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

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), Maximum Likelihood (Fridman and Harris, 1998), 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.

Key Words:

• Laplace approximation
• Nonlinear state space representation
• Stochastic volatility

Mathematics Subject Classification:

• 62M10
• 91B84

Acknowledgment

The first version of this article was written while the second author was visiting the University of Western Ontario. The article was presented at International Symposium on Financial Time Series sponsored by Tokyo Metropolitan University (February, 2004, Tokyo). The authors appreciate the referee for his careful reading of the manuscript and thoughtful comments, which greatly help us improve the article. We are grateful for helpful comments from J. Knight, R. A. Davis, and H. K. van Dijk.

Notes

^aThe Newton–Raphson method has the following three properties: (1) The Newton–Raphson method requires the knowledge of derivatives of the objective function. (2) The iteration algorithm may stop at the local maximum if the iteration starts at the far remote point from the global maximum point. (3) The speed of convergence to the maximum is faster than the simplex method. On the other hand, the simplex method has the opposite properties: (a) The simplex method does not require the knowledge of derivatives. (b) The algorithm may not stop at the local maximum even if the iteration starts at the far remote point from the global maximum. (c) The speed of convergence is slower than the Newton–Raphson method. We choose a strategy of finely tuning the maximum after attaining an approximate maximum point. Hence, the two-stage algorithm may guarantee rapid convergence to the global maximum.

Note: The table shows the mean and the RMSE (in parentheses). These entries are calculated from the K = 500 simulated samples with the T = 500 length of samples. MCL, F&H's ML, and MCMC are respectively, obtained from Table 2 of Sandmann and Koopman (1998), Table 1 of Fridman and Harris (1998), and Table 7 of Jacquier et al. (1994). The RMSE of MCL is calculated from the bias and the standard deviation in Table 2 of Sandmann and Koopman (1998).

Note: The table shows the mean and the RMSE (in parentheses). These entries are calculated from the K = 500 simulated samples with the T = 2000 length of samples for LA, MCL, and MCMC and from the K = 1000 with T = 2000 for NFML. The MCL, NFML, and MCMC are respectively, obtained from Table 3 of Sandmann and Koopman (1998), Table 1 of Watanabe (1999), and Table 9 of Jacquier et al. (1994). The N of NFML stands for the number of segments in numerical integration.

Note: GRMSE×10000 is displayed. These entries are calculated from the K = 500 simulated samples with the T = 500 length of samples. The F&H's ML and the MCMC are, respectively, obtained from Table 3 of Fridman and Harris (1998) and Table 10 of Jacquier et al. (1994).

^bIdeally we should report the estimates of all of the LA, MCL, NFML, and MCMC methods for each of the extended models we apply in this section. However, we do not report the estimates of MCMC for the extended SV models, mainly because it is not easy at present time for us to implement the MCMC algorithm. Dropout of the MCMC from comparison is weakly justified because the LA, MCL, and NFML are based on the maximum likelihood principle from the classical inference point of view, while the MCMC is based on the Bayesian inference. Hence, it may not be unreasonable to restrict the main target of comparison of the LA to the methods based on the likelihood principle, particularly to the MCL which is more broadly applicable method than the NFML.

^cWe estimated the AR models with lag lengths 1 through 4. The SBIC was maximized at the lag length of 1.

Note: White's (1980) heteroskedasticity corrected standard errors are in the parentheses. The last column denotes the heteroskedasticity corrected Ljung–Box statistic for twelve lags of the residual autocorrelations which is calculated from the method of Diebold (1988). Its p-values is 0.081.

Note: The standard errors of estimators are in the parentheses. The standard errors in the classical sampling theory (LA, MCL, NFML) is calculated by the inverse of the asymptotic Fisher information matrix. The standard deviation of MCMC is calculated from the sample draws from Bayesian posterior distribution. The number of draws in the MCL is M = 5. The number of segments in the NFML is N = 100. In the MCMC, we discard the first 1500 sample draws, and we use the after 2500 sample draws to estimate parameters.

^dThe algorithms for the MCL, NFML, and MCMC were written using Fortran 90. We use a single-move procedure by Shephard and Pitt (1997) for the MCMC.

Note: The standard errors are in the parentheses. The number of draws in the MCL is M = 5.The number of segments in the NFML is N = 100.

^eIf Japanese or US market closed on a day, we assume both two markets are closed on that day.

^fWe omit the estimation results of AR(1) model for the NYSE index to save the space.

Note: The standard errors are in the parentheses. The number of draws M = 5 in the MCL.

Note: Standard errors are in the parentheses.