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asi-randomization tests of copula symmetry
Brendan K. Beare1 and Juwon Seo2
1Department of Economics, University of California, San Diego2Department of Economics, National University of Singapore
October 23, 2017
New nonparametric tests of copula exchangeability and radial symmetry areproposed. e novel aspect of the tests is a resampling procedure that exploitsgroup invariance conditions associated with the relevant symmetry hypothesis.ey may be viewed as modied versions of randomization tests, the laer beinginapplicable due to the unobservability of margins. Our tests are simple to com-pute, control size asymptotically, consistently detect arbitrary forms of asym-metry, and do not require the specication of a tuning parameter. Simulationsindicate excellent small sample properties compared to existing procedures in-volving the bootstrap.
1 IntroductionIn this paper we propose statistical tests of the null hypothesis that a copula C issymmetric, based on a sample of independent and identically distributed (iid) pairs ofrandom variables with common copula C . While our approach can be applied morebroadly, we focus on two notions of symmetry that have received particular aentionin the literature: exchangeability and radial symmetry. e copula C is said to beexchangeable when
C (u,v ) = C (v,u) for all (u,v ) [0, 1]2. (1.1)
Exchangeability of C is satised if and only if (U ,V ) D= (V ,U ), where D= signiesequality in law. e copula C is said to be radially symmetric when
C (u,v ) = Cs(u,v ) for all (u,v ) [0, 1]2, (1.2)
whereCs(u,v ) = u+v 1+C (1u, 1v ), the survival copula forC . Radial symmetryofC is satised if and only if (U ,V ) D= (1U , 1V ). See Nelsen (1993, 2006, 2007) forfurther discussion of the exchangeability and radial symmetry properties.
e property of exchangeability plays an important role in various models of eco-nomic interaction. Menzel (2016) writes that exchangeability of a certain form isa feature of almost any commonly used empirical specication for game-theoreticmodels with more than two players. A prominent example is the symmetric com-mon value auction model, which was developed by Milgrom and Weber (1982) underthe assumption that the distribution of signals across bidders is exchangeable. Suchexchangeability has powerful implications for the identication of structural econo-metric models of auctions (Athey and Haile, 2002) and is frequently assumed whenthey are estimated (Li et al., 2000; Hendricks et al., 2003; Tang, 2011). Another exam-ple is the model of product bundling developed by Chen and Riordan (2013), in whichthe exchangeability of the copula describing the dependence between consumer val-uations of dierent products is a central assumption when a multi-product rm com-petes with a single-product rm. Radial symmetry, or rather the lack thereof, has beena subject of interest in empirical nance: researchers have found that the dependencebetween various asset returns, particularly equity portfolios, is markedly stronger indownturns than in upturns (Ang and Chen, 2002; Hong et al., 2007). Radially asym-metric copula functions have proved to be useful for modeling this feature of returndependence (Paon, 2004, 2006; Okimoto, 2008; Garcia and Tsafack, 2011).
Several statistical tests of exchangeability and radial symmetry for bivariate cop-ulas have been proposed in recent literature. Genest et al. (2012) and Genest and
Neslehova (2014) proposed tests of copula exchangeability and radial symmetry re-spectively. Extensions of Genest et al.s exchangeability tests to higher dimensionalcopulas have been provided by Harder and Stadtmuller (2017). Genest and Neslehovasradial symmetry tests extend earlier contributions of Bouzebda and Cher (2012) andDehgani et al. (2013). Tests of copula exchangeability and radial symmetry were alsoproposed by Li and Genton (2013) and by Krupskii (2017). ese tests use xed criticalvalues obtained from the chi-squared and normal distributions. Beare and Seo (2014)also proposed a test of copula exchangeability, but for the somewhat dierent casewhere the copula in question characterizes the serial dependence in a univariate timeseries. Many other authors have considered tests of exchangeability or radial symme-try for multivariate cdfssee, for instance, essy (2016) and references thereinbutsuch tests are typically inapplicable to hypotheses of copula symmetry due to the un-observability of margins.
e new tests of copula symmetry proposed in this paper use the same statisticsas the tests proposed by Genest et al. (2012) and Genest and Neslehova (2014), but adierent method of constructing critical values. Whereas those authors obtain criticalvalues using the bootstrap procedures of Remillard and Scaillet (2009) and Bucher andDee (2010), we instead use a novel resampling procedure motivated by randomiza-tion tests of symmetry hypotheses. Romano (1989, 1990) observed that exact tests ofsymmetry hypotheses on multivariate cdfs could be obtained by applying randomiza-tion procedures that exploit group invariance conditions implied by symmetry. Whilethese tests are not directly applicable to hypotheses of copula symmetry, we showhow a modied randomization procedure may be used to obtain critical values thatproperly account for uncertainty about margins. e justication for our procedure isasymptotic rather than exact, but numerical simulations indicate excellent size controlwith sample sizes as small as n = 30. Simulations also indicate substantially improvedpower compared to the tests of Genest et al. (2012) and Genest and Neslehova (2014).
A recent paper by Canay et al. (2017) is related to ours in that it studies the behav-ior of randomization tests when symmetry is only approximately satised. Supposewe have a sample Z (n) of size n taking values in a sample space Zn. Approximatesymmetry in the sense of Canay et al. (2017) means that for each n there exists a mapSn fromZn to a metric space S such that (i) Sn (Z (n) ) converges in law to a random ele-ment S of S as n , and (ii) (S ) is equal in law to S for all in some nite group oftransformations G. Crucially, S and G cannot depend on n. In our paper, the sample isa collection of iid pairsZ (n) = ((X1,Y1), . . . , (Xn,Yn )) taking values inZn = (R2)n. Ap-proximate symmetry holds in the following sense: if Sn : Zn ([0, 1]2)n is the mapthat transforms our sample to the normalized rank pairs ((Un1,Vn1), . . . , (Unn,Vnn ))dened in equation (2.1) below, then the law of Sn (Z (n) ) is (loosely speaking) approx-
imately that of n iid draws from the copula C . When C is symmetric, such n-tuplesof iid draws are distributionally invariant under a group Gn consisting of 2n distincttransformations from ([0, 1]2)n to itself; we postpone details of Gn to Section 3. Sincethe dimension of ([0, 1]2)n and the number of transformations in Gn grow with n, ourproblem falls outside the scope of the results of Canay et al. (2017), in which S and Gare assumed xed.
Recent results of Chung and Romano (2013, 2016a,b) are also somewhat relatedto the problem studied here. Like us, and unlike Canay et al. (2017), Chung and Ro-mano allow the number of transforms in the group Gn to increase with n. However,whereas in our seing the normalized rank pairs ((Un1,Vn1), . . . , (Unn,Vnn )) are ap-proximately distributionally invariant under Gn whenever the null is satised, in theseing considered by Chung and Romano the data are exactly distributionally invari-ant under Gn on a subset of the null, and not even approximately invariant elsewherein the null. e problems we study are therefore fundamentally dierent. Chung andRomano establish their results by verifying a condition of Hoeding sucient for suit-able convergence of the randomization distribution. We instead take the conditionalapproach to which Chung and Romano (2013, p. 497) refer following their discussionof Hoedings condition. Specically, in place of Hoedings condition we verify thata statistic computed from a random transformation of the normalized rank pairs con-verges weakly to a suitable limit conditional on the data in probability.
Our paper is structured as follows. In Section 2 we dene our test statistics andcharacterize their limit distributions under the null hypothesis of symmetry; this isa review of material from Genest et al. (2012) and Genest and Neslehova (2014). InSection 3 we describe our quasi-randomization procedure for obtaining critical values.In Section 4 we provide results on the asymptotic behavior of our quasi-randomizationprocedure and of our tests based upon it. e results of our numerical simulations arepresented in Section 5. Proofs and supplementary lemmas are collected in Section 6.
2 Test statistics
2.1 Basic setupLetX and Y be random variables with bivariate cumulative distribution function (cdf)H (x ,y) = P(X x ,Y y) and margins F (x ) = P(X x ) and G (y) = P(Y y). Weassume that F and G are continuous. Sklars theorem (Sklar, 1959) then ensures theexistence of a unique copulaC : [0, 1]2 [0, 1] satisfyingC (F (x ),G (y)) = H (x ,y) forall x ,y R. e copula C is the bivariate cdf of the probability integral transformsU = F (X ) and V = G (Y ).
Our data consist of n iid draws (X1,Y1), . . . , (Xn,Yn ) from H . Let Fn, Gn and Hnbe the empirical cdfs corresponding to F , G and H respectively. We use Fn and Gn toconstruct (normalized) ranks
Uni = Fn (Xi ), Vni = Gn (Yi ), i = 1, . . . ,n. (2.1)
From the rank pairs (Uni ,Vni ) we construct the empirical copula
Cn (u,v ) =1n
1 (Uni u,Vni v ) , (u,v ) [0, 1]2. (2.2)
An alternative denition of the empirical copula in common use is
CDn (u,v ) = Hn(Fn (u),G
n (v )
), (u,v ) [0, 1]2, (2.3)
where Fn is the generalized inverse of Fn,