Asymptotic properties of the Bernstein density copula estimator for α -mixing data

Journal of Multivariate Analysis 101 (2010) 1–10

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Asymptotic properties of the Bernstein density copula estimator for α -mixing data Taoufik Bouezmarni a,b , Jeroen V.K. Rombouts c,∗ , Abderrahim Taamouti d a

Département de mathématiques et de statistique, Université de Montréal, C.P. 6128, succursale Centre-ville, Montréal, Canada, H3C 3J7

b

Institute of Statistics, Université catholique de Louvain, Canada

c

Institute of Applied Economics at HEC Montréal, CIRANO, CIRPEE, CORE (Université catholique de Louvain), 3000 Cote Sainte Catherine, Montréal (QC), Canada, H3T 2A7 d Departamento de Economía, Universidad Carlos III de Madrid Calle Madrid, 126 28903 Getafe (Madrid), Spain

article

info

Article history: Received 7 September 2008 Available online 11 March 2009 AMS subject classifications: 62G07 62G20 Keywords: Nonparametric estimation Copula Bernstein polynomial α -mixing Asymptotic properties Boundary bias

abstract Copulas are extensively used for dependence modeling. In many cases the data does not reveal how the dependence can be modeled using a particular parametric copula. Nonparametric copulas do not share this problem since they are entirely data based. This paper proposes nonparametric estimation of the density copula for α -mixing data using Bernstein polynomials. We focus only on the dependence structure between stochastic processes, captured by the copula density defined on the unit cube, and not the complete distribution. We study the asymptotic properties of the Bernstein density copula, i.e., we provide the exact asymptotic bias and variance, we establish the uniform strong consistency and the asymptotic normality. An empirical application is considered to illustrate the dependence structure among international stock markets (US and Canada) using the Bernstein density copula estimator. Crown Copyright © 2009 Published by Elsevier Inc. All rights reserved.

1. Introduction The correlation coefficient of Pearson, Kendall’s tau, and Spearman’s rho are widely used to measure the dependence between variables. Despite their simplicity to implement and interpret, they are not able to capture all forms of dependencies. The copula function has the advantage to model completely the dependence among variables. Further, the above coefficients of dependence can be deduced from the copula. Thanks to [1], the copula function is directly linked to the distribution function. Indeed, the distribution function can be controlled by the marginal distributions, which give the information on each component, and the copula that captures the dependence between components. This fact gives rise to a flexible two step modeling approach where in the first step one models the marginal distributions and in the second step one characterizes the dependence using a copula model. There are several ways to estimate copulas. First, the parametric approach imposes a specific model for the copula that is known up to some parameters. We can estimate the parameters using the maximum likelihood or inference function for margins. This approach is widely used in practice because of its simplicity, see [2,3] for textbook details. Examples are [4] in the context of survival data, [5] investigates several parametric copulas to model the dependence in time series data, [6] studies the dependence among financial variables for US company data. Parametric copulas are also used in nonparametric regression, see for example [7].

∗

Corresponding author. E-mail addresses: [email protected] (J.V.K. Rombouts), [email protected] (A. Taamouti).

0047-259X/$ – see front matter Crown Copyright © 2009 Published by Elsevier Inc. All rights reserved. doi:10.1016/j.jmva.2009.02.014

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T. Bouezmarni et al. / Journal of Multivariate Analysis 101 (2010) 1–10

The second way to estimate copulas is the nonparametric approach. The advantage of this approach is its flexibility to adapt to any kind of dependence structure particular to a given dataset. An important contribution is [8] which suggests the multivariate empirical distribution to estimate the copula function. Empirical copulas are for example used in [9] to test for equality between two copulas. Gijbels and Mielniczuk [10] estimate a bivariate copula using smoothing kernel methods. Also, they suggest the reflection method in order to solve the boundary bias problem of the kernel methods. Chen and Huang [11] propose a bivariate estimator based on the local linear estimator, which is consistent everywhere in the support of the copula function. Rödel [12] uses the orthogonal series method. For independent and identically distributed (i.i.d.) data, Sancetta and Satchell [13] develop a Bernstein polynomial estimator of the copula function and find an upper bound of its asymptotic bias and variance. They also establish the asymptotic normality of the Bernstein density copula estimator, however without an explicit expression of the variance. In this paper, we consider α -mixing dependent data and propose nonparametric estimation of the density copula based on Bernstein polynomials. We suppose that the marginal distributions are estimated by their empirical distribution functions. We focus only on the dependence structure between the stochastic processes, captured by the copula density defined on the unit cube, and not the complete distribution. We study the asymptotic properties of the Bernstein density copula, i.e., we provide the exact asymptotic bias and variance, and we establish the uniform strong consistency and the asymptotic normality of Bernstein estimator for the density copula. In an empirical application, we consider two international stock market indices to illustrate the flexibility of the Bernstein copula by comparing it to parametric copulas. Motivated by Weierstrass theorem, Bernstein polynomials are considered by Lorentz [14] who proves that any continuous function can be approximated by Bernstein polynomials. For density functions, estimation using the Bernstein polynomial is suggested by Vitale [15] and with a slight modification by Grawronski and Stadtmüller [16]. Tenbusch [17] investigates the Bernstein estimator for bivariate density functions and Bouezmarni and Rolin [18] prove the consistency of Bernstein estimator for unbounded density functions. Kakizawa [19,20] consider the Bernstein polynomial to estimate density and spectral density functions, respectively. Tenbusch [21] and Brown and Chen [22] propose estimators of the regression functions based on the Bernstein polynomial. In the Bayesian context, Bernstein polynomials are explored by Petrone [23,24], Petrone and Wasserman [25], and Ghosal [26]. This paper is organized as follows. The Bernstein copula estimator is introduced in Section 2. Section 3 provides the asymptotic properties of the Bernstein density copula estimator, that is the asymptotic bias and variance, the uniform strong consistency, and the asymptotic normality for α -mixing dependent data. Section 4 presents an empirical illustration using financial data and Section 5 concludes the paper. 2. Bernstein copula estimator Let X = {(Xi1 , . . . , Xid )0 , i = 1, . . . , n} be a sample of n observations of α -mixing vectors in Rd , with distribution function F and density function f . A sequence is α -mixing of order h if the mixing coefficient α(h) goes to zero as the order h goes to infinity, where

α(h) =

sup A∈F1t (X ),B∈Ft∞ +h (X )

|P (A ∩ B) − P (A)P (B)|,

F1t (X ) and Ft∞ +h (X ) are the σ -field of events generated by {Xl , l ≤ t } and {Xl , l ≥ t + h}, respectively. The concept of α mixing is omnipresent in time series analysis and is less restrictive than β - and ρ -mixing. The following condition requires an α -mixing coefficient with exponential decay. We assume that the process X is α -mixing such that

α(h) ≤ ρ h ,

h ≥ 1,

(1)

for some constant 0 < ρ < 1. According to Sklar [1], the distribution function of X can be expressed via a copula: F (x1 , . . . , xd ) = C (F1 (x1 ), . . . , Fd (xd )),

(2)

where Fj is the marginal distribution function of the random variable X = {X1j , . . . , Xnj } and C is a copula function which captures the dependence in X . Deheuvels [8] uses a nonparametric approach, based on the empirical distribution function, to estimate the distribution copula. Using Bernstein polynomials, to smooth the empirical distribution, Sancetta and Satchell [13] propose the empirical Bernstein copula which is defined as follows: for s = (s1 , . . . , sd ) ∈ [0, 1]d j

Cˆ (s1 , . . . , sd ) =

k X

υ1 =0

...

k X

Fn

v

υ d =0

1

k

,...,

d vd  Y

k

Pυj ,k (sj ),

(3)

j =1

where k is an integer playing the role of the bandwidth parameter, Fn is the empirical distribution function of X , and Pυj ,k (sj ) is the binomial distribution function: Pυj ,k (sj ) =

  k

υj

υj

sj (1 − sj )k−υj .

T. Bouezmarni et al. / Journal of Multivariate Analysis 101 (2010) 1–10

3

Since the theoretical results in this paper are derived on (0, 1)d , one can also use the rescaled empirical distribution function, n defined as n+ F , instead of Fn . If we derive (2) with respect to (x1 , . . . , xd ), we obtain the density function, say f (x1 , . . . , xd ), 1 n of X that can be expressed as follows: f (x1 , . . . , xd ) = (f1 (x1 ) × · · · × fd (xd )) × c (F1 (x1 ), . . . , Fd (xd )) , where fj is the marginal density of random variable X j and c is the copula density. Hence, the estimation of the joint density function can be done by estimating the univariate marginal densities and the copula density function. In this paper, our interest lies in the estimation of the copula density function using Bernstein polynomials. Indeed, if we derive (3) with respect to (s1 , . . . , sd ) we obtain the Bernstein density copula: cˆ (s1 , . . . , sd ) =

n 1X

n i =1

Kk (s, Si )

(4)

where Si = (Fn1 (Xi1 ), . . . , Fnd (Xid )) , Fnj (.) is the empirical distribution function of X j , Kk (s, Si ) = kd

k−1 X

...

υ 1 =0

k−1 X

ASi ,υ

υ d =0

d Y

Pυj ,k−1 (sj ),

j =1

and ASi ,υ = 1{Si ∈Bυ } ,

 with Bυ =

   υd υd + 1 υ1 υ1 + 1 , × ··· × , . k

k

k

k

In what follows, we will denote Pυj ,k−1 (sj ) by pυj (sj ) and the multiple sums υ1 =0 . . . υd =0 by υ . The Bernstein estimator for the density copula function is simple to implement, non-negative, and integrates to one. Sancetta and Satchell [13] give the upper bounds of the bias and variance of the Bernstein copula density estimator for i.i.d observations. In this paper, we provide the asymptotic properties of the Bernstein copula density for α -mixing dependent data. For such data, we give the exact asymptotic bias and variance, we prove the uniform almost sure (a.s.) convergence of the Bernstein density copula, and we establish its asymptotic normality.

Pk

Pk

P

3. Main results This section studies the asymptotic properties of the Bernstein density copula estimator. We first show that the asymptotic bias of the Bernstein density copula estimator has a uniform rate of convergence. Hence, asymptotically there is no boundary bias problem. Second, we provide the exact asymptotic variance of the estimator in the interior region. Finally, we establish the uniform strong consistency and the asymptotic normality of the Bernstein density copula estimator. We start by studying the bias of the Bernstein density copula estimator. The following proposition gives the exact asymptotic bias. To stress that the bandwidth depends on n, we replace k by kn . Proposition 1 (Asymptotic Bias). Suppose that the density copula function c has a continuous second-order derivative. Let cˆ be the Bernstein density copula estimator of c as defined in (4). Then, for s = (s1 , . . . , sd ) ∈ (0, 1)d , if kn → ∞, we have 1 ∗ −1 E(ˆc (s)) = c (s) + k− n γ (s) + o(kn )

where

γ (s) = ∗

d 1X

(

2 j=1

dc (s) dsj

(1 − 2sj ) +

d2 c (s) ds2j

) sj (1 − sj ) .

Proof. Using a second-order Taylor expansion and various sums we have

E(ˆc (s)) − c (s) = kdn

=

kdn

+

+

X Z υ

(c (u) − c (s))du Bυ

l =1

dul

) (ul − sl )du Bυ

υ

l6=m

dul dum

υ

l =1

du2l

pυj (sj )

) (ul − sl )(um − sm )du Bυ

( Z d kdn X X d2 c (s) 2

d Y j =1

( Z d kdn X X d2 c (s) 2

p υj ( s j )

j =1

( Z d X X dc (s) υ

Y d

pυj (sj )

j =1

) (ul − sl ) du 2

Bυ

d Y

d Y j =1

1 pυj (sj ) + o(k− n )

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T. Bouezmarni et al. / Journal of Multivariate Analysis 101 (2010) 1–10

=

( d 1 −1 X dc 2

kn

duj

j =1

(s)(1 − 2sj ) +

d X d2 c j =1

du2j

) 1 (s)sj (1 − sj ) + o(k− n ).

The last equality is obtained by using the mean and variance of the Binomial distribution and the fact that

( Z d kdn X X d2 c (s) 2

υ

l6=m

dul dum

) (ul − sl )(um − sm )du Bυ

d Y

pυj (sj ) = O(k−2 ). 

j =1

We denote gj (x, y) = f(1,j) (x, y) − f1 (x)fj (y),

x, y ∈ RI d

where f(1,j) is the joint density of (X1 , Xj ), for 2 ≤ j ≤ n. Now, we compute the variance of the Bernstein density copula estimator. This is given by the following proposition. Proposition 2 (Asymptotic Variance). Let cˆ be the Bernstein density copula estimator of c as defined in (4). Assume further that d/2 kgj k∞ is bounded. Then, for s ∈ (0, 1)d , under condition (1) and if n−1 kn → 0, we have Var(ˆc (s)) = n−1 knd/2 V (s) + o(n−1 kdn/2 ) where V (s) = (4π )−d/2 Qd

c (s)

1/2 j=1 (sj (1−sj ))

.

Note that the formula of the variance at s = 0 and s = 1 is given by [13]. We see that the variance increases with dimension d and it increases near the boundary because of the term (sj (1 − sj ))1/2 in the denominator of V (s). Proof. We have Var(ˆc (s)) =

1 n

n−1 2X

Var(Kkn (s, Si )) +

n i=1

(1 − n−1 i) Cov Kkn (s, S1 ), Kkn (s, Si+1 )




= I + II . d/2

First, using Billingsley’s inequality and kKkn (s, Si )k∞ = O(kn ), we have


Cov Kk (s, S1 ), Kk (s, Si+1 ) ≤ 4α(i)kKk (s, S1 )k∞ kKk (s, Si+1 )k∞ n n n n ≤ C1 ρ i kdn .

(5)

Second, because kgj k∞ is bounded,

Z
Cov Kk (s, S1 ), Kk (s, Si+1 ) = Kk (s, u)Kk (s, v)gi (u, v)dudv n n n n Z ≤ kgj k∞ Kkn (s, u)Kkn (s, v)dudv ≤ C2 .

(6)

If we combine (5) and (6), then for some sequence Tn we obtain II ≤

≤

2 n 2 n

( ) Tn n X X −1 −1 i d (1 − n i) C2 + (1 − n i) C1 ρ kn i =1

i=Tn +1

( Tn C2 +

n X

C1 kdn

) ρ

i

.

i=Tn +1

ξ

If we choose Tn = kn (0 < ξ < d/2), then II = o(n−1 kdn/2 ). Now, I =

1 n

Var(Kkn (s, Si )) =

X k2d n n

υ

Var(ASi ,υ )

d Y

p2υj (sj ),

j =1

using the variance of Bernoulli random variable, we have Var(ASi ,υ ) = ps,υ1 ,...,υkn − (ps,υ1 ,...,υkn )2

T. Bouezmarni et al. / Journal of Multivariate Analysis 101 (2010) 1–10

5

where υ1 +1

Z ps,υ1 ···υkn ≡

kn

...

υ1

υkn +1

Z

kn

c (u)du.

υkn

kn

kn

From [13], υ

c ( k1 , . . . , n

ps,υ1 ···υkn =

υd kn

)

d + o(k− n ).

kdn

Therefore, υ

Var(ASi ,υ ) = ps,υ1 ···υkn + O(kn

−2d

)=

c ( k1 , . . . , n

kdn

υd kn

)

d + o(k− n ).

Hence 1 n

Var(Kkn (s, Si )) =

=

n

kdn

" X

n

)

kn

n

υ

k2d n

υd

υ

c ( k1 , . . . ,

X k2d n

! + o(kn−d )

d Y

p2υj (sj )

j =1

# (.) +

υ∈I

X

(.)

υ∈I c

≡ V1 + V2 where

 I =

 υj υ = (υ1 , . . . , υd ); − sj < k−δ , j = 1 , . . . , d1 / 3 < δ < 1 / 2 . n k n

We start with the second term V2 . By considering the notations m = sups (c (s)), and when v ∈ I c this means that there

υ exists d0 , for 1 ≤ d0 ≤ d, elements of vj such that k j − sj > k−δ n , we have n

V2 =

υd

υ1

c( k , . . . ,

X k2d n

kn

n

)

kdn

n υ∈I c

!

≤

≤

n m n

Y

d + o(k− n )

p2υj (sj )

j =1

 mkdn

d

 d0

Y 

 d

X

 Y

j=1 υj kn −sj >k−δ n

p2υj (sj ) 

X

j=d0 +1 υj kn −sj
p2υj (sj )



kdn/2 kn−7/2 d0 = o(n−1 kdn/2 ).




For the last inequality, on the one hand, we use

2

 X

X

p2υj (sj ) ≤ 

|υj /kn −sj |>k−δ n

pυj (sj )

|υj /kn −sj |>k−δ n 4 = O(k− n ),

from [14].

On the other hand, from Laplace’s formula we have 1

kn2 p2υj (sj ) Pυj (sj )

→ 1,

as kn → ∞

where 1/2

P υj ( s j ) =

kn

2π sj (1 − sj )

Z

vj +1 kn

vj kn

 exp −

kn sj ( 1 − sj )

(t − sj )

2



dt .

Let vj0 and vj00 be the smallest and the biggest integers such that |υj /k − sj | ≤ k−δ . Using Laplace’s formula and the change of variables we get

6

T. Bouezmarni et al. / Journal of Multivariate Analysis 101 (2010) 1–10

X

Z

1

pυj (sj ) ≈ 2

kn

where υj1 =

2kn sj (1−sj )

υ

j0

kn



− sj and υj2 =

υj2

Z

1

2 π sj (1 − sj )

q

exp −

kn

−1/2

= p



kn

υj0

2π sj (1 − sj )

|υj /kn −sj |≤k−δ n

υj00

√

2π

q

υj1

kn sj (1 − sj )

exp −y2 /2 dy




υ

2kn sj (1−sj )

 (t − sj )2 dt

 − sj . Note that, when kn → ∞, then j1 → −∞ and j2 → +∞,

j00

kn

because sj − vj0 /k ≤ k−δ , vj00 /k − sj ≤ k−δ , and δ < 1/2. Thus, 1/2 p2υj (sj ) = O(k− ), n

X

as kn → ∞.

|υj /kn −sj |≤k−δ n

Now, for V1 V1 =

=

X k2d n

υ

c ( k1 , . . . , n

n

c (s)

kn

)

kdn

n υ∈I kdn

υd

d X Y j=1 υ∈I

= n−1 kdn/2 (4π )−d/2

! + o(kn ) −d

d Y

p2υj (sj )

j =1

! + o(n−1 kdn/2 ),

pυj (sj ) 2

c (s) d Q

because sj ≈

υj kn

+ o(n−1 kdn/2 ). 

(sj (1 − sj ))1/2

j=1

For α -mixing dependent data, the uniform almost sure convergence of the Bernstein density copula estimator is stated in the following proposition. Proposition 3 (Uniform a.s. Convergence). Suppose that the density copula function c has continuous second-order derivatives and that {Si } is an α -mixing sequence with coefficient α(h) = O(ρ h ), for some 0 < ρ < 1. Let cˆ be the Bernstein density copula d/4 estimator of c as defined in (4). Then, if kn → ∞ such that n−1/2 kn ln(n) → 0, we have for any compact I in (0, 1)d , 1 −1/2 d/4 sup |ˆc (s) − c (s)| = O(k− kn ln(n)), n +n

a.s.

s∈I

Proof. From the bias term and under the assumption that c has a continuous second-order derivative, then bounded on the compact I, we have 1 sup |E(ˆc (s)) − c (s)| = O(k− n ). s∈I

If we denote Yn,s,i =

1 n

{Kkn (s, Si ) − E(Kkn (s, Si ))},

then we can see that, for all i, E|Yn,s,i |2 = Var(Yn,s,i ) = Hence,

1 Var n2

d/2

(Kkn (s, Si )) = 1n Var(ˆc (s)) = O(n−2 kn ) [see Proposition 2].

R2 (n, s) = sup E|Yn2,s,i |1/2 = O(n−1 kdn/4 ). i

d/2

Ck Based on Stirling’s formula, Tenbusch [27] establishes that |Yn,s,i | ≤ n√s(n1−s) := S (n, s), for some positive constant C . Using d/2 an inequality of Bernstein type, [28] states that, if N (n)S (n) = O(n−1 kn ln(n)) = o(1), where N (n) is the integer part of −1 3| ln(ρ) ln(n)|), we have

|ˆc (s) − E(ˆc (s))| = O(n1/2 R2 (n, s) ln(n) + ln2 (n)S (n, s)). Therefore, under the assumption on the bandwidth parameter, the fact that the bounds of R2 (n, s) and |Yn,s,i | are uniform d/4

on a compact set in (0, 1)d and that ln2 (n)S (n, s) = o(n−1/2 kn sup |ˆc (s) − E(ˆc (s))| = O(n s∈I

which concludes the proof.



−1/2 d/4

kn

ln(n)),

ln(n)) we obtain

T. Bouezmarni et al. / Journal of Multivariate Analysis 101 (2010) 1–10

7

The next proposition establishes the asymptotic normality of the Bernstein density copula estimator for α -mixing dependent data. This result can be applied in many contexts. We can use it for example to build copula-based tests of goodness-fit, see for example [29], and conditional independence. Proposition 4 (Asymptotic Normality). Suppose that the density copula function c is twice differentiable and that {Si } is an α mixing sequence with coefficient α(h) = O(ρ h ), for some 0 < ρ < 1. Let cˆ be the Bernstein density copula estimator of c as defined in (4). Then, if kn → ∞ such that kn = O(n2/(4+d) ), for s ∈ (0, 1)d , we have d/4 n1/2 k− n

1 ∗ cˆ (s) − c (s) − k− n γ (s)

√

V (s)

→ N (0, 1).

Remark that if we choose kn = O(n2/(4+d) ), then the bias term disappears. Proof. Based on Proposition 1, we need to show that d/4 n1/2 k− n



cˆ (s) − E(ˆc (s))



→ N (0, 1),

√

V (s)

for s ∈ (0, 1)d .

(7)

If we denote Yi,s =

Kkn (s, Si ) − E(Kkn (s, Si ))

√

V (s)

then



d/4 n1/2 k− n

cˆ (s) − E(ˆc (s))



√

V (s)

d/4 = n−1/2 k− n

n X

Yi,s ≡ n−1/2 In.s .

i=1

We follow Doob’s method to show the asymptotic normality for dependent random vectors, see [30]. We consider the −d/4 −d/4 variables Vi,s = kn (Y(i−1)(p+q)+1,s + · · · + Yip+(i−1)q,s ) and Vi∗,s = kn (Yip+(i−1)q+1,s + · · · + Yi(p+q),s ). For r (p + q) ≤ n ≤ r (p + q + 1), In.s =

r X

Vi,s +

r X

i=1

d/4 Vi∗,s + k− n

i =1

n X

Yi,s .

(8)

i=r (p+q)

We can show that n

−1/2

r X

Vi∗,s

−d/4

+ kn

n X

! Yi,s

P

−→ 0.

i=r (p+q)

i=1

Indeed, if we choose r ∼ na , p ∼ n1−a , and q ∼ nc , where 0 < a < 1 and 0 < c < 1 − a, we get n

−1

Var

r X

! Vi∗,s

= O(n

a+c −1

),

and n

−1 −d/2

kn

Var

n X

! Yi,s

= O(na−1 ).

i=r (p+q)

i=1

The two last terms on the right-hand side of (8) are asymptotically negligible. j Now, we show that Vi,s are asymptotically mutually independent. Let Ui,s = exp(itVi,s ) which is Fi -measurable, where i = (i − 1)(p + q) + 1 and j = ip + (i − 1)q, hence from [31]

!! r r X Y Vi,s − E(exp(itVi,s )) ≤ 16(r − 1)α(q + 1). E exp it i=1 i=1 Lastly, we employ Lyapunov’s theorem for the asymptotic normality of n−1/2 r P

E(|Vi,s |3 )

i=1

(r var(V1,s ))3/2

Pn

i =1

≤ kVi,s k∞ (r var(V1,s ))−1/2 d/4 ≤ p k− kKkn (s, t )k∞ (r var(V1,s ))−1/2 n d+2

= O(n d+4 −a ) = o(1) because kKkn (s, Xi )k∞ = O(kd/2 ). 

2 , we obtain, Vi,s . If we choose a > dd+ +4

8

T. Bouezmarni et al. / Journal of Multivariate Analysis 101 (2010) 1–10

(a) US equity returns sample path.

(b) Canada equity returns sample path.

Fig. 1. US and Canada equity returns (MSCI Equity Indices). The sample covers the period from October 16th 1984 to December 21st 2004 for a total of 1054 observations.

(a) Scatter plot (original data).

(b) Scatter plot (empirical distributions). Fig. 2. Dependence between the US and Canadian stock markets.

4. Empirical illustration We are interested in modeling the dependence structure among international stock markets (US and Canada). The data consists of weekly observations on MSCI equity indices series for US and Canada. The sample goes from October 16th 1984 to December 21st 2004 for a total of 1054 observations. The returns are computed using the standard continuous compounding formula. All returns are derived on a weekly basis from daily prices expressed in US dollars. Both US and Canada return time series are presented in Fig. 1. The time series plots of weekly returns show the familiar volatility clustering effect, along with some occasional large absolute returns. To investigate if the return time series are α -mixing, we fit an autoregressive model for the conditional mean and GARCH model of [32] for the conditional variance. Using the results from [33], the estimated parameters imply α -mixing processes for both time series. To illustrate the dependence between both return series we present in panel (a) of Fig. 2 the scatter plot of the data. We observe a relative high association between the US and Canadian stock markets, which is in fact not surprising since both countries have integrated economies and their stock markets react to similar shocks. Furthermore, there seems to be some tail dependence in the data as can be seen by the fact that both stock markets substantially decreased at the October 1987 crash. In panel (b) of Fig. 2, we display the scatter plot of the US and Canadian empirical distribution functions of returns. These are the data that actually enter the Bernstein density copula estimate in (4). We estimate the Bernstein density copula on a grid of points defined on the unit square. Fig. 3 displays the copula for a bandwidth parameter k equal to 30. We investigated several values for k between 10 (very smooth) and 100 (very unsmooth) and found 30 to be a reasonable balance. The advantage of the Bernstein density copula is its flexibility to adapt to any kind of dependence structure particular to a given dataset. As a comparison, we also estimate the Gaussian and the Gumbel density copula. They are parametric copulas and their densities are defined respectively as follows 1 C ( u1 , u2 ) = √ exp 1 − α2



  2  −(w12 − 2αw1 w2 + w22 ) w1 + w22 exp , 2(1 − α 2 ) 2

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Fig. 3. Bernstein density copula estimate for dependence between the US and Canadian stock markets.

(a) Gaussian copula.

(b) Gumbel copula.

Fig. 4. Dependence structure between the US and Canadian stock markets implied by parametric copulas (Gaussian and the Gumbel density copula).

and exp −(v1 + v2 )1/β {ln(u1 ) ln(u2 )}α−1 {v1 + v2 }1/β + β − 1




C ( u1 , u2 ) =

u1 u2




−{v1 + v2 }2−1/β




,

where αˆ = 0.713 is the estimated correlation coefficient, βˆ = (1 − τn )−1 , with τn = 0.485 the sample Kendall’s tau, wi = Φ −1 (ui ), vi = (− ln(ui ))β and Φ −1 ( ) is the inverse of the normal distribution function. Both copulas are presented in panels (a) and (b) of Fig. 4. The graphs suggest that both the parametric copulas are not able to represent the dependence structure we observe in panel (b) of Fig. 2. For example, we observe that there is a density mass around (1, 0) boundary which is completely ignored by these parametric copulas since they put most of the weight on (0, 0) and (1, 1) boundaries as implied by their estimated parameters. This may lead to severe consequences in terms of international portfolio risk evaluation. The relationships among different stock markets are known to be complex and need to be modeled by flexible copulas such as the Bernstein copula. 5. Conclusion A nonparametric Bernstein polynomial-based estimator of density copula for α -mixing data is provided. The proposed estimator can be applied in several contexts, and we can use it to build copula-based tests of, for example, goodness-fit and conditional independence, see [34]. We provide the exact asymptotic bias and variance of the Bernstein copula density estimator and we establish its uniform strong consistency, and the asymptotic normality. Our results can be extended to the right censored data using smoothed Kaplan–Meier estimator instead of the empirical distribution function. A bandwidth choice in practice remains an open question and existing methods like cross-validation can be investigated. References [1] [2] [3] [4] [5]

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