Full Terms & Conditions of access and use can be found at
http://www.tandfonline.com/action/journalInformation?journalCode=ubes20
Download by: [Universitas Maritim Raja Ali Haji] Date: 11 January 2016, At: 19:19
Journal of Business & Economic Statistics
ISSN: 0735-0015 (Print) 1537-2707 (Online) Journal homepage: http://www.tandfonline.com/loi/ubes20
Smooth Tests of Copula Specifications
Juan Lin & Ximing Wu
To cite this article: Juan Lin & Ximing Wu (2015) Smooth Tests of Copula Specifications, Journal
of Business & Economic Statistics, 33:1, 128-143, DOI: 10.1080/07350015.2014.932696
To link to this article: http://dx.doi.org/10.1080/07350015.2014.932696
Accepted author version posted online: 25 Sep 2014.
Submit your article to this journal
Article views: 232
View related articles
Smooth Tests of Copula Specifications
Juan L
INDepartment of Finance, School of Economics & Wang Yanan Institute for Studies in Economics, Xiamen University, China ([email protected])
Ximing W
UDepartment of Agricultural Economics, Texas A&M University, College Station, TX 77843; School of Economics, Xiamen University, China ([email protected])
We present a family of smooth tests for the goodness of fit of semiparametric multivariate copula models. The proposed tests are distribution free and can be easily implemented. They are diagnostic and construc-tive in the sense that when a null distribution is rejected, the test provides useful pointers to alternaconstruc-tive copula distributions. We then propose a method of copula density construction, which can be viewed as a multivariate extension of Efron and Tibshirani. We further generalize our methods to the semiparamet-ric copula-based multivariate dynamic models. We report extensive Monte Carlo simulations and three empirical examples to illustrate the effectiveness and usefulness of our method.
KEY WORDS: Copula; Dynamic models; Smooth test; Specification test.
1. INTRODUCTION
Copula methods have been extensively used in risk manage-ment, finance, and insurance due to their flexibility in separately modeling the marginal distributions of individual series and their dependence structure; see Cherubini, Luciano, and Vecchiato (2004), Cherubini et al. (2011), McNeil, Frey, and Embrechts (2005), Patton (2012), and references therein. Commonly used copulas are often parameterized by one or two parameters. The Gaussian copula is most popular because it is easy to compute, simulate, and can be easily extended to arbitrary dimensions. The Gaussian copula is also easy to calibrate since it is uniquely defined by the correlation matrix of marginal distributions and therefore only requires calculating pairwise correlations. How-ever, its simplicity and ease of use come with a price. The Gaussian copula has several drawbacks. For instance, it does not allow tail dependence and therefore cannot capture inter-dependence among extreme events. In addition, the Gaussian copula is radially symmetric and thus does not allow asymmet-ric dependence among variables.
It is widely appreciated that specification testings of paramet-ric copula model are of great importance since different para-metric copulas lead to multivariate models that may have very different dependence properties. A number of existing papers have attempted to address this issue, including among others, Chen, Fan, and Patton (2004), Chen and Fan (2005,2006), Fer-manian (2005), Chen (2007), Li and Peng (2009), Prokhorov and Schmidt (2009), Chen et al. (2010), and Manner and Reznikova (2012). For general overviews of copula specification tests, see Genest, R´emillard, and Beaudoin (2009), Berg (2009), and Fer-manian (2012). The majority of tests reviewed in these sur-vey articles are omnibus or “blanket” tests that possess powers against all alternatives.
Although they are known to be consistent, omnibus tests of-ten only possess satisfactory power against certain deviations from the null hypothesis and lack power in specific directions (Janssen2000). Moreover, they might not have good finite sam-ple power properties (Bera, Ghosh, and Xiao2012). Recently,
Chen (2007) proposed moment-based copula tests for multi-variate dynamic models whose marginals and copulas are both parametrically specified. Chen (2007) indicated that “although the moment-based tests have better size performance, they re-quire correctly specified standardized error distributions.” In practice, it is rare that the true marginal distributions are known. Consequently, tests based on parametrically estimated marginal distributions may have size distortions. It is conceivable that moment-based tests constructed upon empirical CDFs should combine the strength of moment-based tests and the robustness of empirical CDFs to offer a viable alternative. In this article, we aim to fill this gap in the literature by proposing a family of moment-based copula tests that use empirical CDFs of the marginal distributions. Our tests are distribution free and can be easily calculated. They can be viewed as a generalization of the smooth test of Neyman (1937). Unlike omnibus tests, moment-based tests face the nontrivial task of moment selection. We base our tests on a series of orthogonal basis functions such that the selection of moments can be proceeded hierarchically. At the same time, the design of tests is flexible and it can be tailored to focus on certain types of hypotheses, such as symmetry, tail dependence, and so on. Results from Monte Carlo simulations demonstrate good finite sample size and power performance of our tests.
The proposed tests have a particularly appealing feature. When a null hypothesis is rejected under a selected set of moment conditions, we can subsequently construct alternative copula functions by augmenting the null distribution with these extra moment conditions in the spirit of Efron and Tibshirani (1996). Therefore, the tests are not only diagnostic but also con-ductive to improved copula specifications. Our numerical ex-amples illustrate the usefulness of this approach: the augmented
© 2015American Statistical Association Journal of Business & Economic Statistics
January 2015, Vol. 33, No. 1 DOI:10.1080/07350015.2014.932696
128
copula functions are shown to provide more satisfactory fitting to the dependence structure among variables of interest.
Copula methods have found most applications in financial economics, wherein the data often exhibit intertemporal depen-dence such as series correlation or clusters of extreme values. To accommodate time series data, we further extend our meth-ods to the semiparametric copula-based multivariate dynamic (SCOMDY) models as in Chen and Fan (2006). A key finding of Chen and Fan (2006) is that the copula parameters are asymp-totically invariant to the estimation of dynamic parameters when the marginal distributions are estimated by the empirical CDFs. This remarkable asymptotic invariance is confirmed in our sim-ulations. We show that our tests extend to the innovations of SCOMDY models directly and possess the same good finite sample performance as in the iid case.
The remainder of this article is organized as follows. In Sec-tion2, we first briefly review Neyman’s smooth test of distri-butions and then present copula smooth tests under simple hy-potheses. In Section3, we present general copula smooth tests with nonparametric marginal distributions and estimated copula parameters. In Section4, we extend the tests to copula-based multivariate dynamic models. Finite sample performance of the proposed tests is investigated in Section5through Monte Carlo simulations. Section6provides three empirical applications and the last section briefly concludes. All proofs are relegated to Ap-pendix A.
Throughout the article, we use upper case letters to denote cumulative distribution functions and corresponding lower case letters to denote the density functions. We use subscriptt to index observations and subscriptjto denote the coordinate of multivariate random vectors.
2. COPULA TESTS UNDER SIMPLE HYPOTHESES
In this section, we present smooth tests of copula specifica-tions under the ideal situation of known marginal distribuspecifica-tions and known copula distributions. Since our tests are motivated by Neyman’s smooth test of distributions, we first provide a brief introduction of Neyman’s test, followed by our smooth tests of copula distributions under simple hypotheses.
2.1 Neyman’s Smooth Test
LetY1, . . . , Ynbe an iid random sample from an unknown dis-tribution. The hypothesis of interest is to test whether the sample is generated from a specific distributionFy. Neyman’s test of distribution is based on the fact that under the nullU1=Fy(Y1),
. . . , Un=Fy(Yn) are distributed according to the standard uni-form distribution. Therefore, the test on a generic distribution
Fy is transformed to a test on uniformity ofU1 . . . , Un. Ney-man (1937) motivated his smooth test with the following smooth alternative to the uniform density
fK(v)=exp grates to unity, andψk’s are normalized Legendre polynomials
on [0,1] given by
which are orthonormal with respect to the standard uniform distribution.
Under the null hypothesis of uniformity, λ≡(λ1,
. . . , λK)T =0 and the test of uniformity is equivalent to a test onλ=0. One can construct a likelihood ratio test on this hy-pothesis. Alternatively one can use a score test, which is asymp-totically equivalent to the likelihood ratio test. The score test is locally optimal and particularly appealing since it does not require the estimation ofλ. Letψ=(ψ1, . . . , ψK)T and
Neyman’s smooth test is constructed as
S=nψˆTψ .ˆ
Under the null hypothesis,Sconverges in distribution to theχ2 distribution withKdegrees of freedom asn→ ∞.
Compared to omnibus tests, smooth tests have attractive finite sample properties both in terms of size and power (see Rayner and Best1990for a comprehensive review on smooth tests). In addition, the smooth test has one particular appealing feature. If the null hypothesisλ=0 is rejected, it is natural to considerfK as a plausible alternative density. In this sense, this test is not only diagnostic but also constructive because it provides useful pointers for subsequent analysis.
2.2 Smooth Tests of Copula Specifications: Simple Hypothesis
To fix this idea, we first present a smooth test of copula spec-ification for the simplest case where the marginal distributions are known and the copula density in question is completely specified.
Let X1=(X11, . . . , Xd1)T, . . . , Xn=(X1n, . . . , Xdn)T be a random sample from a d-dimensional distribution func-tion F(x1, . . . , xd) with continuous marginal distributions
F1(x1), . . . , Fd(xd), whered≥2. According to the theorem of Sklar (1959), there exists a unique copulaCsuch that
F(x1, . . . , xd)=C(F1(x1), . . . , Fd(xd)) of parametric copula density functions characterized by a finite
p-dimensional parameterα. We are interested in the following simple hypothesis:
H0 : Pr(c(v1, . . . , vd)=c0(v1, . . . , vd;α0))=1 (2)
against the alternative hypothesis
H1 : Pr(c(v1, . . . , vd)=c0(v1, . . . , vd;α0))<1
for someα0∈A, a compact subset ofRp.
Let g=(g1, . . . , gK)T be a series of linearly indepen-dent bounded real valued functions defined on [0,1]d. Define
µ(g;α)=g(v1, . . . , vd)c0(v1, . . . , vd;α)dvand
g(v1, . . . , vd;α)=g(v1, . . . , vd)−µ(g;α).
The computation ofµin practice is defined in Section5. Define
Ut =(U1t, . . . , Udt)T withUj t=Fj(Xj t), j =1, . . . , d. Let
α0)T. We construct a smooth test for copula specification as follows:
Q≡ngˆT(α0)[ ˆV(α0)]−1gˆ(α0), (5)
which can be shown to converge in distribution to theχ2 dis-tribution withK degrees of freedom under suitable regularity conditions given in the next section. Compared with Neyman’s smooth test of uniformity, the additional standardization factor in the form of [ ˆV(α0)]−1is necessary sinceg
=(g1, . . . , gK)T is generally not orthonormal with respect to the null copula densityc0.
Similar to Neyman’s smooth test of uniformity, our smooth test of copula specification can be motivated by a smooth alter-native. Consider the following density
cg(v1, . . . , vd;λ) seen that (5) is a score test on this hypothesis.
We note that (6) can be derived as the density that minimizes the Kullback–Leibler information criterion between the target density and the null density subject to moment conditions as-sociated withg(see Efron and Tibshirani1996). In particular, consider the following optimization problem that minimizes the Kullback–Leibler information criterion between the null copula densityc0and a generic densityc: [0,1]d
The solution takes the form
cg(v1, . . . , vd; ˆλ) with the moment conditions (7). Therefore, if the hypothesis
λ=0 is rejected by (5), it is reasonable to consider (8) as an alternative density, which has the appealing interpretation that among all densities satisfying the sample moment conditions (7), it is the unique density within the family of (6) that is closest to the null copula density in terms of the Kullback– Leibler information criterion.
3. SEMIPARAMETRIC SMOOTH COPULA TESTS
In practice, the marginal distributions and copula parameters are usually unknown and have to be estimated. Consequently, the hypothesis of interest becomes a composite one as follows:
H0: Pr(c(v1, . . . , vd)=c0(v1, . . . , vd;α0))
=1for someα0 ∈A (9)
against the alternative hypothesis
H1: Pr(c(v1, . . . , vd)=c0(v1, . . . , vd;α))<1 for allα∈A.
One can estimate the marginal distributions using paramet-ric or nonparametparamet-ric methods. Chen (2007) discussed moment-based tests using parametrically estimated marginal distribu-tions. Although efficient under correct parametric specification, the test can suffer size distortion and power loss under incor-rect distributional assumptions. In this study, we focus on copula specification tests where the marginal distributions are estimated nonparametrically by their empirical distributions. Compared with parametric estimates, the empirical CDFs are always con-sistent. On the other hand, compared with nonparametric esti-mates such as kernel distribution estiesti-mates, the empirical CDFs are free of smoothing parameters.
Lacking a priori guidance on the marginal distributions, we estimateFj by the rescaled empirical distribution function,
˜ ula parameters via the maximum likelihood estimation as fol-lows:
One can envision that a smooth test on the hypothesis (9) can directly applicable here because of the presence of nuisance pa-rameters in (11). There are two sets of nuisance parameters: the finite-dimensional copula parameterαand the marginal distri-butionsFj, j =1, . . . , d, which are infinite-dimensional when estimated nonparametrically.
To construct a test based on (12), we need to account for the influences of the nuisance parameters. We start with the asymptotic distribution of ˆα, which is studied by Gen-est, Ghoudi, and Rivest (1995) and Chen and Fan (2005). Let
l(v1, . . . , vd;α)=logc(v1, . . . , vd;α) and use indices α and
j =1, . . . , dto denote partial derivatives oflwith respect toα
andvj, j=1, . . . , d, respectively. Also define
Wj(Uj t;α)≡E[lαj(U1s, . . . , Uds;α){1(Uj t≤Uj s)−Uj s}|Uj t].
We can then show the following:
Theorem 1. Suppose E [l(U1t, . . . , Udt;α)] has a unique maximum at α0. Under the regularity conditions given in the Appendices, the semiparametric estimator ˆα→p α0 and √
withlααbeing the second derivative oflwith respect toα.
Under the null, we have B=E [lα(U1t, . . . , Udt;α0)
lα(U1t, . . . , Udt;α0)T] by the information matrix equality. In ad-dition, since the conditional expectation oflα(U1t, . . . , Udt;α0) with respect to eitherUj tis zero,lα(U1t, . . . , Udt;α0) is uncorre-lated with eachWj(Uj t;α0). Therefore, the asymptotic variance of√n( ˆα−α0),B−1B−1, can be simplified to
Obviously this asymptotic variance is further reduced toB−1
when the marginal distributions are known. Next let
The asymptotic variance of ˆαcan be consistently estimated by
ˆ
We obtain the following asymptotic distribution for ˆg( ˆα).
Theorem 2. Suppose the regularity conditions given in the
Appendices hold. Under the null hypothesis, ˆg( ˆα)
p
Remark 1. The asymptotic variancereflects the influence
of nuisance parameters in ˆg( ˆα). If the marginal distributions
are known, the terms involving W’s and Z’s can be dropped, resulting in
=var [g(U1t, . . . , Udt;α0)−GTB−1{lα(U1t, . . . , Udt;α0)}].
When the copula parameterαis known, the third term on the right-hand side of (16) can be dropped. When both the marginal distributions and copula parameter are known,is reduced to
V(α0)=var [g(U1t, . . . , Udt;α0)], which is exactly the variance (4) derived in the previous section. Another possible incidence of
=V(α0) is when the true copula is the independence copula, a result readily available from Proposition 2.2 of Genest, Ghoudi, and Rivest (1995).
Next we present a consistent estimator for. Let
ϕt =gU˜1t, . . . ,U˜dt; ˆα
It follows thatcan be estimated consistently by
ˆ
where ¯ϕis the sample mean of{ϕt}nt=1.
We can now construct a smooth test of copula specification as follows.
Theorem 3. Under the regularity conditions given in the Ap-pendices, the semiparametric smooth test of copula specification is given by
ˆ
Q≡ngˆ( ˆα)Tˆ−1gˆ( ˆα). (20)
Under the null hypothesis, ˆQ→d χ2
Kasn→ ∞.
Similar to the results on simple hypotheses, when the com-posite null hypothesis is rejected by test (20), we can construct an alternative copula density according to (8), which provides a plausible approximation to the unknown copula density.
4. EXTENSION TO COPULA-BASED MULTIVARIATE DYNAMIC MODELS
So far we have maintained that our samples are iid realiza-tions of random distriburealiza-tions. One area that copula methods have found most applications is financial economics, wherein the iid assumption usually does not hold. In this section, we extend the proposed smooth test to the semiparametric copula-based mul-tivariate dynamic (SCOMDY) models studied by Chen and Fan (2006) and Chan et al. (2009). Following Chen and Fan (2006), we consider a class of SCOMDY models for ad-dimensional time seriesXt =(X1t, . . . , Xdt), t=1, . . . , T. Denote byFt−1 the information set that contains information from the past and from exogenous variables. The model is given by
Xt =µt(θ)+
is the conditional variance ofXj tgivenFt−1. We assume that the model is correctly parameterized up to a finite-dimensional unknown parameterθ. Moreover,εt =(ε1t, . . . , εdt)T is a se-quence of iid random vectors with a distribution function
F(ε)=C(F1(ε1), . . . , Fd(εd);α), where Fj(·) is the true but unknown continuous marginal distribution ofεj t.
Chen and Fan (2006) proposed a three-step estimator ˆαof the copula parameterα0. In the first step, the dynamic parameters are estimated consistently by ˆθ, usually via the quasi-maximum likelihood estimate (QMLE). In the second step, the distribu-tions of the standardized error terms of the marginal distribudistribu-tions are estimated by their rescaled empirical distributions. Denote by ˜Uj t( ˆθ), j=1, . . . , d the rescaled empirical CDFs thus ob-tained. In the third step, the copula parameter ˆα( ˆθ) are estimated using the MLE based on ( ˜U1t( ˆθ), . . . ,U˜dt( ˆθ)), t=1, . . . , T.
An important contribution of Chen and Fan (2006) is the finding that asymptotic distribution of ˆα( ˆθ) is not affected by the esti-mation of dynamic parameters when the marginal distributions are estimated by the empirical CDFs. Thanks to this invariance result, the asymptotic distribution of ˆg( ˆα,θˆ) is readily available.
Theorem 4. Suppose the regularity conditions given in
the Appendices hold. Under the null hypothesis that
C(F1(ǫ1), . . . , Fd(ǫd))=C0(U1, . . . , Ud;α0), ˆg( ˆα,θˆ)
p
→0 and √
Tgˆ( ˆα,θˆ)→d N(0, ), whereis defined in Equation (16).
It follows immediately that one can construct a smooth test of copula specification for the SCOMDY in a similar manner.
Theorem 5. Suppose the regularity conditions given in the
Appendices hold. The semiparametric smooth test of copula specification is given by
Remark 2. The appealing property that the asymptotic distri-bution of ˆg( ˆα,θˆ) is not affected by the estimation of dynamic
pa-rameters in the SCOMDY models only holds when the marginal distributions are estimated by the empirical CDFs and the copula in question is time invariant (R´emillard2010).
5. MONTE CARLO SIMULATIONS
We conduct a series of Monte Carlo simulations to assess the finite-sample performance of the proposed tests. Since the Gaussian copula andtcopula are the most commonly used cop-ulas in practice, we focus on these two copcop-ulas as the null copula distributions. We generate the marginal distributions from the standard normal distributions and consider six different data-generating copula distributions, falling into three categories:
1. Radially symmetric copulas: the Gaussian (CN) copula, Student’stcopula with four degrees of freedom (Ct4), and Frank
copula (CF).
2. Symmetric but not radially symmetric copulas (also known as “exchangeable copulas”): the Clayton copula (CC) and Gumbel–Hougaard copula (CG).
3. Asymmetric copula: asymmetric Gumbel–Hougaard cop-ula (CAG). Given the exchangeable Gumbel–Hougaard copula, an asymmetric version of it can be constructed using Khoudraji’s device (Khoudraji1995) as follows,
CAG(v1, . . . , vd;λ1, . . . , λd, α)
The dimensiond was set to 2 or 3. For the null distribu-tions, we focus on Gaussian andtcopulas with exchangeable correlation matrix and consider two levels of dependence, corre-sponding to Kendall’s tau atτ =0.3 and 0.6,respectively. The parameters for data-generating copulas are calculated by inver-sion of Kendall’s tau. For the asymmetric Gumbel–Hougaard copula, the parameter α was set to 4. In the bivariate case,
λ=(λ1,0.95) withλ1∈ {0.3479,0.7929}. The corresponding values for Kendall’s tau are about 0.3 and 0.6, respectively. In the trivariate case, following Kojadinovic, Segers, and Yan (2011),λwas set to (0.2,0.4,0.95). For a detailed account of the various copulas considered in this study and their estima-tions, see Nelsen (1993). We run simulations with sample size
n=300,500 and each experiment is repeated 1000 times. One appealing feature of the proposed smooth test is its flexibility, under which one can tailor the moment functions,
g(v1, . . . , vd), according to his research needs. Several gen-eral considerations apply here. First, the functions shall be lin-early independent. Second, as is well known about Neyman’s smooth test, its asymptotic power suffers when the degree of freedom,K, is large. Usually, a small number of terms is used. For instance, Neyman (1937) recommendedK≤4. Thomas and Pierce (1979) suggestedK=2 for composite tests. Third, ideally one shall choose the moment functions that best capture deviations from the null distributions.
Taking all these factors into consideration, when there is no a priori reason to focus on a particular direction of deviation, we select the moment functions from the normalized Legendre polynomials. We focus on the first three Legendre polynomials given by, forv∈[0,1],
ψ1(v)=√3(2v−1),
ψ2(v)=√5(6v2−6v+1),
ψ3(v)=√7(20v3−30v2+12v−1).
Ourgfunctions then consist of various tensor products of these basis functions.Table 1summarizes the various configurations considered in this study.
To save place, we use g1=ψ1ψ1 to denote the function
g1(v1, v2)=ψ1(v1)ψ1(v2), and all other terms are defined sim-ilarly. A few explanations are in order for the configurations listed inTable 1. Consider the tensor product of (ψ1, ψ2, ψ3) as a 3 by 3 matrix with the (i, j)th entry beingψiψj. The entries from the upper triangular part of the matrix, except forψ1ψ3, are considered.ψ1ψ3 is excluded because it does not provide
Table 1. Configurations of moment functions
Group Moment functions Notation
Diagonal g1=ψ1ψ1, g2=ψ2ψ2 gD1 g1=ψ1ψ1, g2=ψ2ψ2, g3=ψ3ψ3 gD2 Others g1=ψ1ψ1, g2=ψ1ψ2, g3=ψ2ψ2 gO1
g1=ψ1ψ1, g2=ψ2ψ2, g3=ψ2ψ3, g4=
ψ3ψ3
gO2
g1=ψ1ψ1, g2=ψ1ψ2, g3=ψ2ψ2, g4=
ψ3ψ3
gO3
g1=ψ1ψ1, g2=ψ1ψ2, g3=ψ2ψ2, g4=
ψ2ψ3, g5=ψ3ψ3
gO4
good powers in most of our experiments. The group “Diago-nal” contains multiple entries on the diagonal and the group “Others” adds additional off-diagonal entries to the “Diagonal” group. We also experiment with “singleton” tests that contain a single moment function. This exercise provides useful insight into how different moment functions capture deviations of al-ternative copulas from the null copula. The details are given in Appendix B. Since the “singleton” tests are generally dominated by tests with multiple moment functions, we exclude them from further consideration. Interestingly, Kallenberg and Ledwina (1999) used a similar strategy to test the independence between random variables. They constructed two alternative exponential families, which contain elements from the tensor products of the orthonormal Legendre polynomials. The first family is restricted to the “diagonal” entries and symmetric inv1andv2, which is similar to the “diagonal” group in our configuration. The second family is the combination of the “diagonal” and “off-diagonal” entries, which is similar to the “Others” group in our study.
Inspired by the idea of C-vine and D-vine copulas, which construct multivariate copulas using a cascade of bivariate cop-ulas, we select three-dimensional moment functions based on the two-dimensional functions listed in Table 1 with the re-striction that the number of the moment functionsK≥2 and all three variable are present in the moment functions. In par-ticular, without loss of generality the first k moment condi-tions are funccondi-tions of (v1, v2), where k=(K+1)/2 if K is an odd number andk=K/2 ifKis an even number, and the rest are functions of (v2, v3). For example, based on gO2, we
use {ψ1(v1)ψ1(v2), ψ2(v1)ψ2(v2), ψ2(v2)ψ3(v3), ψ3(v2)ψ3(v3)} as the moment functions. Our simulation results show that the ordering ofv1, v2, v3in the moment functions do not matter.
To implement our test, one needs to evaluate nu-merically µ(g,αˆ) and ˆG, where µ(g,αˆ)= g(v1, . . . , vd)
c0(v1, . . . , vd; ˆα)dv and Gˆ =
g(v1, . . . , vd)c0α(v1, . . . , vd; ˆ
α)dv, in which cα0(v1, . . . , vd; ˆα) is the derivative of
c0(v1, . . . , vd; ˆα) with respect to α. We use the Smolyak cubature for numerical integrations in our simulations. The quadrature nodes and weights are generated using the func-tionsmolyak.quadin R packagegss. Smolyak cubatures are highly accurate for smooth functions. By thinning out the nodes from the product quadratures, they are suitable for multi-dimensional integration (Gu and Wang2003).
In what follows, we refer to the smooth tests with moment functionsgD1 asQD1 and all other tests are defined similarly. For comparison, we also report the simulation results of the multiplier test by Kojadinovic and Yan (2011). The multiplier test is an omnibus test based on the empirical copula process. Since it uses resampling to calculate critical values, this test can be computationally expensive especially if the sample size is large. In contrast, our test has a chi-squared limit distribution with known degrees of freedom and is computationally much easier.
We report the empirical sizes and powers of the tests ford =2 inTable 2. The results for the multiplier test are reported in the row signified byMn. Entries in bold reflect empirical sizes of the tests, and the rest of the table reflects their powers against various alternatives. Overall, the tests exhibit correct sizes, cen-tering about the nominal value of 5%. As expected, both size and power improve with sample size. As the dependence gets
Table 2. Empirical sizes and powers of the tests in the bivariate case (%)
CN Ct4 CF CC CG CAG
Copula underH0 τ Test n=300 500 n=300 500 n=300 500 n=300 500 n=300 500 n=300 500
CN 0.3 QD1 4.1 5.2 59.7 87.0 39.7 67.0 6.0 7.1 8.7 18.0 11.2 21.1
QD2 4.2 4.4 64.1 88.3 30.8 59.1 12.3 18.3 12.5 23.6 9.8 18.9 QO1 5.2 4.8 55.7 81.6 35.7 60.8 76.5 94.3 31.7 55.2 98.4 99.9 QO2 4.5 4.4 62.3 84.9 28.2 55.2 45.2 73.4 30.3 48.0 97.2 99.9 QO3 6.9 5.5 57.8 88.5 30.8 55.6 72.6 93.9 30.6 56.7 96.6 100 QO4 6.2 4.6 55.9 82.0 26.6 52.6 74.3 95.1 34.0 62.3 99.9 100 Mn 4.1 4.1 10.3 13.2 36.0 58.7 80.6 96.4 40.5 62.5 99.9 100
0.6 QD1 7.6 5.6 22.2 38.6 87.7 99.6 19.8 36.5 2.7 6.4 65.8 91.7
QD2 5.9 5.5 40.0 62.8 83.8 99.6 25.6 53.0 7.3 16.2 92.8 99.9 QO1 6.2 5.8 19.6 33.8 85.3 99.0 100 100 48.6 77.7 97.1 100 QO2 6.5 5.8 37.2 66.1 81.2 99.0 99.6 100 52.4 79.0 100 100 QO3 7.6 6.0 33.9 62.2 83.2 99.3 99.9 100 46.2 77.2 99.2 100 QO4 6.2 4.0 35.3 60.2 80.8 98.9 100 100 57.4 86.8 100 100
Mn 3.8 3.9 7.9 10.7 93.8 99.7 100 100 67.0 87.4 100 100
Ct4 0.3 QD1 64.6 92.8 8.3 7.7 78.7 95.0 44.3 68.7 27.5 42.6 35.4 55.9 QD2 66.4 91.9 8.6 7.3 79.9 96.4 46.9 65.2 27.5 37.6 34.9 57.9 QO1 57.9 88.0 6.5 6.3 81.3 97.9 87.6 98.3 43.9 69.1 99.5 100 QO2 55.3 82.9 6.2 6.3 82.6 98.1 63.7 88.4 34.7 55.1 98.4 100 QO3 53.7 85.8 5.5 4.5 84.5 98.4 83.4 97.1 43.7 63.4 99.4 100 QO4 52.6 83.3 6.4 5.6 78.5 96.9 86.7 99.4 42.4 66.0 100 100 Mn 32.9 52.5 4.0 4.2 72.2 94.6 87.0 98.9 48.5 72.1 100 100
0.6 QD1 39.5 84.1 6.4 5.0 98.8 100 74.4 92.7 13.0 25.6 36.3 63.0
QD2 37.5 79.3 7.0 6.2 98.4 100 87.2 96.4 19.8 27.6 23.9 48.3 QO1 26.1 75.6 4.8 4.6 97.0 100 99.9 100 48.1 76.9 81.7 98.5 QO2 32.1 67.6 6.0 5.6 97.4 100 98.9 100 38.0 66.2 89.8 100 QO3 34.4 73.2 6.6 6.0 97.5 100 99.9 100 53.7 79.4 76.3 96.4 QO4 30.2 66.2 5.4 4.6 97.8 100 100 100 54.0 84.6 90.3 99.8 Mn 28.5 48.8 4.0 4.2 99.4 100 100 100 69.7 89.8 99.5 100
stronger, the power remains comparable or increases in most scenarios. Noticeable exceptions include the cases where the null hypothesis is the Gaussian copula and the data are gen-erated from at copula and where the null hypothesis is the t
copula and the data are generated from a Gaussian copula. This exception is not unexpected because when the correlation is high, Gaussian copula andtcopula are known to be similar.
The “diagonal” group is powerful in distinguishing copula distributions within the radially symmetric copulas. This re-sult is consistent with Kallenberg and Ledwina (1999), who suggested that the “diagonal” family is sufficient to distinguish between dependence and independence for most symmetric dis-tributions occurring in practice. To distinguish the radially sym-metric copulas from other types of copula distributions, we also need the off-diagonal entries. Therefore, it is desirable to use the moment functions from the “others” group. In practice, if researchers are not sure about possible directions of deviations from the null distribution, tests in the “others” group are gener-ally recommended.
It is also noted that the proposed tests are significantly supe-rior to the multiplier testsMnfor the important case of testing null Gaussian copula against the t copula in both the bivari-ate and trivaribivari-ate cases. For example, whenτ =0.3, the power of QO4 test is about 56% for n=300 and 82% forn=500
while the power ofMntest is about 10% and 13%, respectively; when τ =0.6, the power ofQO4 test is about 35% and 60%
forn=300 andn=500 while the power ofMntest is about
8% and 11%, respectively. In most other cases, the proposed tests with multiple moment functions perform comparably to the multiplier testsMn.
Our results suggest that if one function from the singleton group (see Appendix B for details) contributes significantly to the power against a specific alternative, all tests including this term register good powers against this specific alternative. Con-sequently, tests with multiple instruments outperform those with a single instrument in most cases and can provide power against a variety of alternatives. This is important because in many practical cases, researchers have no a priori guidance on how to select a particular direction for specification test. It is encourag-ing that a combination of a small number of terms appears to be a safe testing strategy that delivers satisfactory performance.
Since we use pairwise moment conditions in the trivariate cases, we restrict our attention to tests with at least two moment conditions. Our simulation results for the bivariate case have lent support to this strategy as well.Table 3reports the simulation results for the trivariate cases. The overall results are quanti-tatively similar to those of the bivariate cases, especially the case of testing null Gaussian copulas againstt-copulas, where our tests outperform theMntests by considerable margins. No-ticeable exceptions include the cases where the null hypothesis is the Gaussian copula and the data are generated from the Frank copula. Taken=500 as an example. Whenτ =0.3, the power ofQO4 test is about 53% ford =2 and 18% ford=3.
This exception is not unexpected because the Frank copula with
Table 3. Empirical sizes and powers of the tests ford= 3 (%)
Test CN Ct4 CF CC CG CAG
Copula underH0 τ λ statistics n=300 500 n=300 500 n=300 500 n=300 500 n=300 500 n=300 500
CN 0.3 QD1 7.1 4.3 54.5 80.6 9.1 14.5 6.2 8.1 10.3 17.0
QD2 6.6 6.0 66.3 90.8 11.5 16.3 14.4 24.9 15.6 30.1 QO1 6.5 5.6 50.5 76.9 9.7 14.3 77.7 93.3 28.7 51.3 QO2 6.2 5.9 58.3 86.8 10.9 15.1 49.5 75.2 32.0 49.2 QO3 6.4 4.5 52.6 79.2 9.3 9.5 73.3 93.7 29.9 53.3 QO4 6.2 5.8 57.4 84.6 12.3 18.2 86.7 98.6 44.5 70.5 Mn 3.2 3.2 9.4 15.2 17.1 32.7 95.1 99.9 58.5 83.7
0.6 QD1 8.4 6.8 15.5 27.9 49.0 83.0 10.0 15.0 3.8 4.7
QD2 7.6 5.7 39.5 67.6 62.4 89.7 18.7 35.7 8.9 14.4 QO1 7.2 5.9 12.3 24.3 47.4 81.7 99.7 100 44.6 72.8 QO2 6.8 6.2 35.5 65.0 63.2 86.9 99.9 100 47.2 77.3 QO3 7.6 5.6 24.9 49.0 50.2 78.9 99.9 100 42.2 71.8 QO4 6.9 5.8 34.9 61.1 63.0 86.1 100 100 67.0 93.4 Mn 3.4 3.1 7.0 10.4 90.7 99.4 100 100 65.8 93.5
λ∗ Q
D1 96.4 100
QD2 73.3 94.7
QO1 94.7 99.6
QO2 99.8 100
QO3 92.1 99.3
QO4 99.9 100
Mn 95.8 100
Ct4 0.3 QD1 58.1 82.1 5.7 4.6 81.5 94.2 35.9 55.2 24.0 35.5 QD2 74.3 93.6 4.8 4.7 92.7 99.2 38.0 61.0 23.8 39.7 QO1 49.1 78.5 6.0 5.5 72.5 93.1 91.0 99.2 45.6 72.7 QO2 70.2 92.5 5.6 5.3 89.3 99.3 74.1 93.3 42.7 67.7 QO3 49.2 75.3 5.7 5.4 74.7 92.3 86.6 98.2 47.7 64.1 QO4 68.8 90.3 5.4 5.3 87.9 98.2 93.7 99.7 54.7 80.3 Mn 77.6 96.6 3.4 4.1 77.3 97.0 98.4 100 34.0 59.8
0.6 QD1 32.8 74.5 7.6 7.4 92.2 99.6 51.3 73.2 10.1 16.0
QD2 46.3 72.7 8.1 6.7 98.9 100 81.1 96.2 18.4 26.8 QO1 30.5 69.9 6.1 5.9 89.5 99.7 100 100 53.5 81.1 QO2 36.0 65.2 5.9 4.3 98.8 100 99.8 100 41.2 74.1 QO3 29.1 62.9 5.6 4.9 93.3 99.6 100 100 52.5 77.6 QO4 34.7 64.7 5.9 4.7 98.3 100 100 100 67.6 91.5 Mn 60.0 87.8 3.3 3.4 99.6 100 100 100 65.8 89.3
λ∗ QD
1 41.3 61.7
QD2 69.5 87.5
QO1 54.6 76.3
QO2 100 100
QO3 45.7 73.0
QO4 99.6 100
Mn 93.1 100
NOTE:λ∗=(0.2,0.4,0.95).
moderate dependence is known to be similar to the Gaussian copula (Kojadinovic and Yan2011).
Next we report simulations on SCOMDY models as in Chen and Fan (2006). In this time series setup, we only consider two-dimensional case and Gaussian copula as the null hypothesis copula. Following Chen et al. (2004), we consider the following dynamic data-generating process (DGP),
Xj t =0.01+0.05Xj,t−1+ej t
ej t = !
hj t·−1(Uj t)
hj t =0.05+0.85hj,t−1+0.1e2j t−1, j=1,2,
where −1(
·) is the inverse of the standard normal distribu-tion. The DGP is such that individual variables are condition-ally normal with the same AR(1)-GARCH(1,1) specifications and the copula is generated according to the Gaussian, twith four degrees of freedom, Frank, Clayton, Gumbel-Hougaard, or asymmetric Gumbel–Hougaard copula.
We use a three-step estimator to construct the test statistic. First, we estimate the dynamic parameters of the model via MLE. We then estimate the marginal distributions of the stan-dardized residuals via the rescaled empirical distributions. In the third step, we estimate the coefficients of the null copula
Table 4. Empirical sizes and powers of the tests for SCOMDY models in the bivariate case (%)
Test CN Ct4 CF CC CG CAG
τ statistics n=300 500 n=300 500 n=300 500 n=300 500 n=300 500 n=300 500
0.3 QD1 3.7 5.2 57.3 85.7 38.2 64.7 4.9 6.2 9.6 18.6 12.1 19.6
QD2 3.7 4.2 61.5 86.3 30.1 59.5 10.5 19.2 12.5 24.6 8.4 18.2
QO1 4.1 4.9 52.0 79.2 31.4 59.8 74.1 94.6 31.8 51.8 96.8 99.9
QO2 4.6 4.3 58.6 84.9 26.3 53.5 45.5 70.2 29.3 48.3 96.3 99.8
QO3 3.7 4.2 53.1 83.9 28.9 55.1 69.5 94.2 32.4 53.4 95.6 99.8
QO4 6.5 4.9 54.9 80.6 24.3 51.4 74.8 94.3 37.5 60.3 99.7 100
0.6 QD1 6.5 5.7 20.8 36.3 88.9 99.7 17.2 39.5 3.2 5.9 62.8 91.5
QD2 4.7 5.1 34.8 63.5 83.9 99.5 27.6 53.1 6.3 16.3 90.0 99.8
QO1 6.2 5.7 18.7 32.0 84.6 99.2 99.9 100 46.8 76.2 95.9 99.9
QO2 7.1 5.5 36.7 61.1 78.5 98.9 99.4 100 45.0 76.6 99.5 100
QO3 7.0 5.1 33.2 61.3 83.4 98.9 99.4 100 45.3 73.8 98.9 100
QO4 6.4 7.1 35.6 57.3 79.5 98.5 99.9 100 56.3 84.2 99.8 100
distribution using MLE. The test statistic is then constructed using the rescaled empirical CDFs and the estimated copula co-efficients according to (21). Like the previous experiment, we set sample sizen=300,500 and repeat each experiment 1000 times.
We report the simulation results inTable 4. The overall results are similar to those obtained for iid data in terms of size and power performance. As a matter of fact, the general patterns are essentially identical across these two tables. Our experi-ments confirm the remarkable finding of Chen and Fan (2006) that the asymptotic variance of the MLE of the copula param-eters is not affected by the estimation of dynamic coefficients in the SCOMDY models. Thus, our proposed copula specifica-tion tests can be applied to the dynamic models directly. Given that innovations in these models are properly standardized, the asymptotic results derived under the iid assumption are shown to be valid in these experiments.
6. EMPIRICAL APPLICATION
In this section, we present three illustrative examples to demonstrate the usefulness of our tests. For simplicity, we only consider bivariate examples. We consider two cross-sectional datasets and one time series case. In each example, we first conduct the specification test. When the null hypothesis is re-jected, we proceed to estimate an alternative copula function using diagnostic information provided by the test. Although in our simulations we only consider the Gaussian ortcopula as the null distribution, here we provide an example where the more plausible null distribution is the Gumbel distribution, further demonstrating the flexibility of the proposed method.
6.1 Uranium Exploration Data
Our first example looks at the Uranium exploration data that have been investigated by Cook and Johnson (1981,1986) and Genest, Quessy, and R´emillard (2006). The dataset consists of 655 log-concentrations of seven chemical elements in water samples collected from the Montrose quadrangle of western Colorado (USA). Our interest is to understand the dependence in concentrations between the elements cesium (Cs) and
scan-dium (Sc). This dataset contains a nonnegligible number of ties, which may affect the test results. Following Kojadinovic and Yan (2010), we calculate pseudo-observations by randomly breaking the ties.Figure 1displays the scatterplots of log con-centrations of Cs and Sc and the resulting pseudo-observations. The graphs suggest a strong positive left-tail dependence be-tween the concentration of Cs and Sc, but little right-tail de-pendence. Gaussian copula is expected to be rejected since it is radially symmetric and does not allow tail dependence.
We transformed the samples by their empirical distribution functions and tested the Gaussian copula null hypothesis on these variables. The two-step estimator of the correlation coef-ficient of the Gaussian copula, as defined in Equation (11), is
ˆ
α=0.3362. The p-values of constructed tests are reported in Table 5. Not surprisingly, the null hypothesis of Gaussian copula is rejected at the 5% level by all tests.
As is discussed above, the proposed tests are diagnostic for they provide useful guidance in the construction of alternative copula specifications. When the null hypothesis is rejected, we shall augment the null copula distribution with extra features suggested by the diagnostic tests. This alternative density is obtained by minimizing the Kullback–Leibler information cri-terion between the target density and the null density, subject to the moment conditions associated with the set of features our test fails to reject. With the null distribution being the Gaussian copulaCN, the alternative density takes the form
cg(v1, . . . , vd)=cN(v1, . . . , vd; ˆα) exp ˆ
λTg(v1, . . . , vd)−λˆ0
(22)
with
ˆ
λ0=log
cN(v1, . . . , vd; ˆα) exp ˆ
λTg(v1, . . . , vd)
dv, (23)
where g is a vector of moment functions involved in a given test. For each test that rejects the null hypothesis, we can subse-quently calculate an alternative copula density using the method
Table 5. p-Values of the test for the Uranium exploration data
Test QD1 QD2 QO1 QO2 QO3 QO4
p-Value 9.206e-05 0.0002 3.973e-09 5.421e-08 6.477e-09 8.948e-11
1.5 2.0 2.5
0.4
0
.6
0.8
1.0
1.2
1.4
Cesium and Scandium on a log scale
Cs
Sc
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0
.2
0.4
0.6
0.8
1.0
Pseudo−observations
Cs
Sc
Figure 1. Scatterplots of cesium versus scandium.
of Wu (2010), incorporating the set of moment functions in-volved in the test. For instance, we denote bycD1the alternative
copula density corresponding with testQD1. It follows that
cD1(v1, v2)=cN(v1, v2; ˆα) exp(ˆλ1ψ1(v1)ψ1(v2)
+λˆ2ψ2(v1)ψ2(v2)−λˆ0).
All other densities are constructed in a similar manner. To calcu-late the alternative copula density function, one needs to evaluate numerically ˆλ0defined in (23). We use the Smolyak cubature for numerical integrations. We report the log-likelihood (denoted by logL) and the Akaike information criterion (AIC) value of each estimated copula density inTable 6. For comparison, we also report results on the null distribution. AlthoughcO4has a higher
log-likelihood, the AIC favorscO1. Note thatcO1contains three
moment functions (ψ1ψ1, ψ1ψ2, ψ2ψ2), whilecO4contains two
additional termsψ2ψ3andψ3ψ3. Our results thus suggest that limited information content ofψ2ψ3andψ3ψ3, given that con-veyed by (ψ1ψ1, ψ1ψ2, ψ2ψ2), does not warrant its inclusion in the model. To formally test whether a model favored by an in-formation criterion is significantly better than other candidates, wherein all competing models are allowed to be misspecified, please refer to Chen and Fan (2005,2006).
Figure 2reports the contour plots of the Gaussian copula den-sitycN with the correlation coefficient ˆρ=0.3362 (left-hand panel) and the alternative copula density functioncO1
(right-hand panel). The Gaussian copula is radially symmetric. In con-trast, the alternative copula density captures one salient feature
of the data: the density at the lower-left corner is clearly higher than that at the upper-right corner, reflecting the asymmetric tail behaviors revealed inFigure 1. It is reassuring to see that our test provides useful pointers for subsequent analysis when the null hypothesis is rejected. This is in contrast to omnibus tests, such as the Kolmogorov–Smirnov test or Cramer–von-Mises test, which do not suggest possible directions of alternative specifi-cations.
6.2 Loss and ALAE Data
Our second example concerns the dependence structure of the indemnity payment (Loss) and the allocated loss adjustment ex-pense (ALAE) of insurance companies. The data were collected by the U.S. Insurance Services Office and have been analyzed by Frees and Valdez (1998), Genest, Ghoudi, and Rivest (1995), Klugman and Parsa (1999), Denuit, Purcaru, and Van Keile-gom (2004), and Chen and Fan (2005). Following Chen and Fan (2005), we restrict ourselves to 1466 uncensored data. The scatterplots of the Loss and ALAE data on a log scale and the resulting pseudo-observations (with ties broken randomly) are depicted in Figure 3. There exists apparent positive right-tail dependence between Loss and ALAE: large Losses tend to be associated with large ALAEs. On the other hand, there is rel-atively weaker left-tail dependence between the two variables. As a result, Gumbel–Hougaard copula was chosen as the best-fitting copula by Frees and Valdez (1998), Genest, Ghoudi, and Rivest (1998), Denuit, Purcaru, and Van Keilegom (2004), and
Table 6. Estimation results of copula density functions for Uranium exploration data
density cN cD1 cD2 cO1 cO2 cO3 cO4
logL 39.353 48.160 48.199 58.797 54.616 58.805 60.599
AIC −76.705 −90.320 −88.398 −109.595 −99.232 −107.610 −109.199
0 1 2 3 4 5
0.2 0.4 0.6 0.8 0.2
0.4 0.6 0.8
Gaussian copula c N
0 2 4 6 8 10 12
0.2 0.4 0.6 0.8 0.2
0.4 0.6 0.8
Alternative Copula c O1
Figure 2. Contour plots of the copula density functions for Uranium exploration data.
Chen and Fan (2005). We refer readers to Frees and Valdez (1998) for a detailed description of this dataset and to Chen and Fan (2005) for an overview of the copula model selection results of this dataset in the literature. In keeping with the ex-isting results, we use the Gumbel–Hougaard copula as the null hypothesis.
We employ the same estimation and testing procedure as in the previous example on the current dataset. The only difference is that now the baseline copula is the Gumbel–Hougaard copula. The testing results are reported inTable 7. It transpires that the previous studies have done a rather admirable job in selecting the Gumbel–Houggard copula: all tests fail to reject the null
Table 7. p-Values of the test for the Loss and ALAE data
Test QD1 QD2 QO1 QO2 QO3 QO4
p-Value 0.243 0.218 0.076 0.247 0.095 0.136
hypothesis at the 5% confidence level. There are two tests with
p-values slightly less than 10%. We proceed to construct alter-native copula densities, considering all tests withp-values less than 0.1. The estimation results reported inTable 8suggest that the AIC favors Gumbel–Hougaard copula.
2 4 6 8 10 12 14
468
1
0
1
2
Loss and ALAE on a log scale
Loss
ALAE
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0
.2
0.4
0
.6
0.8
1
.0
Pseudo−observations
Loss
ALAE
Figure 3. Scatterplots of Loss versus ALAE.
−6 −4 −2 0 2
−6
−4
−2
0
2
residSPX and residHSI
residSPX
residHSI
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Pseudo−observations
residSPX
residHSI
Figure 4. Scatterplots of the residuals of SPX returns and HSI returns.
Table 8. Estimation results of the copula density functions for Loss and ALAE data
Density cG cO1 cO3
logL 189.817 191.843 192.813
AIC −377.634 −375.686 −375.625
6.3 Stock Returns Data
Finally, we apply our test to time series data, examining the dynamic relationship between the monthly SP500 Index (SPX) and Hengseng Index of Hong Kong (HSI). Our sample cov-ers the period from January 1970 till October 2012 with 513 observations.
We calculate the log returns for each index and model the log return time series with an AR(0)-GARCH(1,1) model. The standardized residuals from the GARCH(1,1) models are used in subsequent estimation and testing.Figure 4displays the scatter plots of the standardized residuals of two series and the resulting pseudo-observations. One can see that the United States and
Table 9. p-Values of the test for the stock returns data
Test QD1 QD2 QO1 QO2 QO3 QO4
p-Value 0.066 0.062 1.423e-05 0.001 3.871e-05 1.682e-05
the HK markets exhibit considerable left-tail dependence but smaller right-tail dependence.
We transformed the standardized residuals by their empirical distribution functions and tested null hypothesis of the Gaussian copula distribution. The three-step estimator of the Gaussian copula yields ˆα=0.4496. Thep-values of the tests are reported inTable 9. All tests containing off-diagonal entries in the form of
ψ1ψ2orψ2ψ3reject the null hypothesis, which can be explained by the evident asymmetry in the two extreme tails revealed in Figure 4.
We next construct alternative copula density functions asso-ciated with those tests that reject the null hypothesis at the 10% level. The estimation results are reported inTable 10. Although
cO3 andcO4 have slightly higher log-likelihood, the AIC
sug-gestscO1as the preferred specification. Note thatcO1 contains
three moment functions (ψ1ψ1, ψ1ψ2, ψ2ψ2) whilecO3andcO4
contain one or two additional terms. Our results suggest that these extra terms do not contain sufficiently useful information given the terms (ψ1ψ1, ψ1ψ2, ψ2ψ2) already incorporated in
cO1.
The contours of the estimated Gaussian copula ( ˆα=0.4496) andcO1 are reported inFigure 5. Two differences are revealed
by the plots. First, the Gaussian copula fails to capture the radial asymmetry in the data. In contrast, the alternative copula density demonstrates a considerably higher density at the lower tail. Second, the alternative copula density exhibits a narrower and slightly asymmetric “ridge” along the diagonal. Apparently
Table 10. Estimation results of the copula density function for stock return data
density cN cD1 cD2 cO1 cO2 cO3 cO4
logL 57.916 59.072 59.363 63.929 63.438 64.157 65.347
AIC −113.833 −112.144 −110.727 −119.858 −116.876 −118.314 −118.695
0 1 2 3 4 5 6
0.2 0.4 0.6 0.8 0.2
0.4 0.6 0.8
Gaussian copula c N
0 2 4 6 8 10 12
0.2 0.4 0.6 0.8 0.2
0.4 0.6 0.8
Alternative Copula cO1
Figure 5. Contour plots of the copula density functions for stock return data.
these richer features are missed by the Gaussian copula, which is restrictive for it is parameterized by a single parameter. Our confidence in these implied features is supported by the strong statistical evidence conveyed in the reported specification tests and goodness-of-fit measures.
7. CONCLUDING REMARKS
We have proposed moment-based tests of copula specifica-tions for semiparametric copula models. Our tests can be charac-terized as score tests on moment conditions of empirical copula distributions under the null hypothesis. The tests are both flex-ible and easy to implement for it does not require selection of smoothing parameters. It is distribution free and therefore no simulations are needed for the critical values. In addition, our tests are diagnostic: when a set of moment conditions are not rejected by our specification test, they provide useful pointers for the construction of alternative copula densities. Our Monte Carlo simulations and empirical examples demonstrate the effi-cacy and usefulness of our method.
We conclude this study with some possible directions for future study. First, following Chen and Fan (2006) our exten-sion to time series model maintains that the innovations of the SCOMDY models are iid. Further generalizations to accom-modate intertemporally correlated innovations or time varying copulas may be of interest. Second, a common practice in the applications of Neyman’s smooth test is to restrict the number of moments to a small number (usually no greater than 4). Ledwina (1994) and Kallenberg and Ledwina (1995) studied a modified Neyman’s test wherein the number of moments is data-driven. Generalization of our tests along this line of thought is expected to lead to further improvements. Finally, Chen et al. (2010) proposes the pseudo-likelihood ratio tests for semiparametric multivariate copula model selection subject to general censor-ship. Our tests can also be generalized in the same direction. We
present smooth copula specification tests for censored data in a separate article.
APPENDIX A: MATHEMATICAL PROOFS
We first list some conditions needed to establish the large sample properties of the proposed tests.
Condition 1. E [l(U1t, . . . , Udt;α)] has a unique maximum α0∈
int(A), whereAis a compact subset ofRp.
Condition 2. The true (unknown) copula functionC(v1, . . . , vd) has continuous partial derivatives.
Condition 3. For any (v1, . . . , vd)∈(0,1)d, l(v1, . . . , vd;
α) is a continuous function ofα; E [supα∈A|l(U1t, . . . , Udt;α)|]<
∞.
Condition 4. B≡ −E [lαα(U1t, . . . , Udt;α0)] is positive definite;
≡var [lα(U1t, . . . , Udt;α0)+ d
j=1Wj(Uj t;α0)] is finite and
positive definite.
Condition 5. Forj=1, . . . , d, lαj(v1, . . . , vd;α0) is well defined
and continuous in (v1, . . . , vd)∈(0,1)d.
Condition 6. (i) lα(v1, . . . , vd;α0) ≤constant×"dj=1
{vj(1−vj)}−aj for someaj≥0 such that
E
⎡
⎣ d #
j=1
{Uj t(1−Uj t)}−2aj ⎤
⎦<∞;
(ii)lαk(v1, . . . , vd;α0) ≤constant× {vk(1−vk)}−bk "d
j=1,j =k{vj(1−vj)}−ajfor somebk> aksuch that
E
⎡
⎣{Ukt(1−Ukt)}ξk−bk d #
j=1,j =k
{Uj t(1−Uj t)−aj} ⎤
⎦<∞
for someξk∈(0,1/2).
Condition 7. For any (v1, . . . , vd)∈(0,1)d, lαα(v1, . . . , vd;
α) is a continuous function of α in a neighborhood of α0;
E
sup α∈A:α−α0=o(1)
lαα(U1t, . . . , Udt;α)
<∞.
The consistence and asymptotic normality of ˆαis established by Genest, Ghoudi, and Rivest (1995) and Chen and Fan (2005). A sketch of the proof is given below.
Proof of Theorem 1.Under Conditions 1–3, the consistence of ˆαis readily obtained according to Proposition 1 of Chen and Fan (2005). Next, under Conditions 1–7, the asymptotic normality of ˆαis given by Proposition 2 of Chen and Fan (2005). Condition 6 allows the score function and its partial derivatives with respect tovj, j =1, . . . , dto blow up at the boundaries. The condition characterizes many popular copula functions such as the Gaussian,tand Clayton copulas.
Condition 8. For any (v1, . . . , vd)∈(0,1)d, g(v1, . . . , vd) has continuous partial derivatives with respect to v1, . . . , vd; g(v1, . . . , vd;α) is a continuous function of α; a continuous function of α in a neighborhood of α0;
sup
Next Theorem 1 indicates that ˆαcan be expressed as an asymptoti-cally linear estimator such that
ˆ where ˜Cis the rescaled empirical copula function given by
˜
Using Proposition A.1(i) of Genest, Ghoudi, and Rivest (1995) or Lemma 1 of Chen and Fan (2005), we then have un-der Conditions 1–3 and 8, Rˆ →p E [ ˆR]≡E [g(v1, . . . , vd;
α0)−GTB−1lα(v1, . . . , vd;α0)]=0. It follows that gˆ( ˆα)→p
E [g(α0)]=0 asn→ ∞.
Next using Proposition A.1(ii) of Genest, Ghoudi, and Rivest (1995) or Lemma 2 of Chen and Fan (2005), we then have under Conditions 1–10,
is then obtained by applying the variance formula (A.5) to (A.7), using also the following result:
∂
→R. We have by the triangle inequality,
) zero by the dominated convergence theorem. The second term, (A.9) vanishes asymptotically as a consequence of Proposition A.1(i) of Genest, Ghoudi, and Rivest (1995) or Lemma 1 of Chen and Fan (2005). It follows that ˆB estimates B consistently, where ˆB and B are defined in (13) and Theorem 1, respectively. Similarly, ˆG estimate G consistently under Condition 10, where
G and ˆG are defined in (15) and (18), respectively. Finally, take H(v1, . . . , vd;α)=ϕtϕtT withϕt given by (17). Using essentially the same argument, one can show that ˆestimatesconsistently under Condition 9. The result of this theorem then follows readily from the asymptotic normality of ˆg( ˆα) given in Theorem 2.
Proof of Theorems 4 and 5.Under the regularity conditions given in the Appendix of Chen and Fan (2006), these theorems are direct results of Propositions 3.1 and 3.2 in the same article. For brevity, the proofs
are not reproduced here.
Table B.1. Configurations of “Singleton” tests
Group Moment functions Notation
Table B.2. Empirical sizes and powers of the tests in the bivariate case (%)
CN Ct4 CF CC CG CAG
Copula underH0 τ Test n=300 500 n=300 500 n=300 500 n=300 500 n=300 500 n=300 500
CN 0.3 QS1 3.9 4.8 3.7 4.2 47.4 77.3 1.4 2.4 3.4 4.4 0.4 0.6
QS2 4.7 5.3 76.2 91.6 6.5 8.0 10.7 14.9 18.3 28.9 18.9 35.0 QS3 4.6 4.6 36.3 58.7 5.3 7.0 18.6 29.2 18.3 25.9 5.7 6.4 QS4 4.6 4.4 5.1 5.7 4.4 5.9 86.4 98.5 36.7 56.5 99.7 100 QS5 4.0 5.1 4.8 6.5 3.8 3.9 58.6 82.1 30.3 43.0 99.0 100
0.6 QS1 5.7 5.3 3.8 4.2 94.0 99.8 24.5 52.7 2.7 1.8 0.1 0.9
QS2 5.4 4.0 32.0 53.2 16.2 19.7 4.1 4.3 6.2 10.8 78.0 95.8 QS3 3.9 4.8 49.1 71.4 19.4 30.2 14.6 28.2 13.8 27.2 96.4 100 QS4 4.6 4.8 4.4 4.0 4.4 4.9 100 100 66.7 86.2 85.1 97.1 QS5 3.8 4.9 3.8 4.2 5.0 4.6 99.7 100 53.7 79.4 94.8 99.4 Ct4 0.3 QS1 63.9 96.0 5.4 4.9 82.7 99.3 10.5 30.6 10.6 28.0 5.6 14.5 QS2 76.1 93.0 5.3 4.8 90.9 99.1 58.1 81.7 40.2 57.5 32.2 47 QS3 35.5 51.7 3.6 4.5 50.9 70.9 7.4 11.7 8.3 10.5 34.4 55.5 QS4 5.0 4.3 5.8 4.1 5.0 5.3 85.9 98.5 39.6 59.2 99.7 100 QS5 4.2 4.2 4.4 5.5 6.0 4.0 58.4 82.1 27.3 47.6 97.0 100
0.6 QS1 59.1 93.6 5.7 4.2 98.7 100 21.2 48.6 10.8 28.8 44.7 69.6
QS2 11.9 8.4 6.6 5.1 77.7 91.4 85.7 95.6 18.6 19.2 9.4 12.6 QS3 36.1 50.0 5.4 4.1 93.0 99.2 46.5 62.5 16.7 21.3 3.1 5.9 QS4 5.4 5.3 5.6 4.3 5.3 5.2 100 100 62.4 85.1 84.5 97.5 QS5 4.7 5.1 3.7 4.7 4.0 5.2 99.7 100 55.0 81.4 94.8 99.6
APPENDIX B: ADDITIONAL SIMULATION RESULTS
In this Appendix, we consider a group of “Singleton” tests with a single moment function for the bivariate case to motivate the selection of moment functions listed inTable 1of Section5. This group contains five single entries of the upper triangular part of a 3 by 3 matrix.ψ1ψ3
is excluded because it does not provide good powers in most of our experiments. The configuration of this group is listed in Table B.1. The simulation results of this group with sample sizen=300,500 are reported inTable B.2. Each experiment is repeated 1000 times.
The test performance varies across the configuration of moment functions, the types of null copula families, the types of true copula families, and the degree of dependence. Generally speaking, the diago-nal entries,ψiψi,i=1,2,3, are helpful to capture the difference in the tail thickness (or tail dependence) and therefore provide good power to distinguish between the radially symmetric copulas. The off-diagonal entries such asψ1ψ2 andψ2ψ3have zero expectations and therefore
do not have discriminant powers against distributions in the radially symmetric family. This conjecture is confirmed by our experiments: tests with a single moment functionψ1ψ2orψ2ψ3are shown to have
powers close to the nominal size. However, they are useful to detect the asymmetry, and therefore contribute significantly to the power to distinguish the radially symmetric copulas from other types of copula distributions. A closer examination of the power performance of the results offers the following interesting insights.
1. Both Gaussian copula and tcopula belong to elliptical families. The tail thickness is the main difference between these two copulas. When the null hypothesis is Gaussian copula and the data are gen-erated from atcopula, the termψ2ψ2contributes most discriminant
power whileψ3ψ3also show good discriminant powers as the
de-gree of dependence increases; when the null hypothesis istcopula and the data are generated from a Gaussian copula,ψ1ψ1contribute
significantly to the power and the termψ3ψ3 also has nontrivial
power.
2. Compared with Gaussian copula, Frank copula has stronger central dependence but weaker tail dependence. When the null hypothesis is Gaussian copula, the term ψ1ψ1 provides good power, which
increases with the degree of dependence substantially. When the
null hypothesis is t copula, all diagonal terms ψiψi, i=1,2,3 contribute significantly to the power.
3. In contrast to tests against elliptical families, the tests against copu-las that are not radially symmetric owe their most significant powers to off-diagonal terms. When the alternative distributions are Clay-ton, Gumble–Hougaard, or asymmetric Gumble–Hougaard copula, terms likeψ1ψ2 and ψ2ψ3 are shown to have consistently high
discriminant powers.
ACKNOWLEDGMENTS
We thank the associate editor, co-editor, and two anonymous referees whose insightful comments and suggestions helped to greatly improve this article. We are grateful to Songnian Chen, Xiaohong Chen, Yi-Ting Chen, Jin-Chuan Duan, and Yanqin Fan for helpful discussions regarding this article. Lin thanks the Risk Management Institute (RMI), National University of Singapore for its support on this research. Wu acknowledges support from the National Social Science Foundation of China (Major Program 13&ZD148).
[Received December 2012. Revised December 2013.]
REFERENCES
Bera, A. K., Ghosh, A., and Xiao, Z. (2012), “A Smooth Test for the Equality of Distributions,”Econometric Theory, 29, 419–446. [128]
Berg, D. (2009), “Copula Goodness-of-Fit Testing: An Overview and Power Comparison,”European Journal of Finance, 15, 675–701. [128]
Chan, N., Chen, J., Chen, X., Fan, Y., and Peng, L. (2009), “Statistical Inference for Multivariate Residual Copula of GARCH Models,”Statistica Sinica, 19, 53–70. [132]
Chen, X., and Fan, Y. (2005), “Pseudo-Likelihood Ratio Tests for Semiparamet-ric Multivariate Copula Model Selection,”Canadian Journal of Statistics, 33, 389–414. [128,131,137,141]
——— (2006), “Estimation and Model Selection of Semiparametric Copula-Based Multivariate Dynamic Models Under Copula Misspecification,” Jour-nal of Econometrics, 135, 125–154. [128,129,132,135,136,137,140,141] Chen, X., Fan, Y., and Patton, A. (2004), “Simple Tests for Models of
Depen-dence Between Multiple Financial Time Series, With Applications to US Equity Returns and Exchange Rates,” London Economics Financial Markets Group Working Paper No. 483. [128,135]
Chen, X., Fan, Y., Pouzo, D., and Ying, Z. (2010), “Estimation and Model Selection of Semiparametric Multivariate Survival Functions Under General Censorship,” Journal of Econometrics, 157, 129– 142. [128,140]
Chen, Y. (2007), “Moment-Based Copula Tests for Financial Returns,”Journal of Business and Economic Statistics, 25, 377–397. [128,130]
Cherubini, U., Luciano, E., and Vecchiato, W. (2004),Copula Methods in Fi-nance, New York: Wiley. [128]
Cherubini, U., Mulinacci, S., Gobbi, F., and Romagnoli, S. (2011),Dynamic Copula Methods in Finance, New York: Wiley. [128]
Cook, R., and Johnson, M. (1981), “A Family of Distributions for Modelling Non-Elliptically Symmetric Multivariate Data,”Journal of the Royal Statis-tical Society,Series B, 43, 210–218. [136]
——— (1986), “Generalized Burr-Pareto-Logistic Distributions With Appli-cations to a Uranium Exploration Data Set,”Technometrics, 28, 123–131. [136]
Denuit, M., Purcaru, O., and Van Keilegom, I. (2004), “Bivariate Archimedean Copula Modelling for Loss-ALAE Data in Non-Life Insurance,” IS Discus-sion Papers 423, Universit´e Catholique de Louvain. [137]
Efron, B., and Tibshirani, R. (1996), “Using Specially Designed Exponential Families for Density Estimation,”The Annals of Statistics, 24, 2431–2461. [128,130]
Fermanian, J. (2005), “Goodness-of-Fit Tests for Copulas,”Journal of Multi-variate Analysis, 95, 119–152. [128]
——— (2012), “An Overview of the Goodness-of-Fit Test Problem for Copu-las,” Working Paper, CREST. [128]
Frees, E., and Valdez, E. (1998), “Understanding Relationships Using Copulas,”
North American Actuarial Journal, 2, 1–25. [137]
Genest, C., Ghoudi, K., and Rivest, L. (1995), “A Semiparametric Estimation Procedure of Dependence Parameters in Multivariate Families of Distribu-tions,”Biometrika, 82, 543–552. [131,137,141]
——— (1998), Comment on “Understanding Relationships Using Copulas,”
North American Actuarial Journal, 2, 143–149. [137]
Genest, C., Quessy, J., and R´emillard, B. (2006), “Goodness-of-Fit Procedures for Copula Models Based on the Probability Integral Transformation,” Scan-dinavian Journal of Statistics, 33, 337–366. [136]
Genest, C., R´emillard, B., and Beaudoin, D. (2009), “Goodness-of-Fit Tests for Copulas: A Review and a Power Study,”Insurance: Mathematics and Economics, 44, 199–213. [128]
Gu, C., and Wang, J. (2003), “Penalized Likelihood Density Estimation: Direct Cross-Validation and Scalable Approximation,”Statistica Sinica, 13, 811– 826. [133]
Janssen, A. (2000), “Global Power Functions of Goodness of Fit Tests,”The Annals of Statistics, 28, 239–253. [128]
Kallenberg, W., and Ledwina, T. (1995), “Consistency and Monte Carlo Simu-lation of Data Driven Version of Smooth Goodness of Fit Tests,”The Annals of Statistics, 23, 1594–1608. [140]
Kallenberg, W. C., and Ledwina, T. (1999), “Data-Driven Rank Tests for Inde-pendence,”Journal of the American Statistical Association, 94, 285–301. [133,134]
Khoudraji, A. (1995), “Contributions `a L’´etude des Copules et `a la Mod´elisation des Valeurs Extrˆemes Bivari´ees,” Ph.D. dissertation, Universit´e Laval, Qu´ebec, Canada. [132]
Klugman, S., and Parsa, R. (1999), “Fitting Bivariate Loss Distributions With Copulas,”Insurance: Mathematics and Economics, 24, 139–148. [137] Kojadinovic, I., Segers, J., and Yan, J. (2011), “Large-Sample Tests of
Extreme-Value Dependence for Multivariate Copulas,”Canadian Journal of Statis-tics, 39, 703–720. [133]
Kojadinovic, I., and Yan, J. (2010), “Modeling Multivariate Distributions With Continuous Margins Using the Copula R Package,”Journal of Statistical Software, 34, 1–20. [136]
——— (2011), “A Goodness-of-Fit Test for Multivariate Multiparameter Copu-las Based on Multiplier Central Limit Theorems,”Statistics and Computing, 21, 17–30. [133,135]
Ledwina, T. (1994), “Data Driven Version of the Neyman Smooth Test of Fit,”
Journal of American Statistical Association, 89, 1000–1005. [140] Li, D., and Peng, L. (2009), “Goodness-of-Fit Test for Tail Copulas Modeled by
Elliptical Copulas,”Statistics and Probability Letters, 79, 1097–1104. [128] Manner, H., and Reznikova, O. (2012), “A Survey of Time-Varying Copulas: Specification, Simulation and App,”Econometric Reviews, 31, 654–687. [128]
McNeil, A., Frey, R., and Embrechts, P. (2005),Quantitative Risk Management: Concepts, Techniques, and Tools, Princeton: Princeton University Press. [128]
Nelsen, R. B. (1993), “Some Concepts of Bivariate Symmetry,” Journal of Nonparametric Statistics, 3, 95–101. [133]
Neyman, J. (1937), “‘Smooth Test’ for Goodness of Fit,”Skandinaviske Aktua-rietidskrift, 20, 150–199. [128,129,133]
Patton, A. (2012), “A Review of Copula Models for Economic Time Series,”
Journal of Multivariate Analysis, 110, 4–18. [128]
Prokhorov, A., and Schmidt, P. (2009), “Likelihood-Based Estimation in a Panel Setting: Robustness, Redundancy and Validity of Copulas,”Journal of Econometrics, 153, 93–104. [128]
Rayner, J., and Best, D. (1990), “Smooth Tests of Goodness of Fit: An Overview,”International Statistical Review, 58, 9–17. [129]
R´emillard, B. (2010), “Goodness-of-Fit Tests for Copulas of Multivariate Time Series,” Working Paper, HEC Montr´eal, Canada. [132]
Sklar, A. (1959), “Fonctions De rpartition n Dimensions et Leurs Marges,”
Publications de l’Institut de Statistique de l’Universit de Paris, 8, 229–231. [129]
Thomas, D. R., and Pierce, D. A. (1979), “Neyman’s Smooth Goodness-of-Fit Test When the Hypothesis is Composite,”Journal of the American Statistical Association, 74, 441–445. [133]
Wu, X. (2010), “Exponential Series Estimator of Multivariate Densities,” Jour-nal of Econometrics, 156, 354–366. [137]
