본 연구는 연관성 파라미터가 시변하는 확률코퓰러모형 추정을 위해 Langrock et al.(2012)의 HMM근사를 이용한 최우추정을 제안한다. 이산시간 연속상태공간모형에 대한 HMM근사를 이용한 우도함수 근사는 확률코퓰러모형과 같이 상태변수가 마르코프 성질을 충족하는 AR(1)과정을 따르는 비선형비정규상태공간모형 추정을 위한 빠르고 효과적인 수치적분 방법이다. 본 연구는 모의실험을 통해 HMM근사를 이용한 확률코퓰러모형 최우추정 성과가 EIS를 이용한 최우추정 및 MCMC알고리듬을 이용한 베이지언 추론의 성과와 차이가 없을 뿐만 아니라 추정소요시간이 EIS를 이용한 최우추정법의 3.4~9.6% 불과함을 보인다. 또한, 본 연구는 2003년 1월 3일∼2015년 12월 30일 기간의 KOSPI 지수와 HSCE 지수 일간수익률 자료에 대한 실증분석 예시를 통해 상관계수가 시변연관성 파라미터인 가우시언확률코퓰러모형이 모형설정 오류가 없을 뿐만 아니라 비교모형들 가운데 가장 적합성이 좋은 모형이라는 결과를 얻었다.

The association or interdependence between asset returns plays a crucial role in financial and managerial theories and practices, such as asset pricing, portfolio construction, risk management, and derivative pricing. The Pearson correlation coefficient, a representative measure of linear interdependence under the elliptical distribution, has been widely used in theory and practice. However, after the US financial crisis of 2008, various limitations of the correlation coefficient as a measure of interdependence have been exposed. Since Patton (2006a), The copula function, which has been actively researched in the field of financial econometrics in order to overcome the limitations of the linear measure of interdependence, that is, correlation coefficient. Unlike the linear measure of interdependence, the copula function can capture nonlinear, asymmetric, and tail dependence. Various study results have shown that the interdependence between asset returns, such as time-varying conditional variance and asymmetric behavior, varies over time and with different degrees of asymmetry. To capture such asymmetric time-varying interdependence, various conditional dynamic copula models have been proposed, such as Patton's (2006a) Copula-GARCH model, Creal et al's (2011, 2013) GAS model (generalized autoregressive score models), and Hafner and Manner's (2012) stochastic copula model. In the case of the Copula-GARCH and GAS models, similar to the GARCH models, the dynamic equation of association parameter is a deterministic function of parameters and returns of previous periods. In contrast, the stochastic copula model is a nonlinear, non-normal state space model in which the association parameter is a nonlinear function of a latent state variable as in the stochastic volatility model.

As in the maximum likelihood estimation of the volatility model, it is necessary to integrate out unobserved state variables in order to perform maximum likelihood estimation of the Copula model and it is well known that integrating-out of such state variables can be challenging. Efficient Importance Sampling (EIS), proposed by Liesenfeld and Richard (2003) and Richard and Zhang (2007), was used by Hafner and Manner (2012) to obtain the likelihood function required for estimating the stochastic volatility model. Almeida and Czado (2012), Kim and Park (2018), and Kreuzer and Czado (2020), among others, have proposed MCMC algorithms for Bayesian inference on stochastic copula models. Although those simulation-based estimation methods are efficient, they have the disadvantage of taking a considerable amount of time for model estimation. For a small number of state variables, it is possible to calculate the likelihood function of a general discrete-time continuous state-space model using numerical integration. Kitagawa (1987) proposed a method of numerical integration by partitioning continuous state variables for the estimation of discrete-time non-normal state-space models. Similarly, Fridman and Harris (1998), Bartolucci and De Luca (2001, 2003), and others have proposed methods of approximating the likelihood function of stochastic volatility models using quadrature methods. The maximum likelihood estimation using the likelihood function obtained by numerical integration may take a considerable amount of time for numerical integration as the number of observed data and the dimension of the state variable increases due to the curse of dimensionality. Therefore, numerical integration-based maximum likelihood estimation is not commonly used for the estimation of stochastic volatility models. Langrock et al. (2012) used the HMM (hidden Markov model) approximation to obtain an approximate likelihood function of the stochastic volatility model by partitioning the state variable into multiple intervals and approximating the transition probability density function of the discrete-time continuous state variable with the transition probability of the discrete state variable.

We propose to exploit the Hidden Markov Model(HMM) approximation of Langrock et al. (2012) for the maximum likelihood estimation of stochastic copula models. The likelihood function approximation using the discrete-state HMM approximation of discrete-time continuous state-space models with state variables satisfying the Markov property is a fast and effective numerical integration method for estimating nonlinear, nonnormal state space models, such as stochastic copula models, with a state variable following an autoregressive process of order one. Through simulation studies, we demonstrate that the performance of the maximum likelihood estimation of stochastic copula models using the HMM approximation is as good as those of the maximum likelihood estimation using the efficient importance sampling algorithm and the Bayesian inference using the Markov chain Monte Carlo algorithm and that the maximum likelihood estimation using the proposed HMM approximation is at least 10 times faster than that using efficient importance sampling. As an empirical illustration, we analyzed the stochastic copula models for the KOSPI index and the HSCE index daily returns from January 3, 2003 to December 30, 2015. We obtained the empirical results that the Gaussian stochastic copula model has better goodness-of-fit than any other models considered as well as passes the model specification tests.