillard (2004) and Corollary 1, the latter converges to C(j, k){J(x), J(y)}dxdy. = Č (j, k){J(x), J(y)}dxdy. 5 J = N for van der Waerden, J is the distribution of the uniform over [0, √ 12] for Spearman's rho while J is the distribution function of the discrete random variable taking values 0 and 2 with p = 1/2 for Blomqvist's coefficient.
The representations come from the convergence ofĈ n toČ. The proof of the covariance with W can be dealt similarly to the one involving Kendall's tau.
C.2. Proof of Proposition 5. The convergence of C n = √ n(C n − C) follows from Corollary 1 and the joint convergence of (α n , Φ n ) follows from the representation of α n and the estimators of Sections 2.1-2.2. Using the smoothness of c φ , it follows thaṫ C = ∂ φ C φ is continuous and under H 0 , As a result, Genest and Rémillard (2008), the parametric bootstrap approach will work since E ΦW ⊤ = I, as shown in Sections 2.1-2.2.
To show that the multipliers method works, it suffices to note that conditionally on U 1,n , . . . , U n,n , the finite dimensional distributions of α (k) n converge to those of α (k) , an independent copy of α by construction. Next the tightness of α (k) n follows from the tightness of Cn(u) √ n n i=1 ξ i,k and the tightness of 1 √ n n i=1 ξ i,k 1(U i,n ≤ u), using Bickel and Wichura (1971) and the convergence in probability of 1 n ⌊ns⌋ i=1 1(U i,n ≤ u) to the continuous distribution function sK(u) at each (s, u) ∈ [0, 1] d+1 . As a result, (T n , T n > T n is an approximate P -value for T n .
Here are some general families of elliptic distributions. Table 2. Generators of some d-dimensional elliptic distributions.
As a consequence, the density of any marginal distribution of a d-dimensional elliptic distribution with generator g and parameters (0, R) is For example, if g is the generator of the d-dimensional Pearson type VII with parameters (α, ν), then g i is the generator of the d i -dimensional Pearson type VII with parameters (α, ν), i = 1, 2. One can also show that if g is the generator of the d-dimensional Pearson type II with parameter α, then g i is the generator of the d i -dimensional Pearson type II with parameter α + d 3−i , i = 1, 2. In particular, the marginal distributions of a Pearson type VII has density f (x) = Γ(α + 1/2) (πν) 1/2 Γ(α) (1 + x 2 /ν) −α−1/2 .
Note that formula (19) is particularly useful for computing Rosenblatt's transforms.
D.3. Other copula families. As proposed in Dias and Embrechts (2004), one could also consider mixtures of copulas, i.e., consider families of the form with m k=1 π k = 1, π k > 0, and θ = (θ 1 , . . . , θ m ). Copulas C (k) θ k may be part of the same family; for example, one could consider a mixture of Gaussian copulas. One could also take a mixture of different families, e.g. a mixture of Clayton and Gumbel copulas.