STATISTICAL PROPERTIES OF SEMIPARAMETRIC ESTIMATORS ...

INTERNATIONAL JOURNAL OF NUMERICAL ANALYSIS AND MODELING, SERIES B

c 2012 Institute for Scientific

Computing and Information

Volume 3, Number 4, Pages 371–387

STATISTICAL PROPERTIES OF SEMIPARAMETRIC ESTIMATORS FOR COPULA-BASED MARKOV CHAIN VECTORS MODELS WENDE YI AND B. STEPHEN SHAOYI LIAO

Abstract. This paper proposes a method for estimation of a class of copula-based semiparametric stationary Markov vector time series models, namely, the two-stage semiparametric pseudo maximum likelihood estimation (2SSPPMLE). These Markov vector time series models are characterized by nonparametric marginal distributions and parametric copula functions of temporal and contemporaneous dependence, while the copulas capture two classes of dependence relationships of Markov time series. We provide simple estimators of marginal distribution and two classes of copulas parameters and establish their asymptotic properties following conclusions in Chen and Fan (2006) and some easily verifiable conditions. Moreover, we obtain the estimation of conditional moment and conditional quantile functions for the bivariate Markov time series model. Key words. Copula, Semiparametric estimation, Temporal dependence, Contemporaneous dependence, and 2SSPPMLE

1. Introduction In many fields, including international asset pricing, portfolio diversification and risk management, et al., dependence of models on random variables is an interesting topic. In dependence analysis of time series, we must synchronously consider two crucial classes of dependence relationships, namely, temporal dependence and contemporaneous dependence. However, a great deal of research has shown that economic and financial multivariate time series are typically nonlinear, are nonnormally distributed and have nonlinear comovements beyond the first two conditional moments. In recent years, copulas have started to be applied to model the dependence structure of time series in various fields. In particular, modeling and estimating the dependence structure between several univariate time series in the finance and insurance community are of great interest; see Joe [1]and Embrechts et al. [2] for reviews. Chen and Fan [3], Abegaz and Naik-Nimbalkar [4] studied the properties of estimators of semiparametric and parametrical stationary Markov chain models. Embrechts et al. [2] used copula functions to model multivariate distribution of returns. In economic and financial applications, it is interesting to estimate or forecast certain features of time series, such as the value-at-risk (VaR) of a portfolio of assets, which has become routine in risk management. Duffie and Pan [5], Hull and White [6], Engle and Manganelli [7] and Embrechts [8] have done much work on this topic. In Yi and Liao [9], we established a model based on copula functions to study the two classes of dependence relationships (including nonlinear dependence) of time series vectors, proposed a three-stage pseudo maximum likelihood estimation of the model and discussed the properties of parametric Received by the editors August 2, 2012 and, in revised form October 24, 2012. 2000 Mathematics Subject Classification. 62F12. This work has been supported by the National Natural Science Foundation of China (No: 71271227), the Humaniores & Social Science Research Project of the Ministry of Education (No: 11XJC790004) and the GRF grant (No: CityU 147407) of the Research Grant Commission, Hong Kong SAR Government. 371

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estimators. In this paper, we study a class of bivariate copula-based semiparametric stationary Markov models in which temporal copulas and contemporaneous copulas are parameterized, but the marginal distributions are left unspecified. Models in this class are completely characterized by three unknown parameters, namely (1) contemporaneous dependence copula parameters θ∗ (i.e., the finite-dimensional parameters to capture the dependence structure between the two univariate time ∗ series); (2) temporal dependence copula parameters δ ∗ = (δX , δY∗ ) (i.e., the finitedimensional parameters to characterize the temporal dependence of individual time series); and (3) marginal distributions F ∗ (·) and G∗ (·). In estimation of the two classes of parameters of temporal and contemporaneous dependence copulas, our main contribution is that we propose a two-stage semiparameters pseudo maximum likelihood estimation (2SSPPMLE) for bivariate copula-based semiparametric time series vector models based on the two-stage estimator proposed for bivariate and univariate copula models with i.i.d. observations (Genest [10], Shih [11], Chen and Fan [3], and Abegaz and Naik- Nimbalkar [4]). Moreover, we focus on establishing the consistency and asymptotic normality of the resulting estimators of model parameters based on the 2SSPPMLE. The remainder of this paper is organized as follows. Section 2 reviews related research and proposes the models of bivariate stationary Markov time series vectors of order 1. Section 3 proposes the 2SSPPMLE of model parameters and estimations of conditional moment and conditional quantile functions copula-based bivariate Markov chain model. Section 4 presents some assumptions of consistency and asymptotical normality of semiparametric estimators of univariate copula-based time series models. Section 5 discusses the consistency and asymptotic normality of semiparametric estimators of the 2SSPPMLE. Section 6 discusses the problem of copula selection under the three-stage procedure, and the last section consists of concluding remarks. 2. Related Work and Copula-based bivariate Markov chains models of order 1 A copula is defined as a multivariate distribution with standard uniform marginal distributions. The Sklar’s theorem (Sklar [12]) illustrates how to model a multivariate distribution by modeling its marginal distributions and its copula function separately (see Nelsen [13] for details). Sklar’s theorem. Let H (·, ·) be a bivariate function with continuous marginals F (·) and G (·). There exists a unique copula C (·, ·) such that the joint distribution can be written as H (x, y) = C (F (x) , G (y))

(x, y) ∈ R2

Research on dependence of time series based on copulas mainly focuses on two forms of dependence structures. The first is the contemporaneous dependence of multivariate time series, where the focus is on modeling the joint distribution of ′ some random vector, Xt = [X1t , X2t , · · · , Xnt ] , conditional on the information set Ft−1 . Bouy et al. [14] used parametric copulas to model dynamic dependence of time series and provided applications to financial returns and transactions based forex data. They applied the two-step procedure of Genest et al. [15] to estimate the copula dependence parameter. Patton [16, 17] defined a ”conditional copula” as a multivariate distribution of variables distributed as uniform (0,1) conditional on Ft−1 , and estimated parameters of multivariate models for time series of possibly different lengths. The second is the temporal dependence of univariate time series based on stationary Markov chains, where copulas are used to describe the dependence between

STATISTICAL PROPERTIES OF SEMIPARAMETRIC ESTIMATORS

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observations from a given univariate time series, for example, by capturing the dependence between [Xt , Xt+1 , · · · , Xt+n ]′ . Chen and Fan [3, 18] investigated the parametric estimation procedure of a class of copula-based semiparametric stationary Markov time series models and established their asymptotic properties such as root-n consistency and asymptotic normality of estimators. In addition, Abegaz and Naik-Nimbalkar [4] discussed the two-stage pseudo maximum likelihood estimation (2SPMLE) procedure of copula-based parametric stationary Markov time series models and established the asymptotic properties of the 2SPMLE under certain regularity conditions. Yi and Liao [9] proposed the three-stage pseudo maximum likelihood estimation (3SPMLE) for copula-based parametric stationary Markov bivariate time series models and discussed the asymptotic properties of the 3SPMLE. An alternative to parametric specifications of marginal distributions of the multivariate distribution would, of course, be a nonparametric estimate, as in Fermanian and Scaillet [19], which can accommodate all possible distributional forms. However, one common drawback of nonparametric approaches is lack of precision that occurs when the dimension of the distribution of interest is moderately large (say, over four). In this section, based on the copula theory, we establish the dependence model of bivariate time series, combining the temporal dependence of univariate time series in which the marginal distribution is left unspecified with the contemporaneous dependence between time series to investigate the dependence structures in bivariate Markov chains of order 1. Assumption 1 . Let the process {Xt , t = 1, 2, · · ·} and {Yt , t = 1, 2, · · ·} be ∗ two stationary first-order Markov chains generated from (F ∗ (·) , CX (·, ·; δX )) and ∗ ∗ ∗ ∗ (G (·) , CY (·, ·; δY )), respectively, where F (·) and G (·) are true invariant distri∗ butions, and (F ∗ (·) , CX (·, ·; δX )) and (G∗ (·) , CY (·, ·; δY∗ )) satisfy the hypothesis of Assumption 1 in Chen and Fan (2006a, P: 310) [3] (omitting). We note that under Assumption 1, the transformed processes, {Ut : Ut ≡ F ∗ (Xt )} and {Vt : Vt ≡ G∗ (Yt )}, are two stationary parametric Markov processes of order 1 in which the joint distributions of Ut and Ut−1 , andVt and Vt−1 are given ∗ by copulas CX (u0 , u1 ; δX ) and CY (v0 , v1 ; δY∗ ), and the conditional densities of Ut ∗ given Ut−1 = u0 , and Vt given Vt−1 = v0 are fUt |Ut−1 =u0 (u) = cX (u0 , u; δX ) and ∗ gVt |Vt−1 =v0 (v) = cY (v0 , v; δY ), respectively. Chen and Fan [3]showed the following Lemma 1. 2 ∗ Lemma 1. Under Assumption 1, if cX (·, ·; δX ) is positive on (0, 1) , then (i) and (ii) of Proposition 2.1 in Chen and Fan (2006a, P:313-314)[3] hold (omitting). Suppose processes {Xt } and {Yt } follow the conditions of Assumption 1, and the conditional distributions xt |Xt−1 = xt−1 ∼ FX and yt |Yt−1 = yt−1 ∼ GY , then the copula C (·, ·; θ∗ ) of (xt |Xt−1 = xt−1 , yt |Yt−1 = yt−1 ) is the true joint distribution of time series {WXt , t = 1, 2, · · ·} and {WY t , t = 1, 2, · · ·}, where wXt = FX (xt |Xt−1 = xt−1 ) and wY t = GY (yt |Yt−1 = yt−1 ), and the density c (·, ·; θ∗ ) 2 ∗ with respect to the Lebesgue measure on (0, 1) ; δX , δY∗ , θ∗ are true parametrical vectors. We first consider the temporal dependence of the univariate time series under the condition of known variable value at time t − 1 and then construct the contemporaneous dependence model of the time series vector given the variable value at time t − 1 . Using Sklar’s theorem, then,

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H (xt |Xt−1 = xt−1 , yt |Yt−1 = yt−1 ) =C (FX (xt |Xt−1 = . xt−1 ) , GY (yt |Yt−1 = yt−1. ) ; θ∗ )

=C

=C

(2)

∗ ∗ ∂HX (xt−1 ,xt ;δX ) ∂F ∗ (xt−1 ) ∂HY (yt−1 ,yt ;δY ) ∂G∗ (yt−1 ) , ; θ∗ ∂x ∂x ∂y ∂yt−1 t−1 t−1 t−1 ∗ ∗ ∂CX (F ∗ (xt−1 ),F ∗ (xt );δX ) ∂CY (G∗ (yt−1 ),G∗ (yt );δY ) , ; θ∗ ∂F ∗ (xt−1 ) ∂G∗ (yt−1 )

h (xt |Xt−1 = xt−1 , yt |Yt−1 = yt−1 ) = c (FX (xt |Xt−1 = xt−1 ) , GY (yt |Yt−1 = yt−1 ) ; θ∗ ) · ∗ ∗ fX (xt |Xt−1 = xt−1 ; δX ) · gY∗ (yt |Yt−1 = yt−1 ; δY∗ ) = c (FX (xt |Xt−1 = xt−1 ) , GY (yt |Yt−1 = yt−1 ) ; θ∗ ) · ∗ cX (F ∗ (xt−1 ) , F ∗ (xt ) ; δX )· ∗ ∗ f (xt ) · cY (G (yt−1 ) , G∗ (yt ) ; δY∗ ) · g ∗ (yt )

∗ ∗ ∗ where , fX (xt |Xt−1 = xt−1 ; δX ) = f ∗ (xt ) cX (F ∗ (xt−1 ) , F ∗ (xt ) ; δX ), ∗ ∗ ∗ ∗ ∗ ∗ gY (yt |Yt−1 = yt−1 ; δY ) = g (yt ) cY (G (yt−1 ) , G (yt ) ; δY ) are the true conditional density functions of Xt given Xt−1 = xt−1 , and Yt given Yt−1 = yt−1 , respectively. One obvious advantage of the copula approach over the standard approach is that it separates contemporaneous dependence structure from temporal dependence structure, and from the marginal behavior toward each other. The copula approach also allows us to take advantage of numerous existing parametric copulas and nonparametric distributions to construct new conditional density functions via (2), see Joe [1], Nelsen [13] and Chen and Fan [3] for expressions of many commonly used parametric copulas.

3. Estimation about Model 3.1. Two-stage semiparameters pseudo maximum likelihood estimation of model parameters. A semiparametric copula-based Markov chains vectors time series model is de∗ termined by (F ∗ , G∗ , δX , δY∗ , θ∗ ). The unknown marginal distributions F ∗ and G∗ can be estimated by Fn (·) and Gn (·). The rescaled empirical distribution functions are defined as n

(3)

Fn (x) =

1 X I {Xt ≤ x} n + 1 t=1

and Gn (y) =

1 n+1

n P

t=1

I {Yt ≤ y}

Maximum likelihood is the natural estimation procedure for parametric copula models. Hence, if we perfectly know two marginal distributions F ∗ (·) and G∗ (·), then we can estimate the temporal dependence models of the univariate variable according to marginal distributions. In the final stage, we estimate the contemporaneous dependence model. Although estimating all coefficients simultaneously yields the most efficient estimates, the large number of parameters can make numerical maximization of the likelihood function difficult. In this section, we study the estimation of semiparameters of a copula-based bivariate model using the two-stage estimation method. The likelihood for the single realization of Markov chains {x1 , · · · , xn } and {y1 , · · · , yn } using (2)is given by


(4)

375

n Q L (δ, θ) = f ∗ (x1 ) · g ∗ (y1 ) h (xt |Xt−1 = xt−1 , yt |Yt−1 = yt−1 ; δ, θ ) t=2 n n n Q ∗ Q Q = f (xt ) · g ∗ (yt ) · cX (F ∗ (xt−1 ) , F ∗ (xt ) ; δX ) t=1 t=2 nt=1 Q ∗ ∗ · cY (G (Yt−1 ) , G (Yt ) ; δY ) t=2 n Q · c (FX (xt |Xt−1 = xt−1 ; δX ) , GY (yt |Yt−1 = yt−1 ; δY ) ; θ) t=2

where δ = (δX , δY ) is parametric vector. As can be seen from (4), the log-likelihood l (δ, θ), in principle, can be divided into three parts, one for the known marginal (5), another for the marginal temporal dependence conditional copula models (6) and the third for the contemporaneous dependence conditional copula (7) below; i.e., l (δ, θ) = lM (F ∗ , G∗ ) + lT e (δ) + lCo (δ, θ) with (5)

lM (F ∗ , G∗ ) =

n X t=1

and (6) lT e (δ) =

n P

log f ∗ (xt ) +

n X

log g ∗ (yt ) = lM (F ∗ ) + lM (G∗ )

t=1

log cX (F ∗ (xt−1 ) , F ∗ (xt ) ; δX ) +

t=2

n P

log cY (G∗ (yt−1 ) , G∗ (yt ) ; δY )

t=2

= lT e (δX ) + lT e (δY )

and (7) lCo (δ, θ) =

n X t=2

log c (FX (xt |Xt−1 = xt−1 ; δX ) , GY (yt |Yt−1 = yt−1 ; δY ) ; θ)

Joe [1] and Abegaz and Naik-Nimbalkar [4] studied the parametric maximum likelihood estimator of one-stage and two-stage methods, respectively, and established their asymptotic properties under certain regularity conditions. Chen and Fan [3] studied the estimation of copula-based semiparametric time series models and established the asymptotic properties of parametric estimation. Yi and Liao [9] discussed the three-stage pseudo maximum likelihood estimation procedure and their asymptotic properties for Markov chain vectors based on copula models. In the following we discuss the two-stage semiparametric pseudo maximum likelihood estimation procedure (2SSPPMLE). Ignoring (5) and replacing F ∗ with Fn , and G∗ with Gn in (6) motivate the ∗ two-stage semiparametric pseudo maximum likelihood estimator δ˜X of δX , δ˜Y of ∗ ∗ δY and θ˜ of θ , respectively Stage 1. Estimate the marginal temporal dependence conditional copula parameter vector δ by substituting marginal distributions Fn and Gn into (6) and then maximizing the resulting temporal copula pseudo log-likelihood. The pseudo temporal dependence conditional copula score functions are given by . . S˜δX (δX ) = ∂ ˜lT e (δX ) ∂δX S˜δY (δY ) = ∂ ˜lT e (δY ) ∂δY n n P P where ˜lT e (δX ) = log cX (Fn (xt−1 ) , Fn (xt ) ; δX ), ˜lT e (δY ) = log cY t=2

t=2

(Gn (yt−1 ) , Gn (yt ) ; δY ) Solving S˜δX (δX ) = 0 and S˜δY (δY ) = 0 obtains the pseudo

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∗ maximum likelihood estimators δ˜X of δX and δ˜Y of δY∗ , respectively, such that

δ˜X = arg max ˜lT e (δX )

(8)

δX

δ˜Y = arg max ˜lT e (δY ) δY

To ensure that the criterion function and the first order conditions of criterion (8) are well defined for all finite n, we use the rescaled empirical distributions Fn (·) n P and Gn (·) in (3) instead of the standard empirical distribution n−1 I {Zt ≤ ·} , t=1

since partial derivatives of log c (u1 , u2 ; δ) are infinity at ui = 0 or 1 for i = 1, 2 for many copula densities. Stage 2. Estimate the contemporaneous dependence conditional copula parameter θ first by plugging the above Stage 1 parameter estimates δ˜ into (7) and by maximizing the resulting contemporaneous dependence conditional copula pseudo log-likelihood. The pseudo contemporaneous dependence conditional copula score function is given by . ˜ θ = ∂ ˜lCo δ, ˜ θ ∂θ, S˜θ δ, n X ˜lCo δ, ˜θ = log c FX xt Xt−1 = xt−1 ; δ˜X , GY yt Yt−1 = yt−1 ; δ˜Y ; θ

t=2 ˜ θ = 0 gives the pseudo maximum likelihood estimator of ˜(θ) and Solving S˜θ δ, ∗ θ . ˜ θ˜ , referred to as the Thus, from Stage 1 to 2, we obtain the estimators δ, 2SSPPMLE of (δ ∗ , θ∗ ).

3.2. Estimation of conditional moment and conditional quantile function based-copula bivariate Markov chain model. Now suppose values of time series vector at time t − 1 and time series at time t are known. Then we consider the true conditional distribution of the other variable, Xt , and the conditional distribution is denoted as H ∗ (xt |Xt−1 = xt−1 |(Yt = yt |Yt−1 = yt−1 ) ). H ∗ (xt |Xt−1 = xt−1 |(Yt = yt |Yt−1 = yt−1 ) )

(9)

P {Xt ≤ xt |Xt−1 = xt−1 , yt < Yt ≤ yt + ∆y |Yt−1 = yt−1 } ∆y→0 P {yt < Yt ≤ yt + ∆y |Yt−1 = yt−1 } ∗ = CX|Y (FX (xt ; δX ) , GY (yt ; δY∗ ) ; θ∗ )

= lim

where CX|Y (x, y) = (10)

∂C(x,y) ; ∂y

∗ ∗ FX (xt ; δX ) = FX (xt |Xt−1 = xt−1 ; δX ) = P {Xt ≤ xt |Xt−1 = xt−1 } ∗ = CX,2|1 (F ∗ (xt−1 ) , F ∗ (xt ) ; δX )

and (11)

GY (yt ; δY∗ ) = GY (yt |Yt−1 = yt−1 ; δY∗ ) = P {Yt ≤ yt |Yt−1 = yt−1 } = CY,2|1 (G∗ (yt−1 ) , G∗ (yt ) ; δY∗ )

where C·,2|1 (u, v) = (12)

∂C· (u,v) . ∂u

h∗ (xt |Xt−1 = xt−1 |(Yt = yt |Yt−1 = yt−1 ) )

∗ ∗ = c (FX (xt ; δX ) , GY (yt ; δY∗ ) ; θ∗ ) fX (xt ; δX )

∗ ∗ = c (FX (xt ; δX ) , GY (yt ; δY∗ ) ; θ∗ ) · cX (F ∗ (xt−1 ) , F ∗ (xt ) ; δX ) · f ∗ (xt )

Similarly we can obtain the conditional distribution of Yt given Xt and (Xt−1 , Yt−1 ) .


377

In economic and financial applications, one is often interested in estimating or forecasting certain characteristics of a variable at time t given the information of the variable before time t and the information of other dependent variables. These can be easily obtained from the conditional density function h∗ (xt |Xt−1 = xt−1 |(Yt = yt |Yt−1 = yt−1 ) ) of Xt given Xt−1 and Yt = yt |Yt−1 = yt−1 For example, the conditional k th moment Xt given Xt−1 and Yt = yt |Yt−1 = yt−1 can be calculated via E Xtk |Xt−1 = xt−1 |(Yt = yt |Yt−1 = yt−1 ) Z = z k h∗ (z |Xt−1 = xt−1 |(Yt = yt |Yt−1 = yt−1 ) ) dz (13) Z ∗ ∗ ) , GY (yt ; δY∗ ) ; θ) · cX (F ∗ (xt−1 ) , F ∗ (z) ; δX )dF ∗ (z) = z k c (FX (z; δX

More generally, we may be interested in estimating a vector of conditional moment functions E (ψ (Xt ) |Xt−1 = xt−1 |(Yt = yt |Yt−1 = yt−1 ) ), where ψ is a vector of known measurable functions of Xt . (14)

E (ψ (Xt ) |Xt−1 = xt−1 |(Yt = yt |Yt−1 = yt−1 ) ) Z ∗ ∗ )dF ∗ (z) ) , GY (yt ; δY∗ ) ; θ∗ ) · cX (F ∗ (xt−1 ) , F ∗ (z) ; δX = ψ (z) c (FX (z; δX

In the right hand side of the equation, marginal distributions F ∗ (·) and G∗ (·) may be replaced with their empirical distribution functions Fn (·) and Gn (·) and hence (14) can be estimated by the following simple plug-in estimator: (15)

˜ (ψ (Xt ) |Xt−1 = xt−1 |(Yt = yt |Yt−1 = yt−1 ) ) E Z ∗ ∗ = ψ (z) c (FX (z; δX ) , GY (yt ; δY∗ ) ; θ∗ ) · cX (Fn (x) , Fn (z) ; δX )dFn (z)

Another important characteristic of conditional distribution of Xt given Xt−1 and Yt = yt |Yt−1 = yt−1 is the conditional quantile of Xt given Xt−1 and Yt = yt |Yt−1 = yt−1 or the conditional VaR of Xt . In risk management, estimating the conditional VaR of portfolios of assets has become routine but the effect of temporal dependence of time series is seldom considered in this process. In our research we calculate the conditional quantile of Xt given Xt−1 and Yt = yt |Yt−1 = yt−1 under temporal and contemporaneous conditions. For the q-th ∗ conditional quantile of Xt given Xt−1 and Yt = yt |Yt−1 = yt−1 , (i.e. FX (xt ; δX ) ∗ such that q = CX|Y (FX (xt ; δX ) , GY (yt ; δY∗ ) ; θ∗ )), the procedure of inference consists of two stages. First, from (9) we obtain the probability of Xt given Xt−1 and −1 ∗ Yt = yt |Yt−1 = yt−1 , denoted as uX,t = FX (xt ; δX ) = CX|Y (q, GY (yt ; δY∗ ) ; θ∗ ); ∗ ∗ ∗ then from uX,t = CX,2|1 (F (xt−1 ) , F (xt ) ; δX ), we can obtain −1 ∗ F ∗ (xt ) = CX,2|1 (F ∗ (xt−1 ) , uX,t ; δX ), hence the qth conditional quantile of Xt given Xt−1 and Yt = yt |Yt−1 = yt−1 is given by −1 ∗ ∗ ∗−1 (F (x ) , u ; δ ) C (16) QX (X |Y ) = F t−1 X,t X t−1 q X,2|1

−1 (q, GY (yt ; δY∗ ) ; θ∗ ), and the Xt = F −1 (Ut ) is a monotonic where uX,t = CX|Y transformation of Ut . The plug-in estimator of the conditional quantile QX q (Xt−1 |Y ) of Xt given Xt−1 and Yt = yt |Yt−1 = yt−1 can be obtained by replacing F ∗ (·) and G∗ (·) with their empirical distribution functions Fn (·) and Gn (·). −1 −1 − ˜X q, Gn (yt ) ; θ˜ ; δ˜X ) (17) Q q (Xt−1 |Y ) = Fn (CX,2|1 Fn (xt−1 ) , CX|Y

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where Fn− (u) = inf {x : Fn (x) ≥ u}. Similarly, we can obtain the conditional quantile QYq (Yt−1 |X ) . It is well known that the conditional quantile function Qq (·; δ ∗ , θ∗ ) is generally nonlinear. Bouy´ e and Salmon [14] provide explicit expressions of the conditional quantile functions Qq (·; α) for several specific copulas including the Gaussian copula, the Frank copula, and the Clayton copula. Chen and Fan [3] provide expressions of the conditional quantile functions of Yt given Yt−1 . From the joint conditional density function h (· |Xt−1 , · |Yt−1 , ) of Xt and Yt given Xt−1 and Yt−1 , we can calculate the conditional moment of combination of Xt and Yt given Xt−1 and Yt−1 via

(18)

E (φ (Xt , Yt ) |Xt−1 = xt−1 , Yt−1 = yt−1 ) Z Z = φ (x, y) c (FX (x |Xt−1 = xt−1 ) , GY (y |Yt−1 = yt−1 ) ; θ∗ )

∗ ) · cY (G∗ (yt−1 ) , G∗ (y) ; δY∗ ) dF ∗ (x) dG∗ (y) · cX (F ∗ (xt−1 ) , F ∗ (x) ; δX

It can be estimated by the following simple plug-in estimator:

(19)

˜ (φ (Xt , Yt ) |Xt−1 = xt−1 , Yt−1 = yt−1 ) E Z Z = φ (x, y) c FX (x |Xt−1 = xt−1 ) , GY (y |Yt−1 = yt−1 ) ; θ˜ · cX Fn (xt−1 ) , Fn (x) ; δ˜X · cY Gn (yt−1 ) , Gn (y) ; δ˜Y dFn (x) dGn (y)

For the qth conditional quantile of Xt and Yt given Xt−1 and Yt−1 , we can not give the direct expression of the conditional quantile, but may calculate it by simulation. 4. Assumptions for Consistency and Asymptotic Normality Chen and Fan [3] established convergence of the rescaled empirical distribution function Gn (·) in a weighted metric and used it to establish the consistency and asymptotic normality of semiparametric estimator α ˜ of copula C (·, ·; α∗ ) of time series (Yt−1 , Yt ), presented the joint asymptotic distribution of Gn (·) and α ˜ and established asymptotic properties of the conditional moment and conditional quantile estimators. ˜n (v) ≡ Gn G∗−1 (v) (or ≡ Fn F ∗−1 (v) According to Chen and Fan [3], let U ) for v ∈ (0, 1), and W ∗ (·) be a zero-mean tight Gaussian process in D [0, 1] such that W ∗ (0) = W ∗ (1) = 0 , and E {W ∗ (v1 ) W ∗ (v2 )} (20)

= min {v1 , v2 } − v1 v2 +

∞ X

k=2

{Cov [I (U1 ≤ v1 ) , I (Uk ≤ v2 )]

+Cov [I (Uk ≤ v1 ) , I (U1 ≤ v2 )] The following Lemma holds. Lemma 2. (See Lemma 4.1 in Chen and Fan [3]) Suppose {Xt } and {Yt } satisfy Assumption 1 and are β-mixing. Let w (·) be a continuous function with the conditions of Lemma 4.1 in Chen and Fan [3]. By Lemma 4.1, we have the following results: ˜ (v)−v F (x)−F ∗ (x) (1) sup Unw(v) = oa.s. (1), sup nw(F ∗ (x)) = oa.s. (1), x v∈[0,1] Gn (y)−G∗ (y) sup w(G∗ (y)) = oa.s. (1) y


379

√ ˜ ∗ (2) n U n (·) − · /w (·) →dist W (·) /w (·) in D [0, 1], √ √ (x)−F ∗ (x) Gn (y)−G∗ (y) n sup Fnw(F n sup = O (1) = O (1) ∗ (x)) P P w(G∗ (y)) x

y

Define M as the space of probability distributions over the support of Yt . For any G ∈ M we let kG − G∗ kM = supy |{G (y) − G∗ (y)} /w (G∗ (y))| with w (·) satisfying the condition in Lemma 2 (1). Let Mα = {G ∈ M : kG − G∗ kM ≤ α} for a small α > 0. Let Ω = (Ω1 , Ω2 ) ⊂ Rd be the parameter space and let vector parameter be denoted by η = (δ, θ) where δ = (δX , δY ) ∈ (ΩX,1 , ΩY,1 ) = Ω1 and θ ∈ Ω2 are of dimensions d1 and d2 with d1 + d2 = d. Further, let uX,t = F (xt ) , uY,t = G (yt ) , wX,t = FX (xt |Xt−1 = xt−1 ; δX ) and wY,t = GY (yt |Yt−1 = yt−1 ; δY ). We denote lT e (v1 , v2 ; δX ) = log cX (uX,t−1 , uX,t ; δX ) , lT e (v1 , v2 ; δY ) = log cY (uY,t−1 , uY,t ; δY ) ′ lδX (v1 , v2 ; δX ) = ∂lT e (v1 , v2 ; δX )/∂δX , lδX δX (v1 , v2 ; δX ) = ∂ 2 lT e (v1 , v2 ; δX ) /∂δX ∂δX 2 and lδX ,j (v1 , v2 ; δX ) = ∂ lT e (v1 , v2 ; δX ) /∂vj ∂δX for j = 1, 2. lδY , lδY δY and lδY ,j are defined similarly.

lCo (v1 , v2 ; δ, θ) = log c (wX,t , wY,t ; δ, θ) , lθ (v1 , v2 ; δ, θ) = ∂lCo (v1 , v2 ; δ, θ)/∂θ, lθθ , lθ,j , lδθ and lδθ,j are defined similarly. Lemma 3. (See Proposition 4.2 in Chen and Fan [3]) Suppose Assumption 1 ∗ holds and δX and δX of {Xt } and δY and δY∗ of {Yt } satisfy the conditions of Proposition 4.2 in Chen and Fan [3]. We use the serial number A1-A5 instead of the C1-C5 conditions of Proposition 4.2 in Chen and Fan [3] (for simplifying, only giving the of process {Xt } , omitting {Yt }.). We have the following

conditions

˜

˜ ∗ ∗ results: δX − δ = oP (1) and δY − δ = oP (1). X

Y

According to the results and conditions in Chen and Fan [3], we have the follow∗ ing expressions and conditions: Denote Fα = {(δX , F ) ∈ Ω × Mα : kδX − δX k ≤ α} for a small α > 0. Let {Fη : η ∈ [0, 1]} ⊂ Mα be a one-dimensional smooth path in Mα with Fη|η=0 = F ∗ , and {(δX,η , Fη ) : η ∈ [0, 1]} ⊂ Fα be a one-dimensional ∗ smooth path in Fα with (δX,η , Fη ) |η=0 = (δX , F ∗ ). We also define n

(21)

A∗X,n

1 X ∗ ≡ [lδ (Ut−1 , Ut ; δX ) + W1 (Ut−1 ) + W2 (Ut )] n − 1 t=2 X

(22)

A∗Y,n

1 X [lδ (Vt−1 , Vt ; δY∗ ) + W1 (Vt−1 ) + W2 (Vt )] ≡ n − 1 t=2 Y

n

Z

1

Z

1

(23) W1 (Ut−1 ) ≡

1Z

1

(24)

Z

W2 (Ut ) ≡

0

0

∗ ∗ [I {Ut−1 ≤ v1 } − v1 ]lδX ,1 (v1 , v2 ; δX ) c (v1 , v2 ; δX ) dv1 dv2

0

0

∗ ∗ [I {Ut ≤ v2 } − v2 ]lδX ,2 (v1 , v2 ; δX ) c (v1 , v2 ; δX ) dv1 dv2

Chen and Fan [3] show that the following set of conditions were sufficient to √ ensure the n-asymptotic normality of δ˜X (or δ˜Y ) (for simplifying giving conditions of process {Xt } , only omitting {Yt }.): ∗ B1. (i) condition A1 is satisfied with δX ∈ int (Ω); Te ∗ (ii)BX ≡ −E [lδX ,δX (Ut−1 , Ut ; δX )] (BY is defined similarly) is positive definite;

380

W. YI AND B. LIAO

P P √ ∗ (iii) X ≡ lim

n→∞ V ar( nAXn ) (similarly define Y ) is positive definite;

∗ (iv) δ˜X − δX

= oP (1), and supx |{Fn (x) − F ∗ (x)} /w2 (F ∗ (x))| = On n−1/2 ,

where w2 (·) satisfies the condition in Lemma 2(2); 2 B2. lδX ,δX (v1 , v2 ; δX ) is well-defined and continuous in (v1 , v2 ; δX ) ∈ (0, 1) × int (Ω); B3. the interchange of differentiation and integration of lδX (Fη (Xt−1 ) , Fη (Xt ) ; δXη ) with respect to η ∈ (0, 1) is valid; B4. (i){Xt : t = 1, 2, · · ·} is stationary β-mixing with mixing decay rate βt = O t−b n for some b > γ/ (γ − 1) , ino which γ > 1; 2γ

(ii)E kW1 (Ut−1 ) + W2 (Ut )k < ∞; n

o2γ (iii)E sup(δ¯X ,F )∈Fα lδX F (Xt−1 ) , F (Xt ) ; δ¯X 1; o 2 (ii)E kW1 (Ut−1 ) + W2 (Ut )k log [1 + kW1 (Ut−1 ) + W2 (Ut )k] < ∞; n

2 o (iii)E sup(δ¯X ,F )∈Fα lδX F (Xt−1 ) , F (Xt ) ; δ¯X

log 1 + lδX F (Xt−1 ) , F (Xt ) ; δ¯X < ∞; n

o2 B5. E sup(δ¯X ,F )∈Fα lδX ,δX F (Xt−1 ) , F (Xt ) ; δ¯X < ∞; n o2

B6. E sup(δ¯X ,F )∈Fα lδX ,j F (Xt−1 ) , F (Xt ) ; δ¯X w (Ut−2+j ) < ∞ for j=1,2, h i2 w2 (Ut ) 0, there exists α > 0 and m finite integers such that {θ1 , · · · , θm } forms a α-covering of Ω2 , and sup θ∈Ω2 ,kθ−θi k≤α

sup θ∈Ω2 ,kθ−θi k≤α

Hence,

sup θ∈Ω2 ,kθ−θi k≤α

klθ (Wt∗ ; θ) − lθ (Wt∗ ; θi )k ≤ ε,

Co

E {lθ (Wt∗ ; θ) − lθ (Wt∗ ; θi )} ≤ ε

n

1 P

∗ ∗

{lθ (Wt ; θ) − lθ (Wt ; θi )}

≤ ε,

n−1 t=2

sup

θ∈Ω2 ,kθ−θi k≤α

kµn (lθ (Wt∗ ; θ)) − µn (lθ (Wt∗ ; θi ))k n

1 X lθ (Wt∗ ; θ) − E Co [lθ (Wt∗ ; θ)] n − 1 θ∈Ω2 ,kθ−θi k≤α t=2 ( ) n 1 X ∗ Co ∗ − lθ (Wt ; θi ) − E [lθ (Wt ; θi )] k n − 1 t=2

n n

1 X

1 X

∗ ∗ = sup { lθ (Wt ; θ) − lθ (Wt ; θi )

n − 1 n − 1 θ∈Ω2 ,kθ−θi k≤α t=2 t=2

Co

+ E [lθ (Wt∗ ; θ)] −E Co [lθ (Wt∗ ; θi )] } ≤ 2ε =

sup

k

Under conditions A3, C4 and Lemma 3, we have by Theorem 1 and Application 5 in Rio [20], max kµn (lθ (Wt∗ ; θ))k = oP (1)

1≤i≤m

Hence (∗) is valid. We have θ˜ − θ∗ = oP (1),

sup x,y,δX ,δY

˜

Wt − Wt∗ → oP (1).

Proposition 2. Suppose Assumption 1, the assumptions of Lemma 3 and Proposition 1, and the following conditions hold: Co ∗ ∗ D1. (i) condition C1 is satisfied with θ∗ ∈ int (Ω); (ii)I

θ ≡ −E [lθθ (Wt ; θ )]

˜

˜ ∗ ∗ is positive definite; (iii) θ − θ = oP (1), and sup Wt − W → oP (1), and x,y,δX ,δY

t

conditions in Proposition 1 are satisfied; 2 D2. lθθ (v1 , v2 ; θ) is well-defined and continuous in (v1 , v2 ; θ) ∈ (0, 1) × Ω; D3. the differentiation and integration of lθ (Wt,η ; θη ) with respect to any η ∈ (0, 1) can be interchanged;  2 

 D4. E sup lθθ Wt ; θ¯