Estimation of Stochastic Volatility Models : An Approximation to the Nonlinear State Space
The stochastic volatility (SV) models had not been popular as the ARCH (autoregressive conditional heteroskedasticity) models in practical applications until recent years even though the SV models have close relationship to financial economic theories. The main reason is that the likelihood of the SV models is not easy to evaluate unlike the ARCH models. Developments of Markov Chain Monte-Carlo (MCMC) methods have increased the popularity of Bayesian inference in many fields of research including the SV models. After Jacquire et al. (1994) applied a Bayesian analysis for estimating the SV model in their epoch making work, the Bayesian approach has greatly contributed to the research on the SV models. The classical analysis based on the likelihood for estimating the (SV) model has been extensively studied in the recent years. Danielson (1994) approximates the marginal likelihood of the observable process by simulating the latent volatility conditional on the available information. Shephard and Pitt (1997) gave an idea of evaluating likelihood by exploiting sampled volatility. Durbin and Koopman (1997) explored the idea of Shephard and Pitt (1997) and evaluated the likelihood by Monte-Carlo integration. Sandmann and Koopman (1998) applied this method for the SV model. Durbin and Koopman (2000) reviewed the methods of Monte Carlo maximum likelihood from both Bayesian and classical perspectives. The purpose of this paper is to propose the Laplace approximation (LA) method to the nonlinear state space representation, and to show that the LA method is workable for estimating the SV models including the multivariate SV model and the dynamic bivariate mixture (DBM) model. The SV model can be regarded as a nonlinear state space model. The LA method approximates the logarithm of the joint density of current observation and volatility conditional on the past observations by the second order Taylor expansion around its mode, and then applies the nonlinear filtering algorithm. This idea of approximation is found in Shephard and Pitt (1997) and Durbin and Koopmann (1997). The Monte-Carlo Likelihood (MCL: Sandmann and Koopman (1998)) is now a standard classical method for estimating the SV models. It is based on importance sampling technique. Importance sampling is regarded as an exact method for maximum likelihood estimation. We show that the LA method of this paper approximates the weight function by unity in the context of importance sampling. We do not need to carry out the Monte Carlo integration for obtaining the likelihood since the approximate likelihood function can be analytically obtained. If one-step ahead prediction density of observation and volatility variables conditional on the past observations is sufficiently accurately approximated, the LA method is workable. We examine how the LA method works by simulations as well as various empirical studies. We conduct the Monte-Carlo simulations for the univariate SV model for examining the small sample properties and compare them with those of other methods. Simulation experiments reveals that our method is comparable to the MCL, Maximum Likelihood (Fridman and Harris (1998)) and MCMC methods. We apply this method to the univariate SV models with normal distribution or t-distribution, the bivariate SV model and the dynamic bivariate mixture model, and empirically illustrate how the LA method works for each of the extended models. The empirical results on the stock markets reveal that our method provides very similar estimates of coefficients to those of the MCL. As a result, this paper demonstrates that the LA method is workable in two ways: simulation studies and empirical studies. Naturally, the workability is limited to the cases we have examined. But we believe the LA method is applicable to many SV models based on our study of this paper
|Date of creation:||11 Aug 2004|
|Contact details of provider:|| Phone: 1 212 998 3820|
Fax: 1 212 995 4487
Web page: http://www.econometricsociety.org/pastmeetings.asp
More information through EDIRC
Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.:
- Tim Bollerslev, 1986.
"Generalized autoregressive conditional heteroskedasticity,"
EERI Research Paper Series
EERI RP 1986/01, Economics and Econometrics Research Institute (EERI), Brussels.
- Bollerslev, Tim, 1986. "Generalized autoregressive conditional heteroskedasticity," Journal of Econometrics, Elsevier, vol. 31(3), pages 307-327, April.
- Melino, Angelo & Turnbull, Stuart M., 1990. "Pricing foreign currency options with stochastic volatility," Journal of Econometrics, Elsevier, vol. 45(1-2), pages 239-265.
- Andrew Harvey & Esther Ruiz & Neil Shephard, 1994. "Multivariate Stochastic Variance Models," Review of Economic Studies, Oxford University Press, vol. 61(2), pages 247-264.
- Neil Shephard & Michael K Pitt, 1995.
"Likelihood analysis of non-Gaussian parameter driven models,"
15 & 108., Economics Group, Nuffield College, University of Oxford.
- Shephard, N. & Pitt, M.K., 1995. "Likelihood Analysis of Non-Gaussian Parameter-Driven Models," Economics Papers 108, Economics Group, Nuffield College, University of Oxford.
- Torben G. Andersen & Hyung-Jin Chung & Bent E. Sorensen, "undated".
"EMM Estimation of a Stochastic Volatility Model: A Monte Carlo Study,"
Computing in Economics and Finance 1997
6, Society for Computational Economics.
- Andersen, Torben G & Sorensen, Bent E, 1996. "GMM Estimation of a Stochastic Volatility Model: A Monte Carlo Study," Journal of Business & Economic Statistics, American Statistical Association, vol. 14(3), pages 328-352, July.
- Torben G. Andersen & Bent E. Sorensen, 1995. "GMM Estimation of a Stochastic Volatility Model: A Monte Carlo Study," Discussion Papers 95-19, University of Copenhagen. Department of Economics.
- Neil Shephard, 2005. "Stochastic Volatility," Economics Papers 2005-W17, Economics Group, Nuffield College, University of Oxford.
- Clark, Peter K, 1973. "A Subordinated Stochastic Process Model with Finite Variance for Speculative Prices," Econometrica, Econometric Society, vol. 41(1), pages 135-155, January.
- Watanabe, Toshiaki, 2000. "Bayesian Analysis of Dynamic Bivariate Mixture Models: Can They Explain the Behavior of Returns and Trading Volume?," Journal of Business & Economic Statistics, American Statistical Association, vol. 18(2), pages 199-210, April.
- Andersen, Torben G, 1996. " Return Volatility and Trading Volume: An Information Flow Interpretation of Stochastic Volatility," Journal of Finance, American Finance Association, vol. 51(1), pages 169-204, March.
- Jacquier, Eric & Polson, Nicholas G & Rossi, Peter E, 1994.
"Bayesian Analysis of Stochastic Volatility Models,"
Journal of Business & Economic Statistics,
American Statistical Association, vol. 12(4), pages 371-389, October.
- Tom Doan, "undated". "RATS programs to replicate Jacquier, Polson, Rossi (1994) stochastic volatility," Statistical Software Components RTZ00105, Boston College Department of Economics.
- Tauchen, George E & Pitts, Mark, 1983. "The Price Variability-Volume Relationship on Speculative Markets," Econometrica, Econometric Society, vol. 51(2), pages 485-505, March.
- Engle, Robert F, 1982. "Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation," Econometrica, Econometric Society, vol. 50(4), pages 987-1007, July.
- Nelson, Daniel B, 1991. "Conditional Heteroskedasticity in Asset Returns: A New Approach," Econometrica, Econometric Society, vol. 59(2), pages 347-370, March.
- Liesenfeld, Roman, 1998. "Dynamic Bivariate Mixture Models: Modeling the Behavior of Prices and Trading Volume," Journal of Business & Economic Statistics, American Statistical Association, vol. 16(1), pages 101-109, January.
- Sandmann, Gleb & Koopman, Siem Jan, 1998. "Estimation of stochastic volatility models via Monte Carlo maximum likelihood," Journal of Econometrics, Elsevier, vol. 87(2), pages 271-301, September.
When requesting a correction, please mention this item's handle: RePEc:ecm:feam04:611. See general information about how to correct material in RePEc.
For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (Christopher F. Baum)
If references are entirely missing, you can add them using this form.