Estimation of Stochastic Volatility Models: An Approximation to the Nonlinear State Space Representation

Abstract

ABSTRACT The stochastic volatility (SV) model can be regarded as a nonlinear state space model. This article proposes the Laplace approximation method to the nonlinear state space representation and applies it for estimating the SV models. We examine how the approximation works by simulations as well as various empirical studies. The Monte Carlo experiments for the standard SV model indicate that our method is comparable to the Monte-Calro Likelihood (MCL; Durbin and Koopman, 1997 Durbin , J. , Koopman , S. J. ( 1997 ). Monte Carlo maximum likelihood estimation of non-Gaussian state space model . Biometrika 84 : 669 – 684 . [CROSSREF] [Crossref], [Web of Science ®] , [Google Scholar]), Maximum Likelihood (Fridman and Harris, 1998 Fridman , M. , Harris , L. ( 1998 ). A maximum likelihood approach for non-Gaussian stochastic volatility models . J. Bus. Econo. Statist. 16 ( 3 ): 284 – 291 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]), and Markov chain Monte Carlo (MCMC) methods in the sense of mean square error in finite sample. The empirical studies for stock markets reveal that our method provides very similar estimates of coefficients to those of the MCL. We show a relationship of our Laplace approximation method to importance sampling.