Inferences in stochastic volatility models: a new simpler way. (English) Zbl 1407.62329

Sutradhar, Brajendra C. (ed.), Advances and challenges in parametric and semi-parametric analysis for correlated data. Proceedings of the 2015 international symposium in statistics, ISS 2015, St. John’s, Canada, July 6–8, 2015. Cham: Springer. Lect. Notes Stat. 218, 97-131 (2016).
Summary: Two competitive analytical approaches, namely, the generalized method of moments (GMM) and quasi-maximum likelihood (QML) are widely used in statistics and econometrics literature for inferences in stochastic volatility models (SVMs). Alternative numerical approaches such as Monte Carlo Markov chain (MCMC), simulated maximum likelihood (SML) and Bayesian approaches are also available. All these later approaches are, however, based on simulations. In this paper, we revisit the analytical estimation approaches and briefly demonstrate that the existing GMM approach is unnecessarily complicated. Also, the asymptotic properties of the likelihood approximation based QML approach are unknown and the finite sample based QML estimators can be inefficient. We then develop a precise set of moment estimating equations and demonstrate that the proposed method of moments (MM) estimators are easy to compute and they perform well in estimating the parameters of the SVMs in both small and large time series set up. A ‘working’ generalized quasi-likelihood (WGQL) estimation approach is also considered. Estimation methods are illustrated by reanalyzing a part of the Swiss-Franc and U.S. dollar exchange rates data.
For the entire collection see [Zbl 1347.62015].


62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62P05 Applications of statistics to actuarial sciences and financial mathematics
62E15 Exact distribution theory in statistics


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[1] Abramowitz, M., Stegun, N.C.: Handbook of Mathematical Functions. Dover Publications, New York (1970) · Zbl 0171.38503
[2] Ahn, S., Schmidt, P.: Efficient estimation of models for dynamic panel data. J. Econ. 68, 5-28 (1995) · Zbl 0831.62094
[3] Andersen, T.G., Sorensen, B.E.: GMM estimation of a stochastic volatility model: a Monte Carlo study. J. Bus. Econ. Stat. 14, 328-352 (1996)
[4] Andersen, T.G., Sorensen, B.E.: GMM and QML asymptotic standard deviations in stochastic volatility models: comments on Ruiz (1994). J. Econ. 76, 397-403 (1997) · Zbl 0900.62632
[5] Arellano, M., Bond, S.: Some tests of specification for panel data: Monte Carlo evidence and an application to employment equations. Rev. Econ. Stat. 58, 277-298 (1991) · Zbl 0719.62116
[6] Broto, C., Ruiz, E.: Estimation methods for stochastic volatility models: a survey. J. Econ. Surv. 18, 613-649 (2004)
[7] Chamberlain, G.: Sequential moment restrictions in panel data-comment. J. Bus. Econ. Stat. 10, 20-26 (1992)
[8] Danielsson, J.: Stochastic volatility in asset prices: estimation with simulated maximum likelihood. J. Econ. 64, 375-400 (1994) · Zbl 0825.62953
[9] Durbin, J., Koopman, S.J.: Monte Carlo maximum likelihood estimation for non-Gaussian state space models. Biometrika 84, 669-684 (1997) · Zbl 0888.62086
[10] Hansen, L.P.: Large sample properties of generalized method of moments estimators. Econometrica 50, 1029-1054 (1982) · Zbl 0502.62098
[11] Harvey, A.C., Ruiz, E., Shephard, N.: Multivariate stochastic variance models. Rev. Econ. 61, 247-264 (1994) · Zbl 0805.90026
[12] Jacquier, E., Polson, N.G., Rossi, P.E.: Bayesian analysis of stochastic volatility models (with discussion). J. Bus. Econ. Stat. 12, 371-417 (1994)
[13] Keane, M., Runkle, D.: On the estimation of panel data models with serial correlation when instruments are not strictly exogenous. J. Bus. Econ. Stat. 10, 1-9 (1992)
[14] Koopman, S.J., Harvey, A.C., Doornik, J.A., Shephard, N.: Stamp 5.0: Structural Time Series Analyser, Modeller, and Predictor. Chapman and Hall, London (1995)
[15] Lee, K.M., Koopman, S.J.: Estimating stochastic volatility models: a comparison of two importance samplers. Stud. Nonlin. Dyn. Econ. 8, 1-15 (2004) · Zbl 1081.91534
[16] Liesenfeld, R., Richard, J.F.: Univariate and multivariate stochastic volatility models: estimation and diagnostics. J. Empir. Finan. 207, 1-27 (2003)
[17] Melino, A., Turnbull, S.M.: Pricing foreign currency options with stochastic volatility. J. Econ. 45, 239-265 (1990) · Zbl 1126.91374
[18] Mills, T.C.: The Econometric Modelling of Finance Time Series. Cambridge University Press, Cambridge (1999) · Zbl 1145.62402
[19] Rao, R.P., Sutradhar, B.C., Pandit, V.N.: GMM versus GQL inferences in semiparametric linear dynamic mixed models. Braz. J. Probab. Stat. 26, 167-177 (2012) · Zbl 1235.62137
[20] Ruiz, E.: Quasi-maximum likelihood estimation of stochastic volatility models. J. Econ. 63, 289-306 (1994) · Zbl 0825.62949
[21] Ruiz, E.: QML and GMM estimators of stochastic volatility models: response to Anderson and Sorensen. J. Econ. 76, 405 (1997) · Zbl 0900.62633
[22] Shephard, N.: Statistical aspects of ARCH and stochastic volatility. In: Cox, D.R., Hinkley, D.V., Barndorff-Nielsen,O.E. (eds.) Time Series Models in Econometrics, Finance and Other Fields, pp. 1-67. Chapman & Hall, London (1996)
[23] Shephard, N., Pitt, M.K.: Likelihood analysis of non-Gaussian parameter-driven models. Biometrica 84, 653-667 (1997) · Zbl 0888.62095
[24] Sutradhar, B.C.: On exact quasilikelihood inference in generalized linear mixed models. Sankhya B: Indian J. Stat. 66, 261-289 (2004) · Zbl 1192.62176
[25] Taylor, S.J.: Financial returns modelled by the product of two stochastic processes - a study of daily sugar prices, 1961-79. In: Anderson, O.D. (ed.) Time Series Analysis: Theory and Practice, vol. 1. North-Holland Publishing Company, Amsterdam (1982)
[26] Tsay, R.S.: Analysis of Financial Time Series. Wiley Interscience, New York (2005) · Zbl 1086.91054
[27] Usmani, R.: Inversion of a tridiagonal Jacobi matrix. Linear Algebra Appl. 212/213, 413-414 (1994) · Zbl 0813.15001
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