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Randomization tests of copula symmetry
\affil[1]{School of Economics, University of Sydney} \affil[2]{Department of Economics, National University of Singapore}
In this paper we propose statistical tests of the null hypothesis that a copula \(C\) is symmetric, based on a sample of independent and identically distributed (iid) pairs of random variables with common copula $C$. We focus on two notions of symmetry that have received particular attention in the literature: exchangeability and radial symmetry. Let \((U,V)\) be a pair of random variables whose joint distribution is given by the copula \(C\). We say that \(C\) is exchangeable when
Exchangeability of \(C\) is satisfied if and only if \((U,V) \mathrel{\mathop{=}\limits^{ \vbox to0ex{\kern-5\ex@ \hbox{$\scriptstyle\mathcal{D}$}\vss}}}(V,U)\), where \( \mathrel{\mathop{=}\limits^{ \vbox to0ex{\kern-5\ex@ \hbox{$\scriptstyle\mathcal{D}$}\vss}}}\) signifies equality in law. We say that \(C\) is radially symmetric when
where \(C^\mathrm{s}(u,v)=u+v-1+C(1-u,1-v)\), the survival copula for \(C\). Radial symmetry of \(C\) is satisfied if and only if \((U,V) \mathrel{\mathop{=}\limits^{ \vbox to0ex{\kern-5\ex@ \hbox{$\scriptstyle\mathcal{D}$}\vss}}}(1-U,1-V)\). See N93,N06,N07 for further discussion of the exchangeability and radial symmetry properties.
The property of exchangeability plays an important role in various models of economic interaction. M16 writes that “exchangeability of a certain form is a feature of almost any commonly used empirical specification for game-theoretic models with more than two players”. A prominent example is the symmetric common value auction model, which was developed by MW82 under the assumption that the distribution of signals across bidders is exchangeable. Such exchangeability has powerful implications for the identification of structural econometric models of auctions AH02 and is frequently assumed when they are estimated LPV00,HPP03,T11. Another example is the model of product bundling developed by CR13b, in which the exchangeability of the copula describing the dependence between consumer valuations of different products is a central assumption when a multi-product firm competes with a single-product firm. Radial symmetry, or rather the lack thereof, has been a subject of interest in empirical finance: researchers have found that the dependence between various asset returns, particularly equity portfolios, is markedly stronger in downturns than in upturns AC02,HTZ02. Radially asymmetric copula functions have proved to be useful for modeling this feature of return dependence P04,P06,O08,GT11.
Several statistical tests of exchangeability and radial symmetry for bivariate copulas have been proposed in recent literature. GNQ12 and GN14 proposed tests of copula exchangeability and radial symmetry respectively. Extensions of Genest et al.'s exchangeability tests to higher dimensional copulas have been provided by HS17. Genest and Ne\v{s}lehov\'{a}'s radial symmetry tests extend earlier contributions of BC12 and DDU13. Other tests of copula exchangeability and radial symmetry were proposed by LG13, QB13, BQ17, BIW17 and K17. BS14 also proposed a test of copula exchangeability, but for the somewhat different case where the copula in question characterizes the serial dependence in a univariate time series. Many other authors have considered tests of exchangeability or radial symmetry for multivariate cdfs---see, for instance, Q16 and references therein---but such tests are typically inapplicable to hypotheses of copula symmetry due to the unobservability of margins.
The new tests of copula symmetry proposed in this paper combine the test statistics of GNQ12 and GN14 with a new method of constructing critical values. Whereas those authors obtain critical values using the multiplier bootstrap of RS09 and BD10, we instead use a novel resampling procedure motivated by randomization tests of symmetry hypotheses. R89,R90 observed that exact tests of symmetry hypotheses on multivariate cdfs could be obtained by applying randomization procedures that exploit group invariance conditions implied by symmetry. While these tests are not directly applicable to hypotheses of copula symmetry, we show how a feasible randomization procedure may be used to obtain critical values that properly account for uncertainty about margins. The justification for our procedure is asymptotic rather than exact, but numerical simulations indicate excellent size control with sample sizes as small as \(n=30\). Simulations also indicate substantially improved power compared to the tests based on the multiplier bootstrap at smaller sample sizes.
A recent paper by CRS17 is related to ours in that it studies the behavior of randomization tests when symmetry is only approximately satisfied. Suppose we have a sample \(Z^{(n)}\) of size \(n\) taking values in a sample space \(\mathcal Z_n\). Approximate symmetry in the sense of CRS17 means that for each \(n\) there exists a map \(S_n\) from \(\mathcal Z_n\) to a metric space \(\mathcal S\) such that (i) \(S_n(Z^{(n)})\) converges in law to a random element \(S\) of \(\mathcal S\) as \(n\to\infty\), and (ii) \(g(S)\) is equal in law to \(S\) for all \(g\) in some finite group of transformations \(\mathbf G\). Crucially, \(\mathcal S\) and \(\mathbf G\) cannot depend on \(n\). In our paper, the sample is a collection of iid pairs \(Z^{(n)}=((X_1,Y_1),\ldots,(X_n,Y_n))\) taking values in \(\mathcal Z_n=(\mathbf R^2)^n\). Approximate symmetry holds in the following sense: if \(S_n:\mathcal Z_n\to([0,1]^2)^n\) is the map that transforms our sample to the normalized rank pairs \(((U_{n1},V_{n1}),\ldots,(U_{nn},V_{nn}))\) defined in equation (ref) below, then the law of \(S_n(Z^{(n)})\) is (loosely speaking) approximately that of \(n\) iid draws from the copula \(C\). When \(C\) is symmetric, such \(n\)-tuples of iid draws are distributionally invariant under a group \(\mathbf G_n\) consisting of \(2^n\) distinct transformations from \(([0,1]^2)^n\) to itself; we postpone details of \(\mathbf G_n\) to Section (ref). Since the dimension of \(([0,1]^2)^n\) and the number of transformations in \(\mathbf G_n\) grow with \(n\), our problem falls outside the scope of the results of CRS17, in which \(\mathcal S\) and \(\mathbf G\) are assumed fixed.
Recent results of CR13,CR16a,CR16b are also somewhat related to the problem studied here. Like us, and unlike CRS17, Chung and Romano allow the number of transforms in the group \(\mathbf G_n\) to increase with \(n\). However, whereas in our setting the normalized rank pairs \(((U_{n1},V_{n1}),\ldots,(U_{nn},V_{nn}))\) are approximately distributionally invariant under \(\mathbf G_n\) whenever the null is satisfied, in the setting considered by Chung and Romano the data are exactly distributionally invariant under \(\mathbf G_n\) on a subset of the null, and not even approximately invariant elsewhere in the null. The problems we study are therefore fundamentally different. Chung and Romano establish their results by verifying a condition of H52 necessary and sufficient for suitable convergence of the randomization distribution. We instead take the conditional approach to which CR13 refer following their discussion of Hoeffding's condition. Specifically, in place of Hoeffding's condition we verify that a statistic computed from a random transformation of the normalized rank pairs converges weakly to a suitable limit conditional on the data in probability.
Our paper is structured as follows. In Section (ref) we define our test statistics and characterize their limit distributions under the null hypothesis of symmetry. In Section (ref) we describe our feasible randomization procedure for obtaining critical values. In Section (ref) we provide results on the asymptotic properties of tests based on our feasible randomization procedure. The results of our numerical simulations are presented in Section (ref). Some closing remarks are given in Section (ref). Proofs and supplementary lemmas are collected in Appendix (ref).
Let \(X\) and \(Y\) be random variables with bivariate cumulative distribution function (cdf) \(H(x,y)=\mathrm{P}(X\leq x,Y\leq y)\) and margins \(F(x)=\mathrm{P}(X\leq x)\) and \(G(y)=\mathrm{P}(Y\leq y)\). We assume that \(F\) and \(G\) are continuous. Sklar's theorem S59 then ensures the existence of a unique copula \(C:[0,1]^2\to[0,1]\) satisfying \(C(F(x),G(y))=H(x,y)\) for all \(x,y\in\mathbf R\). The copula \(C\) is the bivariate cdf of the probability integral transforms \(U=F(X)\) and \(V=G(Y)\).
Our data consist of \(n\) iid draws \((X_1,Y_1),\ldots,(X_n,Y_n)\) from \(H\). Let \(F_n\), \(G_n\) and \(H_n\) be the empirical cdfs corresponding to \(F\), \(G\) and \(H\) respectively. We use \(F_n\) and \(G_n\) to construct (normalized) ranks
From the rank pairs \((U_{ni},V_{ni})\) we construct the empirical copula
An alternative definition of the empirical copula in common use is
where \(F_n^\leftarrow\) is the generalized inverse of \(F_n\),
and \(G_n^\leftarrow\) is defined similarly. The definitions of the empirical copula given in ((ref)) and ((ref)) are attributed to R76 and D79 respectively. They differ by at most \(2n^{-1}\) almost surely (a.s.); see Lemma (ref) in the Appendix. The R\"{u}schendorf empirical copula is more convenient for computation, while the Deheuvels empirical copula is more convenient to analyze using the delta method.
The asymptotic validity of our proposed testing procedures hinges on weak convergence of the empirical copula process \(\mathbb C_n=\sqrt{n}\left(C_n-C\right)\) in the space \(\ell^\infty([0,1]^2)\) of bounded real valued functions on the unit square equipped with the uniform metric. Such weak convergence is satisfied under the following condition of S12.
We extend the definition of \(\dot{C}_1\) to \([0,1]^2\) by setting
and similarly for \(\dot{C}_2\). When \(C\) is regular, S12 has shown that the empirical copula process satisfies \(\mathbb C_n\rightsquigarrow\mathbb C\) in \(\ell^\infty([0,1]^2)\), where \(\rightsquigarrow\) denotes Hoffmann-J\o rgensen convergence in some metric space K08. The limit \(\mathbb C\) can be written as
with \(\mathbb B\) a centered Gaussian random element of \(\ell^\infty([0,1]^2)\) with continuous sample paths and covariance kernel
An alternative demonstration of the weak convergence \(\mathbb C_n\rightsquigarrow\mathbb C\) based on the delta method was given by B11 and by BV13. Earlier work by FRW04 accomplished the same thing under a condition somewhat stronger than regularity.
For any function \(\theta\in\ell^\infty([0,1]^2)\) we define \(\theta^\top\in\ell^\infty([0,1]^2)\) by \(\theta^\top(u,v)=\theta(v,u)\). The null hypothesis that \(C\) is exchangeable may then be written as \(C=C^\top\). Statistics for testing this null may be constructed from the difference \(C_n-C_n^\top\). Following GNQ12 we consider the three statistics
Equivalent expressions for \(R_n\), \(S_n\) and \(T_n\) more amenable to exact computation, with the integrals replaced by sums and the supremum replaced by a maximum over a finite set, have been given by GNQ12. When \(C\) is regular and exchangeable, GNQ12 have shown that
For any function \(\theta\in\ell^\infty([0,1]^2)\), we define \(\theta^\mathrm{s},\theta^{\scriptscriptstyle\circlearrowleft}\in\ell^\infty([0,1]^2)\) by
The null hypothesis that \(C\) is radially symmetric may be written as \(C=C^\mathrm{s}\). Denote the empirical cdf of rotated rank pairs \((1-U_{ni},1-V_{ni})\) by
The cdf \(D_n\) differs from \(C_n^\mathrm{s}\) by no more than \(4n^{-1}\) a.s.; see Lemma (ref) in the Appendix. This motivates the use of the following statistics for testing radial symmetry:
The statistics \(R_n'\), \(S_n'\) and \(T_n'\) are the same as those of GN14, except that those authors defined \(U_{ni}\) and \(V_{ni}\) in a way that differs from us by an asymptotically negligible factor of \(n/(n+1)\). GN14 provided equivalent expressions for these statistics more amenable to computation, with integrals replaced by sums and the supremum replaced by a maximum over a finite set. When \(C\) is regular and radially symmetric, results of BC12 imply that
We require critical values with which to compare the test statistics defined in ((ref))--((ref)) and ((ref))--((ref)). These should approximate the relevant quantiles of the null limit distributions given in ((ref))-((ref)) and ((ref))-((ref)). The approach taken by GNQ12 and GN14 was to use the multiplier bootstrap of RS09 and BD10 to generate bootstrap versions of the empirical copula process \(\mathbb C_n\), and thereby approximate the null limit distribution of the relevant statistic. We will propose a different resampling scheme similar to randomization tests, narrowly tailored to the symmetry testing problem. It delivers improved small sample performance in simulations reported in Section (ref).
Suppose for a moment that \(F\) and \(G\) are known, so that we observe \(n\) iid pairs \((U_i,V_i)=(F(X_i),G(Y_i))\) whose common bivariate cdf is the copula \(C\). A randomization test of a hypothesis about \(C\) may be possible if there is a finite group \(\mathbf G\) of transformations from \([0,1]^{2}\) to itself such that, when the hypothesis is satisfied, \(g(U,V) \mathrel{\mathop{=}\limits^{ \vbox to0ex{\kern-5\ex@ \hbox{$\scriptstyle\mathcal{D}$}\vss}}}(U,V)\) for all \(g\in\mathbf G\). For testing bivariate exchangeability and radial symmetry it is enough to consider a group of two transformations \(\mathbf{G}=\{\pi^0,\pi^1\}\). The transformations \(\pi^0:[0,1]^2\to[0,1]^2\) and \(\pi^1:[0,1]^2\to[0,1]^2\) are given by
when our hypothesis is exchangeability, and by
when our hypothesis is radial symmetry. It is easy to check that either choice of \(\mathbf G\) is a group under the operation of composition. Using \(\mathbf G\) we can construct a second group \(\mathbf G_n\) of \(2^n\) transformations from \(([0,1]^2)^n\) to itself by setting \(\mathbf G_n=\{g_\tau:\tau=(\tau_1,\ldots,\tau_n)\in\{0,1\}^n\}\), where
Again, it is easy to check that \(\mathbf G_n\) is a group under the operation of composition. Moreover, when the relevant symmetry hypothesis is satisfied, our sample of pairs satisfies
Given an arbitrary test statistic \(W_n=W_n((U_1,V_1),\ldots,(U_n,V_n))\), property (ref) can be used to justify the construction of an exact level \(\alpha\) randomization test of our symmetry hypothesis. The procedure, as described by R90, is as follows.
It can be shown that, when the null hypothesis of symmetry is true, this procedure leads us to reject it with probability exactly equal to \(\alpha\). See R89,R90 and references therein for further details on randomization tests.
Unfortunately a randomization test of the kind just described is not feasible for our hypothesis testing problem because, since \(F\) and \(G\) are not known, we do not observe pairs \((U_i,V_i)\) drawn from \(C\). In their place, we observe rank pairs \((U_{ni},V_{ni})\) based on preliminary estimates of the margins \(F\) and \(G\), but these rank pairs do not satisfy the exact group invariance property (ref). A naive application of Procedure (ref) to these rank pairs will not achieve exact size, and cannot be expected to achieve correct size asymptotically because the effect of estimating the margins upon the distribution of the test statistic is not properly accounted for. We instead propose a feasible randomization procedure which accounts for the estimation of margins and will be shown to deliver asymptotically valid inference.
Let \(W_n=W_n((U_{n1},V_{n1}),\ldots,(U_{nn},V_{nn}))\) be the statistic of interest. Our procedure for computing critical values for \(W_n\) is as follows.
Steps 2 and 3 of Procedure (ref) are what distinguish it from a naive application of Procedure (ref) to the rank pairs \((U_{ni},V_{ni})\). Step 2 involves applying small random perturbations \(n^{-1}\eta_i\) to the transformed rank pairs \((U^\tau_{ni},V^\tau_{ni})\). Since the randomized statistic \(W_n^\tau\) depends only on the rank pairs of the transformed rank pairs \((U_{ni}^\tau,V_{ni}^\tau)\), and the perturbations \(n^{-1}\eta_i\) are smaller than the gap \(n^{-1}\) between consecutive ranks, the sole effect of the perturbations is to randomly break ties in ranks. This effect is asymptotically negligible (see Lemma (ref) below) but we have found that it improves the performance of our procedure with very small sample sizes. It is immaterial that \(\eta_i\) is uniformly distributed; we could equivalently draw each $\eta_i$ from any continuous distribution with support contained in \((0,1)\).
The more important distinction between Procedures (ref) and (ref) is Step 3 of Procedure (ref), in which \((\check{U}_{ni}^\tau,\check{V}_{ni}^\tau)\) is transformed to \((\tilde{U}^\tau_{ni},\tilde{V}^\tau_{ni})\). If we were to drop Step 3 in Procedure (ref) and naively use \((U_{ni}^\tau,V_{ni}^\tau)\) or \((\check{U}_{ni}^\tau,\check{V}_{ni}^\tau)\) in place of \((\tilde{U}_{ni}^\tau,\tilde{V}_{ni}^\tau)\) in Step 4, our method of constructing critical values would not deliver asymptotically valid inference. We illustrate this claim in Figure (ref). In panel (a) we plot smoothed histograms of the simulated sampling distribution of the exchangeability test statistic \(S_n\), of the distribution of randomized versions of \(S_n\) obtained by naively applying Procedure (ref) to the rank pairs \((U_{ni},V_{ni})\), and of the distribution of proper randomized versions of \(S_n\) obtained by applying Procedure (ref) to the rank pairs \((U_{ni},V_{ni})\). It is clear that the naive application of Procedure (ref) to the rank pairs \((U_{ni},V_{ni})\) provides a very poor approximation to the sampling distribution of \(S_n\), whereas the approximation provided by Procedure (ref) is very good. Moreover, we found that the latter approximation does not change appreciably for this sample size if the perturbation in Step 2 of Procedure (ref) is omitted. Panel (b) shows qualitatively similar results for the radial symmetry test statistic \(S_n'\). We obtained qualitatively similar results for the exchangeability test statistics \(R_n\) and \(T_n\), and the radial symmetry test statistics \(R_n'\) and \(T_n'\), hence we do not display them here.
To study the asymptotic properties of tests based on Procedure (ref) it will be useful to consider the behavior of \(W_n^\tau\) when \(\tau\) is drawn randomly from \(\mathbf G_n\). Suppose we were to choose as \(\tau\) an \(n\)-tuple of Bernoulli random variables each taking the values zero and one with equal probabilities, independent of one another and of the data. The corresponding transform \(g_\tau\) would then be a random draw from the uniform distribution over \(\mathbf G_n\), and the critical value \(W_n^{(k)}\) computed in step 4 of Procedure (ref) would be the conditional \((1-\alpha)\)-quantile of \(W_n^\tau\) given the data. That is, \(W_n^{(k)}=Q_n(1-\alpha)\), where
We will show in Section (ref) that, when the null of symmetry is satisfied, the conditional law of \(W_n^\tau\) given the data asymptotically approximates the unconditional law of the test statistic \(W_n\). Consequently, our critical value \(W_n^{(k)}\) approximates the \((1-\alpha)\)-quantile of the law of \(W_n\) under the null, as desired.
In practice it may be computationally burdensome to evaluate all \(2^n\) statistics \(W_n^\tau\) corresponding to the \(2^n\) transforms \(g_\tau\in\mathbf G_n\), even with modest sample sizes. Instead of repeating steps 1--4 of Procedure (ref) \(2^n\) times, we may repeat them some large number of times \(N\). Each time we select a transform \(g_\tau\in\mathbf G_n\) in Step 1, we should do so at random and with replacement, choosing each possible transform with equal probabilities. This amounts to generating \(\tau\) as an independent \(n\)-tuple of Bernoulli random variables, each taking the values zero and one with equal probabilities. In Step 5 of Procedure (ref) we then set our critical value equal to the \(k^\text{th}\) smallest of the \(N\) computed statistics, where \(k=N-\lfloor N\alpha\rfloor\). So long as \(N\) is large, our critical value should be close to the conditional \((1-\alpha)\)-quantile of \(W_n^\tau\) given the data.
Step 6 in Procedure (ref) entails randomizing between rejection and nonrejection when the test statistic $W_n$ and critical value $W_n^{(k)}$ are equal. Applied researchers may be reluctant to randomize in this way, and instead prefer to only reject the null when the test statistic strictly exceeds the critical value. In numerical simulations discussed in Section (ref) we found that the effect of always failing to reject, rather than randomizing, when the test statistic and critical value are equal is negligible when using the exchangeability test statistics $R_n$ and $S_n$ and the radial symmetry test statistics $R_n'$ and $S_n'$, even at sample sizes as small as $n=30$. On the other hand, choosing not to randomize can have a noticeable impact when using the statistics $T_n$ and $T_n'$. This may be because, holding $W_n$ fixed, the range of values taken by $W_n^\tau$ is much coarser when we construct $W_n$ using the uniform norm, leading to $W_n$ being exactly equal to its critical value with greater frequency. Although this discrepancy diminishes at larger sample sizes, applied researchers unwilling to randomize may prefer not to use the statistics $T_n$ and $T_n'$ in small samples.
Though the procedure we have proposed may be viewed as a feasible version of a randomization test, we will study it using techniques most often used to demonstrate the asymptotic validity of bootstrap tests. Such demonstrations generally hinge upon the conditional law of a bootstrapped statistic converging in a suitable sense to a target distribution. The conditional law we refer to here is the law obtained by holding the data fixed and allowing the random weights used to generate the bootstrapped statistic to vary. The analysis of our procedure will be similar, in that it hinges upon suitable convergence of the conditional law of the randomized statistic \(W_n^\tau\). However in this case the source of random variation is not a collection of bootstrap weights, but rather a random \(n\)-tuple \(\tau=(\tau_1,\ldots,\tau_n)\) that indexes transforms in the group \(\mathbf G_n\).
Throughout this section, \(\tau\) represents an \(n\)-tuple of independent Bernoulli random variables each taking the values zero and one with equal probabilities, jointly independent of the data \((X_1,Y_1),\ldots,(X_n,Y_n)\). For \(n\in\mathbf N\), \(\xi^\tau_n\) is an element of a metric space \(\mathbf D\) depending on the data and \(\tau\), and \(\xi_n\) is an element of \(\mathbf D\) depending on the data but not on \(\tau\). The following notion of convergence is from K08.
The second condition in Definition (ref), which asserts a form of conditional asymptotic measurability, allows us to handle cases where \(\xi^\tau_n\) is not Borel measurable. While this level of generality can be useful in other contexts, whenever the symbols \(\rightsquigarrow\) or \( \mathrel{\mathop{ \mathrel{\mathop{\rightsquigarrow}\limits^{ \vbox to0ex{\kern7\ex@ \hbox{$\scriptstyle\tau$}\vss}}}}\limits^{ \vbox to0ex{\kern0.5\ex@ \hbox{$\scriptstyle\mathrm{P}$}\vss}}}\) are used in this paper, each member of the convergent sequence will in fact be Borel measurable. This is true even in the nonseparable space \(\mathbf D=\ell^\infty([0,1]^2)\). To see why, note that the empirical copula \(C_n\) is uniquely determined by the \(n^2\) coordinate projections \(C_n(i/n,j/n)\), \(i,j=1,\ldots,n\), and that each of these projections can only take the values \(0,n^{-1},2n^{-1},\ldots,1\). Thus \(C_n\) can take only finitely many values in \(\ell^\infty([0,1]^2)\), and since the projections \(C_n(i/n,j/n)\) are random variables, \(C_n\) must be a simple map into \(\ell^\infty([0,1]^2)\). Similarly, \(D_n\) is a simple map into \(\ell^\infty([0,1]^2)\). The weakly convergent or conditionally weakly convergent sequences we consider in this paper are all simple, hence Borel measurable.
Theorems (ref) and (ref) below establish that the randomized statistic \(W_n^\tau\) satisfies \(W_n^\tau \mathrel{\mathop{ \mathrel{\mathop{\rightsquigarrow}\limits^{ \vbox to0ex{\kern7\ex@ \hbox{$\scriptstyle\tau$}\vss}}}}\limits^{ \vbox to0ex{\kern0.5\ex@ \hbox{$\scriptstyle\mathrm{P}$}\vss}}} W\), where the law of \(W\) coincides with the weak limit of \(W_n\) when the null of symmetry is satisfied. From this conditional weak convergence and Lemma 10.11 of K08 it follows that
in probability for all continuity points \(c\) of the cdf of \(W\). Consequently, in view of (ref), our feasible randomization tests control size asymptotically and are consistent against arbitrary violations of symmetry.
Recent work on the asymptotic properties of randomization tests by CR13,CR16a,CR16b establishes conditional weak convergence of their randomization distributions by verifying an equivalent condition of H52. Let $\tilde{W}_n(\cdot)$ be the map from randomized rank-pairs $(U_{n1}^\tau,V_{n1}^\tau),\ldots,(U_{nn}^\tau,V_{nn}^\tau)$ to randomized test statistic $W_n^\tau$, so that
The conditional probability in (ref) is equal to
and corresponds to the randomization distribution defined in Equation (5.1) of CR13. Theorem 5.1 of CR13 therefore implies that (ref) is satisfied if and only if
where \(W'\) and \(\tau'\) are independent copies of \(W\) and \(\tau\) respectively. Condition (ref) is called Hoeffding's condition. We do not make use of Hoeffding's condition and instead verify (ref) directly, which we find to be more convenient.
The proofs of our main results rely on repeated applications of conditional versions of two fundamental results on weak convergence: the continuous mapping theorem and the delta method. We will state these here for convenience, and also because we require a version of the conditional delta method that is somewhat different to the usual statement. The following statement of the conditional continuous mapping theorem is Theorem 10.8 of K08. We have dropped Kosorok's measurability condition on \(\tau\mapsto\xi_n^\tau\) because \(\tau\) takes only finitely many values.
The following statement of the conditional delta method is a version of Theorem 12.1 of K08, often referred to as the delta method for the bootstrap. It is unusual in that we allow the weak limits \(\mathbb X_1\) and \(\mathbb X_2\) to differ (and not merely by a scalar multiple \(c\)). We have again dropped the measurability condition on \(\tau\mapsto\xi_n^\tau\).
To obtain Theorem (ref) we set \(c=1\) in the proof of Theorem 12.1 of K08, and when Kosorok introduces two independent copies \(\tilde{\mathbb X}_1\) and \(\tilde{\mathbb X}_2\) of \(\mathbb X\), we instead introduce two independent copies \(\tilde{\mathbb X}_1\) and \(\tilde{\mathbb X}_2\) of \(\mathbb X_1\) and \(\mathbb X_2\) respectively. It is crucial for us to allow the laws of \(\mathbb X_1\) and \(\mathbb X_2\) to differ. In the proofs of Lemmas (ref) and (ref) and Theorems (ref) and (ref) below, we apply the conditional delta method even though the weak limits in (ref) and (ref), in (ref) and (ref), in (ref) and (ref), and in (ref) and (ref), differ.
Asymptotic justifications of bootstrap procedures typically appeal directly or indirectly to a multiplier central limit theorem. For instance, RS09 appeal directly to a multiplier central limit theorem to demonstrate the asymptotic validity of their procedure, whereas BD10 appeal to Theorem 2.6 of K08, which is proved using a multiplier central limit theorem. This approach is less suitable for our problem, because the source of variation in the randomized statistic \(W_n^\tau\) is not a random \(n\)-tuple of independent bootstrap weights (i.e.\ multipliers), but rather a random \(n\)-tuple of transforms drawn independently from \(\mathbf G\). The following lemma plays the role of a multiplier central limit theorem in our analysis, in the sense that it shows how random variation in the draws from \(\mathbf G\) can provide a source of conditional weak convergence to a suitable Gaussian limit. It is proved in Section (ref) by verifying the conditions of a functional central limit theorem of P90.
In this subsection we let \(W_n=W_n((U_{n1},V_{n1}),\ldots,(U_{nn},V_{nn}))\) denote any of the statistics \(R_n\), \(S_n\) and \(T_n\), and let \(\pi^0\) and \(\pi^1\) be as defined in (ref). Let \((\tilde{U}_{ni}^\tau,\tilde{V}_{ni}^\tau)\), \(i=1,\ldots,n\), be constructed as described in Steps 1--3 of Procedure (ref), with \(\tau\) an \(n\)-tuple of independent Bernoulli random variables each taking the values zero and one with equal probabilities. Let \(C_n^\tau\) be the random element of \(\ell^\infty([0,1]^2)\) given by
The randomized statistic \(W_n^\tau=W_n((\tilde{U}^\tau_{n1},\tilde{V}^\tau_{n1}),\ldots,(\tilde{U}^\tau_{nn},\tilde{V}^\tau_{nn}))\) can be expressed in terms of \(C_n^\tau\): we simply substitute \(C_n^\tau\) for \(C_n\) in the relevant formula from (ref)--(ref). The following lemma describes the conditional asymptotic behavior of \(C_n^\tau\).
From Lemma (ref) and the conditional continuous mapping theorem we have
On the other hand, it is apparent from (ref)--(ref) that it is the law of \(\mathbb C-\mathbb C^\top\) which we wish to approximate. In fact, Lemma (ref) below establishes that, when the null of exchangeability is satisfied, \(\mathbb C-\mathbb C^\top\) and \(\mathbb D-\mathbb D^\top\) are equal in law. This leads us to our main result on our procedure for testing copula exchangeability.
We saw in Figure (ref) that the naive application of Procedure (ref) to the pairs \((U_{ni},V_{ni})\) does not provide an acceptable approximation to the sampling distribution of our test statistic. To see why, let \(\tilde{C}_n^\tau\) be given by
where \((U_{ni}^\tau,V_{ni}^\tau)=\pi^{\tau_i}(U_{ni},V_{ni})\) and \(\pi^0\) and \(\pi^1\) are defined as in (ref). Naive application of Procedure (ref) to the rank pairs \((U_{ni},V_{ni})\) amounts to using \(\tilde{C}_n^\tau\) in place of \(C_n^\tau\) in our feasible randomization procedure. By applying the conditional continuous mapping theorem to the convergence (ref) established in the proof of Lemma (ref), we find that
Comparing (ref) and (ref) we see that using \(\tilde{C}_n^\tau\) in place of \(C_n^\tau\) leads to an invalid approximation of the weak limit \(\mathbb C-\mathbb C^\top \mathrel{\mathop{=}\limits^{ \vbox to0ex{\kern-5\ex@ \hbox{$\scriptstyle\mathcal{D}$}\vss}}}\mathbb D-\mathbb D^\top\) appearing in (ref)--(ref). The problem is that \(\mathbb A\) differs from \(\mathbb D\) by the sum of the latter two terms on the right hand side of equality (ref). An analogous problem confounds the naive application of Procedure (ref) in the context of radial symmetry testing.
In this subsection we let \(W_n=W_n((U_{n1},V_{n1}),\ldots,(U_{nn},V_{nn}))\) denote any of the statistics \(R_n'\), \(S_n'\) and \(T_n'\), and let \(\pi^0\) and \(\pi^1\) be as defined in (ref). We let \(C_n^\tau\) be defined in the same fashion as in the previous subsection, and let \(D_n^\tau\) be the random element of \(\ell^\infty([0,1]^2)\) given by
The randomized statistic \(W_n^\tau=W_n((\tilde{U}^\tau_{n1},\tilde{V}^\tau_{n1}),\ldots,(\tilde{U}^\tau_{nn},\tilde{V}^\tau_{nn}))\) can be expressed in terms of \(C_n^\tau\) and \(D_n^\tau\): we simply substitute \(C_n^\tau\) for \(C_n\) and \(D_n^\tau\) for \(D_n\) in the relevant formula from (ref)--(ref). The following lemma, analogous to Lemma (ref), describes the conditional asymptotic behavior of \(C_n^\tau\).
Note that the random element \(\mathbb D\) appearing in the statement of Lemma (ref) is not the same as the random element \(\mathbb D\) appearing in the statement of Lemma (ref). It is apparent from (ref)--(ref) that the law of \(\mathbb C-\mathbb C^{\scriptscriptstyle\circlearrowleft}\) (and also \(C\), if \(W_n=S_n'\)) determines the null limit distribution of \(W_n\). Lemma (ref) below establishes that, when the null of radial symmetry is satisfied, \(\mathbb C-\mathbb C^{\scriptscriptstyle\circlearrowleft}\) and \(\mathbb D-\mathbb D^{\scriptscriptstyle\circlearrowleft}\) are equal in law. This leads us to our main result on our procedure for testing copula radial symmetry.
In Table (ref) we report the results of numerical simulations used to investigate the null rejection rates obtained using our feasible randomization procedure (i.e., Procedure (ref)) at small sample sizes. Rejection frequencies were computed over 10000 experimental replications for the sample sizes \(n=30\) and \(n=50\) at the nominal levels \(\alpha=0.05\) and \(\alpha=0.1\). Alongside the rejection rates for our feasible randomization procedure, we report rejection rates for the infeasible randomization procedure (Procedure (ref)) and for the multiplier bootstrap used by GNQ12 for testing exchangeability and by GN14 for testing radial symmetry. In each replication, critical values were computed using \(N=250\) random draws from \(\mathbf G_n\) for the randomization procedures and \(N=250\) repetitions for the multiplier bootstrap.
The top half of Table (ref) displays results for testing the null of exchangeability using the statistics \(R_n\), \(S_n\) and \(T_n\). Results are reported for five exchangeable copulas: the Gaussian, Clayton, Gumbel, Student and Frank copulas, parameterized to have a rank correlation of 0.5 (and with 3 degrees of freedom for the Student copula). The first three columns of rejection rates correspond to our feasible randomization procedure. These rates are slightly greater than, but very close to, the nominal level, especially at the larger sample size \(n=50\). There is no appreciable difference between the rejection rates obtained using the three statistics \(R_n\), \(S_n\) and \(T_n\). The next three columns of rejection rates correspond to infeasible randomization applied directly to the unobserved pairs \((U_i,V_i)\). This testing procedure is exact, and indeed we see that the computed rejection rates are extremely close to the nominal level, differing only due to Monte Carlo error. The final three columns of rejection rates correspond to the multiplier bootstrap. These rejection rates are much lower than the nominal level. The lowest rejection rates are obtained using the \(T_n\) statistic, and the largest with the \(S_n\) statistic. Simulations reported by GNQ12 indicate that excessive conservatism may continue to be an issue at sample sizes as large as \(n=250\).
The bottom half of Table (ref) displays results for testing the null of radial symmetry using the statistics \(R'_n\), \(S'_n\) and \(T'_n\). Results are reported for five radially symmetric copulas: the Gaussian, Student, Frank, Plackett and Cauchy copulas, parameterized to have a rank correlation of 0.5 (and with 3 degrees of freedom for the Student copula). Qualitatively, the results are similar to those reported for the exchangeability tests. Our feasible randomization procedure tends to overreject a bit more with the Cauchy copula than with the other copulas considered, especially with the statistic \(R_n'\) at the smaller sample size \(n=30\). Again, the rejection rates based on the multiplier bootstrap tend to be much lower than the nominal level, and simulations reported by GN14 indicate that the issue may persist at sample sizes as large as \(n=500\).
In addition to the simulations reported in Table (ref), we ran further simulations to investigate the null rejection rates obtained when Procedure (ref) is naively applied to the rank pairs \((U_{ni},V_{ni})\). We computed a rejection rate of zero for all configurations listed in Table (ref). This may be unsurprising in view of Figure (ref), where we saw that the upper quantiles of the sampling distributions of \(S_n\) and \(S_n'\) when \(C\) is the product copula lie far to the left of those in the (poorly) approximating distributions obtained through a naive application of Procedure (ref).
As discussed in Section (ref), applied researchers may be unwilling to randomize between rejection and nonrejection when the test statistic and critical value are equal, and instead prefer to only reject when the test statistic strictly exceeds the critical value. To investigate the impact of such a decision rule, we recomputed the rejection rates for the feasible and infeasible randomization tests in Table (ref), recording rejections only when a test statistic strictly exceeded the critical value. We found that the rejection rates using the $R_n$, $S_n$, $R_n'$ and $S_n'$ statistics were essentially unchanged. However, the rejection rates using the $T_n$ and $T_n'$ statistics dropped to about 0.04 with nominal level $\alpha=0.05$, and to about 0.08 with nominal level $\alpha=0.1$.
In this section we report numerical evidence on the power of symmetry tests based on feasible randomization and on the multiplier bootstrap. The main finding is that power is greater with feasible randomization. The improved power appears to be driven by our finding in Section (ref) that feasible randomization delivers a rejection rate close to the nominal level even at small sample sizes, whereas the multiplier bootstrap produces excessively conservative tests. We focus below on sample sizes \(n=50,100,250\). The power advantage of tests based on feasible randomization appears to be smaller at larger sample sizes. This is not unexpected in view of results of R89, who showed the asymptotic equivalence of randomization and bootstrap procedures in fairly general settings. Note, however, that simulation results reported by GN14 indicate that the excessive conservatism induced by the multiplier bootstrap may persist at sample sizes as large as \(n=500\).
In Table (ref) we report the results of numerical simulations used to investigate the power of exchangeability tests based on our feasible randomization procedure or on the multiplier bootstrap at small (\(n=50\)) and medium (\(n=100\)) sample sizes, with nominal level \(\alpha=0.05\). As in the previous subsection, we computed rejection frequencies over 10000 experimental replications, and in each replication computed critical values using \(N=250\) random draws from the group \(\mathbf G_n\) or bootstrap repetitions.
The nonexchangeable copulas used to produce the rejection frequencies in Table (ref) were obtained by applying the Khoudraji transform K95 to the Gaussian, Clayton and Gumbel copulas with parameters chosen such that the rank correlation \(\tau\) is equal to 0.5, 0.7 or 0.9. The Khoudraji transform of a copula \(C\) is given by \(C^\mathrm{K}_\delta(u,v)=u^\delta C(u^{1-\delta},v)\), where we allow \(\delta=0.25,0.5,0.75\). GNQ12 also simulate rejection frequencies of their exchangeability tests using these copula specifications; the rejection rates for the multiplier bootstrap with \(n=100\) in Table (ref) are taken directly from their paper.
Cursory examination of the numbers in Table (ref) reveals that our feasible randomization procedure generates more power than the multiplier bootstrap -- often much more. A test using the statistic \(T_n\) with the multiplier bootstrap has essentially no power at the smaller sample size \(n=50\), whereas a test using the same statistic with our feasible randomization procedure has substantial power. This may be unsurprising in view of the extremely low null rejection rates exhibited by the former test in Table (ref). The differences in power with the statistics \(R_n\) and \(S_n\), or at the larger sample size \(n=100\), are less extreme, but nevertheless it is clear that our randomization procedure generates significant power improvements over the multiplier bootstrap at these sample sizes, with the latter procedure hamstrung by its low null rejection rates.
In Table (ref) we report the results of numerical simulations used to investigate the power of radial symmetry tests based on our feasible randomization procedure or on the multiplier bootstrap at small (\(n=50\)), medium (\(n=100\)) and large (\(n=250\)) sample sizes, with nominal level \(\alpha=0.05\). We again used 10000 experimental replications, and \(N=250\) repetitions for the randomization and bootstrap procedures in each replication. Rejection rates were computed for the (radially asymmetric) Clayton and Gumbel copulas, parametrized to have rank correlation \(\tau=0.25,0.5,0.75\). As was the case in Table (ref), it is clear from the numbers in Table (ref) that our feasible randomization procedure generates large improvements in power over the multiplier bootstrap. This is particularly true for the Clayton copula at the smaller sample sizes \(n=50\) and \(n=100\). Rejection rates are lower for the Gumbel copula, which apparently exhibits a more mild form of radial asymmetry than the Clayton copula.
In Section (ref) we mentioned that radially asymmetric dependence between asset returns has been a subject of interest in empirical finance. As is well known, time series of asset returns typically exhibit conditional heteroskedasticity. A bivariate time series of asset returns will therefore not satisfy the iid condition underlying our analysis. We may nevertheless consider applying our symmetry tests to bivariate return series after first filtering out conditional heteroskedasticity or other dependencies in the data. In this section we report the results of some simulations we ran to investigate this possibility.
The design of our simulations is motivated by the semiparametric copula-based multivariate dynamic (SCOMDY) model of CF06. In this model, a pair (or more) of time series are each assumed to evolve according to an ARMA-GARCH model or some other parametric conditional mean-variance specification. Dependence between the two series is generated by linking contemporaneous ARMA-GARCH innovations with a parametric copula function. The marginal distributions of the innovations are left unspecified, making the SCOMDY model semiparametric.
We calibrated our simulations using 658 daily returns on the Hang Seng and Nikkei market indices, running from March 7, 2016, to March 5, 2019. A scatterplot of these returns is displayed in panel (a) of Figure (ref). To each univariate return series we fit a GARCH(1,1) model with iid Student innovations by maximum likelihood. A scatterplot of the fitted innovations for each return series is displayed in panel (b) of Figure (ref). The fitted innovations were transformed to ranks, and divided by sample size to render them between zero and one. A scatterplot of the ranked fitted innovations is displayed in panel (c) of Figure (ref). This last scatterplot provides us with a sense of the shape of the copula linking contemporaneous GARCH innovations. At a glance, it does not appear to provide evidence of nonexchangeability or radial asymmetry. To confirm, we applied our feasible randomization tests of copula exchangeability and radial symmetry to the fitted innovations, using \(N=1000\) repetitions of our feasible randomization procedure. We computed exchangeability test statistics \(R_n=0.0262\), \(S_n=0.0283\) and \(T_n=0.6627\), none of which led us to reject the null at the 0.2 nominal level. We computed radial symmetry test statistics \(R'_n=0.0459\), \(S'_n=0.0581\) and \(T'_n=0.7407\), none of which led us to reject the null at the 0.15 nominal level. We would therefore like to conclude that the GARCH innovations are consistent with copula exchangeability and radial symmetry. However, such a conclusion is not justified by the results of this paper, because the symmetry tests were applied to fitted GARCH innovations, and it has not been established whether the preliminary estimation of GARCH parameters affects the asymptotic null rejection rates of our testing procedures.
To investigate further, we ran a number of simulations based on the SCOMDY model fitted to Hang Seng and Nikkei returns. Rather than using a parametric copula specification, we took the following nonparametric approach: we first obtained a bivariate distribution \(H\) for the GARCH innovations by slightly smoothing the empirical distribution of the fitted innovations in panel (b) of Figure (ref). Let \(F\) and \(G\) be the margins and \(C\) the copula corresponding to \(H\). To impose the null of copula exchangeability or radial symmetry in our simulations, we set \(H^\ast=C^\ast(F,G)\), where \(C^\ast=(C+C^\top)/2\) when we wish to impose exchangeability, and \(C^\ast=(C+C^{\scriptscriptstyle\circlearrowleft})/2\) when we wish to impose radial symmetry. The data generating process for our simulations is then as follows: in each of \(1000\) experimental replications, we generate \(n\) pairs of innovations by iid sampling from \(H^\ast\), and then feed these into the GARCH(1,1) models calibrated to the Hang Seng and Nikkei returns, to produce simulated returns \((X_1,Y_1),\ldots,(X_n,Y_n)\). We then re-estimate the GARCH(1,1) models using these simulated returns, and apply our symmetry tests to the pairs of fitted innovations with \(N=250\) repetitions of our randomization procedure. To provide a point of comparison, we also applied our symmetry tests to the true innovations drawn directly from \(H^\ast\). We used sample sizes \(n\) of 50, 100 and 250, and nominal significance levels \(\alpha\) of 0.05 and 0.1.
The results of our simulations are displayed in Table (ref). Strikingly, we see that the null rejection rates when our symmetry tests are applied to the fitted innovations are not appreciably different to the rejection rates obtained with the true innovations, and are generally close to the nominal level. Thus, the preliminary estimation of GARCH parameters does not appear to be affecting the rejection rate of our tests. A similar phenomenon has been shown to hold in closely related contexts. CF06 showed that, when the copula for contemporaneous innovations belongs to a known parametric class, the asymptotic distribution of the maximum likelihood estimator of the copula parameters is unaffected by the preliminary estimation of GARCH parameters. CCCFP09 showed that the asymptotic distribution of a statistic measuring copula goodness-of-fit is also unaffected by the preliminary estimation of GARCH parameters. Our tests of symmetry appear to benefit from the same property, although we have not shown this formally.
We have proposed a feasible randomization procedure which leads to consistent tests of copula symmetry which control size asymptotically. Numerical simulations indicate that basing tests of copula symmetry on our feasible randomization procedure instead of the multiplier bootstrap leads to substantially improved small sample performance.
A number of avenues for further research remain open. The asymptotic validity of our procedure was demonstrated for the specific statistics \(R_n\), \(S_n\), \(T_n\), \(R'_n\), \(S'_n\) and \(T'_n\), but in principle we may apply our procedure using other statistics measuring copula asymmetry. For instance, BQ17 have recently proposed tests of copula radial symmetry based on a range of statistics computed from the copula characteristic function, with critical values obtained using an adaptation of the multiplier bootstrap. Like the radial symmetry tests of GN14, the tests of Bahraoui and Quessy frequently have null rejection rates well below the nominal level at smaller sample sizes; they note that “this behavior is typical of methods based on the multiplier bootstrap” BQ17. Further investigation of the applicability of our feasible randomization procedure with these and other statistics measuring nonexchangeability or radial asymmetry remains a task for future research.
It may be useful to extend our feasible randomization procedure to other hypothesis testing problems in which the need to estimate margins is a complicating factor. S18 has recently taken this approach to develop a permutation test of the null hypothesis that two independent multivariate samples are drawn from multivariate distributions with the same copula. We might also consider adapting our procedure to obtain tests of forms of copula symmetry other than exchangeability or radial symmetry, such as joint symmetry N06, which is a stronger property than radial symmetry. We leave the pursuit of such adaptations for future research.