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- de Gruyter
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- © 2021 Axel Bücher et al., published by De Gruyter
- ISSN
- 2300-2298
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- 2300-2298
- DOI
- 10.1515/demo-2021-0105
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Abstract
Depend. Model. 2021; 9:121–140 Research Article Open Access Axel Bücher, Miriam Jaser, and Aleksey Min* Detecting departures from meta-ellipticity for multivariate stationary time series https://doi.org/10.1515/demo-2021-0105 Received February 23, 2021; accepted June 15, 2021 Abstract: A test for detecting departures from meta-ellipticity for multivariate stationary time series is pro- posed. The large sample behavior of the test statistic is shown to depend in a complicated way on the under- lying copula as well as on the serial dependence. Valid asymptotic critical values are obtained by a bootstrap device based on subsampling. The nite-sample performance of the test is investigated in a large-scale sim- ulation study, and the theoretical results are illustrated by a case study involving nancial log returns. Keywords: elliptical copula, empirical process, nancial log returns, goodness-of- t test, subsampling boot- strap MSC: 62H15, 62M10 1 Introduction In the recent decades, copula models have been successfully used in a wide range of applications, includ- ing nance, hydrology or risk management, see [18, 30, 31]. In the bivariate case, any of the most commonly applied copula families, including the Gaussian, Clayton, Gumbel, Frank or t-copula, can be identi ed as a member of one of the following large (nonparametric) subclasses: the class of Archimedean copulas, the class of extreme-value copulas or the class of elliptical copulas. The latter class also provides exible parametric families in the higher-dimensional case, while more work is needed to de ne exible models involving the former two classes. Multivariate extreme-value copulas typically arise from max-stable process models [12], while exible copulas involving Archimedean building blocks may be de ned based on certain hierarchi- cal constructions [34]. Next to these approaches, vine copulas provide a versatile concept to connect mostly arbitrary bivariate building blocks into exible multivariate models [1]. More recent approaches in the multi- variate case comprise Archimax copulas [10] or non-central squared copulas [32]. While testing the goodness-of- t of a certain parametric class of copulas has attracted a lot of attention [17, 21, 29], much less work has been devoted to testing whether a copula belongs to any of the large subclasses mentioned above. We refer to [7] for the case of Archimedean copulas, to [5] for tests for extreme-value copu- las, while tests for the simplifying assumption in vine copula models can be found in [14]. Within this paper, we are interested in testing for the null hypothesis that a copula is elliptical; a question that is of particular in- terest in the context of nancial risk management [35, 43], but see also [19] for applications in hydrology. Note that ellipticity of a copula is also referred to as meta-ellipticity of the underlying multivariate distribution, see [15] and [2]. For the case of observing i.i.d. data, respective tests have recently been investigated in [26] and [37], both of which exploit the fact that all bivariate margins of a d-variate elliptical copula exhibit equal val- ues for Kendall’s tau and Blomqvist’s beta. While the former authors work under the unrealistic assumption that marginal distributions are known, the latter author considers suitable rank-based test statistics (see also Axel Bücher: Heinrich-Heine-University Düsseldorf, E-mail: axel.buecher@hhu.de Miriam Jaser: Technical University of Munich, E-mail: miriam.jaser@tum.de *Corresponding Author: Aleksey Min: Technical University of Munich, E-mail: min@tum.de Open Access. © 2021 Axel Bücher et al., published by De Gruyter. This work is licensed under the Creative Commons Attribution alone 4.0 License. 122 Ë Axel Bücher, Miriam Jaser, and Aleksey Min [25], where similar tests were worked out independently). Critical values in [37] are then obtained by a certain multiplier bootstrap procedure. The present paper is motivated by the fact that available observations are often serially dependent time series (in particular in the important context of nancial risk management), such that the tests mentioned in the previous paragraph are not valid anymore. We revisit the large sample theory for the respective test statistics, show that the asymptotic distribution is typically dierent than in the i.i.d. case, and propose a suitable bootstrap approach to calculate valid critical values. The bootstrap scheme relies on subsampling [36], and heavily exploits recent theoretical results in [28] on subsampling empirical copulas. It is important to mention that we believe to also close important gaps in the theoretical results in [37]: while we believe that his results regarding bootstrap validity are correct and provable, the given proofs lack mathematical rigorousness (for instance, in his Appendix A.4, weak limit elds are treated as if they were de ned on the same probability space as the original data; moreover, they are partly considered non-random). The remaining parts of this paper are organized as follows: some mathematical preliminaries on copu- las, elliptical distributions and bivariate association measures are collected in Section 2. The test for meta- ellipticity is de ned in Section 3, with respective large-sample theory and bootstrap results collected in Sec- tion 3.1 and 3.2, respectively. Results from a large-scale Monte Carlo simulation study are presented in Sec- tion 4. A case study on nancial log returns is worked out in Section 5, while Section 6 brie y concludes. Finally, all proofs are postponed to Appendix A and B. 2 Mathematical preliminaries Let X = (X , . . . , X ) 2 R be a d-dimensional random vector with cumulative distribution function (c.d.f.) 1 d F and continuous univariate marginal c.d.f.s F , . . . , F . According to Sklar’s theorem [42], there exists a 1 d d d unique copula C : [0, 1] 7! [0, 1] such that, for all x 2 R , F(x) = C(F (x ), . . . , F (x )) . 1 1 d d The unique copula C may be written as − − d C(u) = F(F (u ), . . . , F (u )), u 2 [0, 1] , 1 1 d where F denotes the generalized inverse of F , k 2 f1, . . . , dg. A copula C is called elliptical if it is the copula of some elliptical distribution that is absolutely continuous with respect to the Lebesgue measure. Recall that a random vector Z 2 R is said to have an elliptical distri- d d×m bution if it admits, for some μ 2 R , some A 2 R with m 2 N, and some non-negative random variable R, the decomposition Z = μ + RAV , where V is a random vector that is independent of R and uniformly distributed on the unit sphere in R . Note that the distribution of Z is absolutely continuous with respect to the Lebesgue measure i R has a Lebesgue density and if Σ = AA is positive de nite (Theorem 2.9 and the discussion on page 46 in [16]); the corresponding Lebesgue density of Z is then given −1/2 −1 d f (z) = jΣj g (z − μ)Σ (z − μ) , z 2 R , for some function g that is in one-to-one correspondence with the density ofR. As suggested by the above con- struction, elliptical copulas are typically not available in closed form, two prime examples being the Gaussian and t-copula. Following [15], a distribution on R with continuous marginal c.d.f.s is called meta-elliptical if its associated copula is elliptical. As explained in the next paragraph, elliptical copulas exhibit a remarkable relationship between two well-known pairwise association measures: Kendall’s tau and Blomqvist’s beta [4, 27]. For k, ` 2 f1, . . . , dg Testing for meta-ellipticity Ë 123 distinct and C non necessarily elliptical, the latter are de ned as τ := E[sgn(X − X )sgn(X − X )] k` k1 k2 `1 `2 = P((X − X )(X − X ) > 0) − P((X − X )(X − X ) < 0) k1 k2 `1 `2 k1 k2 `1 `2 and β := E[sgn(X − ˜x )sgn(X − ˜x )] k` k k ` ` = P((X − ˜x )(X − ˜x ) > 0) − P((X − ˜x )(X − ˜x ) < 0) , k k ` ` k k ` ` where (X , X ) and (X , X ) are independent copies of (X , X ), where sgn denotes the signum function k1 `1 k2 `2 k ` and where ˜x and ˜x denote the population medians of X and X , respectively. It is well-known that the two k ` k ` coe cients are completely determined by the (unique) bivariate copula C of (X , X ), i.e.: k` k ` 1 1 Z Z τ = τ = 4 C (u , u ) dC (u , u ) − 1 k` C k` k ` k` k ` k` 0 0 and β = β = 4C (0.5, 0.5) − 1 . (1) k` k` k` (k`) d Note that C can be retrieved from C, as C (u , u ) = C(u ), where, for u = (u , . . . , u ) 2 [0, 1] and k` k` k ` 1 d (A) d A f1, . . . , dg, the vector u 2 R denotes the vector where all components of u except the components of the index set A are replaced by 1. As a direct consequence of the de nition of an elliptical distribution, all bivariate margins of an elliptical distribution are elliptical as well. As a consequence, the same is true for elliptical copulas. It then follows from Theorem 3.1 in [15] and Proposition 8 in [40] that, for all k, ` 2 f1, . . . , dg with k < `, τ = arcsin(ρ ) = β , (2) k` k` k` where ρ = σ / σ σ . As in [25, 26, 37], the latter will be the basis for the test for ellipticity. It is important k` k` kk `` to note that there exist non-elliptical copulas for which (2) is met: Example 2.1. (i) Consider the bivariate checkerboard copula C with Lebesgue-density c(u, v) = 4 · (1 + 1 + 1 + 1 )(u, v) A B C D 2 2 where A = [0, 1/4] , B = [1/4, 1/2] × [3/4, 1], C = [1/2, 3/4] , D = [3/4, 1] × [1/4, 1/2]. A straightforward calculation shows that τ = β = 0. The same is true for the copula whose induced law is the uniform distribution onf(u, u) : u 2 [0, 1]g[f(u, 1 − u) : u 2 [0, 1]g. (ii) Among the most common bivariate copulas that are non-elliptical are the members from the Gumbel– Hougaard, the Clayton and the Frank copula family (except for some cases at the boundary of the parameter space). In Figure 1, we depict the absolute dierence jβ − τj as a function of τ 2 [0, 1] within the respective families. It can be seen that the dierence is largest for the Frank copula. Quite remarkably, we have τ = β for some non-trivial members from the Clayton and Gumbel family. 124 Ë Axel Bücher, Miriam Jaser, and Aleksey Min 0.06 Copula Family: 0.04 Frank Clayton 0.02 Gumbel 0.00 0.00 0.25 0.50 0.75 1.00 Figure 1: Absolute dierence jτ − βj as a function of τ for the Frank, Clayton and Gumbel–Hougaard family. 3 Testing meta-ellipticity Throughout this section, let X , . . . , X with X = (X , . . . , X ) 2 R be a stretch of a strictly stationary 1 i 1i di time series (X ) of d-dimensional random vectors. The common c.d.f. of X is F, which is assumed to have i i2Z i continuous univariate c.d.f.s F , . . . , F , and its copula is denoted by C. We are going to test for the hypotheses 1 d H : C 2 C vs. H : C 2̸ C , 0 1 elliptical elliptical where C denotes the set of all elliptical copulas. A respective test statistic will be de ned in Section 3.1. elliptical For carrying out the test, we rely on suitable bootstrap approximations, which will be investigated in Sec- tion 3.2. 3.1 The test statistic and its asymptotic behavior By (2), the null hypothesis is equivalent to the fact that τ = β for all k, ` 2 f1, . . . , dg with k < `. For k` k` detecting departures from ellipticity, it hence makes sense to investigate the dierence between empirical counterparts of the two coe cients. It is important to note that, by construction, the test’s ability to detect departures from meta-ellipticity is limited by (1) the fact that it is completely based on investigating bivariate margins, and by (2) the fact that the dierence between Kendall’s tau and Blomquist’s beta may be small even for non-elliptical copulas (see Example 2.1). The classical sample version of Kendall’s tau is de ned as τ = sgn(X − X )sgn(X − X ) . `i `j k`,n ki kj n(n − 1) 1≤i<j≤n Obviously, τ is unbiased in case the underlying sample is serially independent. Under this assumption, k`,n large sample theory dates back to [24] and can be found in classical monographs such as [45], Section 12. In the case of serial dependence, large-sample theory may for instance be deduced from simple multivariate extensions of the results in [13], see also Proposition 2.3 in [6]. Next, a suitable sample version of Blomqvist’s beta motivated by (1) is given by 1 1 1 1 b b b b β = 4C ( , ) − 1 = 1(U ≤ , U ≤ ) − 1, `i k`,n k`,n ki 2 2 2 2 i=1 −1 2 where U = (n + 1) rank(X among X , . . . , X ) and where, for u , u 2 [0, 1] , ki ki k1 kn k ` b b b C (u , u ) = (U ≤ u , U ≤ u ) (3) k`,n k ` ki k k` ` i=1 denotes the empirical copula. Note that, in the case of serial independence, β is in fact an asymptotically k`,n equivalent version of the estimator initially proposed in [4], see [23]. Large sample theory in the case of serial dependence is an immediate consequence of the results in [9]. τ β Testing for meta-ellipticity Ë 125 b b b b For the de nition of suitable test statistics for H , let β = (β , β , . . . , β ) and τ = 0 12,n 13,n n d−1 d,n > d(d−1)/2 b b b (τ , τ , . . . , τ ) denote vectors in R obtained by concatenating all pairwise estimators. 12,n 13,n d−1 d,n Moreover, let b b D = β − τ . (4) n n We will next introduce three suitable conditions that will be su cient to deduce asymptotic normality of D under H . The rst condition concerns the serial dependence of the time series, and is taken from [9]. De ne unob- servable observations U = F (X ) for k 2 f1, . . . , dg and i 2 f1, . . . , ng and let ki k ki > d C (u) = 1fU ≤ u , . . . , U ≤ u g, u = (u , . . . , u ) 2 [0, 1] . (5) n 1 1 1i di d d i=1 ∞ d d d Moreover, let ` ([0, 1] ) denote the set of all bounded, real-valued functions on [0, 1] and let C([0, 1] ) denote the subset of continuous functions, both equipped with the supremum metric. Weak convergence in ∞ d ` ([0, 1] ) is to be understood in the sense of [44] and denoted by ‘ ’. Condition 3.1. The empirical process α = n(C − C) converges weakly towards a tight, centered Gaussian n n eld B concentrated on D , that is C 0 ∞ d α = n(C − C) B in ` ([0, 1] ) , n n where D is given by D = α 2 C([0, 1] ) | α(1, . . . , 1) = 0 and α(u) = 0 if some of the components of u are equal to 0 . The condition is trivially satis ed in the i.i.d. case, in which case the limit is a standard C-brownian bridge on [0, 1] . As outlined in [9], it is also met for the majority of the most common stationary time series models like ARMA and GARCH processes or, more generally, for strongly mixing processes with mixing coe cients n o α(h) := sup jP(A \ B) − P(A)P(B)j : A 2 σ(. . . , X , X ), B 2 σ(X , X , . . . ) (6) −1 0 h h+1 −a of the order α(h) = O(h ) for some a > 1. The covariance kernel of B is then given by Cov(B (u), B (v)) = Cov(1(U ≤ u), 1(U ≤ v)). C C h h2Z Finally, note that there exists an abundance of tests for hypotheses like stationarity, serial independence, or the goodness-of- t of a speci c time series model; all of which may provide empirical evidence for the circumstance that Condition 3.1 is met. The second condition is essentially a further condition on the serial dependence, as it is trivially met for i.i.d. data. It is, however, not met in general for time series, even for continuous stationary c.d.f.s: consider for instance a random repetition process, where, at time t, the previous observation is repeated with positive probability p or a new observation is generated independently with probability 1 − p. Condition 3.2. For any k 2 f1, . . . , dg, the kth component sample X , . . . , X does not contain any ties with k1 kn probability one. The third condition concerns the regularity of C, and is taken from [41]. It is non-restrictive in the sense that it is necessary for weak convergence of the empirical copula process with respect to the supremum distance to a limit with a.s. continuous sample paths. Condition 3.3. For any k 2 f1, . . . , dg, the rst-order partial derivatives ∂ C(u) exist and are continuous on the set U = fu 2 [0, 1] : u 2 (0, 1)g. k k 126 Ë Axel Bücher, Miriam Jaser, and Aleksey Min It is worthwhile to mention that, if one were only interested in weak convergence of β , then it would be su cient to assume existence and continuity of the partial derivatives in a neighbourhood of the pointsfu 2 P P [0, 1] : 1(u = 1/2) = 2, 1(u = 1) = d − 2g only (this follows from a straightforward modi cation k k k k of the arguments in [9], see also Lemma B.1 for the case of ellipticity). However, under such an assumption only, proving our bootstrap consistency results in Section 3.2 would require substantial additional eort. b b Finally, recall the empirical copula C de ned in (3) and let C = n(C −C) denote the empirical copula n n n process. As shown in [9] we have, under the previous conditions, C = n(C − C) G n n ∞ d d in ` ([0, 1] ), where the limiting Gaussian eld G is de ned, for all u 2 [0, 1] , by (k) G (u) = B (u) − ∂ C(u)B (u ) C C k C k=1 (k) with u = (1, . . . , 1, u , 1, . . . , 1). The following theorem is one of the main theoretical results of this paper. Theorem 3.4. Let X , . . . , X be a stretch of a strictly stationary time series (X ) of d-dimensional random 1 i i2Z vectors with common c.d.f F, continuous univariate marginal c.d.f.s F , . . . , F and copula C. If Conditions 3.1, 1 d 3.2 and 3.3 are met, then, for all (k, `) 2 B = f(k, `) 2 f1, . . . , dg : k < `g and as n ! ∞, d,2 n(β − β ) = 4 · C (1/2, 1/2), (7) k`,n k` k`,n n(bτ − τ ) = 8 C (u, v) dC (u, v) + o (1), (8) k`,n k` k`,n k` P where C (u, v) = C(1, . . . , 1, u, 1 . . . , 1, v, 1, . . . , 1) with u and v at the kth and `th position, respectively. k`,n As a consequence, under the null hypothesis of ellipticity, we have n D Z N (0, Σ), (9) d(d−1)/2 where Z = (Z ) with k` (k,`)2B d,2 Z := Ψ (G ) := 4 · G (1/2, 1/2) − 8 G (u, v) dC (u, v). (10) k` k` C k`,C k`,C k` and where Σ = (Σ 0 0 ) 0 0 with Σ 0 0 = Cov(Z , Z 0 0). (k`),(k ` ) (k,`),(k ,` )2B (k`),(k ` ) k` k ` d,2 Remark 3.5. Under slightly more restrictive mixing conditions (see, e.g., [13]) and less restrictive conditions on C, it can be shown that the limiting covariance may alternatively be written as (β) (τ) (β) (τ) Σ = Cov (h − h )(U , U ), (h − h )(U 0 , U 0 ) , (11) 0 0 kh `h k h ` h (k`),(k ` ) k` k` h2Z where, for u, v 2 [0, 1], (β) 1 1 1 1 h (u, v) = 4 · 1(u ≤ , v ≤ ) − 2 · 1(u ≤ ) − 2 · 1(v ≤ ) (12) 2 2 2 2 (τ) h (u, v) = 8C (u, v) − 4u − 4v + 2. (13) k` k` A sketch-proof relying on U-statistic theory for strongly mixing observations is given in Section B. For testing meta-ellipticity, one may use various real-valued functionals of D de ned in (4). Throughout this paper, we opt for the L -type test statistic b b b T := n · D D . (14) n n In the i.i.d. case, related Wald-type statistics have been found to provide worse accuracy, see [26] and [37]. The latter may be explained by the fact that Wald-type statistics involve an estimator for an inverse covariance Testing for meta-ellipticity Ë 127 matrix of possibly small signals. Likewise L - or L -type test statistics have been found to be of comparable quality to the L -statistic, see [37]. Now, Theorem 3.4 and the Continuous Mapping Theorem (see Theorem 1.3.6 in [44]) immediately yield T T := Z Z . The limiting variable can be written as a weighted sum of independent chi-square variables with one degree of freedom, where the weights depend in a complicated, statistically intractable way on the copula C and the serial dependence of the time series. For that purpose, we will introduce a suitable bootstrap scheme in the next section. Remark 3.6. The proposed tests can straightforwardly be adapted to the situation where one is only interested in testing whether some of the bivariate margins are elliptical. 3.2 A subsampling procedure Among the abundance of bootstrap procedures, the subsampling approach [36] has recently attracted atten- tion when working with empirical copulas for a number of practical reasons, see [28]. First of all, in compar- ison to bootstrap schemes that are based on resampling with replacement, the approach does not arti cially introduce ties into the bootstrap samples, thereby avoiding what might be called a ‘tie-bias’. Next, in com- parison to various versions of the multiplier bootstrap [6, 38], subsampling does not require expensive case- by-case implementation of the bootstrap approximation (see also [37] for a multiplier bootstrap for testing ellipticity). Finally, the subsampling approach may easily be modi ed in such a way that it is valid for time series data. Following [28], we de ne two dierent subsampling schemes. The rst one is only valid in the i.i.d. case, while the latter may be applied to a general stationary time series (including the i.i.d. case). In the former case, (iid) n let N = denote the number of subsamples of size b that may be taken from X , . . . , X and denote the b,n subsamples by [m] [m] [m] (iid) X = (X , . . . , X ), m 2 f1, . . . , N g. b 1 b b,n (ts) Under the plain assumption of observing a strictly stationary time series, let N = n − b + 1 denote the b,n number of possible subsamples that consist of b successive observations, and denote them by [m] [m] [m] (ts) X = (X , . . . , X ) = (X , . . . , X ), m 2 f1, . . . , N g. m+b−1 b b b,n Algorithm 3.7. For a given sample X , . . . , X of size n: 1. Compute the statistic T from (14). 2. Choose a number S 2 N of bootstrap replicates and a subsampling size b 2 f1, . . . , ng such that S ≤ N , b,n (iid) (ts) where N = N if the sample is (believed to be) i.i.d. and N = N if the sample is (believed to be) a b,n b,n b,n b,n stationary time series that is not i.i.d. 3. For s 2 f1, . . . , Sg: [I ] [I ] [I ] s,n s,n s,n • Randomly select a subsample X = (X , . . . , X ) of size b by drawing I randomly from s,n b b f1, . . . , N g. b,n [I ] s,n [I ] s,n • Compute the statistic β and τ from the subsample. b,n b,n • Compute the bootstrap statistic [I ] [I ] s,n s,n [I ] [I ] [I ] s,n −1 s,n > s,n b b b b b b b b T = (1 − b/n) b(β − β − τ − τ ) (β − β − τ − τ ). n n b,n n b,n b,n n b b,n 4. An approximate p-value for the test based on T is then given by [I ] s,n b b p = IfT > T g . S,b,n b,n s=1 128 Ë Axel Bücher, Miriam Jaser, and Aleksey Min The following result concerning the validity of the subsampling procedure is the second main theorem. Theorem 3.8. Suppose that X , . . . , X is either i.i.d. or an excerpt from a strongly mixing stationary time −a series with mixing coe cient α(h) = O(h ) for some a > 0, as h ! ∞ (see (6) for the de nition of α(h) and note that, as a consequence, Condition 3.1 is met). Further, assume that Conditions 3.2 and 3.3 are met. If b = b ! ∞, b = o(n) and S = S ! ∞ as n ! ∞, then n n Uniform([0, 1]) , if β = τ for all (k, `) 2 B , d k` k` d,2 bp ! S,b,n 0 , if β ≠ τ for some (k, `) 2 B k` k` d,2 as n ! ∞. In particular, for α 2 (0, 1), the test φ = 1(p ≤ α) is an asymptotic level α for H which is S,b,n S,b,n consistent against all alternatives with β ≠ τ for some (k, `) 2 B . k` k` d,2 It is important to note that, in view of Example 2.1, test φ is not consistent against any non-elliptical S,b,n copula. In practice, it is therefore advisable to complement the above test by suitable nonparametric tests involving other important qualitative features of bivariate elliptical copulas, such as symmetry or radial sym- metry (see [22] and [20], respectively, for the i.i.d. case). 4 Simulation study The nite-sample performance of the proposed test for meta-ellipticity was investigated in a large-scale Monte Carlo simulation study. The study was designed to primarily illustrate the test’s level and power properties for varying (1) sample size, (2) block length parameter, (3) dimension, (4) strength of the serial dependence, and (5) strength of the cross-sectional dependence. We also illustrate that an application of the related test from [37], which is designed for i.i.d. data, can fail in case of serial dependence. 4.1 Setup The aforementioned goals were tackled by considering four dierent copula families (Gaussian and t for H , 5 0 Clayton and Frank for H ), three dierent dimensions d 2 f2, 3, 6g, ve dierent levels of serial dependence, and ve dierent levels of cross-sectional dependence. With respect to the cross-sectional dependence, the respective copula parameters were chosen in such a way that all bivariate margins exhibit the same Kendall’s τ, taken from the setf0.1, 0.25, 0.5, 0.75, 0.9g. With respect to the serial dependence, we opted for the following transformation of a classical Gaussian AR(1)-model. First, starting from d independent AR(1)-models Y = φY + ε , (k 2 f1, . . . , dg, i 2 f1, . . . , ng), k,i k,i−1 k,i with ε i.i.d. N(0, 1) and Y i.i.d. N 0, 1/(1 − φ ) , whose stationary distribution is well-known to be k,i k,0 N 0, 1/(1 − φ ) , we may construct random vectors V = (V , . . . , V ) with independent standard uni- i 1,i d,i 2 1/2 formly distributed coordinates by setting V = Φ (1 − φ ) Y ; Φ the c.d.f. of the standard normal k,i k,i distribution. Next, for some given copula C as speci ed above, the vectors V may be transformed to (se- rially dependent) observations U from C by applying the inverse Rosenblatt transformation [39]. Overall, the serial dependence is controlled by a single parameter φ, which was chosen in such a way that the lag 1 auto-correlation version of Kendall’s tau of (Y ) varies in the setf0, 0.2, 0.4, 0.6, 0.8g. k,i i Finally, the sample size n was chosen to vary inf100, 250, 500, 1000g, while the block length parameter was chosen to vary in f0.05n, 0.1n, . . . , 0.6ng. The number of Monte replications was set to N=1000, the number of subsampling replications to S=300, and all tests were performed at a signi cance level of α = 0.05. Since the plain subsampling approach described in Algorithm 3.7 suers from the fact that observations at the start and at the end of the observation period have a reduced chance of appearing in a randomly selected Testing for meta-ellipticity Ë 129 block of size b, we applied the following slight modi cation: instead of drawing (in step 3) from the blocks starting at observation X with i 2 f1, . . . , n−b + 1g only, we also allow to subsample a block starting at X i n−i with i 2 f0, . . . , b − 2g, with the respective block being de ned as (X , . . . , X , X , . . . , X ) (which n−i 1 b−i+1 is similar in spirit to the circular bootstrap). Since b = o(n), this modi cation does not make a dierence asymptotically, but we observed increased accuracy for nite samples. Finally, for S = 300 ≥ n, there are only n blocks to draw from, whence we did draw each block exactly once, instead of S times with replacement. In terms of computing time, we remark that the subsampling approach with a single xed subsampling size b is advantageous over the multiplier bootstrap from [37], as calculating each bootstrap statistic relies on only b = o(n) observations, compared to n observations for the latter. Within a small experiment with 0.95 b = bn /4c, we found that the relative computing time ‘multiplier/subsampling’ ranges from 2.44 (d = 2, n = 100) up to 61 (d = 6, n = 1000). As a consequence, even evaluating the subsampling approach for various block sizes from a grid does not necessarily make it computationally heavier than the multiplier method. 4.2 Empirical level and empirical power results In this section, we partially report the results from the simulation study, after thoroughly weighing complete- ness against brevity. First of all, Figure 2 shows empirical rejection probabilities for samples from the Gaussian model and the Frank model in dimension d=3 for all chosen sample sizes, block sizes, serial dependencies and cross- sectional dependencies as described in the previous section. Little dots at the left-hand side of each plot refer to the empirical rejection probability of the test from [37] (which is designed for the iid case only). The triangles at the right-hand side will be explained below. In terms of level approximation (upper panel), we see that our test does not show a huge dependence on the choice of the block length in most cases. Moreover, it is slightly conservative in many cases, in particular for small sample sizes and large block sizes. For high levels of serial dependence, the test becomes liberal for small block sizes. In comparison, Quessy’s test does not hold its level for moderate to high levels of serial dependence. Similar results were obtained for dimensions d 2 f2, 6g and for the t-copula. In terms of power (lower panel), we observe very little power for sample size n = 100, but a great in- crease with increasing sample size. Moreover, large block sizes tend to reduce the test’s ability to detect the alternative. The latter may be explained by continuity reasons, observing that in the extreme case, n = b, we (ts) have N = 1, which amounts to no power at all. Qualitatively similar results were obtained for dimensions b,n d 2 f2, 6g and for the Clayton copula. Based on the results summarized in Figure 2, as well as on those that were not reported for the sake of 0.95 brevity, we propose to use b = bn /4c as a general formula for choosing the subsample size (which is in accordance with the assumptions from Theorem 3.8). The empirical rejection probabilities for this block size are displayed as triangles on the right-hand side of each plot in Figure 2. It can be seen that the choice guarantees that the level is not exceeded, while at the same time reaching decent power properties. Next, Tables 1, 2, and 3 report empirical rejection probabilities of our test for xed subsample size b = 0.95 bn /4c, and for the test from [37] based on a multiplier bootstrap (given in parentheses) for dimension d = 2, 3, and 6, respectively. For the sake of brevity, we excluded the case τ = τ(cross) 2 f0.1, 0.9g (for which the results were qualitatively the same) and the sample size n = 100 (for which little to no power was visible). The general ndings are similar as for Figure 2: while our test keeps its level, Quessy’s test fails to do so for moderate to high level of serial dependence. There are almost no dierences between the t-copula and the Gaussian copula models. In terms of detecting alternatives, the Frank copula exhibits much larger rejection probabilities. Moreover, the rejection probabilities are largest for high levels of cross-sectional dependence and small levels of temporal dependence. Finally, note that the Clayton copula may more easily be detected to be non-elliptical by applying a suitable test for radial symmetry (adapted to the time series case). AR = 0 AR = 0 AR = 0.2 AR = 0.2 AR = 0.4 AR = 0.4 AR = 0.6 AR = 0.6 AR = 0.8 AR = 0.8 AR = 0 AR = 0 AR = 0.2 AR = 0.2 AR = 0.4 AR = 0.4 AR = 0.6 AR = 0.6 AR = 0.8 AR = 0.8 130 Ë Axel Bücher, Miriam Jaser, and Aleksey Min Gauss, d= 3 Gauss, d= 3 n = 100 n = 100 n = 250 n = 250 n = 500 n = 500 n = 1000 n = 1000 0.3 0.3 0.2 0.2 0.1 0.1 0.0 0.0 0.3 0.3 0.2 0.2 0.1 0.1 tau (cross): tau (cross): 0.0 0.0 0.3 0.3 0.1 0.1 0.2 0.2 0.25 0.25 0.5 0.5 0.1 0.1 0.75 0.75 0.0 0.0 0.3 0.3 0.9 0.9 0.2 0.2 0.1 0.1 0.0 0.0 0.3 0.3 0.2 0.2 0.1 0.1 0.0 0.0 20 20 40 40 60 60 40 40 80 80 120 120 100 100 200 200 300 300 200 200 400 400 600 600 Block Siz Block Size b e b Frank, d= 3 Frank, d= 3 n = 100 n = 100 n = 250 n = 250 n = 500 n = 500 n = 1000 n = 1000 1.00 1.00 0.75 0.75 0.50 0.50 0.25 0.25 0.00 0.00 1.00 1.00 0.75 0.75 0.50 0.50 0.25 0.25 tau (cross): tau (cross): 0.00 0.00 1.00 1.00 0.1 0.1 0.75 0.75 0.25 0.25 0.50 0.50 0.5 0.5 0.25 0.25 0.75 0.75 0.00 0.00 1.00 1.00 0.9 0.9 0.75 0.75 0.50 0.50 0.25 0.25 0.00 0.00 1.00 1.00 0.75 0.75 0.50 0.50 0.25 0.25 0.00 0.00 20 20 40 40 60 60 40 40 80 80 120 120 100 100 200 200 300 300 200 200 400 400 600 600 Block Siz Block Size b e b Figure 2: Empirical rejection probabilities for the Gaussian model (upper part) and Frank model (lower part) in dimension d=3 against the block size b for n 2 f100, 250, 500, 1000g. AR = c refers to the fact that the marginal AR models were chosen in such a way that the lag 1 auto-Kendall-rankcorrelation equals c 2 f0, .2, .4, .6, .8g. Likewise, τ(cross) 2 f0.1, 0.25, 0.5, 0.75, 0.9g speci es the (pairwise) cross sectional dependencies in terms of Kendall’s tau. Empirical rejection 0.95 probabilities for the test from [37] and for our test with xed block size b = bn /4c are depicted as points on the left side and triangles on the right side, respectively. Empir Empirical Rejection Probability ical Rejection Probability Empir Empirical Rejection Probability ical Rejection Probability Testing for meta-ellipticity Ë 131 Table 1: Dimension d = 2: Empirical level (Panel A) and empirical power (Panel B) of our test for ellipticity based on the sub- 0.95 sampling procedure with block size b = bn /4c and the test from [37] based on a multiplier bootstrap (in parentheses) with signi cance level α = 0.05. The model parameters are as in Figure 2. AR τ n = 250 n = 500 n = 1000 n = 250 n = 500 n = 1000 Panel A: Gaussian t 0 0.25 4.5 (5.6) 5.0 (5.1) 7.1 (5.5) 4.0 (5.5) 5.8 (5.8) 6.4 (5.5) 0.50 4.0 (6.2) 5.2 (5.0) 5.8 (6.0) 3.7 (5.7) 4.9 (4.3) 5.9 (4.4) 0.75 2.4 (4.8) 4.6 (4.8) 4.9 (5.2) 3.4 (4.5) 6.0 (6.2) 5.8 (6.1) 0.2 0.25 4.4 (5.1) 5.5 (6.3) 6.2 (5.5) 3.4 (5.7) 6.3 (6.3) 5.0 (5.0) 0.50 3.9 (6.7) 5.5 (6.1) 4.3 (4.1) 4.7 (6.2) 4.6 (4.6) 4.5 (4.6) 0.75 2.5 (4.5) 4.1 (5.7) 4.8 (5.3) 2.1 (5.8) 4.6 (5.8) 5.1 (5.5) 0.4 0.25 3.5 (6.2) 4.8 (6.7) 5.2 (6.5) 2.4 (4.2) 3.0 (5.1) 4.3 (5.3) 0.50 3.4 (5.2) 4.3 (5.9) 3.9 (4.8) 2.6 (5.4) 3.8 (6.2) 4.1 (5.3) 0.75 1.5 (5.4) 2.6 (5.8) 3.6 (5.0) 1.6 (6.2) 3.5 (5.6) 5.5 (7.3) 0.6 0.25 1.8 (7.8) 3.6 (7.8) 3.9 (6.9) 1.7 (7.8) 3.4 (8.8) 4.8 (8.4) 0.50 1.6 (6.3) 3.3 (9.3) 3.7 (7.8) 1.9 (8.3) 4.0 (8.6) 4.8 (8.9) 0.75 1.5 (9.4) 2.6 (8.0) 3.5 (7.1) 1.4 (6.9) 3.0 (6.7) 4.3 (8.0) 0.8 0.25 2.5 (22.3) 2.7 (23.8) 4.0 (26.5) 3.0 (20.0) 4.5 (25.1) 4.7 (27.7) 0.50 3.3 (23.1) 4.9 (24.3) 4.5 (25.6) 3.1 (18.5) 2.6 (23.0) 4.4 (23.7) 0.75 2.6 (15.7) 4.3 (20.0) 3.6 (19.7) 3.3 (17.0) 3.0 (18.5) 2.8 (19.7) Panel B: Frank Clayton 0 0.25 9.4 (11.0) 18.9 (18.5) 28.9 (29.9) 4.6 (5.3) 5.5 (5.4) 6.8 (4.6) 0.50 25.4 (28.5) 43.0 (44.8) 71.9 (78.4) 4.6 (5.7) 7.0 (7.1) 9.9 (8.0) 0.75 25.0 (33.2) 54.2 (56.9) 80.0 (86.3) 10.5 (14.9) 25.1 (25.6) 45.9 (46.2) 0.2 0.25 8.5 (12.7) 18.9 (18.7) 26.1 (29.1) 3.5 (5.0) 4.9 (5.2) 5.5 (5.1) 0.50 19.7 (25.0) 46.0 (47.1) 67.7 (72.9) 4.3 (6.0) 6.0 (5.4) 8.9 (7.5) 0.75 24.6 (32.9) 55.0 (57.5) 77.0 (84.1) 11.0 (17.9) 23.7 (27.4) 43.2 (45.0) 0.4 0.25 6.4 (10.9) 14.7 (18.6) 27.8 (31.2) 3.2 (5.1) 4.5 (5.7) 5.6 (6.4) 0.50 20.3 (24.9) 39.8 (45.2) 65.8 (73.0) 3.3 (5.5) 5.2 (5.5) 9.7 (8.8) 0.75 20.8 (34.9) 51.2 (59.1) 78.5 (86.3) 7.6 (13.8) 19.8 (23.1) 44.4 (45.8) 0.6 0.25 4.8 (14.1) 12.0 (20.9) 21.3 (30.1) 3.1 (8.3) 3.1 (7.6) 4.5 (7.1) 0.50 12.4 (23.8) 33.4 (44.9) 58.1 (70.2) 2.1 (7.4) 4.9 (9.5) 8.9 (11.0) 0.75 15.4 (31.1) 42.9 (56.2) 69.3 (81.0) 4.6 (15.3) 15.7 (25.7) 35.9 (45.1) 0.8 0.25 4.7 (24.8) 6.5 (29.3) 13.4 (39.7) 3.8 (22.8) 3.8 (21.5) 3.1 (25.3) 0.50 5.8 (28.2) 13.1 (40.8) 31.7 (61.7) 3.1 (21.0) 3.5 (22.2) 5.1 (21.5) 0.75 7.4 (32.0) 18.8 (45.2) 44.7 (71.6) 3.6 (22.1) 7.7 (29.3) 17.0 (39.9) 5 Case Study Elliptical copulas are popular for nancial data due to their tractability, their exibility and in particular their ability to capture the dependence of extreme events. However, recent studies also report an observed non- (meta-)ellipticity of stock returns, see, e.g., [11], [26]. Within this section, we illustrate our method with a case study that allows to further enlighten this topic. More precisely, we reconsider the case study from Section 5 in [26], consisting of a three dimensional data set of 2663 daily log returns of the DAX, the Dow Jones Industrial Average and the Euro Stoxx 50 indices for 11 years in the period January 1, 2006 till December 31, 2016. Unlike in the empirical analysis in that paper, we do not necessarily need to apply ARMA-GARCH- lters to obtain approximately independent observations, 132 Ë Axel Bücher, Miriam Jaser, and Aleksey Min Table 2: Analogue of Table 1 in dimension d = 3. AR τ n = 250 n = 500 n = 1000 n = 250 n = 500 n = 1000 Panel A: Gaussian t 0 0.25 3.6 (4.6) 4.7 (4.7) 4.5 (5.6) 2.4 (3.6) 5.0 (5.3) 4.0 (3.6) 0.50 4.0 (5.7) 4.2 (4.0) 4.7 (4.5) 3.6 (4.8) 5.4 (5.9) 5.0 (5.3) 0.75 3.5 (5.2) 4.5 (5.3) 4.3 (5.7) 2.9 (4.9) 4.9 (6.3) 4.6 (4.7) 0.2 0.25 2.1 (4.4) 4.1 (5.6) 4.4 (5.6) 2.0 (4.2) 4.3 (5.6) 3.7 (4.9) 0.50 3.2 (4.9) 5.7 (6.2) 4.5 (4.1) 2.5 (5.0) 3.2 (4.2) 4.3 (5.4) 0.75 2.9 (5.3) 4.7 (6.1) 4.4 (5.4) 2.0 (4.7) 2.8 (4.7) 3.9 (4.0) 0.4 0.25 1.7 (5.0) 2.8 (5.7) 4.1 (6.1) 2.4 (5.5) 4.3 (6.1) 3.5 (5.3) 0.50 2.1 (5.6) 3.7 (4.6) 3.6 (5.5) 2.6 (6.3) 2.8 (5.1) 3.4 (6.2) 0.75 2.6 (5.9) 2.5 (4.9) 2.9 (5.6) 1.7 (4.3) 3.0 (6.2) 2.4 (5.1) 0.6 0.25 1.7 (9.4) 4.0 (10.2) 3.9 (10.2) 1.8 (9.1) 2.8 (10.1) 3.7 (11.1) 0.50 1.5 (7.5) 2.4 (8.5) 4.5 (10.4) 2.1 (9.4) 1.6 (8.0) 3.5 (9.2) 0.75 0.9 (8.4) 1.5 (6.4) 3.4 (9.4) 1.4 (6.4) 2.2 (7.2) 3.2 (7.3) 0.8 0.25 4.4 (36.3) 4.3 (39.9) 2.7 (44.5) 3.2 (33.0) 3.4 (41.0) 3.9 (43.2) 0.50 3.4 (32.1) 4.5 (35.5) 2.8 (37.2) 4.2 (29.9) 3.0 (36.4) 4.3 (38.9) 0.75 3.6 (20.3) 3.8 (22.6) 3.3 (27.2) 3.9 (23.5) 3.7 (24.6) 2.9 (26.1) Panel B: Frank Clayton 0 0.25 13.4 (16.6) 26.7 (28.3) 46.1 (52.6) 3.2 (4.1) 5.6 (5.0) 5.0 (5.3) 0.50 33.9 (37.4) 66.2 (69.8) 88.4 (93.3) 4.5 (5.1) 8.8 (7.8) 11.0 (9.0) 0.75 36.4 (45.2) 72.1 (76.4) 91.5 (96.4) 14.6 (17.1) 33.9 (33.2) 60.7 (62.0) 0.2 0.25 11.6 (16.0) 22.6 (26.3) 46.7 (54.3) 2.3 (3.8) 4.6 (5.1) 5.7 (6.0) 0.50 32.1 (38.1) 61.9 (65.7) 88.9 (93.7) 5.0 (6.0) 6.8 (6.6) 10.3 (9.8) 0.75 34.2 (43.6) 69.9 (74.3) 91.6 (96.2) 12.0 (16.1) 33.9 (32.1) 58.4 (61.0) 0.4 0.25 8.1 (15.2) 20.7 (26.9) 42.7 (52.1) 3.0 (6.7) 4.1 (6.0) 4.8 (6.3) 0.50 26.2 (36.4) 58.8 (65.6) 88.4 (94.6) 1.9 (5.2) 4.6 (5.8) 9.7 (11.0) 0.75 28.9 (42.7) 65.6 (72.5) 90.9 (95.5) 8.2 (17.3) 28.9 (33.3) 57.4 (61.5) 0.6 0.25 6.1 (18.1) 16.4 (32.7) 34.4 (52.5) 1.8 (9.2) 2.2 (8.9) 3.9 (9.1) 0.50 20.8 (39.5) 47.7 (65.0) 78.2 (89.8) 1.7 (7.1) 4.1 (10.0) 9.1 (14.2) 0.75 23.7 (40.0) 56.4 (71.9) 84.9 (94.0) 5.6 (16.7) 20.8 (32.2) 49.0 (58.6) 0.8 0.25 4.5 (36.3) 8.3 (48.9) 14.3 (61.0) 3.7 (36.9) 4.1 (37.5) 4.7 (42.4) 0.50 9.1 (43.1) 19.2 (60.8) 40.9 (80.4) 4.0 (31.1) 4.0 (34.1) 5.3 (37.0) 0.75 11.1 (39.2) 25.9 (63.0) 58.3 (86.0) 4.1 (25.3) 8.1 (39.3) 27.8 (57.3) but may also investigate the raw data for ellipticity. For the sake of completeness, we chose to analyse both, the raw data and ltered observations. Starting with the former, we may even investigate the six-dimensional data set (x , y , z , x , y , z ) t t t t+1 t+1 t+1 for t 2 f1, . . . , 2662g, where x , y and z is the log return at day t of the DAX, Dow Jones and EURO Stoxx t t t index, respectively. Note that n = 2662. Eight hypotheses are of interest: • Multivariate. Meta-ellipticity of the six-dimensional data set (denoted ‘1:6’) and of the three-dimensional cross-sectional dependence of (x , y , z ) (denoted ‘1:3’). t t t t=1,,...,2662 • Temporal. Meta-ellipticity of the three marginal temporal dependencies of (x , x ) (DAX, denoted t t+1 ‘(1,4)’), of (y , y ) (Dow, denoted ‘(2,5)’), and of (z , z ) (Eurostoxx, denoted ‘(3,6)’). t t+1 t t+1 • Crosssectional Pairs. Meta-ellipticity of the three pairwise dependencies of (x , y ) (DAX-Dow, denoted t t ‘(1,2)’), of (x , y ) (DAX-Eurostoxx, denoted ‘(1,3)’), and of (y , z ) (Dow-Eurostoxx, denoted ‘(2,3)’). t t t t Note that, for consistency, we restrict attention to t 2 f1, . . . , 2662g for the cross-sectional dependencies, despite the fact that we might include the observations for day 2663. Our test has been applied to each of Testing for meta-ellipticity Ë 133 Table 3: Analogue of Table 1 in dimension d = 6. AR τ n = 250 n = 500 n = 1000 n = 250 n = 500 n = 1000 Panel A: Gaussian t 0 0.25 2.1 (3.4) 3.6 (3.8) 3.9 (5.4) 2.2 (3.4) 3.5 (5.3) 2.5 (3.4) 0.50 2.2 (3.5) 3.5 (4.3) 3.9 (5.1) 2.6 (3.6) 4.0 (4.1) 3.9 (5.5) 0.75 2.1 (4.1) 4.8 (4.7) 3.5 (4.4) 2.1 (4.0) 3.0 (4.8) 4.5 (5.6) 0.2 0.25 2.0 (4.9) 1.6 (3.9) 2.7 (4.0) 1.1 (3.3) 3.2 (5.2) 2.3 (4.9) 0.50 2.9 (4.4) 2.9 (4.3) 2.4 (3.9) 1.2 (2.8) 3.3 (4.4) 2.3 (4.0) 0.75 2.3 (4.5) 2.2 (4.7) 2.7 (3.7) 1.9 (4.5) 3.0 (4.8) 3.1 (4.5) 0.4 0.25 1.1 (4.2) 1.7 (5.0) 2.5 (6.5) 1.1 (5.2) 1.5 (5.4) 2.4 (4.9) 0.50 1.2 (4.7) 3.1 (6.8) 2.4 (5.2) 1.4 (5.7) 2.2 (6.1) 2.4 (6.0) 0.75 1.7 (4.8) 2.0 (4.8) 3.0 (4.7) 1.1 (4.5) 2.1 (4.5) 2.8 (5.5) 0.6 0.25 0.5 (12.2) 1.1 (15.4) 1.6 (14.1) 1.1 (11.5) 2.0 (14.3) 2.4 (15.2) 0.50 1.7 (10.5) 2.7 (11.7) 2.5 (12.1) 1.1 (8.8) 2.0 (12.2) 3.3 (12.1) 0.75 1.4 (6.7) 2.2 (8.0) 3.3 (7.8) 1.1 (7.0) 2.3 (9.1) 2.2 (8.6) 0.8 0.25 3.7 (67.6) 2.5 (79.5) 2.7 (87.1) 3.5 (65.7) 3.8 (79.6) 3.3 (84.5) 0.50 4.8 (55.2) 2.8 (66.9) 4.0 (75.2) 4.2 (54.1) 3.8 (66.1) 3.6 (69.7) 0.75 3.5 (30.5) 3.9 (37.4) 2.8 (37.8) 4.0 (29.1) 2.6 (34.9) 2.3 (38.5) Panel B: Frank Clayton 0 0.25 20.8 (28.1) 49.3 (56.2) 78.5 (89.2) 1.9 (2.4) 3.2 (4.1) 4.8 (4.7) 0.50 56.7 (62.6) 91.1 (93.5) 99.6 (100.0) 4.0 (4.4) 8.9 (6.8) 13.6 (11.7) 0.75 52.4 (61.3) 86.5 (90.8) 98.3 (99.6) 20.9 (21.8) 51.6 (47.3) 82.6 (82.8) 0.2 0.25 17.6 (26.8) 45.1 (54.0) 78.2 (88.5) 2.1 (3.9) 2.8 (3.4) 2.7 (3.6) 0.50 53.7 (62.8) 90.2 (93.8) 99.1 (100.0) 3.4 (4.4) 6.7 (7.0) 14.4 (13.6) 0.75 52.1 (60.9) 85.6 (90.1) 99.3 (99.8) 18.8 (20.6) 48.0 (45.4) 79.5 (81.9) 0.4 0.25 12.2 (25.5) 38.5 (52.5) 78.5 (88.7) 1.1 (4.0) 3.4 (6.4) 3.4 (6.6) 0.50 50.0 (62.4) 86.6 (92.2) 99.7 (100.0) 2.5 (4.4) 6.3 (7.7) 10.2 (12.5) 0.75 45.4 (57.7) 84.2 (90.4) 98.2 (99.8) 14.9 (20.4) 45.7 (47.8) 75.7 (81.3) 0.6 0.25 8.7 (30.4) 26.4 (57.3) 65.0 (89.3) 1.0 (10.8) 1.7 (11.5) 2.5 (15.9) 0.50 32.1 (56.3) 75.1 (90.5) 95.7 (99.6) 1.1 (8.5) 3.9 (12.4) 10.7 (21.1) 0.75 35.8 (53.8) 75.7 (87.4) 95.3 (99.5) 10.6 (23.7) 32.6 (45.1) 67.3 (77.6) 0.8 0.25 6.9 (73.5) 10.2 (87.0) 27.3 (95.0) 4.4 (66.3) 2.6 (78.5) 2.6 (85.4) 0.50 15.1 (65.6) 32.3 (88.4) 64.6 (97.5) 4.1 (49.4) 3.1 (61.0) 7.3 (69.7) 0.75 14.7 (52.8) 36.7 (75.8) 67.5 (95.5) 4.2 (31.0) 11.5 (48.8) 37.9 (75.0) the aforementioned situations with subsample sizes b 2 fbcnc : c 2 0.1, 0.11, . . . , 0.39, 0.4g and with 0.95 b = b = bn /4c = 449. The results for b = bcnc are summarized in Figure 3 and in the lower row of opt Table 4, where we state the proportion of signi cant p-values (≤ 0.5). The p-value for b = b can be found opt in the upper row of Table 4. The results can be summarized as follows: p-values for the six-dimensional data set, as well as for two of the temporal dependencies (DAX and Eurostoxx) are clearly signi cant at the ve-percent level, even after a Bonferroni correction. On the other hand, the test did not nd any evidence against meta-ellipticity for the temporal dependence of the Dow Jones index; this dierence to the European indices may possibly be explained by dierences of nancial market regulations in the European Union and the US. Furthermore, the cross-sectional dependency of (x , y , z ) is found to be weakly signi cant, despite the t t t fact that none of the respective pairs is signi cant when considered on its own. The latter is hence an instance of the circumstance that comparably mediocre (pairwise) signals may add up to a strong overall signal. For simplicity ignoring that the data is serially dependent, the ndings may further be supported by standard 134 Ë Axel Bücher, Miriam Jaser, and Aleksey Min 0.95 Table 4: Upper row: p-values for the test with b = b = bn /4c = 449. Lower row: proportion of p-values smaller than 0.05 opt among all tests with b 2 fbcnc : c 2 0.10, 0.11, . . . , 0.39, 0.40g. Dependency 1:6 1:3 (1,4) (2,5) (3,6) (1,2) (1,3) (2,3) P-value(b ) 0.000 0.055 0.009 0.989 0.000 0.062 0.264 0.082 opt Prop. of Rej. 1.000 0.452 0.871 0.000 1.000 0.097 0.000 0.065 model selection procedures. More precisely, for all three bivariate cross-sectional dependencies, the family of t-copulas has been selected among 37 bivariate candidate models based on AIC and BIC model selection. However, the estimated degrees of freedom are equal to 2.808 (standard error 0.236), 3.733 (0.398) and 2.900 (0.247), respectively, which are incompatible with a three-variate t-copula. Moreover, among the candidate three-variate models, a (non-elliptical) d-vine model (with t-pair copulas) has been selected over the family of t-copulas based on AIC and BIC. Overall, the obtained results yield additional evidence for the non-(meta-)ellipticity of stock returns. However, we would like to stress once again that our empirical ndings and interpretations should be treated with caution when the null hypothesis cannot be rejected: we only test for the equality between Kendall’s tau and Blomqvist’s beta, which is not a characterizing property of elliptical copulas (see Example 2.1). Finally, following [26], we have also applied our test to the three-dimensional data set of sample size n = 2663 obtained from independently tting ARMA-GARCH-models to the margins of (x , y , z ) and calculating t t t respective standardized residuals (see the last-named paper for precise model speci cations). For simplicity, we only report the result for the null hypothesis of three-dimensional cross-sectional meta-ellipticity, which gets rejected at the 5% level with a p-value of 0.011. This is in line with the ndings of [26] who obtained a p-value of 0.030. Multivariate Temporal Pairs Crosssectional Pairs 1.00 0.75 0.50 0.25 0.00 400 600 800 1000 400 600 800 1000 400 600 800 1000 Subsample Size b 1:6 (1,4) (3,6) (1,3) Dependency: 1:3 (2,5) (1,2) (2,3) Figure 3: P-values as a function of the block size b for the eight hypotheses described in Section 5. 6 Conclusion A test for detecting departures from meta-ellipticity for multivariate stationary time series has been proposed. Carrying out the test requires (approximate) critical values of a complex asymptotic distribution, which were obtained using the subsampling bootstrap. Large-sample validity was proven. The test was found to perform well for moderate sample sizes in a simulation study. An application to nancial log returns p−value Testing for meta-ellipticity Ë 135 provided evidence for their non-(meta-)ellipticity. Acknowledgements: Axel Bücher’s work has been supported by the Collaborative Research Center “Sta- tistical modeling of nonlinear dynamic processes” (SFB 823) of the German Research Foundation, which is gratefully acknowledged. The authors would like to thank Arnold Janssen for providing them with the sec- ond example in Example 2.1 (i). They are also grateful to Jean-François Quessy for providing them with his MATLAB code on the test in [37], which was very useful for the simulation study performed in R. Finally, the authors are grateful to two unknown referees and an associate editor for their constructive and helpful comments on an earlier version of this article. A Proofs of the main results Proof of Theorem 3.4. The assertion in (7) is obvious. For the proof of (8), we simplify the notation by occa- b b b b sionally omitting the index k, `; i.e., we write τ = τ , C = C etc. Then, n n k`,n k`,n Z Z p p 1 (a) −1/2 b b n(bτ − τ) = n C dC − CdC + O (n ) n n n p Z Z Z −1/2 b b = C dC + n CdC − CdC + O (n ) n n n p Z Z (b) −1/2 = C dC + C dC + O (n ) n n n p (c) = 2 C dC + o (1), (15) which is (8). Explanations: (a) Note that τ = 2U − 1, where n n U = 1f(X − X )(X − X ) > 0g ki kj `i `j n(n − 1) 1≤i<j≤n = 1(X > X , X > X ) `i `j ki kj n(n − 1) 1≤i≠j≤n −1 = 1(X ≥ X , X ≥ X ) + O (n ). ki kj `i `j i,j=1 Further, n n X X 2 2 b b b b b b 1(X ≥ X , X ≥ X ) = 1(U ≥ U , U ≥ U ) = 2 C dC . n n ki kj `i `j ki kj `i `j 2 2 n n i,j=1 i,j=1 R R b b (b) It is su cent to show that CdC = C dC, which is related to the arguments given in the proof of n n Theorem 5.1.1 in [33]. For the ease of reading, we give a self-contained proof. Conditional on (X , . . . , X ) consider independent random vectors (U, V) C and (U , V ) C . We may then write n n n C dC = Pr(U ≤ U, V ≤ V) n n n and Z CdC = Pr(U ≥ U, V ≥ V). n n n 1 n Furthermore, by Condition 3.2, the distribution of U is uniform onf , . . . , g, whence n+1 n+1 Z Z Z i 1 1 Pr(U ≤ U ) = dF (u)dF (u ) = u dF (u ) = = . n n n n U U U n n n + 1 n 2 i=1 [0,1] [0,u ] [0,1] n 136 Ë Axel Bücher, Miriam Jaser, and Aleksey Min This implies Pr(U ≤ U, V ≤ V) = 1 − Pr(U > U) − Pr(V > V) + Pr(U > U, V > V) n n n n n n = 1 − Pr(U ≥ U) − Pr(V ≥ V) + Pr(U ≥ U, V ≥ V) n n n n = Pr(U ≥ U, V ≥ V), n n and hence (b). (c) We have to show that C d(C − C) = o (1). This is direct consequence of the continuous mapping n n P theorem and Lemma C.8 in [3]. The convergence result in (9) is a direct consequence of (7),(8), the continuous mapping theorem and the fact that, under ellipticity, p p p b b b b n(β − τ ) = n(β − β ) − n(τ − τ ). k`,n k`,n k`,n k` k`,n k` Finally, normality of the limit in (9) follows from the fact that G is Gaussian and Ψ de ned in (10) is linear. k` Proof of Theorem 3.8. We only need to prove the weak convergence result for p . For simplicity, we only S,b,n consider the strong mixing case, the proof for the i.i.d. case is essentially the same. For the weak convergence result under the null hypothesis, it is su cient to show that [I ] [I ] [1] [2] 1,n 2,n b b b (T , T , T ) ! (T, T , T ) (16) b,n b,n [1] [2] where T , T are i.i.d. copies of T, the weak limit of T . Indeed, the assertion then follows from Corollary 4.3 in [8], observing that T has a continuous c.d.f. For the proof of (16), recall that, by Theorem 3.4, T = fΨ(C ) + o (1)g fΨ(C ) + o (1)g with n n n P P 0 1 1 1 4 · C ( , ) − 8 · C (u , u )dC (u , u ) 12,n 2 12,n 1 2 12 1 2 2 2 [0,1] B 1 1 C 4 · C ( , ) − 8 · C (u , u )dC (u , u ) B 13,n 2 13,n 1 3 13 1 3 C 2 2 [0,1] B C Ψ(C ) = . B C @ . A 1 1 4 · C ( , ) − 8 · C (u , u )dC (u , u ) d−1,d,n 2 d−1,d,n d−1 d d−1,d d−1 d 2 2 [0,1] Suppose we have shown that [I ] [I ] > [I ] s,n s,n s,n b b b T = Ψ(C ) + o (1) Ψ(C ) + o (1) , (17) P P b,n b b [I ] [I ] [I ] [I ] s,n s,n s,n s,n b b b b where C = b(C − C ) with C the empirical copula based on the subsample X . The assertion b b b b in (16) then follows from the continuous mapping theorem and [I ] [I ] 1,n 2,n [1] [2] ∞ d 3 b b (C , C , C ) (C , C , C ) inf` ([0, 1] )g ; (18) b b C C the latter convergence being a consequence of Theorem 3.3 in [28]. It remains to show (17), which follows from [I ] [I ] s,n s,n 1 1 b b b(β − β ) = 4 · C ( , ) k`,n 2 2 k`,b b and Z Z p p 1 (a) [I ] [I ] [I ] s,n s,n s,n −1/2 b b b b b b b(τ − τ ) = b C dC − C dC + O (b ) k`,n k`,n k`,n P k`,b k`,b k`,b Z Z Z [I ] [I ] [I ] s,n s,n s,n −1/2 b b b b b b = C dC + b C dC − C dC + O (b ) k`,n k`,n k`,n P k`,b k`,b k`,b Z Z (b) [I ] [I ] [I ] s,n s,n s,n 1/2 −1/2 b b b b = C dC + C dC + O(b /n) + O (b ) k`,n P k`,b k`,b k`,b (c) [I ] s,n = 2 C dC + o (1). k`,b Whence it remains to explain (a), (b) and (c) in the latter equation. For that purpose, as in the proof of Theo- rem 3.4, we will omit the index k, `. Testing for meta-ellipticity Ë 137 (a) This follows by the same arguments as for the proof of (a) in (15). (b) Conditional on (X , . . . , X ) and I , consider independent random vectors (U , V ) C and n s,n n n n [I ] s,n (U , V ) C . We may then rewrite b b Z Z [I ] [I ] s,n s,n b b b b C dC = Pr(U ≤ U , V ≤ V ), C dC = Pr(U ≥ U , V ≥ V ). n n n n n n b b b b b b [I ] s,n Under the no-ties condition in Condition 3.2, the subsample X does not contain ties either, whence X X 1 1 Pr(V < V ) = Pr(U < U ) = n n b b n b j=1 i: < n+1 b+1 n o 1 j(n + 1) n + 1 1 1 1 = + O(1) = + O = + O nb b + 1 2n n 2 n j=1 and 1 1 Pr(V = V ) = Pr(U = U ) = Pr(U = ) ≤ . n n n b b b+1 b n j=1 These two equations imply Pr(U ≤ U , V ≤ V ) = 1 − Pr(U > U ) − Pr(V > V ) + Pr(U > U , V > V ) n n n n n n b b b b b b = Pr(U ≥ U , V ≥ V ) + O(1/n), n n b b where the O-terms are not depending on (X , . . . , X ) and I . This implies (b). n s,n R R [I ] [I ] [I ] s,n s,n s,n b b b b (c) We have to show that C d(C −C) = o (1) and C d(C −C) = o (1). This is a direct consequence P P b b b of (18), the continuous mapping theorem and Lemma C.8 in [3]. Finally, consider the alternative. By the same arguments as under the null hypothesis, we have [I ] [I ] d 1,n 2,n [1] [2] b b (T , T ) ! (T , T ), b,n b,n [1] [2] where T and T are as in (16). By Lemma 2.3 in [8], we have [I ] 1,s sup 1(T ≤ x) − F (x) = o (1), T P b,n x2R s=1 where F denotes the c.d.f. of T. As a consequence, bp = 1 − F (T ) + o (1) = o (1), S,b,n T P P where the last equality follows from T ! ∞ in probability under the alternative. B Sketch-proof of Remark 3.5 Lemma B.1. Under the null hypothesis of ellipticity, and if each C has continuous partial derivatives in a k` neighbourhood of (1/2, 1/2), we have sup nfβ − β g − B = o (1), n ! ∞, k`,n k` k`,n P k≠` where, for k, ` 2 f1 . . . , dg with k ≠ `, n o (β) (β) B = p h (U , U ) − E[h (U , U )] , k`,n ki `i ki `i i=1 (β) and where h is de ned in (12). 138 Ë Axel Bücher, Miriam Jaser, and Aleksey Min Proof. Fix k, ` 2 f1, . . . , dg with k ≠ `. By ellipticity of C , we have C (u, v) = u + v − 1 + C (1 − u, 1 − v), k` k` k` which implies ∂ C (u, v) = 1 − ∂ C (1 − u, 1 − v) for j 2 f1, 2g; in particular ∂ C (1/2, 1/2) = 1/2. A j k` j k` j k` straightforward modi cation of Corollary 2.5 in [9] then implies 1 1 1 1 1 1 1 1 1 nfC ( , ) − C ( , )g = α ( , ) − fα ( , 1) + α (1, )g + o (1) k`,n k` k`,n k`,n k`,n P 2 2 2 2 2 2 2 2 2 = B + o (1), k`,n P p p where α = n(C − C) with C as de ned in (5) and where the o (1)-term is uniform in k, `. Since n(β − n n n P k` 1 1 1 1 β ) = 4 nfC ( , ) − C ( , )g, we obtain the assertion. k` k`,n k` 2 2 2 2 Lemma B.2. For k, ` 2 f1 . . . , dg with k ≠ `, let n o (τ) (τ) T = p h (U , U ) − E[h (U , U )] , k`,n ki `i ki `i k` k` i=1 (τ) where h is de ned in (13). Then, under suitable mixing conditions (e.g., Theorem 2.1 in [13]), k` sup nfτ − τ g − T = o (1), n ! ∞. k`,n k` k`,n P k≠` It is important to note that no regularity condition on C is needed (see also [45] for the i.i.d. case). Proof. 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Journal
Dependence Modeling – de Gruyter
Published: Jan 1, 2021
Keywords: elliptical copula; empirical process; financial log returns; goodness-of-fit test; subsampling bootstrap; 62H15; 62M10