07350015%2E2014%2E932696
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.
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Date: 11 January 2016, At: 19:19
Smooth Tests of Copula Specifications
Juan LIN
Department of Finance, School of Economics & Wang Yanan Institute for Studies in Economics,
Xiamen University, China ([email protected])
Ximing WU
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Department 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 constructive in the sense that when a null distribution is rejected, the test provides useful pointers to alternative
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 semiparametric 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 management, 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. However, 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 interdependence among extreme events. In addition, the Gaussian
copula is radially symmetric and thus does not allow asymmetric dependence among variables.
It is widely appreciated that specification testings of parametric copula model are of great importance since different parametric 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), Fermanian (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 Fermanian (2012). The majority of tests reviewed in these survey articles are omnibus or “blanket” tests that possess powers
against all alternatives.
Although they are known to be consistent, omnibus tests often only possess satisfactory power against certain deviations
from the null hypothesis and lack power in specific directions
(Janssen 2000). Moreover, they might not have good finite sample power properties (Bera, Ghosh, and Xiao 2012). Recently,
Chen (2007) proposed moment-based copula tests for multivariate dynamic models whose marginals and copulas are both
parametrically specified. Chen (2007) indicated that “although
the moment-based tests have better size performance, they require 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, momentbased 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 conductive to improved copula specifications. Our numerical examples illustrate the usefulness of this approach: the augmented
128
© 2015 American Statistical Association
Journal of Business & Economic Statistics
January 2015, Vol. 33, No. 1
DOI: 10.1080/07350015.2014.932696
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Lin and Wu: Smooth Tests of Copula Specifications
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 dependence such as series correlation or clusters of extreme values.
To accommodate time series data, we further extend our methods 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 asymptotically 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 simulations. 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 Section 2, we first briefly review Neyman’s smooth test of distributions and then present copula smooth tests under simple hypotheses. In Section 3, we present general copula smooth tests
with nonparametric marginal distributions and estimated copula
parameters. In Section 4, we extend the tests to copula-based
multivariate dynamic models. Finite sample performance of the
proposed tests is investigated in Section 5 through Monte Carlo
simulations. Section 6 provides three empirical applications and
the last section briefly concludes. All proofs are relegated to Appendix 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 subscript t to
index observations and subscript j to denote the coordinate of
multivariate random vectors.
129
on [0, 1] given by
ψk (v) =
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
on λ = 0. One can construct a likelihood ratio test on this hypothesis. Alternatively one can use a score test, which is asymptotically equivalent to the likelihood ratio test. The score test
is locally optimal and particularly appealing since it does not
require the estimation of λ. Let ψ = (ψ1 , . . . , ψK )T and
n
1
ψˆ =
ψ(Fy (Yt )).
n t=1
Neyman’s smooth test is constructed as
ˆ
S = nψˆ T ψ.
Under the null hypothesis, S converges in distribution to the χ 2
distribution with K degrees of freedom as n → ∞.
Compared to omnibus tests, smooth tests have attractive finite
sample properties both in terms of size and power (see Rayner
and Best 1990 for a comprehensive review on smooth tests). In
addition, the smooth test has one particular appealing feature. If
the null hypothesis λ = 0 is rejected, it is natural to consider fK
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
2.
COPULA TESTS UNDER SIMPLE HYPOTHESES
In this section, we present smooth tests of copula specifications under the ideal situation of known marginal distributions
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
Let Y1 , . . . , Yn be an iid random sample from an unknown distribution. The hypothesis of interest is to test whether the sample
is generated from a specific distribution Fy . Neyman’s test of
distribution is based on the fact that under the null U1 = Fy (Y1 ),
. . . , Un = Fy (Yn ) are distributed according to the standard uniform distribution. Therefore, the test on a generic distribution
Fy is transformed to a test on uniformity of U1 . . . , Un . Neyman (1937) motivated his smooth test with the following smooth
alternative to the uniform density
K
λk ψk (v) − λ0 , v ∈ [0, 1],
(1)
fK (v) = exp
k=1
where λ0 = log exp( K
k=1 λk ψk (v))dv ensures that f integrates to unity, and ψk ’s are normalized Legendre polynomials
√
2k + 1 d k 2
(v − v)k ,
k!
dv k
Smooth Tests of Copula Specifications: Simple
Hypothesis
To fix this idea, we first present a smooth test of copula specification 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 function F (x1 , . . . , xd ) with continuous marginal distributions
F1 (x1 ), . . . , Fd (xd ), where d ≥ 2. According to the theorem of
Sklar (1959), there exists a unique copula C such that
F (x1 , . . . , xd ) = C(F1 (x1 ), . . . , Fd (xd ))
for all x1 , . . . , xd ∈ R. Suppose C is differentiable, there
exists a corresponding copula density c(v1 , . . . , vd ) ≡
∂ d /∂v1 . . . ∂vd C(v1 , . . . , vd ).
Let c0 (v1 , . . . , vd ; α), where (v1 , . . . , vd ) ∈ [0, 1]d , be a class
of parametric copula density functions characterized by a finite
p-dimensional parameter α. We are interested in the following
simple hypothesis:
H0 : Pr(c(v1 , . . . , vd ) = c0 (v1 , . . . , vd ; α0 )) = 1
against the alternative hypothesis
H1 : Pr(c(v1 , . . . , vd ) = c0 (v1 , . . . , vd ; α0 )) < 1
for some α0 ∈ A, a compact subset of Rp .
(2)
130
Journal of Business & Economic Statistics, January 2015
Let g = (g1 , . . . , gK )T be a series of linearly independent bounded
defined on [0, 1]d . Define
real valued functions
0
µ(g; α) = g(v1 , . . . , vd )c (v1 , . . . , vd ; α)dv and
g(v1 , . . . , vd ; α) = g(v1 , . . . , vd ) − µ(g; α).
The computation of µ in practice is defined in Section 5. Define
Ut = (U1t , . . . , Udt )T with Uj t = Fj (Xj t ), j = 1, . . . , d. Let
n
1
g(U1t , . . . , Udt ; α0 ),
gˆ (α0 ) =
n t=1
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(4)
Q ≡ nˆgT (α0 )[Vˆ (α0 )]−1 gˆ (α0 ),
cg (v1 , . . . , vd ; λ)
λk gk (v1 , . . . , vd ) − λ0 ,
(6)
where
λ0 = log
0
c (v1 , . . . , vd ; α0 ) exp
K
k=1
ˆ
cg (v1 , . . . , vd ; λ)
K
k=1
ˆλk gk (v1 , . . . , vd ) − λˆ 0 , (8)
where λˆ = (λˆ 1 , . . . , λˆ K )T are Lagrangian multipliers associated
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.
(5)
which can be shown to converge in distribution to the χ 2 distribution with K 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 [Vˆ (α0 )]−1 is necessary since g = (g1 , . . . , gK )T
is generally not orthonormal with respect to the null copula
density c0 .
Similar to Neyman’s smooth test of uniformity, our smooth
test of copula specification can be motivated by a smooth alternative. Consider the following density
(7)
The solution takes the form
= c (v1 , . . . , vd ; α0 ) exp
√
p
d
It follows that under the null, gˆ (α0 ) → 0 and nˆg(α0 ) →
N (0, V (α0 )) as n → ∞
under suitable regularity conditions.
Next let Vˆ (α0 ) = n1 nt=1 g(U1t , . . . , Udt ; α0 )g(U1t , . . . , Udt ;
α0 )T . We construct a smooth test for copula specification as
follows:
k=1
gk (v1 , . . . , vd )c(v1 , . . . , vd )dv = gˆ k , k = 1, . . . , K.
0
V (α0 ) = var [g(U1t , . . . , Udt ; α0 )].
= c0 (v1 , . . . , vd ; α0 ) exp
(3)
and
K
subject to
c(v1 , . . . , vd )dv = 1,
λk gk (v1 , . . . , vd ) dv,
in which dv = dv1 . . . dvd . Under the assumption that
(v1 , . . . , vd ) are distributed according to a distribution in the
family of (6), hypothesis (2) is equivalent to that λ = 0. It is
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 associated with g (see Efron and Tibshirani 1996). In particular,
consider the following optimization problem that minimizes the
Kullback–Leibler information criterion between the null copula
density c0 and a generic density c : [0, 1]d → R
c(v1 , . . . , vd )
dv,
min c(v1 , . . . , vd ) log 0
c
c (v1 , . . . , vd ; α0 )
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 some α0 ∈ A
(9)
against the alternative hypothesis
H1 : Pr(c(v1 , . . . , vd ) = c0 (v1 , . . . , vd ; α)) < 1 for all α ∈ A.
One can estimate the marginal distributions using parametric or nonparametric methods. Chen (2007) discussed momentbased tests using parametrically estimated marginal distributions. Although efficient under correct parametric specification,
the test can suffer size distortion and power loss under incorrect 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 consistent. On the other hand, compared with nonparametric estimates such as kernel distribution estimates, the empirical CDFs
are free of smoothing parameters.
Lacking a priori guidance on the marginal distributions, we
estimate Fj by the rescaled empirical distribution function,
n
F˜j (x) =
1
1(Xj t ≤ x), j = 1, . . . , d,
n + 1 t=1
(10)
where 1(·) is the indicator function. Let U˜ t ≡ (U˜ 1t , . . . , U˜ dt )T
with U˜ j t = F˜j (Xj t ), j = 1, . . . , d. We proceed to estimate copula parameters via the maximum likelihood estimation as follows:
n
1
0 ˜
˜
(11)
log c U1t , . . . , Udt ; α .
αˆ = arg max
α
n t=1
Lin and Wu: Smooth Tests of Copula Specifications
131
One can envision that a smooth test on the hypothesis (9) can
be constructed based on
n
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gˆ (α)
ˆ ≡
1 ˜
g(U1t , . . . , U˜ dt ; α),
ˆ
n t=1
(12)
which is analogous to (3) with Uj t replaced by U˜ j t , j = 1, . . . , d
ˆ However, the test (5) for simple hypotheses is not
and α0 by α.
directly applicable here because of the presence of nuisance parameters in (11). There are two sets of nuisance parameters: the
finite-dimensional copula parameter α and the marginal distributions Fj , 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 α,
ˆ which is studied by Genest, Ghoudi, and Rivest (1995) and Chen and Fan (2005). Let
l(v1 , . . . , vd ; α) = log c(v1 , . . . , vd ; α) and use indices α and
j = 1, . . . , d to denote partial derivatives of l with respect to α
and vj , j = 1, . . . , d, respectively. Also define
The asymptotic variance of αˆ can be consistently estimated by
⎧
⎞T ⎫
⎞⎛
⎛
⎪
⎪
d
d
n
⎨1
⎬
−
−
ˆ
ˆ
ˆ
ˆ
⎠
⎠
⎝
⎝
Bˆ − ,
Wj t
Wj t
B +B
⎪
⎪
⎩ n t=1 j =1
⎭
j =1
ˆ
where Bˆ − is the generalized inverse of B.
Let g(j ) (v1 , . . . , vd ) = ∂g(v1 , . . . , vd )/∂vj . Further define
G≡
∂
µ(g; α0 ),
∂α
(15)
Zj (Uj t ) ≡ E [g(j ) (U1s , . . . , Uds ){1(Uj t ≤ Uj s ) − Uj s }|Uj t ].
We obtain the following asymptotic distribution for gˆ (α).
ˆ
Theorem 2. Suppose the regularity conditions given in the
p
Appendices hold. Under the null hypothesis, gˆ (α)
ˆ → 0 and
√
d
nˆg(α)
ˆ → N (0, ), where
d
= var g(U1t , . . . , Udt ; α0 ) +
Zj (Uj t )
j =1
Wj (Uj t ; α) ≡ E[lαj (U1s , . . . , Uds ; α){1(Uj t ≤ Uj s ) − Uj s }|Uj t ].
−GT B −1 {lα (U1t , . . . , Udt ; α0 ) +
We can then show the following:
d
j =1
Wj (Uj t ; α0 )} .
(16)
Theorem 1. Suppose E [l(U1t , . . . , Udt ; α)] has a unique
maximum at α0 . Under the regularity conditions given in
p
the Appendices, the semiparametric estimator αˆ → α0 and
√
d
n(αˆ − α0 ) → N (0, B −1 B −1 ), where
B ≡ −E [lαα (U1t , . . . , Udt ; α0 )],
≡ var [lα (U1t , . . . , Udt ; α0 ) +
d
Remark 1. The asymptotic variance reflects the influence
of nuisance parameters in gˆ (α).
ˆ If the marginal distributions
are known, the terms involving W’s and Z’s can be dropped,
resulting in
= var [g(U1t , . . . , Udt ; α0 ) − GT B −1 {lα (U1t , . . . , Udt ; α0 )}].
Wj (Uj t ; α0 )],
j =1
with lαα being the second derivative of l with respect to α.
Under the null, we have B = E [lα (U1t , . . . , Udt ; α0 )
lα (U1t , . . . , Udt ; α0 )T ] by the information matrix equality. In addition, since the conditional expectation of lα (U1t , . . . , Udt ; α0 )
with respect to either Uj t is zero, lα (U1t , . . . , Udt ; α0 ) is uncorrelated
√ with each Wj (Uj t ; α0 ). Therefore, the asymptotic variance
of n(αˆ − α0 ), B −1 B −1 , can be simplified to
⎤
⎡
d
Wj (Uj t ; α0 )⎦ B −1 .
B −1 + B −1 var ⎣
j =1
Obviously this asymptotic variance is further reduced to B
when the marginal distributions are known.
Next let
−1
n
1
lα (U˜ 1t , . . . , U˜ dt ; α)l
ˆ α (U˜ 1t , . . . , U˜ dt ; α)
ˆ T,
Bˆ =
n t=1
(13)
n
1
Wˆ j t =
lαj (U˜ 1s , . . . , U˜ ds ; α){1(
ˆ
U˜ j t ≤ U˜ j s ) − U˜ j s }.
n s=1,s =t
(14)
When the copula parameter α is 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 = g U˜ 1t , . . . , U˜ dt ; αˆ
d
d
T ˆ−
ˆ
˜
˜
ˆ
Wˆ j t ,
ˆ +
Zj t − G B lα (U1t , . . . , Udt ; α)
+
j =1
j =1
(17)
where
ˆ = ∂ µ(g; α),
G
ˆ
(18)
∂α
n
1
g(j ) (U˜ 1s , . . . , U˜ ds ){1(U˜ j t ≤ U˜ j s ) − U˜ j s }.
Zˆ j t =
n s=1,s =t
It follows that can be estimated consistently by
n
ˆ = 1
(ϕt − ϕ)(ϕ
¯ t − ϕ)
¯ T,
n t=1
(19)
132
Journal of Business & Economic Statistics, January 2015
where ϕ¯ is the sample mean of {ϕt }nt=1 .
We can now construct a smooth test of copula specification
as follows.
Replacing αˆ in Equation (12) with α(
ˆ θˆ ), we have under the
SCOMDY framework
Theorem 3. Under the regularity conditions given in the Appendices, the semiparametric smooth test of copula specification
is given by
(21)
ˆ ≡ nˆg(α)
ˆ −1 gˆ (α).
ˆ
Q
ˆ T
(20)
d
ˆ → χ 2 as n → ∞.
Under the null hypothesis, Q
K
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Similar to the results on simple hypotheses, when the composite 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 realizations of random distributions. 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 multivariate 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 a d-dimensional
time series Xt = (X1t , . . . , Xdt ), t = 1, . . . , T . Denote by Ft−1
the information set that contains information from the past and
from exogenous variables. The model is given by
!
Xt = µt (θ ) + Ht (θ )εt ,
T
1 ˜ ˆ
ˆ − µ(g; α(
ˆ
ˆ θ)).
g U1t (θ ), . . . , U˜ dt (θ)
gˆ (α,
ˆ θˆ ) =
T t=1
An important contribution of Chen and Fan (2006) is the finding
ˆ is not affected by the estithat asymptotic distribution of α(
ˆ θ)
mation of dynamic parameters when the marginal distributions
are estimated by the empirical CDFs. Thanks to this invariance
result, the asymptotic distribution of gˆ (α,
ˆ θˆ ) is readily available.
Theorem 4. Suppose the regularity conditions given in
the Appendices hold. Under the null hypothesis that
p
ˆ θˆ ) → 0 and
C(F1 (ǫ1 ), . . . , Fd (ǫd )) = C0 (U1 , . . . , Ud ; α0 ), gˆ (α,
√
d
T gˆ (α,
ˆ θˆ ) → N (0, ), where is 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
ˆ ≡ T gˆ (α,
ˆ −1 gˆ (α,
Q
ˆ θˆ )T
ˆ θˆ ),
ˆ is defined as (19) with U˜ 1t , . . . , U˜ dt and αˆ replaced by
where
d
˜
ˆ →
ˆ
ˆ and α(
ˆ Under the null hypothesis, Q
U1t (θ ), . . . , U˜ dt (θ)
ˆ θ).
χK2 as n → ∞.
Remark 2. The appealing property that the asymptotic distribution of gˆ (α,
ˆ θˆ ) is not affected by the estimation of dynamic parameters 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´emillard 2010).
where
µt (θ ) = (µ1t (θ ), . . . , µdt (θ ))′ = E[Xt |Ft−1 ]
is the conditional mean of Xt given Ft−1 , and
Ht (θ ) = diag(h1t (θ ), . . . , hdt (θ )),
in which
hj t (θ ) = E[(Xj t − µj t (θ ))2 |Ft−1 ], j = 1, . . . , d
is the conditional variance of Xj t given Ft−1 . We assume that
the model is correctly parameterized up to a finite-dimensional
unknown parameter θ . Moreover, εt = (ε1t , . . . , εdt )T is a sequence 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 of εj t .
Chen and Fan (2006) proposed a three-step estimator αˆ of the
copula parameter α0 . In the first step, the dynamic parameters
are estimated consistently by θˆ , usually via the quasi-maximum
likelihood estimate (QMLE). In the second step, the distributions of the standardized error terms of the marginal distributions
are estimated by their rescaled empirical distributions. Denote
ˆ j = 1, . . . , d the rescaled empirical CDFs thus obby U˜ j t (θ),
tained. In the third step, the copula parameter α(
ˆ θˆ ) are estimated
ˆ . . . , U˜ dt (θˆ )), t = 1, . . . , T .
using the MLE based on (U˜ 1t (θ),
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 and t copula are the most commonly used copulas in practice, we focus on these two copulas as the null copula
distributions. We generate the marginal distributions from the
standard normal distributions and consider six different datagenerating copula distributions, falling into three categories:
1. Radially symmetric copulas: the Gaussian (CN ) copula,
Student’s t copula 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 copula (CAG ). Given the exchangeable Gumbel–Hougaard copula,
an asymmetric version of it can be constructed using Khoudraji’s
device (Khoudraji 1995) as follows,
CAG (v1 , . . . , vd ; λ1 , . . . , λd , α)
= v11−λ1 . . . vd1−λd CG v1λ1 , . . . , vdλd ; α
for all v1 . . . vd ∈ [0, 1]d and an arbitrary choices of λ =
(λ1 , . . . , λd ) ∈ (0, 1)d such that λi = λj for some 1 ≤ i, j ≤ d.
Downloaded by [Universitas Maritim Raja Ali Haji] at 19:19 11 January 2016
Lin and Wu: Smooth Tests of Copula Specifications
133
The dimension d was set to 2 or 3. For the null distributions, we focus on Gaussian and t copulas with exchangeable
correlation matrix and consider two levels of dependence, corresponding to Kendall’s tau at τ = 0.3 and 0.6, respectively. The
parameters for data-generating copulas are calculated by inversion 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) with λ1 ∈ {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), λ was set to (0.2, 0.4, 0.95). For a detailed account of
the various copulas considered in this study and their estimations, 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 general considerations apply here. First, the functions shall be linearly 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) recommended K ≤ 4. Thomas
and Pierce (1979) suggested K = 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, for v ∈ [0, 1],
√
ψ1 (v) = 3(2v − 1),
√
ψ2 (v) = 5(6v 2 − 6v + 1),
√
ψ3 (v) = 7(20v 3 − 30v 2 + 12v − 1).
Our g functions then consist of various tensor products of these
basis functions. Table 1 summarizes the various configurations
considered in this study.
To save place, we use g1 = ψ1 ψ1 to denote the function
g1 (v1 , v2 ) = ψ1 (v1 )ψ1 (v2 ), and all other terms are defined similarly. A few explanations are in order for the configurations
listed in Table 1. Consider the tensor product of (ψ1 , ψ2 , ψ3 ) as
a 3 by 3 matrix with the (i, j )th entry being ψi ψj . The entries
from the upper triangular part of the matrix, except for ψ1 ψ3 ,
are considered. ψ1 ψ3 is excluded because it does not provide
Table 1. Configurations of moment functions
Group
Diagonal
Others
Moment functions
g1
g1
g1
g1
= ψ1 ψ1 , g2 = ψ2 ψ2
= ψ1 ψ1 , g2 = ψ2 ψ2 , g3
= ψ1 ψ1 , g2 = ψ1 ψ2 , g3
= ψ1 ψ1 , g2 = ψ2 ψ2 , g3
ψ3 ψ3
g1 = ψ1 ψ1 , g2 = ψ1 ψ2 , g3
ψ3 ψ3
g1 = ψ1 ψ1 , g2 = ψ1 ψ2 , g3
ψ2 ψ3 , g5 = ψ3 ψ3
= ψ3 ψ3
= ψ2 ψ2
= ψ2 ψ3 , g4 =
Notation
gD1
gD2
gO1
gO2
= ψ2 ψ2 , g4 =
gO3
= ψ2 ψ2 , g4 =
gO4
good powers in most of our experiments. The group “Diagonal” 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 alternative 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 in v1 and v2 , 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 copulas, we select three-dimensional moment functions based on
the two-dimensional functions listed in Table 1 with the restriction that the number of the moment functions K ≥ 2 and
all three variable are present in the moment functions. In particular, without loss of generality the first k moment conditions are functions of (v1 , v2 ), where k = (K + 1)/2 if K is
an odd number and k = K/2 if K is 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 of v1 , v2 , v3 in the moment functions do not matter.
To implement our test, one needs to evaluate nuˆ where µ(g, α)
merically µ(g, α)
ˆ and G,
ˆ = g(v1 , . . . , vd )
ˆ = g(v1 , . . . , vd )cα0 (v1 , . . . , vd ;
c0 (v1 , . . . , vd ; α)dv
ˆ
and G
α)dv,
ˆ
in which cα0 (v1 , . . . , vd ; α)
ˆ is the derivative of
0
c (v1 , . . . , vd ; α)
ˆ with respect to α. We use the Smolyak
cubature for numerical integrations in our simulations. The
quadrature nodes and weights are generated using the function smolyak.quad in R package gss. Smolyak cubatures
are highly accurate for smooth functions. By thinning out the
nodes from the product quadratures, they are suitable for multidimensional integration (Gu and Wang 2003).
In what follows, we refer to the smooth tests with moment
functions gD1 as QD1 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 for d = 2
in Table 2. The results for the multiplier test are reported in the
row signified by Mn . 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, centering about the nominal value of 5%. As expected, both size
and power improve with sample size. As the dependence gets
134
Journal of Business & Economic Statistics, January 2015
Table 2. Empirical sizes and powers of the tests in the bivariate case (%)
CN
Copula under H0
CN
Downloaded by [Universitas Maritim Raja Ali Haji] at 19:19 11 January 2016
CF
CC
CG
CAG
τ
Test
n = 300
500
n = 300
500
n = 300
500
n = 300
500
n = 300
500
n = 300
500
0.3
QD1
QD2
QO1
QO2
QO3
QO4
Mn
QD1
QD2
QO1
QO2
QO3
QO4
Mn
QD1
QD2
QO1
QO2
QO3
QO4
Mn
QD1
QD2
QO1
QO2
QO3
QO4
Mn
4.1
4.2
5.2
4.5
6.9
6.2
4.1
7.6
5.9
6.2
6.5
7.6
6.2
3.8
64.6
66.4
57.9
55.3
53.7
52.6
32.9
39.5
37.5
26.1
32.1
34.4
30.2
28.5
5.2
4.4
4.8
4.4
5.5
4.6
4.1
5.6
5.5
5.8
5.8
6.0
4.0
3.9
92.8
91.9
88.0
82.9
85.8
83.3
52.5
84.1
79.3
75.6
67.6
73.2
66.2
48.8
59.7
64.1
55.7
62.3
57.8
55.9
10.3
22.2
40.0
19.6
37.2
33.9
35.3
7.9
8.3
8.6
6.5
6.2
5.5
6.4
4.0
6.4
7.0
4.8
6.0
6.6
5.4
4.0
87.0
88.3
81.6
84.9
88.5
82.0
13.2
38.6
62.8
33.8
66.1
62.2
60.2
10.7
7.7
7.3
6.3
6.3
4.5
5.6
4.2
5.0
6.2
4.6
5.6
6.0
4.6
4.2
39.7
30.8
35.7
28.2
30.8
26.6
36.0
87.7
83.8
85.3
81.2
83.2
80.8
93.8
78.7
79.9
81.3
82.6
84.5
78.5
72.2
98.8
98.4
97.0
97.4
97.5
97.8
99.4
67.0
59.1
60.8
55.2
55.6
52.6
58.7
99.6
99.6
99.0
99.0
99.3
98.9
99.7
95.0
96.4
97.9
98.1
98.4
96.9
94.6
100
100
100
100
100
100
100
6.0
12.3
76.5
45.2
72.6
74.3
80.6
19.8
25.6
100
99.6
99.9
100
100
44.3
46.9
87.6
63.7
83.4
86.7
87.0
74.4
87.2
99.9
98.9
99.9
100
100
7.1
18.3
94.3
73.4
93.9
95.1
96.4
36.5
53.0
100
100
100
100
100
68.7
65.2
98.3
88.4
97.1
99.4
98.9
92.7
96.4
100
100
100
100
100
8.7
12.5
31.7
30.3
30.6
34.0
40.5
2.7
7.3
48.6
52.4
46.2
57.4
67.0
27.5
27.5
43.9
34.7
43.7
42.4
48.5
13.0
19.8
48.1
38.0
53.7
54.0
69.7
18.0
23.6
55.2
48.0
56.7
62.3
62.5
6.4
16.2
77.7
79.0
77.2
86.8
87.4
42.6
37.6
69.1
55.1
63.4
66.0
72.1
25.6
27.6
76.9
66.2
79.4
84.6
89.8
11.2
9.8
98.4
97.2
96.6
99.9
99.9
65.8
92.8
97.1
100
99.2
100
100
35.4
34.9
99.5
98.4
99.4
100
100
36.3
23.9
81.7
89.8
76.3
90.3
99.5
21.1
18.9
99.9
99.9
100
100
100
91.7
99.9
100
100
100
100
100
55.9
57.9
100
100
100
100
100
63.0
48.3
98.5
100
96.4
99.8
100
0.6
Ct4
Ct4
0.3
0.6
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 generated from a t 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 and t copula are known to be similar.
The “diagonal” group is powerful in distinguishing copula
distributions within the radially symmetric copulas. This result is consistent with Kallenberg and Ledwina (1999), who
suggested that the “diagonal” family is sufficient to distinguish
between dependence and independence for most symmetric distributions occurring in practice. To distinguish the radially symmetric 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 generally recommended.
It is also noted that the proposed tests are significantly superior to the multiplier tests Mn for the important case of testing
null Gaussian copula against the t copula in both the bivariate and trivariate cases. For example, when τ = 0.3, the power
of QO4 test is about 56% for n = 300 and 82% for n = 500
while the power of Mn test is about 10% and 13%, respectively;
when τ = 0.6, the power of QO4 test is about 35% and 60%
for n = 300 and n = 500 while the power of Mn test is about
8% and 11%, respectively. In most other cases, the proposed
tests with multiple moment functions perform comparably to
the multiplier tests Mn .
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. Consequently, 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 encouraging 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 3 reports the simulation
results for the trivariate cases. The overall results are quantitatively similar to those of the bivariate cases, especially the
case of testing null Gaussian copulas against t-copulas, where
our tests outperform the Mn tests by considerable margins. Noticeable exceptions include the cases where the null hypothesis
is the Gaussian copula and the data are generated from the
Frank copula. Take n = 500 as an example. When τ = 0.3, the
power of QO4 test is about 53% for d = 2 and 18% for d = 3.
This exception is not unexpected because the Frank copula with
Lin and Wu: Smooth Tests of Copula Specifications
135
Table 3. Empirical sizes and powers of the tests for d = 3 (%)
Copula under H0
CN
τ
λ
0.3
Downloaded by [Universitas Maritim Raja Ali Haji] at 19:19 11 January 2016
0.6
λ∗
Ct4
0.3
0.6
λ∗
CN
Ct4
CF
Test
statistics n = 300 500 n = 300 500 n = 300 500
QD1
QD2
QO1
QO2
QO3
QO4
Mn
QD1
QD2
QO1
QO2
QO3
QO4
Mn
QD1
QD2
QO1
QO2
QO3
QO4
Mn
QD1
QD2
QO1
QO2
QO3
QO4
Mn
QD1
QD2
QO1
QO2
QO3
QO4
Mn
QD1
QD2
QO1
QO2
QO3
QO4
Mn
7.1
6.6
6.5
6.2
6.4
6.2
3.2
8.4
7.6
7.2
6.8
7.6
6.9
3.4
58.1
74.3
49.1
70.2
49.2
68.8
77.6
32.8
46.3
30.5
36.0
29.1
34.7
60.0
4.3
6.0
5.6
5.9
4.5
5.8
3.2
6.8
5.7
5.9
6.2
5.6
5.8
3.1
82.1
93.6
78.5
92.5
75.3
90.3
96.6
74.5
72.7
69.9
65.2
62.9
64.7
87.8
54.5
66.3
50.5
58.3
52.6
57.4
9.4
15.5
39.5
12.3
35.5
24.9
34.9
7.0
5.7
4.8
6.0
5.6
5.7
5.4
3.4
7.6
8.1
6.1
5.9
5.6
5.9
3.3
80.6
90.8
76.9
86.8
79.2
84.6
15.2
27.9
67.6
24.3
65.0
49.0
61.1
10.4
4.6
4.7
5.5
5.3
5.4
5.3
4.1
7.4
6.7
5.9
4.3
4.9
4.7
3.4
9.1
11.5
9.7
10.9
9.3
12.3
17.1
49.0
62.4
47.4
63.2
50.2
63.0
90.7
81.5
92.7
72.5
89.3
74.7
87.9
77.3
92.2
98.9
89.5
98.8
93.3
98.3
99.6
14.5
16.3
14.3
15.1
9.5
18.2
32.7
83.0
89.7
81.7
86.9
78.9
86.1
99.4
94.2
99.2
93.1
99.3
92.3
98.2
97.0
99.6
100
99.7
100
99.6
100
100
CC
CG
n = 300
500
6.2
14.4
77.7
49.5
73.3
86.7
95.1
10.0
18.7
99.7
99.9
99.9
100
100
8.1
24.9
93.3
75.2
93.7
98.6
99.9
15.0
35.7
100
100
100
100
100
35.9
38.0
91.0
74.1
86.6
93.7
98.4
51.3
81.1
100
99.8
100
100
100
55.2
61.0
99.2
93.3
98.2
99.7
100
73.2
96.2
100
100
100
100
100
CAG
n = 300 500 n = 300
10.3
15.6
28.7
32.0
29.9
44.5
58.5
3.8
8.9
44.6
47.2
42.2
67.0
65.8
24.0
23.8
45.6
42.7
47.7
54.7
34.0
10.1
18.4
53.5
41.2
52.5
67.6
65.8
500
17.0
30.1
51.3
49.2
53.3
70.5
83.7
4.7
14.4
72.8
77.3
71.8
93.4
93.5
96.4
73.3
94.7
99.8
92.1
99.9
95.8
100
94.7
99.6
100
99.3
100
100
41.3
69.5
54.6
100
45.7
99.6
93.1
61.7
87.5
76.3
100
73.0
100
100
35.5
39.7
72.7
67.7
64.1
80.3
59.8
16.0
26.8
81.1
74.1
77.6
91.5
89.3
NOTE: λ∗ = (0.2, 0.4, 0.95).
moderate dependence is known to be similar to the Gaussian
copula (Kojadinovic and Yan 2011).
Next we report simulations on SCOMDY models as in Chen
and Fan (2006). In this time series setup, we only consider twodimensional 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 + 0.1ej2t−1 , j = 1, 2,
where −1 (·) is the inverse of the standard normal distribution. The DGP is such that individual variables are conditionally normal with the same AR(1)-GARCH(1,1) specifications
and the copula is generated according to the Gaussian, t with
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 standardized residuals via the rescaled empirical distributions. In
the third step, we estimate the coefficients of the null copula
136
Journal of Business & Economic Statistics, January 2015
Table 4. Empirical sizes and powers of the tests for SCOMDY models in the bivariate case (%)
τ
0.3
Downloaded by [Universitas Maritim Raja Ali Haji] at 19:19 11 January 2016
0.6
Test
statistics
QD1
QD2
QO1
QO2
QO3
QO4
QD1
QD2
QO1
QO2
QO3
QO4
CN
Ct4
CF
CG
CAG
n = 300
500
n = 300
500
n = 300
500
n = 300
500
n = 300
500
n = 300
500
3.7
3.7
4.1
4.6
3.7
6.5
6.5
4.7
6.2
7.1
7.0
6.4
5.2
4.2
4.9
4.3
4.2
4.9
5.7
5.1
5.7
5.5
5.1
7.1
57.3
61.5
52.0
58.6
53.1
54.9
20.8
34.8
18.7
36.7
33.2
35.6
85.7
86.3
79.2
84.9
83.9
80.6
36.3
63.5
32.0
61.1
61.3
57.3
38.2
30.1
31.4
26.3
28.9
24.3
88.9
83.9
84.6
78.5
83.4
79.5
64.7
59.5
59.8
53.5
55.1
51.4
99.7
99.5
99.2
98.9
98.9
98.5
4.9
10.5
74.1
45.5
69.5
74.8
17.2
27.6
99.9
99.4
99.4
99.9
6.2
19.2
94.6
70.2
94.2
94.3
39.5
53.1
100
100
100
100
9.6
12.5
31.8
29.3
32.4
37.5
3.2
6.3
46.8
45.0
45.3
56.3
18.6
24.6
51.8
48.3
53.4
60.3
5.9
16.3
76.2
76.6
73.8
84.2
12.1
8.4
96.8
96.3
95.6
99.7
62.8
90.0
95.9
99.5
98.9
99.8
19.6
18.2
99.9
99.8
99.8
100
91.5
99.8
99.9
100
100
100
distribution using MLE. The test statistic is then constructed
using the rescaled empirical CDFs and the estimated copula coefficients according to (21). Like the previous experiment, we
set sample size n = 300, 500 and repeat each experiment 1000
times.
We report the simulation results in Table 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 experiments confirm the remarkable finding of Chen and Fan (2006)
that the asymptotic variance of the MLE of the copula parameters is not affected by the estimation of dynamic coefficients
in the SCOMDY models. Thus, our proposed copula specification 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.
CC
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 rejected, we proceed to estimate an alternative copula function
using diagnostic information provided by the test. Although in
our simulations we only consider the Gaussian or t copula 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 1 displays the scatterplots of log concentrations of Cs and Sc and the resulting pseudo-observations.
The graphs suggest a strong positive left-tail dependence between the concentration of Cs and Sc, but little right-tail dependence. 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 coefficient of the Gaussian copula, as defined in Equation (11), is
αˆ = 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 criterion 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
copula CN , the alternative density takes the form
ˆ exp λˆ T g(v1 , . . . , vd ) − λˆ 0
cg (v1 , . . . , vd ) = cN (v1 , . . . , vd ; α)
(22)
with
λˆ 0 = log
cN (v1 , . . . , vd ; α)
ˆ exp λˆ T g(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 subsequently 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
Lin and Wu: Smooth Tests of Copula Specifications
137
Pseudo−observations
Sc
0.0
0.4
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0.2
0.6
0.4
0.8
Sc
1.0
0.6
1.2
0.8
1.4
1.0
Cesium and Scandium on a log scale
1.5
2.0
2.5
0.0
0.2
0.4
Cs
0.6
0.8
1.0
Cs
Figure 1. Scatterplots of cesium versus scandium.
of Wu (2010), incorporating the set of moment functions involved in the test. For instance, we denote by cD1 the alternative
copula density corresponding with test QD1 . It follows that
cD1 (v1 , v2 ) = cN (v1 , v2 ; α)
ˆ exp(λˆ 1 ψ1 (v1 )ψ1 (v2 )
+ λˆ 2 ψ2 (v1 )ψ2 (v2 ) − λˆ 0 ).
All other densities are constructed in a similar manner. To calculate the alternative copula density function, one needs to evaluate
numerically λˆ 0 defined in (23). We use the Smolyak cubature for
numerical integrations. We report the log-likelihood (denoted by
log L) and the Akaike information criterion (AIC) value of each
estimated copula density in Table 6. For comparison, we also
report results on the null distribution. Although cO4 has a higher
log-likelihood, the AIC favors cO1 . Note that cO1 contains three
moment functions (ψ1 ψ1 , ψ1 ψ2 , ψ2 ψ2 ), while cO4 contains two
additional terms ψ2 ψ3 and ψ3 ψ3 . Our results thus suggest that
limited information content of ψ2 ψ3 and ψ3 ψ3 , given that conveyed by (ψ1 ψ1 , ψ1 ψ2 , ψ2 ψ2 ), does not warrant its inclusion in
the model. To formally test whether a model favored by an information 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 2 reports the contour plots of the Gaussian copula density cN with the correlation coefficient ρˆ = 0.3362 (left-hand
panel) and the alternative copula density function cO1 (righthand panel). The Gaussian copula is radially symmetric. In contrast, the alternative copula density captures one salient feature
of the data: the density at the lower-left corner is clearly higher
than that a
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
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Date: 11 January 2016, At: 19:19
Smooth Tests of Copula Specifications
Juan LIN
Department of Finance, School of Economics & Wang Yanan Institute for Studies in Economics,
Xiamen University, China ([email protected])
Ximing WU
Downloaded by [Universitas Maritim Raja Ali Haji] at 19:19 11 January 2016
Department 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 constructive in the sense that when a null distribution is rejected, the test provides useful pointers to alternative
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 semiparametric 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 management, 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. However, 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 interdependence among extreme events. In addition, the Gaussian
copula is radially symmetric and thus does not allow asymmetric dependence among variables.
It is widely appreciated that specification testings of parametric copula model are of great importance since different parametric 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), Fermanian (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 Fermanian (2012). The majority of tests reviewed in these survey articles are omnibus or “blanket” tests that possess powers
against all alternatives.
Although they are known to be consistent, omnibus tests often only possess satisfactory power against certain deviations
from the null hypothesis and lack power in specific directions
(Janssen 2000). Moreover, they might not have good finite sample power properties (Bera, Ghosh, and Xiao 2012). Recently,
Chen (2007) proposed moment-based copula tests for multivariate dynamic models whose marginals and copulas are both
parametrically specified. Chen (2007) indicated that “although
the moment-based tests have better size performance, they require 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, momentbased 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 conductive to improved copula specifications. Our numerical examples illustrate the usefulness of this approach: the augmented
128
© 2015 American Statistical Association
Journal of Business & Economic Statistics
January 2015, Vol. 33, No. 1
DOI: 10.1080/07350015.2014.932696
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Lin and Wu: Smooth Tests of Copula Specifications
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 dependence such as series correlation or clusters of extreme values.
To accommodate time series data, we further extend our methods 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 asymptotically 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 simulations. 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 Section 2, we first briefly review Neyman’s smooth test of distributions and then present copula smooth tests under simple hypotheses. In Section 3, we present general copula smooth tests
with nonparametric marginal distributions and estimated copula
parameters. In Section 4, we extend the tests to copula-based
multivariate dynamic models. Finite sample performance of the
proposed tests is investigated in Section 5 through Monte Carlo
simulations. Section 6 provides three empirical applications and
the last section briefly concludes. All proofs are relegated to Appendix 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 subscript t to
index observations and subscript j to denote the coordinate of
multivariate random vectors.
129
on [0, 1] given by
ψk (v) =
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
on λ = 0. One can construct a likelihood ratio test on this hypothesis. Alternatively one can use a score test, which is asymptotically equivalent to the likelihood ratio test. The score test
is locally optimal and particularly appealing since it does not
require the estimation of λ. Let ψ = (ψ1 , . . . , ψK )T and
n
1
ψˆ =
ψ(Fy (Yt )).
n t=1
Neyman’s smooth test is constructed as
ˆ
S = nψˆ T ψ.
Under the null hypothesis, S converges in distribution to the χ 2
distribution with K degrees of freedom as n → ∞.
Compared to omnibus tests, smooth tests have attractive finite
sample properties both in terms of size and power (see Rayner
and Best 1990 for a comprehensive review on smooth tests). In
addition, the smooth test has one particular appealing feature. If
the null hypothesis λ = 0 is rejected, it is natural to consider fK
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
2.
COPULA TESTS UNDER SIMPLE HYPOTHESES
In this section, we present smooth tests of copula specifications under the ideal situation of known marginal distributions
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
Let Y1 , . . . , Yn be an iid random sample from an unknown distribution. The hypothesis of interest is to test whether the sample
is generated from a specific distribution Fy . Neyman’s test of
distribution is based on the fact that under the null U1 = Fy (Y1 ),
. . . , Un = Fy (Yn ) are distributed according to the standard uniform distribution. Therefore, the test on a generic distribution
Fy is transformed to a test on uniformity of U1 . . . , Un . Neyman (1937) motivated his smooth test with the following smooth
alternative to the uniform density
K
λk ψk (v) − λ0 , v ∈ [0, 1],
(1)
fK (v) = exp
k=1
where λ0 = log exp( K
k=1 λk ψk (v))dv ensures that f integrates to unity, and ψk ’s are normalized Legendre polynomials
√
2k + 1 d k 2
(v − v)k ,
k!
dv k
Smooth Tests of Copula Specifications: Simple
Hypothesis
To fix this idea, we first present a smooth test of copula specification 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 function F (x1 , . . . , xd ) with continuous marginal distributions
F1 (x1 ), . . . , Fd (xd ), where d ≥ 2. According to the theorem of
Sklar (1959), there exists a unique copula C such that
F (x1 , . . . , xd ) = C(F1 (x1 ), . . . , Fd (xd ))
for all x1 , . . . , xd ∈ R. Suppose C is differentiable, there
exists a corresponding copula density c(v1 , . . . , vd ) ≡
∂ d /∂v1 . . . ∂vd C(v1 , . . . , vd ).
Let c0 (v1 , . . . , vd ; α), where (v1 , . . . , vd ) ∈ [0, 1]d , be a class
of parametric copula density functions characterized by a finite
p-dimensional parameter α. We are interested in the following
simple hypothesis:
H0 : Pr(c(v1 , . . . , vd ) = c0 (v1 , . . . , vd ; α0 )) = 1
against the alternative hypothesis
H1 : Pr(c(v1 , . . . , vd ) = c0 (v1 , . . . , vd ; α0 )) < 1
for some α0 ∈ A, a compact subset of Rp .
(2)
130
Journal of Business & Economic Statistics, January 2015
Let g = (g1 , . . . , gK )T be a series of linearly independent bounded
defined on [0, 1]d . Define
real valued functions
0
µ(g; α) = g(v1 , . . . , vd )c (v1 , . . . , vd ; α)dv and
g(v1 , . . . , vd ; α) = g(v1 , . . . , vd ) − µ(g; α).
The computation of µ in practice is defined in Section 5. Define
Ut = (U1t , . . . , Udt )T with Uj t = Fj (Xj t ), j = 1, . . . , d. Let
n
1
g(U1t , . . . , Udt ; α0 ),
gˆ (α0 ) =
n t=1
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(4)
Q ≡ nˆgT (α0 )[Vˆ (α0 )]−1 gˆ (α0 ),
cg (v1 , . . . , vd ; λ)
λk gk (v1 , . . . , vd ) − λ0 ,
(6)
where
λ0 = log
0
c (v1 , . . . , vd ; α0 ) exp
K
k=1
ˆ
cg (v1 , . . . , vd ; λ)
K
k=1
ˆλk gk (v1 , . . . , vd ) − λˆ 0 , (8)
where λˆ = (λˆ 1 , . . . , λˆ K )T are Lagrangian multipliers associated
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.
(5)
which can be shown to converge in distribution to the χ 2 distribution with K 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 [Vˆ (α0 )]−1 is necessary since g = (g1 , . . . , gK )T
is generally not orthonormal with respect to the null copula
density c0 .
Similar to Neyman’s smooth test of uniformity, our smooth
test of copula specification can be motivated by a smooth alternative. Consider the following density
(7)
The solution takes the form
= c (v1 , . . . , vd ; α0 ) exp
√
p
d
It follows that under the null, gˆ (α0 ) → 0 and nˆg(α0 ) →
N (0, V (α0 )) as n → ∞
under suitable regularity conditions.
Next let Vˆ (α0 ) = n1 nt=1 g(U1t , . . . , Udt ; α0 )g(U1t , . . . , Udt ;
α0 )T . We construct a smooth test for copula specification as
follows:
k=1
gk (v1 , . . . , vd )c(v1 , . . . , vd )dv = gˆ k , k = 1, . . . , K.
0
V (α0 ) = var [g(U1t , . . . , Udt ; α0 )].
= c0 (v1 , . . . , vd ; α0 ) exp
(3)
and
K
subject to
c(v1 , . . . , vd )dv = 1,
λk gk (v1 , . . . , vd ) dv,
in which dv = dv1 . . . dvd . Under the assumption that
(v1 , . . . , vd ) are distributed according to a distribution in the
family of (6), hypothesis (2) is equivalent to that λ = 0. It is
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 associated with g (see Efron and Tibshirani 1996). In particular,
consider the following optimization problem that minimizes the
Kullback–Leibler information criterion between the null copula
density c0 and a generic density c : [0, 1]d → R
c(v1 , . . . , vd )
dv,
min c(v1 , . . . , vd ) log 0
c
c (v1 , . . . , vd ; α0 )
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 some α0 ∈ A
(9)
against the alternative hypothesis
H1 : Pr(c(v1 , . . . , vd ) = c0 (v1 , . . . , vd ; α)) < 1 for all α ∈ A.
One can estimate the marginal distributions using parametric or nonparametric methods. Chen (2007) discussed momentbased tests using parametrically estimated marginal distributions. Although efficient under correct parametric specification,
the test can suffer size distortion and power loss under incorrect 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 consistent. On the other hand, compared with nonparametric estimates such as kernel distribution estimates, the empirical CDFs
are free of smoothing parameters.
Lacking a priori guidance on the marginal distributions, we
estimate Fj by the rescaled empirical distribution function,
n
F˜j (x) =
1
1(Xj t ≤ x), j = 1, . . . , d,
n + 1 t=1
(10)
where 1(·) is the indicator function. Let U˜ t ≡ (U˜ 1t , . . . , U˜ dt )T
with U˜ j t = F˜j (Xj t ), j = 1, . . . , d. We proceed to estimate copula parameters via the maximum likelihood estimation as follows:
n
1
0 ˜
˜
(11)
log c U1t , . . . , Udt ; α .
αˆ = arg max
α
n t=1
Lin and Wu: Smooth Tests of Copula Specifications
131
One can envision that a smooth test on the hypothesis (9) can
be constructed based on
n
Downloaded by [Universitas Maritim Raja Ali Haji] at 19:19 11 January 2016
gˆ (α)
ˆ ≡
1 ˜
g(U1t , . . . , U˜ dt ; α),
ˆ
n t=1
(12)
which is analogous to (3) with Uj t replaced by U˜ j t , j = 1, . . . , d
ˆ However, the test (5) for simple hypotheses is not
and α0 by α.
directly applicable here because of the presence of nuisance parameters in (11). There are two sets of nuisance parameters: the
finite-dimensional copula parameter α and the marginal distributions Fj , 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 α,
ˆ which is studied by Genest, Ghoudi, and Rivest (1995) and Chen and Fan (2005). Let
l(v1 , . . . , vd ; α) = log c(v1 , . . . , vd ; α) and use indices α and
j = 1, . . . , d to denote partial derivatives of l with respect to α
and vj , j = 1, . . . , d, respectively. Also define
The asymptotic variance of αˆ can be consistently estimated by
⎧
⎞T ⎫
⎞⎛
⎛
⎪
⎪
d
d
n
⎨1
⎬
−
−
ˆ
ˆ
ˆ
ˆ
⎠
⎠
⎝
⎝
Bˆ − ,
Wj t
Wj t
B +B
⎪
⎪
⎩ n t=1 j =1
⎭
j =1
ˆ
where Bˆ − is the generalized inverse of B.
Let g(j ) (v1 , . . . , vd ) = ∂g(v1 , . . . , vd )/∂vj . Further define
G≡
∂
µ(g; α0 ),
∂α
(15)
Zj (Uj t ) ≡ E [g(j ) (U1s , . . . , Uds ){1(Uj t ≤ Uj s ) − Uj s }|Uj t ].
We obtain the following asymptotic distribution for gˆ (α).
ˆ
Theorem 2. Suppose the regularity conditions given in the
p
Appendices hold. Under the null hypothesis, gˆ (α)
ˆ → 0 and
√
d
nˆg(α)
ˆ → N (0, ), where
d
= var g(U1t , . . . , Udt ; α0 ) +
Zj (Uj t )
j =1
Wj (Uj t ; α) ≡ E[lαj (U1s , . . . , Uds ; α){1(Uj t ≤ Uj s ) − Uj s }|Uj t ].
−GT B −1 {lα (U1t , . . . , Udt ; α0 ) +
We can then show the following:
d
j =1
Wj (Uj t ; α0 )} .
(16)
Theorem 1. Suppose E [l(U1t , . . . , Udt ; α)] has a unique
maximum at α0 . Under the regularity conditions given in
p
the Appendices, the semiparametric estimator αˆ → α0 and
√
d
n(αˆ − α0 ) → N (0, B −1 B −1 ), where
B ≡ −E [lαα (U1t , . . . , Udt ; α0 )],
≡ var [lα (U1t , . . . , Udt ; α0 ) +
d
Remark 1. The asymptotic variance reflects the influence
of nuisance parameters in gˆ (α).
ˆ If the marginal distributions
are known, the terms involving W’s and Z’s can be dropped,
resulting in
= var [g(U1t , . . . , Udt ; α0 ) − GT B −1 {lα (U1t , . . . , Udt ; α0 )}].
Wj (Uj t ; α0 )],
j =1
with lαα being the second derivative of l with respect to α.
Under the null, we have B = E [lα (U1t , . . . , Udt ; α0 )
lα (U1t , . . . , Udt ; α0 )T ] by the information matrix equality. In addition, since the conditional expectation of lα (U1t , . . . , Udt ; α0 )
with respect to either Uj t is zero, lα (U1t , . . . , Udt ; α0 ) is uncorrelated
√ with each Wj (Uj t ; α0 ). Therefore, the asymptotic variance
of n(αˆ − α0 ), B −1 B −1 , can be simplified to
⎤
⎡
d
Wj (Uj t ; α0 )⎦ B −1 .
B −1 + B −1 var ⎣
j =1
Obviously this asymptotic variance is further reduced to B
when the marginal distributions are known.
Next let
−1
n
1
lα (U˜ 1t , . . . , U˜ dt ; α)l
ˆ α (U˜ 1t , . . . , U˜ dt ; α)
ˆ T,
Bˆ =
n t=1
(13)
n
1
Wˆ j t =
lαj (U˜ 1s , . . . , U˜ ds ; α){1(
ˆ
U˜ j t ≤ U˜ j s ) − U˜ j s }.
n s=1,s =t
(14)
When the copula parameter α is 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 = g U˜ 1t , . . . , U˜ dt ; αˆ
d
d
T ˆ−
ˆ
˜
˜
ˆ
Wˆ j t ,
ˆ +
Zj t − G B lα (U1t , . . . , Udt ; α)
+
j =1
j =1
(17)
where
ˆ = ∂ µ(g; α),
G
ˆ
(18)
∂α
n
1
g(j ) (U˜ 1s , . . . , U˜ ds ){1(U˜ j t ≤ U˜ j s ) − U˜ j s }.
Zˆ j t =
n s=1,s =t
It follows that can be estimated consistently by
n
ˆ = 1
(ϕt − ϕ)(ϕ
¯ t − ϕ)
¯ T,
n t=1
(19)
132
Journal of Business & Economic Statistics, January 2015
where ϕ¯ is the sample mean of {ϕt }nt=1 .
We can now construct a smooth test of copula specification
as follows.
Replacing αˆ in Equation (12) with α(
ˆ θˆ ), we have under the
SCOMDY framework
Theorem 3. Under the regularity conditions given in the Appendices, the semiparametric smooth test of copula specification
is given by
(21)
ˆ ≡ nˆg(α)
ˆ −1 gˆ (α).
ˆ
Q
ˆ T
(20)
d
ˆ → χ 2 as n → ∞.
Under the null hypothesis, Q
K
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Similar to the results on simple hypotheses, when the composite 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 realizations of random distributions. 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 multivariate 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 a d-dimensional
time series Xt = (X1t , . . . , Xdt ), t = 1, . . . , T . Denote by Ft−1
the information set that contains information from the past and
from exogenous variables. The model is given by
!
Xt = µt (θ ) + Ht (θ )εt ,
T
1 ˜ ˆ
ˆ − µ(g; α(
ˆ
ˆ θ)).
g U1t (θ ), . . . , U˜ dt (θ)
gˆ (α,
ˆ θˆ ) =
T t=1
An important contribution of Chen and Fan (2006) is the finding
ˆ is not affected by the estithat asymptotic distribution of α(
ˆ θ)
mation of dynamic parameters when the marginal distributions
are estimated by the empirical CDFs. Thanks to this invariance
result, the asymptotic distribution of gˆ (α,
ˆ θˆ ) is readily available.
Theorem 4. Suppose the regularity conditions given in
the Appendices hold. Under the null hypothesis that
p
ˆ θˆ ) → 0 and
C(F1 (ǫ1 ), . . . , Fd (ǫd )) = C0 (U1 , . . . , Ud ; α0 ), gˆ (α,
√
d
T gˆ (α,
ˆ θˆ ) → N (0, ), where is 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
ˆ ≡ T gˆ (α,
ˆ −1 gˆ (α,
Q
ˆ θˆ )T
ˆ θˆ ),
ˆ is defined as (19) with U˜ 1t , . . . , U˜ dt and αˆ replaced by
where
d
˜
ˆ →
ˆ
ˆ and α(
ˆ Under the null hypothesis, Q
U1t (θ ), . . . , U˜ dt (θ)
ˆ θ).
χK2 as n → ∞.
Remark 2. The appealing property that the asymptotic distribution of gˆ (α,
ˆ θˆ ) is not affected by the estimation of dynamic parameters 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´emillard 2010).
where
µt (θ ) = (µ1t (θ ), . . . , µdt (θ ))′ = E[Xt |Ft−1 ]
is the conditional mean of Xt given Ft−1 , and
Ht (θ ) = diag(h1t (θ ), . . . , hdt (θ )),
in which
hj t (θ ) = E[(Xj t − µj t (θ ))2 |Ft−1 ], j = 1, . . . , d
is the conditional variance of Xj t given Ft−1 . We assume that
the model is correctly parameterized up to a finite-dimensional
unknown parameter θ . Moreover, εt = (ε1t , . . . , εdt )T is a sequence 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 of εj t .
Chen and Fan (2006) proposed a three-step estimator αˆ of the
copula parameter α0 . In the first step, the dynamic parameters
are estimated consistently by θˆ , usually via the quasi-maximum
likelihood estimate (QMLE). In the second step, the distributions of the standardized error terms of the marginal distributions
are estimated by their rescaled empirical distributions. Denote
ˆ j = 1, . . . , d the rescaled empirical CDFs thus obby U˜ j t (θ),
tained. In the third step, the copula parameter α(
ˆ θˆ ) are estimated
ˆ . . . , U˜ dt (θˆ )), t = 1, . . . , T .
using the MLE based on (U˜ 1t (θ),
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 and t copula are the most commonly used copulas in practice, we focus on these two copulas as the null copula
distributions. We generate the marginal distributions from the
standard normal distributions and consider six different datagenerating copula distributions, falling into three categories:
1. Radially symmetric copulas: the Gaussian (CN ) copula,
Student’s t copula 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 copula (CAG ). Given the exchangeable Gumbel–Hougaard copula,
an asymmetric version of it can be constructed using Khoudraji’s
device (Khoudraji 1995) as follows,
CAG (v1 , . . . , vd ; λ1 , . . . , λd , α)
= v11−λ1 . . . vd1−λd CG v1λ1 , . . . , vdλd ; α
for all v1 . . . vd ∈ [0, 1]d and an arbitrary choices of λ =
(λ1 , . . . , λd ) ∈ (0, 1)d such that λi = λj for some 1 ≤ i, j ≤ d.
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Lin and Wu: Smooth Tests of Copula Specifications
133
The dimension d was set to 2 or 3. For the null distributions, we focus on Gaussian and t copulas with exchangeable
correlation matrix and consider two levels of dependence, corresponding to Kendall’s tau at τ = 0.3 and 0.6, respectively. The
parameters for data-generating copulas are calculated by inversion 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) with λ1 ∈ {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), λ was set to (0.2, 0.4, 0.95). For a detailed account of
the various copulas considered in this study and their estimations, 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 general considerations apply here. First, the functions shall be linearly 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) recommended K ≤ 4. Thomas
and Pierce (1979) suggested K = 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, for v ∈ [0, 1],
√
ψ1 (v) = 3(2v − 1),
√
ψ2 (v) = 5(6v 2 − 6v + 1),
√
ψ3 (v) = 7(20v 3 − 30v 2 + 12v − 1).
Our g functions then consist of various tensor products of these
basis functions. Table 1 summarizes the various configurations
considered in this study.
To save place, we use g1 = ψ1 ψ1 to denote the function
g1 (v1 , v2 ) = ψ1 (v1 )ψ1 (v2 ), and all other terms are defined similarly. A few explanations are in order for the configurations
listed in Table 1. Consider the tensor product of (ψ1 , ψ2 , ψ3 ) as
a 3 by 3 matrix with the (i, j )th entry being ψi ψj . The entries
from the upper triangular part of the matrix, except for ψ1 ψ3 ,
are considered. ψ1 ψ3 is excluded because it does not provide
Table 1. Configurations of moment functions
Group
Diagonal
Others
Moment functions
g1
g1
g1
g1
= ψ1 ψ1 , g2 = ψ2 ψ2
= ψ1 ψ1 , g2 = ψ2 ψ2 , g3
= ψ1 ψ1 , g2 = ψ1 ψ2 , g3
= ψ1 ψ1 , g2 = ψ2 ψ2 , g3
ψ3 ψ3
g1 = ψ1 ψ1 , g2 = ψ1 ψ2 , g3
ψ3 ψ3
g1 = ψ1 ψ1 , g2 = ψ1 ψ2 , g3
ψ2 ψ3 , g5 = ψ3 ψ3
= ψ3 ψ3
= ψ2 ψ2
= ψ2 ψ3 , g4 =
Notation
gD1
gD2
gO1
gO2
= ψ2 ψ2 , g4 =
gO3
= ψ2 ψ2 , g4 =
gO4
good powers in most of our experiments. The group “Diagonal” 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 alternative 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 in v1 and v2 , 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 copulas, we select three-dimensional moment functions based on
the two-dimensional functions listed in Table 1 with the restriction that the number of the moment functions K ≥ 2 and
all three variable are present in the moment functions. In particular, without loss of generality the first k moment conditions are functions of (v1 , v2 ), where k = (K + 1)/2 if K is
an odd number and k = K/2 if K is 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 of v1 , v2 , v3 in the moment functions do not matter.
To implement our test, one needs to evaluate nuˆ where µ(g, α)
merically µ(g, α)
ˆ and G,
ˆ = g(v1 , . . . , vd )
ˆ = g(v1 , . . . , vd )cα0 (v1 , . . . , vd ;
c0 (v1 , . . . , vd ; α)dv
ˆ
and G
α)dv,
ˆ
in which cα0 (v1 , . . . , vd ; α)
ˆ is the derivative of
0
c (v1 , . . . , vd ; α)
ˆ with respect to α. We use the Smolyak
cubature for numerical integrations in our simulations. The
quadrature nodes and weights are generated using the function smolyak.quad in R package gss. Smolyak cubatures
are highly accurate for smooth functions. By thinning out the
nodes from the product quadratures, they are suitable for multidimensional integration (Gu and Wang 2003).
In what follows, we refer to the smooth tests with moment
functions gD1 as QD1 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 for d = 2
in Table 2. The results for the multiplier test are reported in the
row signified by Mn . 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, centering about the nominal value of 5%. As expected, both size
and power improve with sample size. As the dependence gets
134
Journal of Business & Economic Statistics, January 2015
Table 2. Empirical sizes and powers of the tests in the bivariate case (%)
CN
Copula under H0
CN
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CF
CC
CG
CAG
τ
Test
n = 300
500
n = 300
500
n = 300
500
n = 300
500
n = 300
500
n = 300
500
0.3
QD1
QD2
QO1
QO2
QO3
QO4
Mn
QD1
QD2
QO1
QO2
QO3
QO4
Mn
QD1
QD2
QO1
QO2
QO3
QO4
Mn
QD1
QD2
QO1
QO2
QO3
QO4
Mn
4.1
4.2
5.2
4.5
6.9
6.2
4.1
7.6
5.9
6.2
6.5
7.6
6.2
3.8
64.6
66.4
57.9
55.3
53.7
52.6
32.9
39.5
37.5
26.1
32.1
34.4
30.2
28.5
5.2
4.4
4.8
4.4
5.5
4.6
4.1
5.6
5.5
5.8
5.8
6.0
4.0
3.9
92.8
91.9
88.0
82.9
85.8
83.3
52.5
84.1
79.3
75.6
67.6
73.2
66.2
48.8
59.7
64.1
55.7
62.3
57.8
55.9
10.3
22.2
40.0
19.6
37.2
33.9
35.3
7.9
8.3
8.6
6.5
6.2
5.5
6.4
4.0
6.4
7.0
4.8
6.0
6.6
5.4
4.0
87.0
88.3
81.6
84.9
88.5
82.0
13.2
38.6
62.8
33.8
66.1
62.2
60.2
10.7
7.7
7.3
6.3
6.3
4.5
5.6
4.2
5.0
6.2
4.6
5.6
6.0
4.6
4.2
39.7
30.8
35.7
28.2
30.8
26.6
36.0
87.7
83.8
85.3
81.2
83.2
80.8
93.8
78.7
79.9
81.3
82.6
84.5
78.5
72.2
98.8
98.4
97.0
97.4
97.5
97.8
99.4
67.0
59.1
60.8
55.2
55.6
52.6
58.7
99.6
99.6
99.0
99.0
99.3
98.9
99.7
95.0
96.4
97.9
98.1
98.4
96.9
94.6
100
100
100
100
100
100
100
6.0
12.3
76.5
45.2
72.6
74.3
80.6
19.8
25.6
100
99.6
99.9
100
100
44.3
46.9
87.6
63.7
83.4
86.7
87.0
74.4
87.2
99.9
98.9
99.9
100
100
7.1
18.3
94.3
73.4
93.9
95.1
96.4
36.5
53.0
100
100
100
100
100
68.7
65.2
98.3
88.4
97.1
99.4
98.9
92.7
96.4
100
100
100
100
100
8.7
12.5
31.7
30.3
30.6
34.0
40.5
2.7
7.3
48.6
52.4
46.2
57.4
67.0
27.5
27.5
43.9
34.7
43.7
42.4
48.5
13.0
19.8
48.1
38.0
53.7
54.0
69.7
18.0
23.6
55.2
48.0
56.7
62.3
62.5
6.4
16.2
77.7
79.0
77.2
86.8
87.4
42.6
37.6
69.1
55.1
63.4
66.0
72.1
25.6
27.6
76.9
66.2
79.4
84.6
89.8
11.2
9.8
98.4
97.2
96.6
99.9
99.9
65.8
92.8
97.1
100
99.2
100
100
35.4
34.9
99.5
98.4
99.4
100
100
36.3
23.9
81.7
89.8
76.3
90.3
99.5
21.1
18.9
99.9
99.9
100
100
100
91.7
99.9
100
100
100
100
100
55.9
57.9
100
100
100
100
100
63.0
48.3
98.5
100
96.4
99.8
100
0.6
Ct4
Ct4
0.3
0.6
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 generated from a t 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 and t copula are known to be similar.
The “diagonal” group is powerful in distinguishing copula
distributions within the radially symmetric copulas. This result is consistent with Kallenberg and Ledwina (1999), who
suggested that the “diagonal” family is sufficient to distinguish
between dependence and independence for most symmetric distributions occurring in practice. To distinguish the radially symmetric 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 generally recommended.
It is also noted that the proposed tests are significantly superior to the multiplier tests Mn for the important case of testing
null Gaussian copula against the t copula in both the bivariate and trivariate cases. For example, when τ = 0.3, the power
of QO4 test is about 56% for n = 300 and 82% for n = 500
while the power of Mn test is about 10% and 13%, respectively;
when τ = 0.6, the power of QO4 test is about 35% and 60%
for n = 300 and n = 500 while the power of Mn test is about
8% and 11%, respectively. In most other cases, the proposed
tests with multiple moment functions perform comparably to
the multiplier tests Mn .
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. Consequently, 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 encouraging 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 3 reports the simulation
results for the trivariate cases. The overall results are quantitatively similar to those of the bivariate cases, especially the
case of testing null Gaussian copulas against t-copulas, where
our tests outperform the Mn tests by considerable margins. Noticeable exceptions include the cases where the null hypothesis
is the Gaussian copula and the data are generated from the
Frank copula. Take n = 500 as an example. When τ = 0.3, the
power of QO4 test is about 53% for d = 2 and 18% for d = 3.
This exception is not unexpected because the Frank copula with
Lin and Wu: Smooth Tests of Copula Specifications
135
Table 3. Empirical sizes and powers of the tests for d = 3 (%)
Copula under H0
CN
τ
λ
0.3
Downloaded by [Universitas Maritim Raja Ali Haji] at 19:19 11 January 2016
0.6
λ∗
Ct4
0.3
0.6
λ∗
CN
Ct4
CF
Test
statistics n = 300 500 n = 300 500 n = 300 500
QD1
QD2
QO1
QO2
QO3
QO4
Mn
QD1
QD2
QO1
QO2
QO3
QO4
Mn
QD1
QD2
QO1
QO2
QO3
QO4
Mn
QD1
QD2
QO1
QO2
QO3
QO4
Mn
QD1
QD2
QO1
QO2
QO3
QO4
Mn
QD1
QD2
QO1
QO2
QO3
QO4
Mn
7.1
6.6
6.5
6.2
6.4
6.2
3.2
8.4
7.6
7.2
6.8
7.6
6.9
3.4
58.1
74.3
49.1
70.2
49.2
68.8
77.6
32.8
46.3
30.5
36.0
29.1
34.7
60.0
4.3
6.0
5.6
5.9
4.5
5.8
3.2
6.8
5.7
5.9
6.2
5.6
5.8
3.1
82.1
93.6
78.5
92.5
75.3
90.3
96.6
74.5
72.7
69.9
65.2
62.9
64.7
87.8
54.5
66.3
50.5
58.3
52.6
57.4
9.4
15.5
39.5
12.3
35.5
24.9
34.9
7.0
5.7
4.8
6.0
5.6
5.7
5.4
3.4
7.6
8.1
6.1
5.9
5.6
5.9
3.3
80.6
90.8
76.9
86.8
79.2
84.6
15.2
27.9
67.6
24.3
65.0
49.0
61.1
10.4
4.6
4.7
5.5
5.3
5.4
5.3
4.1
7.4
6.7
5.9
4.3
4.9
4.7
3.4
9.1
11.5
9.7
10.9
9.3
12.3
17.1
49.0
62.4
47.4
63.2
50.2
63.0
90.7
81.5
92.7
72.5
89.3
74.7
87.9
77.3
92.2
98.9
89.5
98.8
93.3
98.3
99.6
14.5
16.3
14.3
15.1
9.5
18.2
32.7
83.0
89.7
81.7
86.9
78.9
86.1
99.4
94.2
99.2
93.1
99.3
92.3
98.2
97.0
99.6
100
99.7
100
99.6
100
100
CC
CG
n = 300
500
6.2
14.4
77.7
49.5
73.3
86.7
95.1
10.0
18.7
99.7
99.9
99.9
100
100
8.1
24.9
93.3
75.2
93.7
98.6
99.9
15.0
35.7
100
100
100
100
100
35.9
38.0
91.0
74.1
86.6
93.7
98.4
51.3
81.1
100
99.8
100
100
100
55.2
61.0
99.2
93.3
98.2
99.7
100
73.2
96.2
100
100
100
100
100
CAG
n = 300 500 n = 300
10.3
15.6
28.7
32.0
29.9
44.5
58.5
3.8
8.9
44.6
47.2
42.2
67.0
65.8
24.0
23.8
45.6
42.7
47.7
54.7
34.0
10.1
18.4
53.5
41.2
52.5
67.6
65.8
500
17.0
30.1
51.3
49.2
53.3
70.5
83.7
4.7
14.4
72.8
77.3
71.8
93.4
93.5
96.4
73.3
94.7
99.8
92.1
99.9
95.8
100
94.7
99.6
100
99.3
100
100
41.3
69.5
54.6
100
45.7
99.6
93.1
61.7
87.5
76.3
100
73.0
100
100
35.5
39.7
72.7
67.7
64.1
80.3
59.8
16.0
26.8
81.1
74.1
77.6
91.5
89.3
NOTE: λ∗ = (0.2, 0.4, 0.95).
moderate dependence is known to be similar to the Gaussian
copula (Kojadinovic and Yan 2011).
Next we report simulations on SCOMDY models as in Chen
and Fan (2006). In this time series setup, we only consider twodimensional 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 + 0.1ej2t−1 , j = 1, 2,
where −1 (·) is the inverse of the standard normal distribution. The DGP is such that individual variables are conditionally normal with the same AR(1)-GARCH(1,1) specifications
and the copula is generated according to the Gaussian, t with
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 standardized residuals via the rescaled empirical distributions. In
the third step, we estimate the coefficients of the null copula
136
Journal of Business & Economic Statistics, January 2015
Table 4. Empirical sizes and powers of the tests for SCOMDY models in the bivariate case (%)
τ
0.3
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0.6
Test
statistics
QD1
QD2
QO1
QO2
QO3
QO4
QD1
QD2
QO1
QO2
QO3
QO4
CN
Ct4
CF
CG
CAG
n = 300
500
n = 300
500
n = 300
500
n = 300
500
n = 300
500
n = 300
500
3.7
3.7
4.1
4.6
3.7
6.5
6.5
4.7
6.2
7.1
7.0
6.4
5.2
4.2
4.9
4.3
4.2
4.9
5.7
5.1
5.7
5.5
5.1
7.1
57.3
61.5
52.0
58.6
53.1
54.9
20.8
34.8
18.7
36.7
33.2
35.6
85.7
86.3
79.2
84.9
83.9
80.6
36.3
63.5
32.0
61.1
61.3
57.3
38.2
30.1
31.4
26.3
28.9
24.3
88.9
83.9
84.6
78.5
83.4
79.5
64.7
59.5
59.8
53.5
55.1
51.4
99.7
99.5
99.2
98.9
98.9
98.5
4.9
10.5
74.1
45.5
69.5
74.8
17.2
27.6
99.9
99.4
99.4
99.9
6.2
19.2
94.6
70.2
94.2
94.3
39.5
53.1
100
100
100
100
9.6
12.5
31.8
29.3
32.4
37.5
3.2
6.3
46.8
45.0
45.3
56.3
18.6
24.6
51.8
48.3
53.4
60.3
5.9
16.3
76.2
76.6
73.8
84.2
12.1
8.4
96.8
96.3
95.6
99.7
62.8
90.0
95.9
99.5
98.9
99.8
19.6
18.2
99.9
99.8
99.8
100
91.5
99.8
99.9
100
100
100
distribution using MLE. The test statistic is then constructed
using the rescaled empirical CDFs and the estimated copula coefficients according to (21). Like the previous experiment, we
set sample size n = 300, 500 and repeat each experiment 1000
times.
We report the simulation results in Table 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 experiments confirm the remarkable finding of Chen and Fan (2006)
that the asymptotic variance of the MLE of the copula parameters is not affected by the estimation of dynamic coefficients
in the SCOMDY models. Thus, our proposed copula specification 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.
CC
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 rejected, we proceed to estimate an alternative copula function
using diagnostic information provided by the test. Although in
our simulations we only consider the Gaussian or t copula 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 1 displays the scatterplots of log concentrations of Cs and Sc and the resulting pseudo-observations.
The graphs suggest a strong positive left-tail dependence between the concentration of Cs and Sc, but little right-tail dependence. 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 coefficient of the Gaussian copula, as defined in Equation (11), is
αˆ = 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 criterion 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
copula CN , the alternative density takes the form
ˆ exp λˆ T g(v1 , . . . , vd ) − λˆ 0
cg (v1 , . . . , vd ) = cN (v1 , . . . , vd ; α)
(22)
with
λˆ 0 = log
cN (v1 , . . . , vd ; α)
ˆ exp λˆ T g(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 subsequently 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
Lin and Wu: Smooth Tests of Copula Specifications
137
Pseudo−observations
Sc
0.0
0.4
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0.2
0.6
0.4
0.8
Sc
1.0
0.6
1.2
0.8
1.4
1.0
Cesium and Scandium on a log scale
1.5
2.0
2.5
0.0
0.2
0.4
Cs
0.6
0.8
1.0
Cs
Figure 1. Scatterplots of cesium versus scandium.
of Wu (2010), incorporating the set of moment functions involved in the test. For instance, we denote by cD1 the alternative
copula density corresponding with test QD1 . It follows that
cD1 (v1 , v2 ) = cN (v1 , v2 ; α)
ˆ exp(λˆ 1 ψ1 (v1 )ψ1 (v2 )
+ λˆ 2 ψ2 (v1 )ψ2 (v2 ) − λˆ 0 ).
All other densities are constructed in a similar manner. To calculate the alternative copula density function, one needs to evaluate
numerically λˆ 0 defined in (23). We use the Smolyak cubature for
numerical integrations. We report the log-likelihood (denoted by
log L) and the Akaike information criterion (AIC) value of each
estimated copula density in Table 6. For comparison, we also
report results on the null distribution. Although cO4 has a higher
log-likelihood, the AIC favors cO1 . Note that cO1 contains three
moment functions (ψ1 ψ1 , ψ1 ψ2 , ψ2 ψ2 ), while cO4 contains two
additional terms ψ2 ψ3 and ψ3 ψ3 . Our results thus suggest that
limited information content of ψ2 ψ3 and ψ3 ψ3 , given that conveyed by (ψ1 ψ1 , ψ1 ψ2 , ψ2 ψ2 ), does not warrant its inclusion in
the model. To formally test whether a model favored by an information 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 2 reports the contour plots of the Gaussian copula density cN with the correlation coefficient ρˆ = 0.3362 (left-hand
panel) and the alternative copula density function cO1 (righthand panel). The Gaussian copula is radially symmetric. In contrast, the alternative copula density captures one salient feature
of the data: the density at the lower-left corner is clearly higher
than that a