關聯結構下橢球形多變量模式之貝式推論

行政院國家科學委員會專題研究計畫成果報告

關聯結構下橢球型多變量模型之貝氏推論(1/2)

Bayesian Inference for coupla-based elliptic multivariate models

計畫編號： NSC98－2118－M－006－002-MY2

執行期限：98 年 8 月 1 日至 100 年 7 月 31 日

主持人：任眉眉 國立成功大學統計系

計畫參與人員：詹嘉豪 國立成功大學統計系

一、中文摘要

本研究主要考慮 Fang et al. (2002)所提出的 橢球形分配中有關相關性參數的估計問題，

Genest et al.(1995)曾提出半參數估計多變量 族群的相關性，在本研究中我們提出以貝氏 方法來估計具有相同相關性橢球形分配 (equi-correlated meta-elliptical distribution)的

相關性參數估計問題，我們將採用的轉換

Beta 先驗分配，並且採用兩種多變量上的測

量d維連續隨機變數關聯結構下的兩個測度

multivariate Kendall's tau (_d,C) and

multivariate Blomqvist's beta (_d,C)，來做成 貝氏估計。我們將採用 Markov chain Monte Carlo (MCMC) 抽樣方法來計算，在平方誤 差損失函數下，上述二測度的事後平均值。

同時我們也考慮採用不同d-dimensional t _ copulas 的資料配適模擬結果，進而針對 S&P 500 and Dow Jones 兩資料進行實證分析。

關鍵詞：橢球形分配, 橢球關聯結構函數, MCMC, Kendall's tau (_d,C), Blomqvist's beta (_d,C).

Abstract

Consider the dependence parameter estimation for the meta-elliptical distributions introduced

by Fang et al. (2002). Genest et al.(1995) proposed a semi-parametric estimation procedure of the dependence parameters in a family of multivariate distributions. In this study, we will propose a Bayesian approach to estimate the dependence parameter in an equi-correlated meta-elliptical distribution. A transformed beta prior distribution for  is suggested and employ two multivariate versions of measure of association for

d-variate continuous random variables with the associated copula C, multivariate Kendall's tau (_d,C) and multivariate Blomqvist's beta (_d,C), to obtain values of parameters in the transformed beta prior distribution by our proposed empirical Bayes method. Then, we use the Markov chain Monte Carlo (MCMC) sampling scheme to compute the posterior mean under the squared-error loss.

Furthermore, we will give the simulation results for different correlation coefficient in

d-dimensional t copulas and provide the _ empirical results for the index datasets of S&P 500 and Dow Jones.

Keywords: meta-elliptical distribution, elliptical copula, Kendall's tau, Blomqvist's beta, empirical Bayes method.

二、主要結果已寫成論文，投稿中，見附件。

Multivariate Meta-elliptical Distributions: a Bayesian

Approach Based on Copulas

Mei-Mei Zen and Chia-Hao Chan

Department of Statistics, National Cheng-Kung University

Abstract: On the subject of the dependence parameter estimation for meta-elliptical distributions introduced by Fang et al. (2002), Genest et al. (1995) had proposed a semiparametric estimation procedure to estimate the dependence parameters in a fam- ily of multivariate distributions. In this paper, we propose a Bayesian approach to estimate the dependence parameter in an equicorrelated meta-elliptical distribution.

We suggest a transformed beta prior distribution for the correlation coefficient in the associated equicorrelated elliptical copula, and the employment of two multivariate versions for the measure of association of the d-variate continuous random variables with the associated copula C, the multivariate Kendall’s tau (τd,C) and the multivari- ate Blomqvist’s beta (βd,C), in order to obtain the values of the parameters in the transformed beta prior distribution with our proposed empirical Bayes method. Then, we use the Markov chain Monte Carlo (MCMC) sampling scheme to compute the pos- terior mean under the squared-error loss. Further, we prent the simulation results for different correlation coefficients in d-dimensional tν copulas and provide the empirical results for the index datasets of the S&P 500 and the Dow Jones.

Key words and phrases: meta-elliptical distribution, elliptical copula, Kendall’s tau, Blomqvist’s beta, empirical Bayes method.

1 Introduction

Let X1, . . . , Xd be random variables with marginal distributions F1, . . . , Fd, respec- tively, and a joint distribution F . A d-dimensional copula C is any distribution function restricted to [0, 1]^d having a standard uniform margin; that is

C(u1, . . . , ud) = P (U1 ≤ u¹, . . . , Ud≤ u^d),

where Ui is U(0, 1) for i = 1, . . . , d. From Theorem 2.10.11 in Nelsen (2006), there exists a copula C satisfying

F (x1, . . . , xd) = C (F1(x1), . . . , Fd(xd)) .

If F1, . . . , Fd are all continuous, C is unique. The copula C associated with a joint distribution F is used to describe the dependence structure among random variables X1, . . . , Xd; see Joe (1997) and Nelsen (2006) for details. In recent times, copulas have frequently been applied in the fields of econometrics and finance, and especially in risk management. Moreover, Fang et al. (2002) introduced meta-elliptical distributions to fit the distribution of risk factors in a complex financial system; some related depen- dence properties of meta-elliptical distributions were provided in Abdous et al. (2005).

In general, it is to use copulas to model a portfolio of d assets or risks with marginal distributions. Thus, parameter estimation plays an important role while dealing with Value-at-Risk (VaR). The estimated parameters of a copula-based model are separated into two parts: the marginal parameters and the dependence parameters of the copula C.

The estimation of parameters usually employs the maximum likelihood (ML) principle- a technique that involves enormous computation time. Hence, Joe and Xu (1996) pro- posed a two-stage estimation procedure, calling it the Inference Functions for Margins (IFM) method. The first stage involves the computation of the maximum likelihood for each univariate margin, and the second stage calculates the maximum likelihood of the dependence parameters with the univariate parameters held fixed from the first stage;

see Joe (1997) for details. In addition, for estimating dependence parameters only, Genest et al. (1995) proposed a semiparametric estimation procedure called the Canon- ical Maximum Likelihood (CML) method. The CML method is used to estimate the marginal distribution by the empirical distribution as non-parametric estimator sub- stituted in pseudo log-likelihood function and then solving the pseudo log-likelihood function to obtain the dependence estimators. Using the asymptotic covariance ma- trix, Xu (1996), Joe (2005) and Zen and Chan (2007a) studied the asymptotic relative efficiency of the IFM estimator (IFME) w.r.t. the ML estimator (MLE) for their corre- sponding multivariate copula-based models. For estimating the copula parameter, Kim et al. (2007) showed, using an extensive simulation study, that both the ML and IFM

methods were vulnerable against specification errors within the marginal distributions, and that the CML method performed better than both the ML and IFM methods.

As mentioned above, the CML method is commonly used to estimate the copula dependence parameters; in addition, the development of semiparametric methodology can be seen, for example, in Tsukahara (2005), Chen and Fan (2006), Chen et al.

(2006), and Hoff (2007). In this paper, however, we would like to estimate the de- pendence parameters of an elliptical copula by using a Bayesian approach. Pitt et al.

(2006) provided a general Bayesian approach for estimating a Gaussian copula model that can handle any combination of discrete and continuous margins. Lambert (2007) used a flexible B-splines specification for the generation of Archimedean copulas and estimated the associated parameters in a Bayesian framework. In order to model de- pendence in the field of operational risks, Valle (2008) provided and used an Inverse Wishart distribution as the prior for the correlation matrices of the Gaussian and tν

copulas to estimate the parameters.

Using the probabilities of concordance and exceedance, Press and Gokhale (1982) carried out an experiment to assess a prior distribution for the correlation coefficient in a bivariate normal distribution. Based on a similar technique, in this paper, we employ two multivariate versions for the measure of association of the d-variate continuous random variables with the associated copula C–the multivariate Kendall’s tau (τd,C) and the multivariate Blomqvist’s beta (βd,C)–to obtain the parameters in a suggested prior distribution using our proposed empirical Bayes method for the equicorrelated meta-elliptical copula models. Moreover, we prove that both the multivariate Kendall’s tau and multivariate Blomqvist’s beta are equal for an elliptical copula.

The paper is organized as follows. In section 2, we briefly introduce the elliptical copulas and the meta-elliptical distributions. Then we introduce the two measures of association of the multivariate continuous random variables and provide a functional relationship between the correlation coefficients and each of the measures of association.

In section 3, we suggest a transformed beta prior distribution for the correlation coeffi- cient of an exchangeable correlation matrix in an elliptical copula. Further, we describe our proposed empirical Bayes method to obtain the estimators in the transformed beta prior distribution and use the Markov chain Monte Carlo (MCMC) sampling scheme

to obtain the posterior mean under the squared-error loss. In section 4, we use our proposed Bayesian approach on the simulation data generated from tν copulas and assume that margins are unknown. In section 5, we apply our proposed approach on index datasets of the S&P 500 and the Dow Jones. Finally, we outline the conclusions.

2 Elliptical Copulas and Dependence

2.1 Elliptical Copulas

As proposed by Fang et al. (2002), a d-dimensional random vector Z = (Z1, . . . , Zd)^′ is said to have an elliptically contoured distribution with parameters µ and Σ with notation Z ∼ EC^d(µ, Σ, g) if it has the stochastic representation

Z = µ + rAU,^d

where r ≥ 0 is a random variable, U is uniformly distributed on the unit sphere in R^d and is independent of r, AA^′ = Σ is the Cholesky decomposition of Σ, and the sign=^d means that both sides of the equality have the same distribution. In particular, if r has a density, then the density of Z is of the form

|Σ|⁻¹²g (z − µ)^′Σ⁻¹(z − µ)
,

where g(·) is a scale function uniquely determined by the distribution of r. See Fang and Anderson (1990) and Fang et al. (1990) for details.

Without the loss of generality, in an approach similar to Fang et al. (2002), con- sidering the random vector Z = (Z1, . . . , Zd)^′ ∼ EC^d(0, R, g), where R = {r^ij, rii = 1, −1 < r^ij < 1 for i 6= j, r^ij = rji; i, j = 1, . . . , d}, and all margins of Z are identical, the probability density function (pdf) is

q_g(z_i) = π^(d−1)/2 Γ((d − 1)/2)

Z ^∞

z²_i y − zi²

(d−1)/2−1

g(y)dy

and the cumulative distribution function (cdf) Qg(zi) = 1

2+ π^(d−1)/2 Γ ((d − 1)/2)

Z zi

0

Z ^∞

u² y − u²
(d−1)/2−1

g(y)dydu

for i = 1, . . . , d. Next, suppose that Ui = Qg(Zi), i = 1, . . . , d, and the Jacobian of the transformation is

|J| =

d

Y

i=1

1 qg Q⁻_g¹(Ui)
. Then, the pdf of the random vector U = (U1, . . . , Ud)^′ is

c(u1, . . . , ud) = |R|⁻¹²h(u1, . . . , ud)/

d

Y

i=1

qg(Q⁻_g¹(ui)),

where h(z1, . . . , zd) = g (z^′R⁻¹z) and Q⁻_g¹ is the inverse cdf of Qg, and the cdf of U is

C(u1, . . . .ud) =

Z Q⁻¹g (u1)

−∞

· · ·

Z Q⁻¹g (ud)

−∞

|R|⁻¹²g z^′R⁻¹z
dzd· · · dz¹. (1) A copula C is called as an elliptical copula if its form is given by (1). If a d-dimensional random vector X has an elliptical copula associated with a joint distribution F and margin Fi, i = 1, . . . , d, X is said to have a meta-elliptical distribution with the notation X ∼ ME^d(0, R, g; F1, . . . , Fd). See Fang et al. (2002) for details. Two examples of meta-elliptical distributions are given as follows.

Example 1. Let X = (X1, . . . , Xd)^′ have a d-dimensional meta-Gaussian distribution with marginal distribution Fi, i = 1, . . . , d. The cdf of X is of the form

F (x1, . . . , xd) =

Z Φ⁻¹(F1(x1))

−∞

· · ·

Z Φ⁻¹(Fd(xd))

−∞

1

(2π)^d/2|R|^1/2e^{−12z′R−1z}dzd· · · dz¹, and the associated Gaussian copula is

C^Ga(u1, . . . , ud) =

Z Φ⁻¹(u1)

−∞

· · ·

Z Φ⁻¹(ud)

−∞

1

(2π)^d/2|R|^1/2e^{−12z′R−1z}dzd· · · dz¹,

where Φ⁻¹ is the functional inverse of the standard normal cdf Φ. The associated scale function g is given by

g(r) = 1

(2π)^d/2e^−12r.

Example 2. Let X = (X1, . . . , Xd)^′ have a d-dimensional meta-tν distribution with marginal distribution Fi, i = 1, . . . , d. The cdf of X is of the form

F (x1, . . . , xd) =

Z t⁻¹ν (F1(x1))

−∞

· · ·

Z t⁻¹ν (Fd(xd))

−∞

Γ ^ν+d₂
Γ ^ν₂

1 (πν)^d/2|R|^1/2



1+z^′R⁻¹z ν

^−ν+d₂

dzd· · · dz¹,

and the associated tν copula is

C^tν(u1, . . . , ud) =

Z t⁻¹ν (u1)

−∞

· · ·

Z t⁻¹ν (ud)

−∞

Γ ^ν+d₂
Γ ^ν₂

1 (πν)^d/2|R|^1/2



1 + z^′R⁻¹z ν

^−ν+d₂

dzd· · · dz¹,

where tν is the cdf of the t distribution with degrees of freedom ν, and t⁻_ν¹ is the functional inverse of tν. The associated scale function g is given by

g(r) = Γ ^ν+d₂
Γ(^ν₂)(πν)^d/2

1 + r ν

^−ν+d₂ .

2.2 Dependence

The two measures of association of d-variate continuous random variables with the associated copula C, employed in this paper, are the Kendall’s tau (τd,C) and the Blomqvist’s beta (βd,C). Kendall’s tau of bivariate continuous random variables (τ2,C) is defined as the probability of concordance minus the probability of discordance.

Blomqvist’s beta, or the medial correlation coefficient, for bivariate continuous random variables (β2,C) describes the proportion of data which fall into the first or third quad- rants of a two-way contingency table, with the cutting points being the sample medians.

Both the measures of association can be evaluated in term of copulas. For the multi- variate versions of both measures of association, if X1, . . . Xd are continuous random variables with a copula C, then

τd,C = 2^d−1 2^d−1− 1

 2

Z 1 0 · · ·

Z 1 0

C(u1, . . . , ud)dC(u1, . . . , ud) − 2^1−d

 , and

βd,C = 2^d−1 2^d−1− 1

 C 1

2, . . . ,1 2



+ ¯C 1

2, . . . ,1 2



− 2^1−d

 ,

where ¯C is the survival copula of the copula C. See Nelsen (2002) and Schmid and Schmidt (2007), respectively, for details. If d = 2,

τ2,C = 4 Z 1

0

Z 1 0

C (u1, u2) dC(u1, u2) − 1, and

β2,C = 4C 1 2,1

2



− 1.

See Nelsen (2006) for details.

Suppose that there are n independent samples; let Xj = (X1j, . . . , Xdj), j = 1, . . . , n and let Uj = (U1j, . . . , Udj), where uij = Fi(xij), i = 1, . . . , d, and j = 1, . . . , n. If the margins are unknown, use the empirical distribution ˆuij = _n+1¹ Pn

k=1I(Xik ≤ X^ij) to estimate uij, i = 1, . . . , d, and j = 1, . . . , n. The nonparametric estimators of the two measures of association, τd,C and βd,C, are of the forms

ˆ

τd,C = 2^d−1 2^d−1− 1

( 21

n

n

X

j=1

1 n

n

X

k=1 d

Y

i=1

I(uik ≤ u^ij) − 2^1−d )

, (2)

and

βˆd,C = 2^d−1 2^d−1− 1

(1 n

n

X

j=1

" _d Y

i=1

I



uij ≤ 1 2

 +

d

Y

i=1

I



uij > 1 2

#

− 2^1−d )

, (3)

as have been demonstrated in Barbe et al. (1996) and Schmid and Schmidt (2007), respectively. If the margins are unknown, then instead of uij, the empirical distribution ˆ

uij, i = 1, . . . , d and j = 1, . . . , n, is employed.

Assume that X ∼ ME^d(0, R, g; F1, . . . , Fd). For d = 2, Schmid and Schmidt (2007) proved τ2,C coincides with β2,C. That is τ2,C = β2,C = _π² sin⁻¹ρ. For d-dimensional elliptical copulas, C ¹₂, . . . ,¹₂
= ¯C ¹₂, . . . ,¹₂
, and then

βd,C = 2^d−1 2^d−1− 1

 2C 1

2, . . . ,1 2



− 2^1−d

 .

Proposition 2.1. Suppose that X ∼ ME^d(0, R, g; F1, . . . , Fd). The multivariate ver- sion of Kendall’s tau is the same as the multivariate version of Blomqvist’s beta, that is,

τd,C = βd,C = 2^d−1 2^d−1− 1

"

2 Z ^∞

−∞

d

Y

i=1

Φ − δit p1 − δi²

!

φ(t)dt − 2^1−d

# ,

where rij = δiδj, |δⁱ| ≤ 1, and Φ and φ are the cdf and the pdf of standard normal.

Proof. Let X ∼ ME^d(0, R, g; F1, . . . , Fd) and Z = (Z1, . . . , Zd)^′ ∼ EC^d(0, R, g). By Theorem 4.1 in Nelsen (2002), the multivariate version of Kendall’s tau for Z is

τd,C = 2^d−1

2^d−1− 12P (Z1 < 0, . . . , Zd< 0) − 2^1−d .

By Theorem 2.22 in Fang et al. (1990), τd,C = 2^d−1

2^d−1− 12P (Z1 < 0, . . . , Zd< 0) − 2^1−d

= 2^d−1

2^d−1− 12P (X₁ < 0, . . . , X_d < 0) − 2^1−d . By Proposition 8 in Schmid and Schmidt (2007),

βd,C = 2^d−1

2^d−1− 12P (X1 < 0, . . . , Xd< 0) − 2^1−d

= 2^d−1 2^d−1− 1

"

2 Z ^∞

−∞

d

Y

i=1

Φ − δit p1 − δi²

!

φ(t)dt − 2^1−d

# .

In this paper, we consider X ∼ ME^d(0, Rρ, g; F1, . . . , Fd), where Rρis an exchange- able correlation matrix with ρ, as, −1/(d − 1) < ρ < 1. Therefore, we have

for d = 3 : τ3,C(ρ) = β3,C(ρ) = 2

πsin⁻¹ρ, for d = 4 : τ_4,C(ρ) = β_4,C(ρ) = 12

7πsin⁻¹ρ + 24 7π²

Z ρ 0

sin⁻¹

 t

1 + 2t

 dt

p1 − ρ², for d = 5 : τ5,C(ρ) = β5,C(ρ) = 4

3πsin⁻¹ρ + 24 3π²

Z ρ 0

sin⁻¹

 t

1 + 2t

 dt

p1 − ρ², as is shown in Schmid and Schmidt (2007).

3 Main Results

Let X ∼ ME^d(0, Rρ, g; F1, . . . , Fd), −d−1¹ < ρ < 1. A Bayesian approach is proposed to estimate the correlation coefficient ρ. In addition, we propose an empirical Bayes method to obtain the values in the given transformed beta prior distribution, employing two multivariate versions of the measure of association, using the MCMC sampling scheme to compute the posterior mean under the squared-error loss.

3.1 The Bayesian Approach

As in the CML method, the dependence parameters for meta-elliptical distributions can be estimated only by the associated elliptical copulas. For a d-dimensional elliptical

copula, therefore, we suggest a transformed beta prior distribution for the correlation coefficient ρ in Rρ. The prior is of the form

π(ρ) = Γ(a + b) Γ(a)Γ(b)

(d − 1)^b

d^a+b−1 (1 + (d − 1)ρ)^a−1(1 − ρ)^b−1, − 1

d − 1 < ρ < 1, with

E^π(ρ) = a

a + b − 1 d − 1

b a + b and

V ar^π(ρ) =

 d

d − 1

2

ab

(a + b)²(a + b + 1).

Given n independent samples Xj and Uj, j = 1, . . . , n, the posterior density function is

π(ρ|u1, . . . , u_n) ∝ (1 + (d − 1)ρ)^a−1(1 − ρ)^b−1

n

Y

j=1

c(u_1j, . . . , u_dj|ρ)

∝ (1 + (d − 1)ρ)^a−n2−1(1 − ρ)^{b−n(d−1)2 −1}

n

Y

j=1

g (1 + (d − 2)ρ)Pd

i=1Q⁻_g¹(uij)²− 2ρP

m>iQ⁻_g¹(uij)Q⁻_g¹(umj) (1 − ρ)(1 + (d − 1)ρ)

! ,

where uj = (u1j, . . . , udj) with uij = Fi(xij), i = 1, . . . , d and j = 1, . . . , n. If the margins are unknown, use the empirical distribution ˆuij instead of uij, i = 1, . . . , d and j = 1, . . . , n. Note that we consider the squared-error loss function L(ρ, ˆρ) = (ρ − ˆρ)² in this paper, where ˆρ is an estimator for ρ.

By Proposition 2.1, both the multivariate versions of Kenall’s tau and Blomqvist’s beta, for an exchangeable correlation matrix Rρ, are given by

τd,C(ρ) = βd,C(ρ) = 2^d−1 2^d−1− 1

"

2 Z ^∞

−∞

 Φ



−

√ρt

√1 − ρ

d

φ(t)dt − 2^1−d

# ,

where Φ and φ are the cdf and pdf of the standard normal. By Taylor’s expansion for τ_d,C(ρ) in powers of (ρ−E^π(ρ)), we assume as approximation, E^π[τ_d,C(ρ)] ≈ τd,C[E^π(ρ)]

and V ar^π[τd,C(ρ)] ≈ τd,C^′ [E^π(ρ)]²V ar^π(ρ). Similarly, assume E^π[βd,C(ρ)] ≈ β^d,C[E^π(ρ)]

and V ar^π[βd,C(ρ)] ≈ βd,C^′ [E^π(ρ)]²V ar^π(ρ). We use the notations µρ = E^π(ρ), σ_ρ² = V ar^π(ρ), µK,π = E^π[τd,C(ρ)], σ_K,π² = V ar^π[τd,C(ρ)], µB,π = E^π[βd,C(ρ)] and σ_B,π² = V ar^π[β_d,C(ρ)]. The values of (a, b) can be obtained in terms of µ_ρ and σ_ρ², as follows:

a = 1 + (d − 1)µ^ρ (d − 1)(1 − µ^ρ)b

and

b = (1 + (d − 1)µρ)(1 − µρ)²

d σ²_ρ − d − 1

d (1 − µ^ρ) .

3.2 The Empirical Bayes Method and Posterior Inference

In order to obtain the values of (a, b), we propose an empirical Bayes method to estimate the expectation and variance of the multivariate measures of association, that is, µK,π and σ_K,π² , and µB,π and σ²_B,π. Based on our assumption, we have the estimators of µρ and σ_ρ². A bootstrap method is employed to estimate µK,π and σ_K,π² . The bootstrap algorithm is given below:

1. Select S independent bootstrap samples, each consisting of n data values drawn with replacement.

2. Evaluate the bootstrap replication corresponding to each bootstrap sample, ˆτ_d,C^s calculated by (2), s = 1, . . . , S.

The estimators of µK,π and σ_K,π² are the sample mean and the sample variance, respectively, of ˆτ_d,C^s , and are found given as follows:

ˆ µ_K,π =

PS s=1τˆ_d,C^s

S and

ˆ σ_K,π² =

PS

s=1 τˆ_d,C^s − ˆµK,π

2

S − 1 .

By assumption, the estimators of µρ and σ_ρ² are given as follows ˆ

µρ,K = τ_d,C⁻¹(ˆµK,π) and

ˆ

σ_ρ,K² = ˆσ_K,π² τ_d,C^′ (ˆµρ,K)². Then,

ˆ

aK = 1 + (d − 1)ˆµ^ρ,K (d − 1)(1 − ˆµ^ρ,K)ˆbK

and

ˆbK = (1 + (d − 1)ˆµ^ρ,K)(1 − ˆµ^ρ,K)²

d ˆσ_ρ,K² −d − 1

d (1 − ˆµρ,K).

Similarly, we can not only use the bootstrap algorithm to obtain ˆβ_d,C^s , as calculated by (3), s = 1, . . . , S, but also obtain both of estimators of a and b, as given by

ˆaB = 1 + (d − 1)ˆµ^ρ,B (d − 1)(1 − ˆµρ,B)ˆbB

and

ˆbB = (1 + (d − 1)ˆµ^ρ,B)(1 − ˆµ^ρ,B)²

d ˆσ_ρ,B² − d − 1

d (1 − ˆµ^ρ,B).

In the above,

ˆ

µρ,B = β_d,C⁻¹(ˆµB,π) and

ˆ

σ_ρ,B² = σˆ²_B,π β_d,C^′ (ˆµρ,B)² with

ˆ µB,π =

PS s=1βˆ_d,C^s

S and

ˆ σ_B,π² =

PS

s=1 ˆβ_d,C^s − ˆµ^B,π2

S − 1 .

Under the squared-error loss, the posterior mean is the Bayesian point estimator for ρ. Since it is difficult to obtain the posterior mean of ρ, the MCMC sampling scheme is employed to perform the computation. The values of a and b in the given prior distribution are obtained by the proposed empirical Bayes method. Thus, the posterior density function for both estimators of a and b is

π(ρ|u¹, . . . , un) ∝ (1 + (d − 1)ρ)^ˆa−n2−1(1 − ρ)^{ˆb−n(d−1)2 −1}

n

Y

j=1

g (1 + (d − 2)ρ)Pd

i=1Q⁻_g¹(uij)²− 2ρP

m>iQ⁻_g¹(uij)Q⁻_g¹(umj) (1 − ρ)(1 + (d − 1)ρ)

! ,

for i = 1, . . . , d and j = 1, . . . , n, where ˆa and ˆb are obtained with the proposed empirical Bayes method with the employed measure of association.

The MCMC sampling scheme consists of the following steps. We start with an arbitrary ρ0; at each iteration t = 1, . . . , T .

• Step 1 Sample ηt from q(·), where q(·) is a proposal density function and ρ0 is the initial value.

• Step 2 Generate u ∼ Unif(0, 1).

• Step 3 With probability T R = min



1,π(ηt|u¹, . . . , un)q(ρt−1) π(ρt−1|u¹, . . . , un)q(ηt)

 , if u ≤ T R, set

ρt= ηt, else put

ρ_t= ρ_t−1.

The acceptance rate is evaluated in Step 3, which is called as Metropolis-Hastings step. We set the initial value ρ0 as the sample correlation coefficient of any two ran- dom variables Xi and Xj, for i 6= j. In this paper, we propose a transformed beta distribution as a proposal density function similar to a prior distribution with ˆa and ˆb obtained by the proposed method. Through the MCMC sampling scheme, therefore, the posterior mean with the corresponding dependence for the squared-error loss can be computed. The poster mean with Kendall’s tau is denoted as ˆρK and the other version is represented as ˆρB.

4 Simulation Study

Using a simulation study, we compare the two measures of association employed in our proposed Bayesian approach. In practice, tν copulas are primarily used to describe the dependence structure among several risk factors or assets. Thus, simulation data consisting of n independent samples is generated from a tν copula with Rρ, and margins are assumed to be unknown. The empirical distribution ˆuij is used instead of uij, for i = 1, . . . , d and j = 1, . . . , n. The pdf of a tν copula with Rρ is of the form

c^tν(u1, . . . , ud|ρ)

= Γ(^ν+d₂ )Γ(^ν₂)^d−1

Γ(^ν+1₂ )^d|R^ρ|¹² 1+(1+(d−2)ρ)Pd

i=1t⁻_ν¹(ui)²−2ρP

m>it⁻_ν¹(ui)t⁻_ν¹(um) ν(1 − ρ)(1 + (d − 1)ρ)

!^−ν+d₂

d

Y

i=1



1+ t⁻_ν¹(ui)² ν

^ν+1₂

tau beta cml

0.840.860.880.900.92

boxplot(t copulas with rho=0.9)

tau beta cml

0.420.440.460.480.500.520.540.56

boxplot(t copulas with rho=0.5)

tau beta cml

−0.10−0.050.000.050.10

boxplot(t copulas with rho=0)

(a) d = 2

tau beta cml

0.8800.8850.8900.8950.9000.9050.910

boxplot(t copulas with rho=0.9)

tau beta cml

0.440.460.480.500.520.54

boxplot(t copulas with rho=0.5)

tau beta cml

−0.06−0.04−0.020.000.020.040.06

boxplot(t copulas with rho=0)

(b) d = 3

tau beta cml

0.8850.8900.8950.9000.9050.9100.915

boxplot(t copulas with rho=0.9)

tau beta cml

0.460.480.500.52

boxplot(t copulas with rho=0.5)

tau beta cml

−0.03−0.010.000.010.020.03

boxplot(t copulas with rho=0)

(c) d = 5

Figure 1: Box-plots of three estimators with ρ = 0.9, 0.5, 0 for d = 2, 3, 5.

where |R^ρ| = (1 − ρ)^d−1(1 + (d − 1)ρ); t^ν is the cdf of the t distribution with degrees of freedom ν, and t⁻_ν¹ is the functional inverse of t_ν. Considering d = 2, 3, 5, the corresponding τd,C and βd,C are given as follows:

τ_2,C(ρ) = β_2,C(ρ) = 2

πsin⁻¹ρ, τ3,C(ρ) = β3,C(ρ) = 2

πsin⁻¹ρ, and

τ5,C(ρ) = β5,C(ρ) = 4

3πsin⁻¹ρ + 24 3π²

Z ρ 0

sin⁻¹

 t

1 + 2t

 dt

p1 − ρ².

When d = 5, ρ is solved by the Newton method for both the τ5,C(ρ) and the β5,C(ρ).

In the simulation study, we set ν = 3, n = 1000, S = 1000, and T = 10000, and obtain the CML estimator, which is denoted by ˆρCM L. The comparison of the three

Table 1: Numerical results of three estimators with ρ = 0.9, 0.5, 0 for d = 2, 3, 5

d = 2 d = 3 d = 5

ρ = 0.9 ρ = 0.5 ρ = 0 ρ = 0.9 ρ = 0.5 ρ = 0 ρ = 0.9 ρ = 0.5 ρ = 0 L(ρ, ˆρK) 0.0658 0.7054 1.3257 0.0371 0.4797 0.4985 0.0307 0.2667 0.1736 L(ρ, ˆρ_B) 0.0722 0.7152 1.4021 0.0410 0.5110 0.5417 0.0327 0.2895 0.1648 L(ρ, ˆρCM L) 0.6485 1.8603 1.0998 0.0403 0.5084 0.4885 0.0309 0.2787 0.1570 rate (τd,C) 0.6006 0.7252 0.7693 0.5712 0.7089 0.7504 0.5133 0.6757 0.6606 rate (βd,C) 0.3325 0.5446 0.6221 0.3282 0.5588 0.6224 0.3035 0.5481 0.4922

* L(ρ, ˆρ) × 10⁻³

estimators for ρ–ˆρK, ˆρB and ˆρCM L–is repeated 100 times. Figure 1 provides the box- plots of the three estimators with ρ = 0.9, 0.5, 0 for d = 2, 3, 5. Table 1 reports the means of squared-error loss for the three estimators and the means of acceptance rates for the corresponding measures of association.

From Figure 1, the distributions of the three estimators are similar with fixed correlations for d = 3, 5. But, for d = 2, the distribution of ˆρ_{CM L} is different from the other estimators, and both of the third quartile values for ˆρCM L are smaller than the real ρ with ρ = 0.9, 0.5. In Table 2, all means of L(ρ, ˆρ_K) are smaller the than the means of L(ρ, ˆρB), except for when d = 5 with ρ = 0. Moreover, all means of the acceptance rates corresponding to the τ_d,C are greater than the means of acceptance rates corresponding to the βd,C. For ρ = 0, all means of L(ρ, ˆρCM L) are smaller than the other means of the squared-error loss. Consequently, it may be concluded from our simulation study that the version of our proposed Bayesian approach that employed the multivariate Kendall’s tau is better than the vesion that employed the multivariate Blomqvist’s beta.

5 Empirical Application

In this section, we use our proposed Bayesian approach on two stock market indices–

the S&P 500 and the Dow Jones. Zen and Chan (2007b) suggested that a bivariate tν

copula is the one best-suited for the log-returns of both the indices. Therefore, taking the log-returns of both the indices, the dependence parameter ρ of a tν copula can be

Table 2: Results about estimators for two situations

First Case Second Case

τ2,C β2,C τ2,C β2,C

ˆ

µA,π 0.7863301 0.7841250 0.7859455 0.7889600 ˆ

σ_A,π² 0.0000320 0.0001477 0.0000331 0.0001350 ˆ

µρ,A 0.9442024 0.9430558 0.9440032 0.9455549 ˆ

σ_ρ,A² 0.0000086 0.0000403 0.0000089 0.0000353 ˆaA 12315.7500 2664.1070 11895.2100 2920.1930 ˆbA 353.45570 78.07563 342.64000 81.71965

ˆ

ρA 0.9425727 0.9418614 0.9425438 0.9422701 ˆ

ρCM L 0.9417675 0.9421761

rate 0.5615 0.3817 0.5768 0.3384

* A is a measure of association estimated by our approach.

The two daily index datasets have been collected between 1997/1/2 and 2007/6/11 and are denoted by Sij, i = 1, 2, j = 1, . . . , 2, 626; with i = 1 for the S&P 500 and i = 2 for the Dow Jones. We use the notation Rik = log(Si(k+1)/Sik) as the log-return for i = 1, 2 and k = 1, . . . , 2, 625. The pdf of a tν copula is

c^tν(u1, u2|ρ)= Γ(^ν+2₂ )Γ(^ν₂) Γ(^ν+1₂ )²p1 − ρ²



1+t⁻_ν¹(u1)²−2ρt⁻ν¹(u1)t⁻_ν¹(u2)+t⁻_ν¹(u2)² ν(1 − ρ²)

^−ν+2₂



1 + t⁻_ν¹(u₁)² ν

^ν+1₂ 

1 + t⁻_ν¹(u₂)² ν

^ν+1₂

We take ν = 2.44, which was the IFME estimated by Zen and Chan (2007b) and consider two situations for the data. In the first case, we assume that both the margins are unknown and, further, CML estimator for ρ is solved. In the second case, we assume that both the margins are t margins, which have location and scale parameters; in addition, ρ is also solved with the known margins. In the second case, both the location and scale parameters are provided by Zen and Chan (2007b).

Table 2 reports the results in relation to all estimators corresponding to the τd,C

and βd,C, respectively. From Table 2, it is evident that the acceptance rates for the corresponding τd,C are greater than those for the corresponding βd,C, for the two cases.

6 Conclusions

For meta-elliptical distributions, the multivariate version of the Kendall’s tau is the same as the multivariate version of the Blomqvist’s beta, and both of these have a functional relationship with the correlation coefficients. Considering an exchangeable correlation matrix with ρ, the functional relationship with both the measures of associ- ation is demonstrated in subsection 2.2. In this paper, we propose a Bayesian approach to estimate ρ in an equicorrelated meta-elliptical distribution under the squared-error loss.

We suggested a transformed beta prior distribution for ρ in an exchangeable correla- tion matrix. An empirical Bayes method was proposed in order to obtain the estimators of the parameters in the transformed beta prior distribution. Based on our assumption, we employed the multivariate version of Kendall’s tau and that of Blomqvist’s beta.

Moreover, a transformed beta distribution with its parameters estimated by our pro- posed empirical Bayes method was used as a proposal density function in the sampling scheme. From the results of our simulation study, we can conclude that the multivariate Kendall’s tau functions better within our proposed Bayesian approach, as opposed the multivariate Blomqvist’s beta, as was evident from the acceptance rate corresponding to the τd,C being greater thoughout than the acceptance rate corresponding to the βd,C.

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