On testing for independence between the innovations of several time series

Bruno Remillard

1 The Canadian Journal of Statistics Vol. xx, No. yy, 2012, Pages 1–32 La revue canadienne de statistique On testing for independence between the innovations of several time series Pierre DUCHESNE1 *, Kilani GHOUDI2 and Bruno RÉMILLARD3 1 Département de mathématiques et de statistique, Université de Montréal, Montréal, Canada Department of Statistics, College of Business and Economics, United Arab Emirates University, Al Ain, United Arab Emirates 3 Service d’enseignement des méthodes quantitatives de gestion, HEC Montréal, Montréal, Canada 2 Key words and phrases: Copula; Cramér–von Mises test statistic; cross-correlation; independence; Kolmogorov–Smirnov test statistic; multivariate lag; rank-based test statistic; time series. MSC 2010: Primary 60F05; secondary 62M10 Abstract: Test statistics for checking the independence between the innovations of several time series are developed. The time series models considered allow for general specifications for the conditional mean and variance functions that could depend on common explanatory variables. In testing for independence between more than two time series, checking pairwise independence does not lead to consistent procedures. Thus a finite family of empirical processes relying on multivariate lagged residuals are constructed, and we derive their asymptotic distributions. In order to obtain simple asymptotic covariance structures, Möbius transformations of the empirical processes are studied, and simplifications occur. Under the null hypothesis of independence, we show that these transformed processes are asymptotically Gaussian, independent, and with tractable covariance functions not depending on the estimated parameters. Various procedures are discussed, including Cramér–von Mises test statistics and tests based on non-parametric measures. The ranks of the residuals are considered in the new methods, giving test statistics which are asymptotically margin-free. Generalized cross-correlations are introduced, extending the concept of cross-correlation to an arbitrarily number of time series; portmanteau procedures based on them are discussed. In order to detect the dependence visually, graphical devices are proposed. Simulations are conducted to explore the finite sample properties of the methodology, which is found to be powerful against various types of alternatives when the independence is tested between two and three time series. An application is considered, using the daily log-returns of Apple, Intel and Hewlett-Packard traded on the Nasdaq financial market. The Canadian c 2012 Statistical Society of Canada Journal of Statistics xx: 1–32; 2012 ⃝ Résumé: Des statistiques de test pour vérifier l’indépendance entre les innovations de plusieurs séries chronologiques sont développées. Les modèles de séries chronologiques considérés permettent des spécifications générales pour les fonctions de moyenne et de variance conditionnelle qui pourraient dépendre de variables explicatives communes. Pour tester l’indépendance entre plus de deux séries chronologiques, vérifier l’indépendance deux à deux ne conduit pas à des procédures convergeantes. Ainsi, une famille finie de processus empiriques reposant sur plusieurs résidus décalés sont construits, et nous dérivons leurs distributions asymptotiques. Afin d’obtenir des structures simples de covariances asymptotiques, des transformations de Möbius des processus empiriques sont étudiées, et des simplifications se produisent. Sous l’hypothèse nulle d’indépendance, nous montrons que ces processus transformés sont asymptotiquement gaussiens, indépendants, et avec des fonctions de covariance commodes qui ne sont pas fonction de l’estimation des paramètres. Différentes procédures sont discutées, incluant les statistiques de test de Cramér-von Mises et les tests basés sur des mesures non paramétriques. Les rangs des résidus sont considérés dans les nouvelles méthodes, donnant des statistiques de test qui ne sont asymptotiquement pas fonction des marges. Des corrélations croisées généralisées sont introduites, fournissant une extension du concept de corrélation croisée à un nombre arbitraire de séries temporelles; des procédures portemanteaux basées sur elles sont discutées. Afin de détecter la dépendance visuellement, des méthodes graphiques sont proposées. Des simulations sont réalisées afin d’étudier les propriétés pour des échantillons finis de la c 2012 Statistical Society of Canada / Société statistique du Canada ⃝ CJS ??? 2 DUCHESNE, GHOUDI AND RÉMILLARD Vol. xx, No. yy méthodologie, qui se trouve être puissante contre divers types de contre-hypothèses lorsque l’indépendance est testée entre deux et trois séries chronologiques. Une application est considérée, en utilisant les logrendements quotidiens d’Apple, d’Intel et de Hewlett-Packard négociées sur le marché financier Nasdaq. La c 2012 Société statistique du Canada revue canadienne de statistique xx: 1–32; 2012 ⃝ 1. INTRODUCTION The existence of relationships between several time series represents a natural question in many practical applications. When two time series need to be analyzed, the question arises of describing the possible interrelationships existing between them. In economics, causality analysis may be a central issue in a forecasting context; see, e.g., McLeod (1979). Consequently, this emphasizes the importance to have methods which reach high power against plausible alternatives. Haugh’s (1976) procedure represents to our knowledge the first attempt at developing test statistics for checking the non-correlation between two stationary autoregressive-moving average (ARMA) models. His approach was based on residual cross-correlations and portmanteau type test statistics similar in spirit to the Box-Ljung-Pierce test statistics; see, e.g., Li (2004). Pierce (1977) and Geweke (1981) studied the power properties of Haugh’s test statistic and a generalized Haugh test statistic has been proposed by Hong (1996), using a spectral approach. Hong’s (1996) procedure is also of the portmanteau type, but each squared cross-correlation is multiplied by a non-negative weight determined by a kernel function and a smoothing parameter. In many situations, the flexible weighting provides more powerful test statistics than Haugh’s test statistic. However, Hong’s (1996) procedure also relies on residual cross-correlations, and will be typically powerful only if cross-correlations with low lag orders are significant. Another extension of Haugh’s procedure is given in Koch & Yang (1986), who introduced a modification of Haugh’s portmanteau test that allows for a potential pattern in the residual cross-correlation function. Li & Hui (1994) proposed a robustified cross-correlation function between two time series and employed it to develop a robust version of Haugh’s (1976) and McLeod’s (1979) test statistics for checking independence. Duchesne & Roy (2003) extended further Hong’s test statistics in a robust framework. A non-parametric approach for checking the independence between two autoregressive time series using autoregressive rank scores was studied by Hallin et al. (1999). Recently, Shao (2009) extended the applicability of test statistics for independence between two stationary time series to the long memory case. Residual cross- and autocorrelations have been found useful for identifying time series models and for checking model adequacy. However, autocorrelations and portmanteau test statistics based on them reach power only in the presence of linear dependence and they are truly tests of independence only for Gaussian processes. Test statistics based on usual residual crosscorrelations merely measure the linear dependence between time series and consequently do not check necessarily independence. In our framework, this means that all test statistics based on residual cross-correlations display high power only if the relationships between the possibly non Gaussian time series are intrinsically linear, and they may not have power under more complicated forms of dependence. That behavior is well illustrated by the so-called Tent map copula (see Genest & Rémillard, 2004), where there is strong dependence between components while almost all parametric and non-parametric measures of dependence give (theoretically) the same value as in the case of independence (see Section 4). In order to propose test statistics which check independence for non Gaussian stochastic processes, not only non-correlation, Kim & Lee (2005) constructed Cramér–von Mises test statistics based on the least squares residuals from ARMA models. Extending the results of Lee & Wei * Author to whom correspondence may be addressed. E-mail: duchesne@dms.umontreal.ca The Canadian Journal of Statistics / La revue canadienne de statistique DOI: 2012 3 (1999), they studied the residual empirical process of the residual bivariate time series, and from their results they proposed test statistics at individual lags. They found that the Cramér–von Mises test statistics based on residuals offer the same limiting distribution as the one of the test statistics relying on the true but unknown innovations. They also proposed portmanteau procedures based on the summation of one-lag test statistics, weighted summation and on the maximum of Cramér–von Mises test statistics. In a related framework, Skaug & Tjøstheim (1993) adopted the summation criterion when testing for independence, and weighted test statistics constructed from one-lag Cramér–von Mises test statistics are advocated in Hong (1998). Haugh’s (1976) or Hong’s (1996) test statistics based on cross-correlations or the omnibus tests relying on empirical processes developed by Kim & Lee (2005) are suitable for checking independence between two time series. When testing for independence between more than two time series, a natural approach consists in testing for pairwise independence, using the previous portmanteau test statistics based either on cross-correlations or Cramér–von Mises test statistics. However, the nominal levels of the individual tests for independence between two time series need to be adjusted, since computing several test statistics at the 5% nominal level, say, does not give a statistical procedure with an overall nominal level of 5%. Obviously, Bonferroni’s procedure or similar adjustments could be performed. However, a fundamental objection to this kind of statistical procedure is that testing pairwise independence is not equivalent to global independence, and it is rather easy to construct examples involving three time series which are pairwise but not jointly independent. On the technical side, as demonstrated in the present article, testing for independence between more than two time series represents a nontrivial generalization. In the bivariate case, we show that important simplifications occur in the asymptotic limit and we retrieve the results of Kim & Lee (2005). However, and somewhat surprisingly, it appears that the asymptotic distributions of the empirical processes needed for testing independence between three and more time series are untractable for practical purposes. Inspired by the work of Ghoudi, Kulperger & Rémillard (2001), Genest & Rémillard (2004) and Genest, Rémillard, & Beaudoin (2009), and resorting to the Möbius transformation of the empirical processes, we are, however, able to derive new test statistics which should be useful in practical applications. The time series models considered here are very general: general specifications for the conditional mean and variance functions are allowed. Classical specifications such as ARMA models are possible, but also non-linear models with conditional heteroscedasticity, together with stochastic explanatory variables. That explains why the proposed methods represent tests for independence between several series of (unknown) innovations rather than tests for independence between several time series. The function determining the dependence between the innovations is given by the so-called copula, which does not take into account the marginal distribution of the innovations. See Joe (1997), Nelsen (2006) and Drouet-Mari & Kotz (2001) for comprehensive reviews, amongst others. The empirical copula, which is the empirical distribution function of the ranks of the residuals, represents a natural tool to develop marginal free test statistics for independence. It has been introduced in a series of papers by Rüschendorf (1974, 1976) and Deheuvels (1979, 1980, 1981a,b,c) and later studied in full generality by Stute (1984) and Gänßler & Stute (1987). Genest & Rémillard (2004) extended these results to copulas for testing independence in consecutive observations of a univariate time series. In order to obtain more robust test statistics, the empirical processes and test statistics based on the ranks of the residuals have been also proposed by the preceding authors. In the particular case consisting of testing independence between two time series, the limiting distribution of the empirical copula process extends in several directions the technical results obtained by Kim & Lee (2005). Under the null hypothesis of independence, it is proven that the limiting process has the same distribution, whether or not the parameters are estimated, that is the limiting empirical copula process based on the ranks of the innovations is exactly the same as the limiting empirical copula process based on the ranks of the DOI: The Canadian Journal of Statistics / La revue canadienne de statistique 4 DUCHESNE, GHOUDI AND RÉMILLARD Vol. xx, No. yy residuals. Using these results, several test statistics are proposed, such as Kolmogorov–Smirnov and Cramér–von Mises test statistics. Non-parametric measures are also studied. Nevertheless, cross-correlations are still useful tools for checking the independence between two time series. In view of that, the concept of cross-correlation is generalized for more than two time series, yielding generalized cross-correlations. Test statistics based on these measures are also proposed and their asymptotic distribution are studied, which have convenient normal distribution asymptotically, generalizing a theorem due to Haugh (1976). In order to propose portmanteau test statistics, we consider sums of one-lag test statistics and we show how to combine optimally their P -values (according to the Bahadur relative efficiency criterion). We demonstrate that for bivariate time series, combining the P -values of generalized cross-correlations reduces essentially to the test statistics of Haugh (1976). Using graphical devices similar to those elaborated by Genest & Rémillard (2004), it is shown how to describe the dependence visually. The paper is organized as follows. In Section 2, we present the general class of models and establish the asymptotic distribution of the empirical processes based on the residuals for testing independence between innovations of several time series, together with the associated Möbius decomposition of these empirical processes. Replacing the residuals by their respective ranks, that is, considering the empirical copula process, it is shown that the limiting process is parameter free and marginal free. This is particularly important since tests based on cross-correlations can be affected severely by the marginal distribution of the innovations; this point is illustrated in Section 4. Using these building blocks, parametric and non-parametric test statistics for testing independence are proposed in Section 3. Kolmogorov–Smirnov and Cramér–von Mises test statistics are studied, and we also introduce test statistics based on generalized cross-correlations. We conclude the section showing how to combine optimally the P -values of the test statistics. Graphical procedures are developed, making these test statistics easy to interpret and to implement. Their finite sample properties are studied in Section 4 through Monte Carlo simulations. Data generating processes (DGPs) involving two and three time series are simulated, and exact levels and powers of our methods are calculated. The effects of the marginal distributions on test statistics based on generalized cross-correlations are also investigated. The limitations of pairwise statistics when examining the dependence between three time series are illustrated. In Section 5, an application with real data is considered, using the daily log-returns of Apple, Intel and Hewlett-Packard traded on the Nasdaq financial market. Concluding remarks are offered in Section 6. 2. EMPIRICAL PROCESSES FOR TESTING INDEPENDENCE BETWEEN SEVERAL TIME SERIES 2.1. Preliminaries { } Let Xt = (X1t , . . . , Xdt )⊤ , t ∈ Z , be a d-dimensional stochastic process whose components satisfy: Xjt = µjt (θ 0 ) + σjt (θ 0 )εjt , j = 1, . . . , d, (1) where θ 0 represents a p-dimensional vector of unknown parameters belonging to an open subset O ⊂ Rp , and where: µjt (θ) = µj (θ; Zt , {(Xt−i , Zt−i ), i > 0}), σjt (θ) = σ j (θ; Zt , {(Xt−i , Zt−i ), i > 0}) > 0. Thus the conditional means and variances can depend possibly on stochastic explanatory variables {Zt , t ∈ Z}. It is assumed that the innovation process {εt , t ∈ Z} corresponds to a dThe Canadian Journal of Statistics / La revue canadienne de statistique DOI: 2012 5 dimensional strong white noise, where εt = (ε1t , . . . , εdt )⊤ , that is {εt } forms a sequence of independent and identically distributed (iid) random vectors, such that εjt has density fj (·) and distribution function Fj (·). The random variable εjt has mean zero and variance one. In addition, εt is presumed independent of Zt and {Xt−i , Zt−i ; i > 0}. We also suppose that {Xt } corresponds to a non anticipative strictly stationary and ergodic solution of (1). See, e.g., Brockwell & Davis (1991) for ARMA models, and Bougerol & Picard (1992a,b) for [general] conditional heteroscedasticity ([G]ARCH) time series models. A recent textbook is Francq & Zakoı̈an (2010). The nonlinear models in (1) represent a very general class of time series models with general specifications for the error term. Note that the models may be different for each time series. That framework includes the classical ARMA time series with strong white noises, but also the more general AR(∞) models considered in Kim & Lee (2005). Exogenous variables are allowed, as the ARMAX time series models discussed in Hannan & Deistler (1988). Time series models with GARCH error terms are allowed, such as the class of ARMA-GARCH time series∑models. For example, one can take the conditional mean ∑ rj s µjt = µj + k=1 ajk Xj,t−k + k=0 bjk Zj,t−k , containing autoregressive components of or⊤ der rj and exogenous common components Z∑ t = (Z1t , . . . , Zdt ) , and the conditional variance ∑ qj pj 2 2 2 σjt = ωj + k=1 αjk (Xj,t−k − µj,t−k ) + k=1 βjk σj,t−k , modeled as a GARCH(pj , qj ), for j = 1, . . . , d. Nonlinear models such as threshold autoregressive models (TAR), self-exciting TAR models, and smooth versions of TAR models are other examples of possible specifications. A similar model has been studied in Chen & Fan (2006), which also specifies the conditional mean and variance of a multivariate time series parametrically, with parametric copula model evaluated at nonparametric marginal distributions of the standardized innovations. Tong (1990) and Granger & Teräsvirta (1993) give detailed treatment of univariate nonlinear models. The null hypothesis H0 of interest here consists in the independence between the d univariate stochastic processes {ε1t }, . . . , {εdt } against the alternative of dependence of arbitrary form. When µjt and σjt depend only on {Xj,t−i ; i > 0}, then the null hypothesis is equivalent to the independence of the time series {X1t }, . . . , {Xdt }. Thus, in order to test independence between {εjt }, j = 1, . . . , d, a natural procedure is achieved by first prewhitening each time series in a first step and then testing for independence using the residuals. That approach was studied by Haugh (1976) and Hong (1996). Given n observations X1 , . . . , Xn , and the initial values Xi = xi , i ≤ 0, the model (1) is fitted and an estimator θ̂ n of the p-dimensional vector θ 0 = (θ10 , . . . , θp0 )⊤ is calculated. The residuals et,n = (e1t,n , . . . , edt,n )⊤ , t = 1, . . . , n, are defined by: −1 (θ̂ n )(Xjt − µjt (θ̂ n )), j = 1, . . . , d. ejt,n = σjt (2) When the time series have zero mean, a natural choice for the initial values is to specify xi = 0, i ≤ 0. Let σ t be a diagonal matrix with diagonal elements σ1t , . . . , σdt and inverse σ −1 t . Set 1/2 Θn = n (θ̂ n − θ). In vector notation, the residuals satisfy: et,n = σ −1 t (θ̂ n )(Xt − µt (θ̂ n )). ∑d We introduce dt,n = εt − et,n − (γ 0t Θn + k=1 εkt γ 1kt Θn )/n1/2 , where γ 0t = σ −1 t µ̇t and σ̇ , with the derivatives satisfying: γ 1kt = σ −1 kt t (µ̇t )jl = ∂θl µjt , DOI: (σ̇kt )jl = ∂θl σjkt = ∂θl (σ t )jk , j, k = 1, . . . , d, l = 1, . . . , p. The Canadian Journal of Statistics / La revue canadienne de statistique 6 DUCHESNE, GHOUDI AND RÉMILLARD Vol. xx, No. yy We also consider:   d d n ∏ 1 ∑ ∏ uj  , u = (u1 , . . . , ud )⊤ ∈ [0, 1]d . αn (u) = 1/2 I{Fj (εjt ) ≤ uj } − n t=1 j=1 j=1 The following technical assumptions are needed. n n 1∑ 1∑ Pr Pr −→ Γ1k , where Γ0 and Γ1k are determinisγ 0t −→ Γ0 , Γ1k,n = γ n t=1 n t=1 1kt tic, k = 1, . . . , d. n n ) ) 1∑ ( 1∑ ( E ∥γ 0t ∥k and E ∥γ 1jt ∥k are bounded, for k = 1, 2. (A2) n t=1 n t=1 ∑ (A3) There exists a sequence of positive terms rt > 0 so that t≥1 rt < ∞ and such that the sequence max ∥dt,n ∥/rt is tight. (A1) Γ0,n = 1≤t≤n (A4) max1≤t≤n ∥γ 0t ∥/n1/2 = oP (1) and max1≤t≤n |εjt |∥γ 1jt ∥/n1/2 = oP (1). (A5) (αn , Θn ) (α, Θ) in D([−∞, ∞]d ) × Rp . (A6) The applications xj 7→ xj fj (xj ) and xj 7→ fj (xj ) are bounded and continuous on their support. Bai (2003) showed that for independent time series including ARMA-GARCH models, Assumptions (A1)–(A6) hold true. Furthermore, note that (A1) and (A2) are trivially satisfied if the sequences {γ 0t } and {γ 1kt }, k = 1, . . . , d are stationary, ergodic and square integrable. Also, in that case, (A4) is satisfied if fourth order moments exist for the innovations εjt . Generally, (A3) follows from the smoothness of µjt and σjt with respect to θ. Note that (A3) includes our assumptions on the effect of the initial values. See also Bai (2003, Assumptions A4 and B4). From hypothesis (A5), when testing for H0 , we suppose that θ 0 is estimated by a n1/2 -efficient procedure, that is, Θn = n1/2 (θ̂ n − θ 0 ) converges in distribution to some random vector Θ. If a given time series is estimated by an ARMA model, the asymptotic properties of least-squares or maximum likelihood estimators are well-known; these estimators are n1/2 -consistent; see, e.g., Brockwell & Davis (1991). For more general nonlinear models, quasi-maximum likelihood and nonlinear least squares methods represent possible estimation procedures. Klimko & Nelson (1978) studied general properties of conditional least squares estimators in univariate time series models. Other references are Potscher & Prucha (1997) or Taniguchi & Kakizawa (2000), among others. When the model for the jth time series corresponds to a pure GARCH or lies in the ARMA-GARCH class of models, maximum likelihood estimation can be considered, and asymptotic properties are studied under weak assumptions in Francq & Zakoı̈an (2004). Estimation results are also given in Chen & Fan (2006). Finally, hypothesis (A6) is satisfied when fj (·) is continuous since εjt has mean zero and unit variance. Test statistics based on residual cross-correlations are powerful to detect linear dependence between the time series but they are not consistent under more general forms of dependence (see Kim & Lee (2005)). In the terminology of Ghoudi & Rémillard (2004), it is useful to note that the residuals defined by (2) fall into the category of pseudo-observations. A growing interest emerged in recent years in the study of these quantities, see Ghoudi & Rémillard (1998, 2004), Ghoudi, Kulperger & Rémillard (2001), Genest & Rémillard (2004) and van der Vaart & Wellner (2007), amongst others. Another example of pseudo-observations are the rank of residuals, which may prove to be very useful in deriving test statistics which offer more robustness to outlying residuals. Thus, in order to develop tests of independence, it appears useful to study the empirical processes based on these lagged residuals. The Canadian Journal of Statistics / La revue canadienne de statistique DOI: 2012 7 For bivariate processes, Kim & Lee (2005) considered test statistics based on the empirical distribution of the random vector (ε1,t−l , ε2,t )⊤ , using methods of Skaug & Tjøstheim (1993) and results of Carlstein (1988). For more than two time series, we introduce multivariate lags as follows. Let ℓ = (l1 , . . . , ld )⊤ ∈ Zd be a d-vector of time lags. That d-dimensional vector represents a multivariate lag and it specifies naturally the time lags between the d time series. For example, in the classical case d = 2, the vector ℓ = (0, −l)⊤ , where l is a positive integer, allows us to consider the usual l-dependence between the two random variables ε1,t and ε2,t−l . In the trivariate case d = 3, ℓ = (0, 0, 0)⊤ gives the indexes to study simultaneous dependence among the three time series, while ℓ = (0, 0, −l)⊤ corresponds to describe the dependence in the random vector (ε1,t , ε2,t , ε3,t−l )⊤ , and so on. Define the empirical distribution function of the residuals indexed by ℓ by: Hℓ,n (x) = n−1 n ∑ I{et ≤ x} = n−1 t=1 n ∏ d ∑ { } I ej,t+lj ≤ xj , x ∈ Rd , (3) t=1 j=1 ∏d where et+n = et ≡ et,n for all t ∈ Z. The distribution function H(x) = j=1 Fj (xj ) corresponds to the (unknown) distribution of εt under the null hypothesis of independence. The definition (3) relies on circular residuals, which are convenient because the residuals are well-defined for all values of t. Since only a finite number of residuals are modified using that definition, that extension does not affect the asymptotic behavior of the processes. In the particular case d = 2, ℓ = (−l, 0)⊤ , l ∈ Z, and we retrieve the empirical process studied in Kim & Lee (2005) . The empirical copula indexed by ℓ ∈ Z is also defined by: Cℓ,n (u) = n −1 n ∏ d ∑ } { I (n + 1)−1 Rj,t+lj ≤ uj , u = (u1 , . . . , ud )⊤ ∈ [0, 1]d , t=1 j=1 where Rj,t+lj = nFjn (ej,t+lj ) = n ∑ I{ejk ≤ ej,t+lj } k=1 represents the rank of ej,t+lj amongst the ejk , k = 1, . . . , n. In order to have well-defined ranks for all t, we also introduce circular ranks, defined as Rjt = Rj,t+n , t ∈ Z, j = 1, . . . , d. Most measures of interdependence are based on the empirical distribution functions Hℓ,n or Cℓ,n (Genest & Rémillard, 2004). For example, the cross-correlations can be expressed in terms of Hℓ,n , while non-parametric measures such as Spearman’s rho or Kendall’s tau can be expressed as a function of Cℓ,n . In our testing framework for independence between innovations, it makes sense to use the empirical copula since the marginal distribution of the random variable εjt is unknown. However, the empirical copula process relies on the ranks of residuals, therefore adding a significant source of complexity. Before stating the main result, we introduce the processes: Dℓ,n (x) = n1/2 DOI:    Hℓ,n (x) − d ∏ j=1 Fjn (xj )    , ℓ ∈ Z d , x ∈ Rd , The Canadian Journal of Statistics / La revue canadienne de statistique 8 DUCHESNE, GHOUDI AND RÉMILLARD Vol. xx, No. yy and Eℓ,n (u) = n1/2   Cℓ,n (u) −  d ∏ Vn (uj ) j=1   , ℓ ∈ Zd , u ∈ [0, 1]d ,  where Vn represents the distribution function of the uniform variable over the set {(n + 1)−1 , . . . , n(n + 1)−1 }, i.e., Vn (s) = n−1 min{n, ⌊(n + 1)s⌋}, s ∈ [0, 1], and ⌊a⌋ denotes the largest integer inferior to a, a ∈ R. Note that Vn (s) − s goes uniformly to zero as n tends to infinity. The reason for introducing Vn is that it improves generally the speed of convergence of the empirical processes Eℓ,n (Genest & Rémillard, 2004). Before establishing the asymptotic behavior of Dℓ,n and Eℓ,n , define, for any ℓ ∈ Zd ,   d d n ∏ ∏ ∑ 1  uj  , u ∈ [0, 1]d . I{Fj (εj,t+lj ) ≤ uj } − αℓ,n (u) = 1/2 n t=1 j=1 j=1 Theorem 1. Let L be a finite subset of Zd . Under Assumptions (A1)–(A6) and under the null hypothesis of independence between the innovations, the sequence of processes {(αℓ,n , Eℓ,n , Dℓ,n ), ℓ ∈ L} converges jointly in D([0, 1]d ) × D([0, 1]d ) × D([−∞, ∞]d ) to continuous centered Gaussian processes {(αℓ , Eℓ , Dℓ ), ℓ ∈ L}, where Dℓ (x) = Eℓ {F1 (x1 ), . . . , Fd (xd )} and   d ∏ ∑ uk  . βj (uj )  (4) Eℓ (u) = αℓ (u) − j=1 k̸=j In (4), αℓ (·) is a d-dimensional Brownian bridge, i.e., a continuous centered Gaussian process with covariance function given by the following expression: cov(αℓ (u), αℓ (v)) = d ∏ j=1 min(uj , vj ) − d ∏ uj vj , u, v ∈ [0, 1]d , j=1 while β1 (u1 ) = αℓ (u1 , 1, . . . , 1), . . ., βd (ud ) = αℓ (1, . . . , 1, ud ) are (independent) onedimensional Brownian bridges which are also independent of ℓ ∈ L. In Theorem 1, the processes Eℓ are the limiting copula processes of sequences of independent uniform variates on [0, 1]d (Gänßler & Stute, 1987), so they can be easily simulated. However, the main interest of the previous result is the remarkable property that the asymptotic distributions of Dℓ,n and Eℓ,n do not depend on the fact that the model parameters were estimated. This result was obtained in the case of two AR(∞) models by Kim & Lee (2005). The same result holds for the asymptotic distribution of Eℓ,n . Theorem 1 could be used to develop test statistics for testing the null hypothesis of independence. However, it appears that the limiting processes Dℓ and Eℓ admit a complicated covariance structure that appears of limited use for practical applications (with the notable exception d = 2). For example, the covariance function of Dℓ is given in Ghoudi, Kulperger & Rémillard (2001, Corollary 2.1). The covariance between Dℓ (u) and Dℓ′ (v) could be obtained from the limiting covariance between αℓ (u) and αℓ′ (v), which is also complicated, with limited use. In the next section, we consider the Möbius transformations of the empirical processes that offer a serious advantage of simplifying considerably the covariance structure. From these transformed proThe Canadian Journal of Statistics / La revue canadienne de statistique DOI: 2012 9 cesses, it is then possible to propose test statistics which are simpler to study theoretically and easier to implement in practical applications. 2.2. Möbius transformations Following the approaches of Ghoudi, Kulperger & Rémillard (2001) and Genest & Rémillard (2004), the combinatorial formula of Möbius (Spitzer, 1974) can be applied, leading to new transformed processes based on Dℓ,n and Eℓ,n , which admit, however, considerably simpler asymptotic covariances. Furthermore, Genest & Rémillard (2004) and Genest, Quessy, & Rémillard (2007) studied test statistics based on the original and transformed empirical processes using the formula of Möbius. From their results, it appears that the test statistics based directly on the processes Dℓ,n or Eℓ,n display under certain conditions less power than those based on the Möbius transformations of the empirical processes, adding additional arguments in favor of considering the use of these particular transformations. When testing for independence between d series of innovations, it is useful to introduce subsets of Sd = {1, . . . , d}. Each subset A ⊂ Sd of cardinality |A| includes the indices i1 , . . . , i|A| identifying which subsets {εi1 ,t }, . . . , {εi|A| ,t } are actually used for testing independence. Proceeding as in Genest & Rémillard (2004), the Möbius transformation of the processes Hℓ,n and Cℓ,n , indexed by the vector of time lags ℓ ∈ Zd , are given by: RA,ℓ,n (x) = n−1/2 n ∏ ∑ [ { } ] I ej,t+lj ≤ xj − Fjn (xj ) , t=1 j∈A and GA,ℓ,n (u) = n−1/2 n ∏ ∑ [ { } ] I Rj,t+lj ≤ (n + 1)uj − Vn (uj ) , t=1 j∈A ∏ respectively, adopting the convention k∈∅ = 1. When |A| = 1, RA,ℓ,n → 0 and GA,ℓ,n → 0; consequently only sets A satisfying |A| > 1 need to be considered, which is quite natural in our testing framework. For d = 2 time series, this means that only the set A ≡ S2 = {1, 2} is relevant. Interestingly, in that particular case we have the relation RA,ℓ,n = Dℓ,n . Consequently, for bivariate processes there is no need to consider transformed processes. It is remarkable that simplifications occur in the particular case d = 2. That sheds new light on the theoretical results of Kim & Lee (2005) and it explains convincingly why a Möbius transformation is not needed in the particular case d = 2. Finally, it can be easily verified that when ℓ ≡A ℓ′ , that is lj − lj′ = l1 − l1′ for all j ∈ A, then RA,ℓ,n − RA,ℓ′ ,n goes to zero in probability. Before stating the main result of this section, define A = {A ⊂ Sd ; |A| > 1}. Theorem 2. Let L be a finite subset of Zd . Under Assumptions (A1)–(A6) and under the null hypothesis of independence between the innovations, the sequence of processes {GA,ℓ,n , A ∈ A, ℓ ∈ L} converges in D([0, 1]d ) to continuous centered processes {GA,ℓ , A ∈ A, ℓ ∈ L} that are jointly Gaussian, where the covariance function ΓA,ℓ,B,ℓ′ between GA,ℓ and GB,ℓ′ is given by: ΓA,ℓ,B,ℓ′ (u, v) = cov {GA,ℓ (u), GB,ℓ′ (v)} , {∏ ′ j∈A {uj ∧ vj − uj vj } , if B = A and ℓ ≡A ℓ, = 0, otherwise, DOI: The Canadian Journal of Statistics / La revue canadienne de statistique 10 DUCHESNE, GHOUDI AND RÉMILLARD Vol. xx, No. yy where u, v ∈ [0, 1]d . Furthermore, the sequence of processes {RA,ℓ,n , A ∈ A, ℓ ∈ Zd } converges in D([−∞, ∞]d ) to continuous centered processes {RA,ℓ , A ∈ A, ℓ ∈ Zd }, where for x ∈ Rd we have RA,ℓ (x) = GA,ℓ {F1 (x1 ), . . . , Fd (xd )}, and the covariance satisfies: cov {RA,ℓ (x), RB,ℓ′ (y)} = {∏ j∈A {Fj (xj ∧ yj ) − Fj (xj ) Fj (yj )} , if B = A and ℓ′ ≡A ℓ, 0, otherwise. From Theorem 2, it follows that when testing for independence between three or more series of innovations, Möbius transformations offer important simplifications in the covariance structure, compared with the results presented in Theorem 1. In the next section, Theorem 2 is used to define appropriate test statistics for checking independence between the innovations of the models defined by (1). 3. TEST STATISTICS FOR CHECKING INDEPENDENCE BETWEEN SEVERAL TIME SERIES 3.1. Kolmogorov–Smirnov and Cramér–von Mises test statistics Consider the set A ∈ A and the multivariate lag index ℓ ∈ Zd . The Kolmogorov–Smirnov test statistic is defined by: KSA,ℓ,n = sup |RA,ℓ,n (x)| . (5) x∈Rd Using the result stating that supx∈Rd |RA,ℓ,n (x)| = supu∈[0,1]d |GA,ℓ,n (u)| + oP (1), it follows from Theorem 2 that these test statistics converge jointly in distribution to the random variables: KSA,ℓ = sup |GA,ℓ (u)| . (6) u∈[0,1]d However, the calculation of test statistics such as (5) necessitates to take the supremum of |RA,ℓ,n (x)| or |GA,ℓ,n (u)| over a d-dimensional space, which may be particularly difficult to evaluate accurately in practice, even in the simplest case d = 2. Consequently, for convenience reasons, we do not focus on this type of test statistics in the following. Similarly, we can define Cramér–von Mises test statistics, indexed by the set A and the multivariate lag ℓ: CVMA,ℓ,n = ∫ [0,1]d =n −1 G2A,ℓ,n (u)du, n ∑ n ∏ ∑ t=1 s=1 j∈A { { } 2n + 1 Rj,t+lj Rj,t+lj − 1 + 6n 2n(n + 1) (7) (8) { } )} ( Rj,s+lj Rj,s+lj − 1 max Rj,t+lj , Rj,s+lj + . − 2n(n + 1) n+1 Note that a natural test statistic for checking independence can be based on the integrated measure: ∫ R2A,ℓ,n (x)dF1n (x1 ) · · · dFdn (xd ). Rd The Canadian Journal of Statistics / La revue canadienne de statistique DOI: 2012 11 This proposal reduces essentially to a statistic studied in Kim & Lee (2005). However, that test statistic turns out to be asymptotically equivalent to the Cramér–von Mises test statistic CVMA,ℓ,n . The asymptotic distribution of (7) follows easily from Theorem 2, since these test statistics are continuous functionals of the empirical processes GA,ℓ,n . That result is precisely stated in the following proposition. Proposition 1. Under the conditions given in Theorem 2, and under the null hypothesis of independence, the sequences {CVMA,ℓ,n , A ∈ A, ℓ ∈ Zd } converge in distribution to random variables {CVMA,ℓ , A ∈ A, ℓ ∈ Zd } which are defined as: CVMA,ℓ = ∫ [0,1]d G2A,ℓ (u)du. (9) The random variables CVMA,ℓ and CVMB,ℓ are independent if A ̸= B or if A = B and ℓ′ ̸≡A ℓ. Moreover, CVMA,ℓ = CVMA,ℓ′ if ℓ′ ≡A ℓ. Theorem 2 facilitates grandly the construction of omnibus test statistics, since a large number of asymptotically independent random variables can be constructed. Interestingly, for any A ∈ A of cardinality |A|, CVMA,ℓ shares the same distribution as that of the random variable: ∑ ξ|A| = (i1 ≥1,...,i|A| ≥1) 1 Z2 , π 2|A| (i1 · · · i|A| )2 i1 ,··· ,i|A| (10) where Zi1 ,...,i|A| are independent N (0, 1) random variables. That alternative representation has important practical consequences, since the critical values of the distribution ξ|A| can be determined. More computational details are provided in Ghoudi, Kulperger & Rémillard (2001), amongst others. However, the convergence to ξ|A| can be quite slow. Thus, in practice, it is suggested to compute P -values for CV MA,ℓ,n using Monte Carlo methods, by generating N = 5000 independent permutations of the ranks for a given sample size. Then P -values for all ℓ and all set A with the same cardinality can be computed at the same time. In our case, to speed up computations in the simulation experiments, we constructed tables for n = 100, 300, using N = 100, 000 replications. 3.2. Generalized cross-correlations Residual cross-correlations between two time series have been found useful in practical applications to check adequacy of time series. The concept can be generalized to an arbitrary number of time series as follows. For any set A ∈ A, ℓ ∈ Zd , define: n1/2 γ̂A,ℓ,n = (−1)|A| ∫ RA,ℓ,n (x(A) )dx(A) = n−1/2 R|A| n ∏ ∑ ( ) ej,t+lj − ēj , (11) t=1 j∈A where et,n ≡ et = (e1t , . . . , edt )⊤ , t = 1, . . . , n, are the residuals, ēj = ∑nwe recall that −1 (A) n is the vector of dimension |A| whose components are those of x lot=1 ejt , and x cated at positions given in A. These generalized cross-correlations are appropriate measures of the linear dependence between the residual time series. Note that the identity (11) follows from Hoeffding’s formula. See also Ghoudi, Kulperger & Rémillard (2001) for additional details. When A = {j, k}, lj = 0, lk = l, then γ̂A,ℓ,n reduces ∑n to the sample lag-l cross-covariance between the jth and kth time series γ̂{j,k},l,n = n−1 t=1 (ejt − ēj )(ek,t+l − ēk ). Thus the generalized cross-correlations enjoy all the usual properties shared by the classical correlations. From DOI: The Canadian Journal of Statistics / La revue canadienne de statistique 12 DUCHESNE, GHOUDI AND RÉMILLARD Vol. xx, No. yy this, we can define cross-correlation coefficients indexed by a set A ∈ A and the multivariate time lag ℓ ∈ Zd : γ̂A,ℓ,n r̂A,ℓ,n = ∏ , j∈A sj,n (12) ∑n where s2j,n = n−1 t=1 e2jt converges in probability to one. Having introduced the generalized cross-correlations (12), it is natural to derive their asymptotic behavior. However, the mapping RA,ℓ,n 7→ n1/2 γ̂A,ℓ,n is not continuous and as a result, the arguments leading to the asymptotic behavior of the Kolmogorov–Smirnov or Cramér–von Mises test statistics do not apply directly. Fortunately, it can be approximated by continuous mappings, by restricting the integration over compact sets. Since the second moment of the innovation process {εt } exists, it can be shown that the difference between the two expressions can be made arbitrarily small. For additional details and an idea of the proof in a different context, see Genest & Rémillard (2004). The following result is then a direct consequence of Theorem 2. Proposition 2. Under the same conditions as in Theorem 2 and under the null hypothesis of independence, the sequences {n1/2 r̂A,ℓ,n , A ∈ A, ℓ ∈ Zd } converge in distribution to standard Gaussian variables {rA,ℓ , A ∈ A, ℓ ∈ Zd } with the covariance structure: cov (rA,ℓ , rB,ℓ′ ) = { 1, if A = B and ℓ′ ≡A ℓ, 0, otherwise. Proposition 2 can be used to test the null hypothesis of zero (generalized) cross-correlation. In the special case |A| = 2 and for classical cross-correlations between the residuals of two time series, the proposition provides an alternative proof and sheds new light on a well-known result of Haugh (1976). In fact, we show that its theorem remains valid under considerably more general assumptions. When |A| > 2, some properties shared by the cross-correlations are lost in general. For example, the generalized cross-correlations are not necessarily bounded by one in absolute value, which may complicate their interpretation in practice. Note that in our testing framework that last property is far from being critical: under the null hypothesis of independence the gener2 , say, we reject the alized cross-correlation is null, and using the asymptotic distribution of nr̂A,ℓ,n 2 null hypothesis if that test statistic is larger than the quantile χ1,1−α , where χ21,1−α , corresponds to the 1 − α quantile from a χ2 distribution with one degree of freedom. However, the generalized cross-correlations suffer of the same limitations than classical cross-correlations: these quantities adequately measure linear dependence, and consequently small generalized crosscorrelations do not imply small dependence. The simulation results in Section 4 demonstrate that the test statistics based on generalized cross-correlations may be powerful for Gaussian processes, suggesting the merits of generalized cross-correlations in applied work. In general, the Cramér–von Mises test statistics introduced in the previous section may be more appropriate if an analyst desires to test independence, when the marginal distribution of the innovations displays fatter tails than the Gaussian distribution. These important issues will be discussed in the Monte Carlo experiments presented in Section 4. 3.3. Non-parametric measures Non-parametric measures of dependence are often used in the time series literature. See Hallin & Puri (1992) for a survey. To our knowledge, the principal non-parametric approach to test independence between two time series is the one of Hallin et al. (1999) valid for autoregressive time series models. In this section, we consider non-parametric test statistics for testing the null The Canadian Journal of Statistics / La revue canadienne de statistique DOI: 2012 13 hypothesis of independence between the innovations of two or more time series. These rank test statistics are expected to offer more robustness than their parametric counterparts. Let G1 , . . . , Gd be distribution functions with finite variance and left-continuous inthe sample means and samverse Jj (uj ) = G−1 j (uj ). For j = 1, . . . , d we introduce ∑n ple variances of these new variables, that is J¯j = n−1 t=1 Jj (t/(n + 1)) and σJ2j n = }2 ∑n { n−1 t=1 Jj (t/(n + 1)) − J¯j , j = 1, . . . , d. A rank-based measure of dependence is naturally defined by: } { ∑n ∏ ¯ t=1 j∈A Jj (Rj,t+lj /(n + 1)) − Jj −1 ∏ ρA,ℓ,n = n , j∈A σJj,n and it can be written as a linear function of the process GA,ℓ,n . As a result, the following proposition can be deduced from Genest & Rémillard (2004) and Theorem 2. Proposition 3. Under the same conditions as in Theorem 2 and under the null hypothesis of independence, the random variables {n1/2 ρA,ℓ,n , A ∈ A, ℓ ∈ Zd } converge jointly in distribution to centered Gaussian random variables {SA,ℓ , A ∈ A, ℓ ∈ Zd } with the following covariance structure: { 1, if A = B and ℓ′ ≡A ℓ, cov (SA,ℓ , SB,ℓ′ ) = 0, otherwise. Several possibilities for the Jj (·) functions lead to dependence measures with their own merits. For example, Jj (u) = u, j = 1, . . . , d allows us to define Spearman rho test statistics. In that situation, Gj represents the distribution function of the uniform random variable on the interval (0, 1) and n1/2 ρA,ℓ,n = n−1/2 ( 12(n + 1) n−1 )|A|/2 ∑ n ∏ ( t=1 j∈A Rj,t+lj 1 − n+1 2 ) . Another popular example is obtained by considering Jj (u) = Φ−1 (u), j = 1, . . . , d, leading to the so-called van der Waerden rho statistic. In this case Gj = Φ corresponds to the distribution function of the standard normal random variable. Consequently, the test statistic is given by: ( ) ∑n ∏ −1 Rj,t+lj /(n + 1) t=1 j∈A Φ 1/2 −1/2 ∏ . n ρA,ℓ,n = n j∈A σJj,n When |A| = 2, that test statistic is similar in spirit to the rank-based correlation coefficient studied in Hallin et al. (1999). Note that as for the generalized cross-correlations, the rank-based dependence measures are appropriate for detecting linear dependence. Based on Theorem 2, several test statistics have been proposed, indexed by A and the lag ℓ. These one-lag test statistics are all asymptotically independent, suggesting that the different sources of dependence can be separated adequately. In practice, these test statistics need to be computed for several lags, and a natural question concerns how to combine the one-lag test statistics. This is discussed in the next section. 3.4. Combining test statistics For a given lag, a test statistic can detect a particular source of dependence but will not be consistent for all alternatives. On the other hand, if an unnecessarily large number of one-lag test DOI: The Canadian Journal of Statistics / La revue canadienne de statistique 14 DUCHESNE, GHOUDI AND RÉMILLARD Vol. xx, No. yy statistics are considered, inefficient procedures are typically obtained. These considerations are well-known with portmanteau test statistics such as the popular Box-Pierce-Ljung test statistic or with the consistent test statistics of Hong; see, e.g., Hong (1996) and Li (2004). Consequently, several one-lag test statistics need to be computed and a natural question concerns how to combine these dependence measures efficiently in order to obtain powerful omnibus procedures. For classical portmanteau test statistics and for testing independence between two series of innovations, the sum of squared cross-correlations is typically considered, for lags −M, . . . , M , where M denotes the maximum lag order. The choice of M is left to the analyst. Alternative approaches are possible, such as those relying on spectral density methods (see Hong, 1996, 1998); with those test statistics, smoothing parameters or maximum lag orders need also to be specified. We discuss here how to combine one-lag test statistics. Inspired by the pioneer work of Fisher (1950, pp. 99-101), Littell & Folks (1973) proposed a way to combine the P −values of asymptotically independent test statistics. Under certain conditions, they developed a combination rule which is asymptotically optimal in the Bahadur sense. When testing for independence and randomness, Genest & Rémillard (2004) proposed a way to combine optimally the P -values of the considered test statistics using the fact that their test statistics are asymptotically independent. They demonstrated that their approach leads to powerful test procedures. Here, we apply these ideas for testing independence between the innovations of several time series. More formally, let TA,ℓ,n , A ∈ A, be test statistics, indexed by the multivariate time lag ℓ ∈ Zd , so that they jointly converge to TA,ℓ , where TA,ℓ and TB,ℓ′ are independent if A ̸= B, or A = B and ℓ̸≡A ℓ′ . Many test statistics studied in the previous section could be considered. For ease of exposition, we focus on Cramér–von Mises test statistics and generalized crosscorrelations. Let pA,ℓ,n = Pr(ζ|A| > TA,ℓ,n ) be the P -value of the test statistics TA,ℓ,n , A ∈ A, ℓ ∈ Zd , where ζ|A| represents the asymptotic distribution of TA,ℓ,n . For each subset A ∈ A, a finite set of multivariate lags DA can be created so that for any ℓ, ℓ′ ∈ DA one has ℓ ≡A ℓ′ if and only if ℓ = ℓ′ . If there is no a priori knowledge on which lags could explain the sources of dependence, it is natural to impose DA = DB if |A| = |B|. For example, when d = 3, and A = {1, 2}, {1, 3} or {2, 3} one takes DA = {(0, l); |l| ≤ M2 }, for a given maximum lag order M2 . We first define test statistics based on the Cramér–von Mises statistics. Let A be an arbitrarily set such that A ∈ A. Define: ∑ {CVMA,ℓ,n − B(n, |A|)}, (13) WA,n = ℓ∈DA where B(n, |A|) represents a bias term; in fact, it corresponds to the difference between the expectations of CVMA,ℓ,n and ξd when |A| = d and it is given by: B(n, d) = ( n−1 6n )d − 1 + (n − 1) 6d ( −1 6n )d . (14) Based on the P -values of CVMA,ℓ,n , we also define: FA,n = −2 ∑ log (pA,ℓ,n ) . (15) ℓ∈DA Under the null hypothesis of independence, it follows that the test statistics WA,n , A ∈ A jointly converge in distribution to independent variables WA which have the same distribution as the sum of |DA | independent copies of ξ|A| , while the test statistics FA,n , A ∈ A jointly converge The Canadian Journal of Statistics / La revue canadienne de statistique DOI: 2012 15 in distribution to independent variables FA having a chi-square distribution with 2|DA | degrees of freedom. In order to define test statistics based on generalized cross-correlations, note that the relation log{1 − Φ(a)} = −a2 /2{1 + o(1)} for large a, as shown in Bahadur (1960), demonstrates that the general approach which consists in combining the P -values reduces essentially to Haugh’s test statistic. That suggests defining: ∑ 2 r̂A,ℓ , A ∈ A. (16) HA,n = n ℓ∈DA Under the null hypothesis of independence, it follows that the test statistics HA,n , A ∈ A jointly converge in distribution to independent variables HA having a chi-square distribution with |DA | degrees of freedom. Interestingly, the general omnibus test statistic can depend on sets A of different cardinality. For example, when testing between the innovations of three time series, it appears desirable to include sets of size |A| = 2, 3. Inspired by the work of Genest, Quessy, & Rémillard (2007) on local efficiency for multivariate data, the combined test statistics are defined by: ∑ π 2(|A|−2) WA,n , (17) Wn = A∈A Fn = ∑ FA,n = −2 ∑ HA,n = n A∈A log (pA,ℓ,n ) , (18) A∈A ℓ∈DA A∈A Hn = ∑ ∑ ∑ ∑ 2 r̂A,ℓ . (19) A∈A ℓ∈DA Since ∑ the random variables WA , A ∈ A, are independent, the test statistic Wn converges in law to A∈A π 2(|A|−2) WA , under the null hypothesis of independence. As a result, when the number of variables CVMA,ℓ is large, the Edgeworth expansion (using the first six cumulants) is expected to yield a satisfactory approximation of the P -value of Wn . That is confirmed in the Monte Carlo simulations presented in Section 4, at least for the chosen experiments. Similarly, under the null hypothesis of independence, the test statistic Fn converges in distribution towards a chi-square ∑ distribution with 2 A∈A |DA | degrees ∑of freedom, while the test statistic Hn converges in law towards a chi-square distribution with A∈A |DA | degrees of freedom. 3.5. Graphical representation: The dependogram and the cross-correlogram When testing for independence between two time series, a natural graphical representation consists in representing the cross-correlations r̂{1,2},ℓ,n versus the lag order l, where ℓ = (0, l)⊤ , with |l| ≤ M2 . That graphical device makes possible to identify visually the lags leading to the rejection of the null hypothesis of independence between the two series of innovations. That approach is in the line of considering one-lag test statistics; see, e.g., Ansley & Newbold (1979), McLeod (1978) and Duchesne & Roy (2003), amongst others. In the more general situation of several time series, the ideas leading to the so-called dependogram, originally proposed by Genest & Rémillard (2004), can be generalized in our context. We focus on the case of d = 3 time series, but the extension to higher dimensions does not cause additional conceptual difficulties. First, three graphs representing the possible subsets of |A| = 2 elements can be constructed. Second, an additional graph for the situation |A| = 3 is necessary. Note that for d > 3 a lexicographical ordering by size may be necessary, starting with |A| = 2, 3, 4, . . .; see Genest & Rémillard (2004) for additional details. In the case |A| = 2, the lags |l| ≤ M2 determine the x-axis, giving a total of 2M2 + 1 lags for the three graphs, and the DOI: The Canadian Journal of Statistics / La revue canadienne de statistique 16 DUCHESNE, GHOUDI AND RÉMILLARD Vol. xx, No. yy corresponding P -values pA,ℓ,n associated with the test statistics TA,ℓ,n , ℓ ∈ DA , A ∈ A, are then represented by vertical bars. Cross-correlograms can also be considered. In the case |A| = 3, the multivariate lags ℓ = (0, l2 , l3 ), |l2 | ≤ M3 , |l3 | ≤ M3 can be considered, giving a total of (2M3 + 1)2 lags. In order to detect lags ℓ for which the null hypothesis of independence is rejected, an horizontal line corresponding to the 5% nominal level can also be added to the dependogram. For the cross-correlograms, the lines ±2/n1/2 can be represented in the graph, and rejection of the null hypothesis is decided when cross-correlations are outside these significance limits. That practice is often advocated in the time series literature when examining residual autocorrelations (see, e.g., Brockwell & Davis, 2002). Note that in Genest & Rémillard (2004) the test statistics are given, not the P -values, but these two graphical tools display essentially the same information. Other specialized graphs can be constructed. For example, it may be informative to represent the P -values of the statistics WA,n in terms of the subsets A ∈ A. This kind of graphs may be particularly useful when the dimension d is large, and can be used to detect departure from independence in particular subsets of the data. Finally, another possibility consists in plotting the P -values of the test statistics TA,ℓ,n , in terms of a selected subset of lags, which can be used to find the specific lags explaining the dependence. 4. SIMULATION EXPERIMENTS In the previous section we introduced one-lag and portmanteau test procedures, which should prove useful in testing for independence between several time series. It is natural to inquire for the finite sample properties of the proposed methods. Here, we present the empirical results conducted in order to study their exact levels and powers when testing independence between the innovations of two and three time series. The test statistics Wn , Fn and Hn were included in the experiments described below. In order to compute the test procedures, the set D{1,2} = {(0, j); |j| ≤ M2 } with M2 = 5 has been used when testing independence between two time series. In the case of three time series, the sets D{1,2} = D{1,3} = D{2,3} = {(0, j); |j| ≤ M2 } and D{1,2,3} = {(0, j, k); |j| ≤ M3 , |k| ≤ M3 } were considered with M2 = 5 and M3 = 2. Note that in the experiments described in the next subsections, the true models under the null hypothesis of independence have been fitted. In the simulation experiments and in the application with real data in Section 5, the R software has been used. Computer codes written in C have been implemented for the new methods and interfaced with the R environment. GARCH models have been fitted using the fGarch package available from CRAN. All the computer codes are freely available from the authors. 4.1. Test of independence between the innovations of two series When testing for independence between two time series Xt and Yt , t = 1, . . . , n, four DGPs were simulated. The processes {Xt } and {Yt } have been chosen such that the degrees of dependence vary between the lagged innovations. The DGPs are defined as follows: DGP1 : Xt = 0.5Xt−1 + εt , Yt = 0.5Yt−1 + ηt ; DGP2 : Xt = 0.5Xt−1 + εt + 0.5ε ( t−1 , Yt = 0.5Y ) t−1 + ηt + 0.5ηt−1 ; DGP3 : Xt = h(t εt , h2t = 10−6) + 0.6 + 0.2ε2t−1 h2t−1 , Yt = vt ηt , 2 2 vt2 = 10−6 + 0.6 + 0.2ηt−1 vt−1 ; ( ) DGP4 : Xt = 0.5Xt−1 + ht εt + 0.5ht−1 εt−1 , h2t =(10−6 + 0.6 +) 0.2ε2t−1 h2t−1 , 2 2 vt−1 Yt = 0.5Yt−1 + vt ηt + 0.5vt−1 ηt−1 , vt2 = 10−6 + 0.6 + 0.2ηt−1 . Thus, the processes generated under DGP1 and DGP2 fall in the class of ARMA models. For the third scenario DGP3 , pure GARCH(1,1) processes were considered. In the last scenario, The Canadian Journal of Statistics / La revue canadienne de statistique DOI: 2012 17 DGP4 is composed of ARMA-GARCH(1,1) models. In these four cases, the innovations {εt } and {ηt } were iid white noises with zero means and unit variances. Under the null hypothesis H0 , the innovations of the two time series are independent, while under the alternatives H1 , the dependence between εt and ηt+l , is modelled by a bivariate copula in the Gaussian, Clayton and Tent map families; we considered l ∈ {0, 2}. For the Gaussian and the Clayton copulas, the Kendall’s tau is such that τ ∈ {0.1282, 1/3}. For the Gaussian copula with Gaussian marginals, these two values of τ correspond to correlations ρ between εt and ηt+l equal to ρ = 0.2 and ρ = 0.5, respectively. For the sake of comparison, the same autoregressive and moving average parameters than those of Kim & Lee (2005) were included in our experiments. Under the alternative hypothesis, we retained also the same correlation parameters. Note that in Kim & Lee (2005) the true models were also estimated. The results are reported in Table 1 for DGP1 , for the sample sizes n = 100 and 300. It appears that the empirical levels of the three test statistics are reasonably close to the nominal levels, even for a sample size as low as n = 100. The test statistics Wn and Fn have rather comparable empirical powers. Interestingly, when the dependence structure of the innovations exhibits correlation, the test statistics Hn reach high power and dominate the procedures Wn and Fn , confirming the importance of cross-correlations in empirical work. For dependent non-correlated innovations, such as the Tent map alternative, the test statistics Hn do not attain large power, but Wn and Fn remain very powerful. TABLE 1: Rejection levels under DGP1 for the test statistics Wn , Fn and Hn defined by (17), (18) and (19), respectively, with D{1,2} = {(0, j); |j| ≤ M2 }, M2 = 5. Both DGPs in DGP1 have standard normal errors with the specified joint copula for different Kendall’s tau. The Monte Carlo results rely on 10000 iterations. Copula of (εt , ηt )⊤ n 100 300 Test statistics Independence Wn 4.32 Normal Clayton Tent map 0.1282 1/3 0.1282 1/3 13.81 88.90 15.10 89.72 99.96 Fn 4.31 13.21 84.85 14.20 85.93 99.63 Hn 4.80 18.95 97.09 21.27 95.80 8.33 Wn 4.54 48.71 100.00 50.34 100.00 100.00 Fn 4.87 46.19 100.00 47.51 100.00 100.00 Hn 4.32 62.79 100.00 66.03 100.00 8.29 Cross-correlations are expected to be powerful for Gaussian innovations and linear dependence. In order to explore that issue, a complementary experiment for the AR(1) processes has been conducted to appreciate the potential effects of the marginal distributions of the innovations on the behavior of the proposed methods. To this aim, we choose the Clayton model to describe the dependence and several marginal distributions for the random variables εt and ηt have been studied. In addition to the Gaussian white noise given in Table 1, we generated centered standard exponential and the centered Pareto with tail parameter equal to six, that is, we considered G(x) = 1 − (x + 6/5)−6 , x ≥ −1/5. The results for these last two situations are provided in Table 2 for the sample sizes n = 100, 300. Interestingly, the empirical results show that the empirical powers of Wn and Fn are robust to the choice of the marginal distributions: very similar results for Wn and Fn are displayed in Tables 1 and 2. This is not the case for the test statistic Hn , which is heavily affected by the marginal distributions. In fact, the empirical powers appeared inferior to those of Wn and Fn , and they deteriorated when the tails of the innovation distributions became heavier. Thus, test statistics based on cross-correlations are expected to be DOI: The Canadian Journal of Statistics / La revue canadienne de statistique 18 DUCHESNE, GHOUDI AND RÉMILLARD Vol. xx, No. yy consistent only under linear dependence, and powerful for Gaussian processes. These results suggest that test statistics based on cross-correlations should be used with care when non-linear dependence structure and non-Gaussian innovations are suspected. TABLE 2: Rejection levels under DGP1 for the test statistics Wn , Fn and Hn defined by (17), (18) and (19), respectively, with D{1,2} = {(0, j); |j| ≤ M2 }, M2 = 5. The innovations in DGP1 follow a Clayton copula with different Kendall’s tau and different marginal distributions. The Monte Carlo results rely on 10000 iterations. Marginal distributions Exponential Pareto Kendall’s tau Kendall’s tau n Test statistics 0.1282 1/3 0.1282 1/3 100 Wn 14.38 87.54 13.69 85.29 300 Fn 13.54 83.03 13.05 80.88 Hn 9.67 44.09 9.80 31.49 Wn 48.41 100.00 48.58 100.00 Fn 45.27 100.00 45.69 100.00 Hn 18.67 95.44 14.65 74.17 To study the impact that the nature of the DGPs has on the behavior of the proposed test statistics, DGP2 , DGP3 and DGP4 were simulated. Only sample size n = 100 and Gaussian marginal innovations were considered. The test statistics were applied to the residuals of the fitted models. For the ARMA model we considered Gaussian, Clayton and Tent map dependence copulas. For the GARCH and ARMA-GARCH models we only simulated dependence following the Gaussian copula. The summarized results are presented in Tables 3–4. Generally, the empirical levels and powers are not affected by the DGPs and are thus robust to the model assumptions. These findings were expected and they illustrate the theoretical results given in Theorems 1 and 2: The asymptotic behaviors of the test statistics do not depend on the model parameters and should be the same whether the parameters are estimated or assumed to be known. TABLE 3: Rejection levels under DGP2 for the test statistics Wn , Fn and Hn defined by (17), (18) and (19), respectively, with D{1,2} = {(0, j); |j| ≤ M2 }, M2 = 5. Both ARMA models have standard normal errors with the specified joint copula for different Kendall’s tau. The Monte Carlo results rely on 10000 iterations. Copula of (εt , ηt )⊤ n 100 Test statistics Independence Wn 4.04 Normal Clayton Tent map 0.1282 1/3 0.1282 1/3 13.34 87.82 13.93 90.05 99.45 Fn 4.12 12.81 83.17 13.13 85.91 98.15 Hn 5.03 18.63 96.57 19.95 95.99 7.74 We analyzed the effect of the lagged dependence, when the processes {Xt } and {Yt } are generated according to AR(1) processes with standard Gaussian innovations, but such that the dependence between εt and ηt+2 is governed by the specified copula. Table 5 summarizes the findings. The results are quite similar to those presented in Tables 1 and 3. This is explained because the test statistics rely on M2 = 5, and thus they are consistent for alternatives involving The Canadian Journal of Statistics / La revue canadienne de statistique DOI: 2012 19 TABLE 4: Rejection levels under DGP3 and DGP4 for the test statistics Wn , Fn and Hn defined by (17), (18) and (19), respectively, with D{1,2} = {(0, j); |j| ≤ M2 }, M2 = 5. The random vector (εt , ηt )⊤ has a bivariate normal distribution with degree of dependence specified by Kendall’s tau. The Monte Carlo results rely on 1000 iterations. DGP3 n Test statistics 100 DGP4 Kendall’s tau Kendall’s tau 0 0.1282 1/3 0 0.1282 1/3 Wn 4.5 16.4 89.9 4.5 13.9 87.2 Fn 5.0 15.9 90.2 7.2 19.3 88.3 Hn 5.5 21.1 95.4 5.2 18.1 94.4 delays up to five temporal lags. TABLE 5: Rejection levels under DGP1 for the test statistics Wn , Fn and Hn defined by (17), (18) and (19), respectively, with D{1,2} = {(0, j); |j| ≤ M2 }, M2 = 5. Both DGPs have standard normal errors with the specified joint copula for different Kendall’s tau. The Monte Carlo results rely on 10000 iterations. Copula of (εt , ηt+2 )⊤ n 100 Test statistics Independence Wn 4.04 Normal Clayton Tent map 0.1282 1/3 0.1282 1/3 13.69 87.84 14.27 88.89 99.91 Fn 4.05 12.89 83.20 13.36 84.70 99.20 Hn 4.54 18.50 96.13 20.69 94.74 7.87 4.2. Test of independence between the innovations of three time series When testing for independence between three time series Xt , Yt and Zt , t = 1, . . . , n, four DGPs were studied. Under DGP5 and DGP6 , only raw data were used, relying on our theoretical results on the robustness to the fitted model. For finite n and in a three time series setting, we investigated under DGP7 and DGP8 the potential effects of model estimation: AR(1) models were simulated for each time series. More precisely, the DGPs are defined in the following: DGP5 : Xt = εt , Yt = ηt , Zt = υt ; DGP6 : Xt = |εt |sign(ηt υt ), Yt = ηt , Zt = υt ; DGP7 : Xt = 0.5Xt−1 + εt , Yt = 0.5Yt−1 + ηt , Zt = 0.5Zt−1 + υt ; DGP8 : Xt = 0.5Xt−1 + |εt |sign(ηt υt ), Yt = 0.5Yt−1 + ηt , Zt = 0.5Zt−1 + υt . The processes {εt }, {ηt } and {υt } were iid standard Gaussian white noises. Under DGP5 , the copula corresponding to independence is chosen and the three processes are mutually independent. Under DGP6 , a specific alternative has been generated. Under that alternative, {Xt }, {Yt } and {Zt } are pairwise but not jointly independent (Romano & Siegel, 1986); we call this the Romano-Siegel scenario. That scenario is an adaptation to a time series framework of the well-known result stating that pairwise independence is not equivalent to global independence. Under DGP5 and DGP6 , the sample size n = 100 has been selected. Under DGP7 , three AR(1) time series models were generated. For the random vector (εt , ηt , υt )⊤ , we considered the mulDOI: The Canadian Journal of Statistics / La revue canadienne de statistique 20 DUCHESNE, GHOUDI AND RÉMILLARD Vol. xx, No. yy tivariate normal distribution with the following covariance matrix:  1 ρ12 ρ13    Σ =  ρ12 1 ρ23  ρ13 ρ23 1 2 {arcsin(ρ12 ) + arcsin(ρ13 ) + arcsin(ρ23 )}. For which yields a Kendall’s tau equals to τ = 3π simplicity, we chose ρ ≡ ρ12 = ρ13 = ρ23 . We considered the case of independence ρ = 0, and the alternatives ρ ∈ {0.2, 0.5}. Thus τ ∈ {0, 0.1282, 1/3}. We also used the multivariate Clayton copula with generator ψ(t) = (1 + θt)−1/θ , whose Kendall’s tau for any dimension d is given in Genest, Nešlehová, & Ben Ghorbal (2011) and is equal to: τ= 1 2d−1 − 1 { d 2 d−1 ∏( p=0 1 + pθ 2 + pθ ) } −1 . We also considered the case where the innovations follow jointly a Clayton copula with different Kendall’s tau and different marginal distributions. More precisely, we investigated as in Section 4.1 the centered exponential and Pareto distributions. Under DGP8 , AR(1) time series models were considered but the innovations satisfied the Romano-Siegel scenario similar as the one under DGP6 . Sample sizes n = 100 and n = 300 have been considered under DGP7 and DGP8 . TABLE 6: Rejection levels under DGP5 and DGP6 with three time series, for the test statistics Wn , Fn , Hn defined by (17), (18) and (19), respectively, with D{1,2} = D{1,3} = D{2,3} = {(0, j); |j| ≤ M2 } and D{1,2,3} = {(0, j, k); |j| ≤ M3 , |k| ≤ M3 }, with (M2 , M3 ) = (5, 2); and also W{1,2},n , F{1,2},n and H{1,2},n . The sample size is n = 100. The Monte Carlo results rely on 10000 iterations. Test statistics DGP Wn Fn Hn W{1,2},n F{1,2},n H{1,2},n DGP5 4.1 3.74 6.35 4.7 4.68 5.15 DGP6 99.6 96.01 66.96 4.51 4.43 5.13 From the results presented in Table 6, the empirical levels for the test statistics W{1,2},n , F{1,2},n and H{1,2},n were reasonably close to the nominal levels under the null hypothesis of independence (DGP5 ) and under the alternative (DGP6 ). That finding was expected, since {Xt } and {Yt } are independent under both DGPs. Interestingly, the test statistics Wn and Fn reach more empirical powers in detecting that particular form of dependence than the test statistics Hn based on correlation measures. We now discuss the results given in Table 7. The results in the case of independence shows that the empirical levels are reasonably close to the nominal levels for the three test statistics Wn , Fn and Hn . When comparing the empirical powers in the normal and Clayton cases, empirical behaviors similar as the ones observed in the bivariate case corresponding to DGP1 have been observed, see Table 1. Under the Romano-Siegel scenario in DGP8 , Wn and Fn were more powerful than Hn when n = 100, as observed under DGP6 , see Table 6. We now discuss the results given in Table 8. Interestingly, the empirical levels and powers of Wn and Fn display robustness to the marginal distributions, since the results are largely similar to those observed in Table 7. However, the empirical powers of Hn were inferior: as in the bivariate case, test statistics relying on cross-correlation measures are expected to be powerful only under normality. Under the Romano-Siegel scenario, Hn seemed less powerful under DGP8 than under DGP6 . Note The Canadian Journal of Statistics / La revue canadienne de statistique DOI: 2012 21 that in unreported results we simulated ARMA(1,1) models using n = 100 and the same model parameters than in DGP2 . We used the same assumptions for the joint copula of the innovations (εt , ηt , ξt )⊤ than under DGP7 and DGP8 . The empirical results were similar to those reported in Table 7. TABLE 7: Rejection levels under DGP7 and DGP8 for the test statistics Wn , Fn and Hn defined by (17), (18) and (19), respectively, with M2 = 5 and M3 = 2. All the DGPs in DGP7 have standard normal errors with the specified joint copula for different Kendall’s tau. The Monte Carlo results rely on 10000 iterations. Copula of (εt , ηt , ξt )⊤ DGP7 n 100 300 Test statistics Independence Wn 3.59 DGP8 Normal Clayton Romano-Siegel 0.1282 1/3 0.1282 1/3 scenario 18.1 98.05 19.73 98.46 98.99 Fn 3.38 16.64 96.67 17.88 97.17 93.56 Hn 5.65 29.05 99.87 40.04 99.72 63.26 Wn 4.74 67.58 100 72.49 100 100 Fn 6.69 68.62 100 73.14 100 99.99 Hn 5.74 83.78 100 90.21 100 100 TABLE 8: Rejection levels under DGP7 with three time series for the test statistics Wn , Fn and Hn defined by (17), (18) and (19), respectively, with M2 = 5 and M3 = 2. The innovations (εt , ηt , ξt )⊤ follow a Clayton copula with different Kendall’s tau and different marginal distributions. The Monte Carlo results rely on 10000 iterations. Marginal distributions Exponential Pareto Kendall’s tau Kendall’s tau n Test statistics 0 0.1282 1/3 0 0.1282 1/3 100 Wn 4.49 18.98 97.45 4.41 19.14 96.67 Fn 4.2 17.39 95.82 4.09 17.69 94.54 300 Hn 11.11 17.64 60.78 13.26 17.67 42.81 Wn 4.9 71.03 100 4.94 71.62 100 Fn 6.89 72.26 100 6.87 72.62 100 Hn 9.21 44.97 99.56 11.91 28.23 86.05 Overall, the results suggest that the empirical levels were reasonably close to the nominal levels. As in the bivariate situation, the test statistics Wn and Fn offered high power for the considered alternatives. The test statistic Hn may also prove to be useful under normality and linear forms of dependence. These results demonstrate that the proposed test statistics offer power for complicated forms of dependence when testing for independence between three time series. To test for independence between more than two time series may be complicated to visualize in empirical work and graphical devices may help. The use of the dependogram may well serve that purpose and we illustrate its use here, based on a single realization of DGP6 . The P -values of the combined test statistics for this particular realization are given in Table 9. The associated dependogram using the test statistics WA,n is shown in Figure 1. The cross-correlograms are DOI: The Canadian Journal of Statistics / La revue canadienne de statistique 22 DUCHESNE, GHOUDI AND RÉMILLARD Vol. xx, No. yy given in Figure 2. The graphs demonstrate that the dependence is detected with sets of cardinality three and not with those with cardinality two. The graphs demonstrate convincingly that there is instantaneous dependence between {Xt }, {Yt } and {Zt }. TABLE 9: P -values (in percentage) from a single realization of {(Xt , Yt , Zt )⊤ = (|εt |sign(ηt υt ), ηt , υt )⊤ }. The sample size is n = 100. Test statistics included: WA,n , FA,n and HA,n for the specified sets A, and the combined test statistics Wn , Fn , Hn defined by (17), (18) and (19), respectively. A Test statistics Combined {1, 2} {1, 3} {2, 3} {1, 2, 3} test statistics 64.85 29.73 62.75 0.00 0.003 FA,n 64.1 26.76 64.31 0.00 0.006 HA,n 34.15 50.03 76.15 0.37 4.02 WA,n Dependogram of CVM{1,3} tests 0.2 0.4 P−value of CVM 0.6 0.4 0.0 0.0 0.2 P−value of CVM 0.6 0.8 0.8 Dependogram of CVM{1,2} tests −4 −3 −2 −1 0 1 2 3 4 5 −5 −4 −3 −2 −1 0 1 2 3 Lags Lags Dependogram of CVM{2,3} tests Dependogram of CVM{1,2,3} tests 4 5 0.6 −5 −4 −3 −2 −1 0 Lags 1 2 3 4 5 { −2 , −2 } { −2 , −1 } { −2 , 0 } { −2 , 1 } { −2 , 2 } { −1 , −2 } { −1 , −1 } { −1 , 0 } { −1 , 1 } { −1 , 2 } { 0 , −2 } { 0 , −1 } {0,0} {0,1} {0,2} { 1 , −2 } { 1 , −1 } {1,0} {1,1} {1,2} { 2 , −2 } { 2 , −1 } {2,0} {2,1} {2,2} 0.0 0.0 0.2 0.4 P−value of CVM 0.4 0.2 P−value of CVM 0.6 0.8 0.8 −5 F IGURE 1: Dependograms for the single realization of {(Xt , Yt , Zt )⊤ = (|εt |sign(ηt υt ), ηt , υt )⊤ }. The Canadian Journal of Statistics / La revue canadienne de statistique DOI: 2012 23 0.1 0.0 Cross−correlations −0.2 −0.1 0.0 −0.2 −0.1 Cross−correlations 0.1 0.2 Cross−correlogram for A={1,3} 0.2 Cross−correlogram for A={1,2} −4 −3 −2 −1 0 1 2 3 4 5 −5 −4 −3 −2 −1 0 1 2 3 Lags Lags Cross−correlogram for A={2,3} Cross−correlogram for A={1,2,3} 4 5 0.2 Cross−correlations 0.0 0.0 −5 −4 −3 −2 −1 0 Lags 1 2 3 4 5 { −2 , −2 } { −2 , −1 } { −2 , 0 } { −2 , 1 } { −2 , 2 } { −1 , −2 } { −1 , −1 } { −1 , 0 } { −1 , 1 } { −1 , 2 } { 0 , −2 } { 0 , −1 } {0,0} {0,1} {0,2} { 1 , −2 } { 1 , −1 } {1,0} {1,1} {1,2} { 2 , −2 } { 2 , −1 } {2,0} {2,1} {2,2} −0.2 −0.2 −0.1 Cross−correlations 0.1 0.4 0.2 −5 F IGURE 2: Cross-correlograms for the single realization of {(Xt , Yt , Zt )⊤ = (|εt |sign(ηt υt ), ηt , υt )⊤ }. 5. APPLICATION WITH FINANCIAL DATA FROM THE NASDAQ INDEX In Section 4, we conducted a simulation study in order to investigate the empirical properties of the proposed test statistics. Here, we apply the test procedures using real data: the log-returns of three major companies traded on the Nasdaq market, that is Apple, Intel and Hewlett-Packard. The observation period covers January 2nd 2009 to February 25th, 2011, giving a total sample size equals to n = 452. For a given company, the log-returns of the asset versus the log-returns in the associated market were modelled using the so-called market model. That model postulates a linear relationship between the expected returns of a particular asset, in terms of the expected returns of the underlying index. The slope coefficient can be interpreted as a standardized measure of risk (Reilly & Brown, 2006). Thus the market model represents a particular case of the general model (1) using the Nasdaq index as an explanatory variable. Let AAPL, INTEL and HPQ denote the log-returns of Apple, Intel and Hewlett-Packard, respectively; and let NDAQ be DOI: The Canadian Journal of Statistics / La revue canadienne de statistique 24 DUCHESNE, GHOUDI AND RÉMILLARD Vol. xx, No. yy the log-returns of the Nasdaq index. Thus the models for the assets are given by: AAPLt = α1 + β1 NDAQt + ε1t , INTELt = α2 + β2 NDAQt + ε2t , HPQt = α3 + β3 NDAQt + ε3t . Classical descriptive analyzes have been performed for each time series. While the log-returns of the time series INTELt and HPQt seemed compatible with simple white noise models for the residuals, it appeared that some dependencies were present in the Apple time series. Empirical evidence suggested fitting an autoregressive of order six for the residuals of Apple. Several models have been adjusted for each time series, and using the AIC criteria, it has been decided to fit an AR(6) model for Apple, and white noise models in the residuals of Intel and Hewlett-Packard. To verify if high-order dependence was still present in the residuals of each time series, we calculated the test statistics proposed by Genest, Ghoudi, & Rémillard (2007). More precisely, we computed the BDS test statistics In and In∗ . Rejection of the null hypothesis by the In∗ test statistic, together with the non-rejection of the null hypothesis for the In test statistic lead to the conclusion that the remaining serial dependence is best explained by a GARCH model (Genest, Ghoudi, & Rémillard, 2007). Consequently, a stationary AR(6)-GARCH(1,1) model has been fitted on the residuals of the Apple time series, while pure GARCH(1,1) models were considered for the residuals of Intel and Hewlett-Packard. It can be seen that all the resulting models can be expressed in the form of model (1). We then applied the global test statistics Wn , Fn , Hn , and also WA,n , FA,n and HA,n , for sets A of cardinality two and three, using the maximal lag orders M2 = 5 and M3 = 2. The results are presented in Table 10. All the P -values are larger than any reasonable nominal level, suggesting that the innovations from the fitted time series are not dependent. TABLE 10: P -values (in percentage) for the data set consisting of the log-returns of the Appel, Intel and Hewlett-Packard, traded on the Nasdaq market. The sample size is n = 452. Test statistics included: WA,n , FA,n and HA,n for the specified sets A, and the combined test statistics Wn , Fn , Hn defined by (17), (18) and (19), respectively. A Test statistics Combined {1, 2} {1, 3} {2, 3} {1, 2, 3} test statistics WA,n 76.23 33.89 94.53 70.47 88.53 FA,n 77.43 34.18 92.11 60.26 83.61 HA,n 47.34 72.86 56.16 30.92 54.63 Thus global test statistics do not suggest that the innovations are incompatible with the assumption of independence. To check if some local dependence not detected at the global level is still remaining, test statistics at individual lags were computed. Thus, dependograms for the sets A = {1, 2}, {1, 3}, {2, 3} and A = {1, 2, 3} are presented in Figure 3. The P -values are in general larger than the 5% nominal level, except for instantaneous dependence between Apple and Hewlett-Packard where the P -value is about 1.6%, which is not significant at the 1% level. Another exception is CVM{1,2,3} , which displayed for the lag (0, 2, 0) a P -value of 4.58%. Cross-correlograms for sets A of cardinality two and three are also presented in Figure 4. Slight instantaneous dependence has been detected between Apple and Hewlett-Packard and also between Intel and Hewlett-Packard, with some cross-correlations slightly outside the 5% significance regions. Given the number of test statistics involved and the rather large sample size, we concluded that the significant P -values in Figure 3 or the cross-correlations outside the The Canadian Journal of Statistics / La revue canadienne de statistique DOI: 2012 25 Dependogram of CVM{1,3} tests 0.2 0.0 0.0 −4 −3 −2 −1 0 1 2 3 4 5 −5 −4 −3 −2 −1 0 1 2 3 Lags Lags Dependogram of CVM{2,3} tests Dependogram of CVM{1,2,3} tests 4 5 0.6 0.4 −5 −4 −3 −2 −1 0 Lags 1 2 3 4 5 { −2 , −2 } { −2 , −1 } { −2 , 0 } { −2 , 1 } { −2 , 2 } { −1 , −2 } { −1 , −1 } { −1 , 0 } { −1 , 1 } { −1 , 2 } { 0 , −2 } { 0 , −1 } {0,0} {0,1} {0,2} { 1 , −2 } { 1 , −1 } {1,0} {1,1} {1,2} { 2 , −2 } { 2 , −1 } {2,0} {2,1} {2,2} 0.0 0.0 0.2 0.2 0.4 P−value of CVM 0.6 0.8 0.8 1.0 1.0 −5 P−value of CVM 0.4 P−value of CVM 0.4 0.2 P−value of CVM 0.6 0.6 0.8 0.8 Dependogram of CVM{1,2} tests F IGURE 3: Dependograms for the data set consisting of the log-returns of the Appel, Intel and HewlettPackard, traded on the Nasdaq market. 5% significance bands did not suggest strong inadequacies. While a more complicated model could probably improve the description of these variables, we found that the simple market models seemed reasonable for these variables. Interestingly, our results suggest that the stochastic explanatory variable corresponding to the Nasdaq market may prove useful in explaining the common dependence. Incidently, we performed similar analyzes in the simpler models excluding the market variable NDAQt . Without including that explanatory variable, the null hypothesis of independence was strongly rejected by the global test statistics for the variables AAPL, INTEL and HPQ, suggesting that the innovations were not independent. Stochastic explanatory variables are often available with financial time series data. The data analysis presented in this section suggests that to have methods which are valid under these realistic conditions may be useful in practice. DOI: The Canadian Journal of Statistics / La revue canadienne de statistique 26 DUCHESNE, GHOUDI AND RÉMILLARD Cross−correlogram for A={1,3} 0.05 −0.05 −4 −3 −2 −1 0 1 2 3 4 5 −5 −4 −3 −2 −1 0 1 2 Lags Lags Cross−correlogram for A={2,3} Cross−correlogram for A={1,2,3} 3 4 5 0.00 Cross−correlations 0.00 −5 −4 −3 −2 −1 0 1 2 Lags 3 4 5 { −2 , −2 } { −2 , −1 } { −2 , 0 } { −2 , 1 } { −2 , 2 } { −1 , −2 } { −1 , −1 } { −1 , 0 } { −1 , 1 } { −1 , 2 } { 0 , −2 } { 0 , −1 } {0,0} {0,1} {0,2} { 1 , −2 } { 1 , −1 } {1,0} {1,1} {1,2} { 2 , −2 } { 2 , −1 } {2,0} {2,1} {2,2} −0.10 −0.05 −0.05 0.05 0.05 −5 Cross−correlations 0.00 Cross−correlations 0.00 −0.05 Cross−correlations 0.05 Cross−correlogram for A={1,2} Vol. xx, No. yy F IGURE 4: Cross-correlograms for the data set consisting of the log-returns of the Appel, Intel and HewlettPackard, traded on the Nasdaq market. 6. CONCLUDING REMARKS In this article, we have developed test statistics for checking the independence between the univariate innovations of several possibly non Gaussian time series. The techniques described in Ghoudi, Kulperger & Rémillard (2001) have been extended, by constructing a finite family of empirical processes relying on multivariate lagged residuals, and we derived their asymptotic distributions. Möbius transformations of the empirical processes were considered, in order to obtain simpler asymptotic covariance structures. Under the null hypothesis of independence, we have proven that these transformed processes are asymptotically Gaussian, independent, and with tractable covariance functions not depending on the estimated parameters. Considering the rank of the residuals, we have demonstrated that the limiting empirical copula process admits the same distribution than the one based on the true but non observable innovations. Several test statistics based on the residuals or the rank of the residuals have been proposed, including Cramér–von Mises test statistics and test statistics based on non-parametric measures. We also considered test statistics based on the concept of generalized cross-correlation. Using graphical devices similar The Canadian Journal of Statistics / La revue canadienne de statistique DOI: 2012 27 to those elaborated by Genest & Rémillard (2004), we have shown how to detect dependence graphically. Simulations were conducted to explore the finite sample properties of the new test statistics, which were found to be powerful against various types of alternatives when the independence has been tested between two and also three time series. A data analysis involving three financial time series from the Nasdaq market demonstrated that our methods are useful in studying the independence between series of innovations of time series models involving explanatory variables. Research projects include generalization of the present methodology to multivariate innovations and time series. Also, empirical studies are needed to study how robust are the proposed test statistics to model misspecification. We let these challenging research avenues for future studies. ACKNOWLEDGEMENTS This research was supported by grants from the Natural Sciences and Engineering Research Council of Canada. The authors would like to thank the Editor, an Associate Editor and two referees for constructive suggestions, which led to an improved paper. BIBLIOGRAPHY Ansley, C. F. & Newbold, P. (1979). On the finite sample distribution of residual autocorrelations in autoregressive-moving average models. Biometrika, 66, 547–553. Bahadur, R. R. (1960). Stochastic comparison of tests. Annals of Mathematical Statistics, 31, 276–295. Bai, J. (2003). Testing parametric conditional distributions of dynamic models. The Review of Economics and Statistics, 85, 531–549. Bickel, P. J. & Wichura, M. J. (1971). Convergence criteria for multiparameter stochastic processes and some applications. Annals of Mathematical Statistics, 42, 1656–1670. Bougerol, P. & Picard, N. (1992a). Stationarity of GARCH processes and of some nonnegative time series. Journal of Econometrics, 52, 115–127. Bougerol, P. & Picard, N. (1992b). Strict stationarity of generalized autoregressive processes. Annals of Probability, 20, 1714–1730. Brockwell, P. J. & Davis, R. A. (1991). Time Series: Theory and Methods. Springer Series in Statistics. Springer-Verlag, New York, second edition. Brockwell, P. J. & Davis, R. A. (2002). Introduction to Time Series and Forecasting. Springer Series in Statistics. Springer-Verlag, New York, second edition. Carlstein, E. (1988). Nonparametric change-point estimation. The Annals of Statistics, 16, 188–197. Chen, X. & Fan, Y. (2006). Estimation and model selection of semiparametric copula-based multivariate dynamic models under copula misspecification. Journal of Econometrics, 135, 125 – 154. Deheuvels, P. (1979). La fonction de dépendance empirique et ses propriétés: Un test non paramétrique d’indépendance. Académie royale de Belgique, Bulletin de la classe des sciences, 5ième série, 65, 274– 292. Deheuvels, P. (1980). Non parametric tests of independence. In Nonparametric Asymptotic Statistics (Proceedings of the Conference held in Rouen in 1979), pp. 95–107. Lecture Notes in Mathematics No 821, Springer, New York. Deheuvels, P. (1981a). An asymptotic decomposition for multivariate distribution-free tests of independence. Journal of Multivariate Analysis, 11, 102–113. Deheuvels, P. (1981b). A Kolmogorov–Smirnov type test for independence and multivariate samples. Revue roumaine de mathématiques pures et appliquées, 26, 213–226. Deheuvels, P. (1981c). A non parametric test for independence. Publications de l’Institut de statistique de l’Université de Paris, 26, 29–50. Drouet-Mari, D. & Kotz, S. (2001). Correlation and Dependence. Imperial College Press, London. DOI: The Canadian Journal of Statistics / La revue canadienne de statistique 28 DUCHESNE, GHOUDI AND RÉMILLARD Vol. xx, No. yy Duchesne, P. & Roy, R. (2003). Robust tests for independence of two time series. Statistica Sinica, 13, 827–852. Fisher, R. A. (1950). Statistical Methods for Research Workers. Oliver and Boyd, London, 11h edition. Francq, C. & Zakoı̈an, J.-M. (2004). Maximum likelihood estimation of pure GARCH and ARMA-GARCH processes. Bernoulli, 10, 605–637. Francq, C. & Zakoı̈an, J.-M. (2010). GARCH models: Structure, Statistical Inference and Financial Applications. John Wiley & Sons, New York. Gänßler, P. & Stute, W. (1987). Seminar on Empirical Processes, volume 9 of DMV Seminar. Birkhäuser Verlag, Basel. Genest, C., Ghoudi, K., & Rémillard, B. (2007). Rank-based extensions of the Brock Dechert Scheinkman test for serial dependence. Journal of the American Statistical Association, 102, 1363–1376. Genest, C., Nešlehová, J., & Ben Ghorbal, N. (2011). Estimators based on Kendall’s tau in multivariate copula models. Australian and New Zealand Journal of Statistics, 53, 157–177. Genest, C., Quessy, J.-F., & Rémillard, B. (2007). Asymptotic local efficiency of Cramér-von Mises tests for multivariate independence. The Annals of Statistics, 35, 166–191. Genest, C. & Rémillard, B. (2004). Tests of independence or randomness based on the empirical copula process. Test, 13, 335–369. Genest, C., Rémillard, B., & Beaudoin, D. (2009). Omnibus goodness-of-fit tests for copulas: A review and a power study. Insurance: Mathematics & Economics, 44, 199–213. Geweke, J. (1981). A comparison of tests of the independence of two covariance-stationary time series. Journal of the American Statistical Association, 76, 363–373. Ghoudi, K., Kulperger, R. J., & Rémillard, B. (2001). A nonparametric test of serial independence for time series and residuals. Journal of Multivariate Analysis, 79, 191–218. Ghoudi, K. & Rémillard, B. (1998). Empirical processes based on pseudo-observations. In Asymptotic Methods in Probability and Statistics (Ottawa, ON, 1997), pp. 171–197. North-Holland, Amsterdam. Ghoudi, K. & Rémillard, B. (2004). Empirical processes based on pseudo-observations. II. The multivariate case. In Asymptotic Methods in Stochastics, volume 44 of Fields Inst. Commun., pages 381–406. Amer. Math. Soc., Providence, RI. Granger, C. & Teräsvirta, T. (1993). Modelling Non-linear Economic Relationships. Oxford University Press, Oxford. Hallin, M., Jurevcková, J., Picek, J., & Zahaf, T. (1999). Nonparametric tests of independence of two autoregressive time series based on autoregression rank scores. Journal of Statistical Planning and Inference, 75, 319–330. Hallin, M. & Puri, M. L. (1992). Rank tests for time series analysis: A survey. In Brillinger, D. R., Parzen, E., & Rosenblatt, M., editors, New Directions in Time Series Analysis, Part I, pp. 111–153. Springer, New York. Hannan, E. J. & Deistler, M. (1988). The Statistical Theory of Linear Systems. Wiley & Sons, New York. Haugh, L. D. (1976). Checking the independence of two covariance-stationary time series: a univariate residual cross-correlation approach. Journal of the American Statistical Association, 71, 378–385. Hong, Y. (1996). Testing for independence between two covariance stationary time series. Biometrika, 83, 615–625. Hong, Y. (1998). Testing for pairwise serial independence via the empirical distribution function. Journal of the Royal Statistical Society, Series B, 60, 429–453. Joe, H. (1997). Multivariate models and dependence concepts, volume 73 of Monographs on Statistics and Applied Probability. Chapman & Hall, London. Kim, E. & Lee, S. (2005). A test for independence of two stationary infinite order autoregressive processes. Annals of the Institute of Statistical Mathematics, 57, 105–127. Klimko, L. A. & Nelson, P. I. (1978). On conditional least squares estimation for stochastic processes. The Annals of Statistics, 6, 629–642. The Canadian Journal of Statistics / La revue canadienne de statistique DOI: 2012 29 Koch, P. D. & Yang, S. S. (1986). A method for testing the independence of two time series that accounts for a potential pattern in the cross-correlation function. Journal of the American Statistical Association, 81, 533–544. Lee, S. & Wei, C. Z. (1999). On residual empirical processes of stochastic regression models with applications to time series. The Annals of Statistics, 27, 237–261. Li, W. K. (2004). Diagnostic Checks in Time Series. Chapman & Hall/CRC, New York. Li, W. K. & Hui, Y. V. (1994). Robust residual cross correlation tests for lagged relations in time series. Journal of Statistical Computation and Simulation, 49, 103–109. Littell, R. C. & Folks, J. L. (1973). Asymptotic optimality of Fisher’s method of combining independent tests. II. Journal of the American Statistical Association, 68, 193–194. McLeod, A. I. (1978). On the distribution of residual autocorrelations in Box-Jenkins models. Journal of the Royal Statistical Society, Series B, 40, 296–302. McLeod, A. I. (1979). Distribution of the residual cross-correlation in univariate ARMA time series models. Journal of the American Statistical Association, 74, 849–855. Nelsen, R. B. (2006). An Introduction to Copulas, volume 139 of Lecture Notes in Statistics. SpringerVerlag, New York, 2nd edition. Pierce, D. A. (1977). Relationships – and the lack thereof – between economic time series, with special reference to money and interest rates. Journal of the American Statistical Association, 72, 11–26. Potscher, B. M. & Prucha, I. R. (1997). Dynamic Nonlinear Econometric Models. Springer, Berlin. Reilly, F. & Brown, K. (2006). Investment Analysis and Portfolio Management. Thomson South-Western, Mason, OH, 8th edition. Rémillard, B. (2010). Goodness-of-fit tests for copulas of multivariate time series. SSRN working paper series No. 1729982. Romano, J. P. & Siegel, A. F. (1986). Counterexamples in Probability and Statistics. Wadsworth, London. Rüschendorf, L. (1974). On the empirical process of multivariate, dependent random variables. Journal of Multivariate Analysis, 4, 469–478. Rüschendorf, L. (1976). Asymptotic distributions of multivariate rank order statistics. The Annals of Statistics, 4, 912–923. Shao, X. (2009). A generalized portmanteau test for independence between two stationary time series. Econometric Theory, 25, 195–210. Skaug, H. J. & Tjøstheim, D. (1993). A nonparametric test of serial independence based on the empirical distribution function. Biometrika, 80, 591–602. Spitzer, F. L. (1974). Introduction aux processus de Markov à parametre dans Zν . In École d’Été de Probabilités de Saint-Flour, III–1973, pp. 114–189. Lecture Notes in Math., Vol. 390. Springer, Berlin. Stute, W. (1984). The oscillation behavior of empirical processes: the multivariate case. The Annals of Probability, 12, 361–379. Taniguchi, M. & Kakizawa, Y. (2000). Asymptotic Theory of Statistical Inference for Time Series. Springer, New York. Tong, H. (1990). Non-linear Time Series: A Dynamical System Approach, volume 6 of Oxford Statistical Science Series. Oxford Science Publications. van der Vaart, A. & Wellner, J. (2007). Empirical processes indexed by estimated functions. In Asymptotics: Particles, Processes and Inverse Problems, volume 55 of IMS Lectures Notes – Monograph Series, pages 234–252. Institute of Mathematical Statistics. APPENDIX For any x ∈ R̄d , set Ǩn (x) = αn {F(x)} and Ǩ(x) = α{F(x)}, where F(x) = ⊤ (F1 (x1 ), . . . , Fd (xd )) . The following result is a particular case of a theorem proven in Rémillard (2010, Theorem 1). DOI: The Canadian Journal of Statistics / La revue canadienne de statistique 30 DUCHESNE, GHOUDI AND RÉMILLARD Theorem 3. Under Assumptions (A1)–(A6), Kn K(x) = Ǩ(x) + d ∑ Vol. xx, No. yy K, with representation given by: fj (xj )Gj (x) {(Γ0 Θ)j + xj (Γ1j Θ)j } , x ∈ Rd , j=1 where Gj (x) = ∏ k̸=j Fk (xk ). Furthermore, for all j = 1, . . . , d, Fjn Fj , where Fj (xj ) = βj ◦ Fj (xj ) + fj (xj ) {(Γ0 Θ)j + xj (Γ1j Θ)j } , and β1 (u1 ) = α(u1 , 1, . . . , 1), . . . , βd (ud ) = α(1, . . . , 1, ud ), u ∈ [0, 1]d . Proof of Theorem 1. For any finite set L ⊂ Zd , it is well-known that the processes {αℓ,n ; ℓ ∈ L} converge jointly in D([0, 1]d ) to continuous centered Gaussian processes {αℓ ; ℓ ∈ L} having the same distribution that α, which is a d-dimensional Brownian bridge, see (Bickel & Wishura, 1971). By construction, it follows that: β1 (u1 ) = αℓ (u1 , . . . , 1), . . . , βd (ud ) = αℓ (1, . . . , 1, ud ), u ∈ [0, 1]d . ∏d Let Hℓ,n be defined by (3) and H(x) = j=1 Fj (xj ). As a result, for any finite set L ⊂ Zd , it follows from Theorem 3 that the processes Kℓ,n = n1/2 (Hℓ,n − H), ℓ ∈ L, converge jointly in D([−∞, ∞]d ) to processes Kℓ having the representation: Kℓ (x) = Ǩℓ (x) + d ∑ fj (xj )Gj (x) {(Γ0 Θ)j + xj (Γ1j Θ)j } , x ∈ Rd , j=1 where Ǩℓ (x) = αℓ {F(x)}. Next, using the multinomial formula given by: d ∏ ∑ (aj + bj ) = j=1 A⊂Sd   ∏ j∈A ( aj  ∏ bk k∈Ac ) , (1) it follows from Theorem 3: n 1/2  d ∏  j=1 Fjn (xj ) − d ∏ j=1 Fj (xj )    =n =n 1/2 1/2   d { ∏ Fj (xj ) + n j=1 ∑ A⊂Sd , A̸=∅ = d ∑ Fjn (xj ) j=1 which converges in D([−∞, ∞]d ) to ∑d j=1 −1/2  ∏  j∈A  ∏  Fk (xk ) k̸=j Fj (xj )Gj (x). The Canadian Journal of Statistics / La revue canadienne de statistique n −1/2   } Fjn (xj ) − Fjn (xj ) d ∏ j=1 Fj (xj ) , {  ∏   k∈Ac } Fk (xk ) , + OP (n−1/2 ),  DOI: 2012 31 { } ∏d Therefore, for all ℓ ∈ B, the processes Dℓ,n (x) = n1/2 Hℓ,n (x) − j=1 Fjn (xj ) con- verge jointly in D([−∞, ∞]d ) to processes Dℓ having the representation: Dℓ (x) = Kℓ − d ∑ Fj (xj )Gj (x), j=1 = Ǩℓ (x) + ∑ fj (xj )Gj (x) {(Γ0 Θ)j + xj (Γ1j Θ)j } , j=1 − d ∑ Gj (x) [βj ◦ Fj (xj ) + fj (xj ) {(Γ0 Θ)j + xj (Γ1j Θ)j }] , j=1 = Ǩℓ (x) − d ∑ βj ◦ Fj (xj )Gj (x) = Eℓ {F1 (x1 ), . . . , Fd (xd )} , j=1 where Eℓ (u) = αℓ (u) − d ∑ j=1  βj (uj )  ∏ k̸=j  uk  , u ∈ [0, 1]d . (2) The convergence of Eℓ,n follows using Genest, Ghoudi, & Rémillard (2007, Proposition A.1) and Theorem 3, or alternatively invoking Rémillard (2010, Corollary 1). For any ℓ ∈ Zd , it follows from the multinomial formula (1) that ∏ ∑ Fkn (xk ) , (−1)|A\B| Dℓ,n (xB ) RA,ℓ,n (x) = Proof of Theorem 2. B⊂A k∈A\B where (xB )j = { xj , if j ∈ B, ∞, otherwise. Then Theorem 1 together with the continuous mapping theorem yield that the processes RA,ℓ,n converge jointly in D([−∞, ∞]d ) to processes RA,ℓ having representation: RA,ℓ (x) = = ∑ (−1)|A\B| Dℓ (xB ) Fk (xk ) , B⊂A ∏ k∈A\B ∑ (−1)|A\B| Ǩℓ (xB ) ∏ Fk (xk ) , ∑ (−1)|A\B| αℓ ◦ F(xB ) B⊂A = (3) k∈A\B B⊂A ∏ Fk (xk ) , (4) k∈A\B where (3) follows from the property that for any j ∈ {1, . . . , d}, ∑ (−1)|A\B| βj ◦ Fj {(xB )j }Gj (xB ) B⊂A DOI: ∏ Fk (xk ) ≡ 0. k∈A\B The Canadian Journal of Statistics / La revue canadienne de statistique 32 DUCHESNE, GHOUDI AND RÉMILLARD Vol. xx, No. yy Similarly, the processes GA,ℓ,n converge jointly in D([0, 1]d ) to processes GA,ℓ having representation ∏ ∑ uk , (−1)|A\B| Eℓ (uB ) GA,ℓ (u) = B⊂A k∈A\B ∑ ∏ = (−1)|A\B| αℓ (uB ) B⊂A uk . k∈A\B Consequently, RA,ℓ = GA,ℓ ◦ F. Furthermore, from the multinomial formula (1): αA,ℓ,n (u) = = n 1 ∑∏[ n1/2 ∑ t=1 j∈A ] I{Fj (εj,t+lj ) ≤ uj } − uj , (−1)|A\B| αℓ,n (uB ) B⊂A ∏ uk . k∈A\B Thus, the processes αA,ℓ,n converge jointly to the processes GA,ℓ . Consequently, for any u, v ∈ [0, 1]d , the covariance reduces to: ΓA,B (u, v) = cov {GA,ℓ (u), GB,ℓ′ (v)} , = lim cov {αA,ℓ,n (u), αB,ℓ′ ,n (u)} , n→∞ {∏ ′ j∈A (uj ∧ vj − uj vj ), A = B and ℓ ≡A ℓ, = 0, otherwise. We deduce that the processes GA,ℓ and GB,ℓ′ are independent if A ̸= B or A = B and ℓ′ ̸≡A ℓ. This completes the proof of the Theorem. Received 9 July 2011 Accepted 8 April 2012 The Canadian Journal of Statistics / La revue canadienne de statistique DOI:

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