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Robust Bayesian analysis of heavy-tailed stochastic volatility models using scale mixtures of normal distributions. (English) Zbl 1284.91579
Summary: A Bayesian analysis of stochastic volatility (SV) models using the class of symmetric scale mixtures of normal (SMN) distributions is considered. In the face of non-normality, this provides an appealing robust alternative to the routine use of the normal distribution. Specific distributions examined include the normal, student-\(t\), slash and the variance gamma distributions. Using a Bayesian paradigm, an efficient Markov chain Monte Carlo (MCMC) algorithm is introduced for parameter estimation. Moreover, the mixing parameters obtained as a by-product of the scale mixture representation can be used to identify outliers. The methods developed are applied to analyze daily stock returns data on S&P500 index. Bayesian model selection criteria as well as out-of-sample forecasting results reveal that the SV models based on heavy-tailed SMN distributions provide significant improvement in model fit as well as prediction to the S&P500 index data over the usual normal model.

91G70 Statistical methods; risk measures
62F35 Robustness and adaptive procedures (parametric inference)
62F15 Bayesian inference
91B70 Stochastic models in economics
Full Text: DOI
[1] Abanto-Valle, C.A., Migon, H.S., Lopes, H.F., 2009. Bayesian modeling of financial returns: A relationship between volatility and trading volume. Applied Stochastic Modeling in Business and Industry, in press (doi:10.1002/asmb.789) · Zbl 1224.91179
[2] Abramowitz, M.; Stegun, N., Handbook of mathematical functions, (1970), Dover Publications, Inc New York
[3] Ando, T., Bayesian inference for nonlinear and non-Gaussian stochastic volatility model wit leverage effect, Journal of Japan statistical society, 36, 173-197, (2006) · Zbl 1110.62141
[4] Ando, T., Bayesian predictive information criterion for the evaluation of hierarchical Bayesian and empirical Bayes models, Biometrika, 94, 443-458, (2007) · Zbl 1132.62005
[5] Andrews, D.F.; Bickel, P.J.; Hampel, F.R.; Huber, P.J.; Rogers, W.H.; Tukey, J., Robust estimates of location: survey and advances, (1972), Princeton University Press Princeton, NJ · Zbl 0254.62001
[6] Andrews, D.F.; Mallows, S.L., Scale mixtures of normal distributions, Journal of the royal statistical society, series B, 36, 99-102, (1974) · Zbl 0282.62017
[7] Barndorff-Nielsen, O.; Shephard, N., Econometric analysis of realised volatility and its use in estimating stochastic volatility models, Journal of the royal statistical society, series B, 64, 253-280, (2001) · Zbl 1059.62107
[8] Berg, A.; Meyer, R.; Yu, J., Deviance information criterion for comparing stochastic volatility models, Journal of business and economic statistics, 22, 107-120, (2004)
[9] Bollerslev, T., Generalized autoregressive conditional heteroskedasticity, Journal of econometrics, 31, 307-327, (1986) · Zbl 0616.62119
[10] Carnero, M.A.; Peña, D.; Ruiz, E., Persistence and kurtosis in GARCH and stochastic volatility models, Journal of financial econometrics, 2, 319-342, (2004)
[11] Carter, C.K.; Kohn, R., On Gibbs sampling for state space models, Biometrika, 81, 541-553, (1994) · Zbl 0809.62087
[12] Celeux, G.; Forbes, F.; Robert, C.P.; Titterington, D.M., Deviance information criteria for missing data models, Bayesian analysis, 1, 651-674, (2006) · Zbl 1331.62329
[13] Chen, C.W.S.; Liu, F.C.; So, M.K.P., Heavy-tailed-distributed threshold stochastic volatility models in financial time series, Australian & New Zealand journal of statistics, 50, 29-51, (2008)
[14] Chib, S., Marginal likelihood from the Gibbs output, Journal of the American statistical association, 90, 1313-1321, (1995) · Zbl 0868.62027
[15] Chib, S.; Nardari, F.; Shepard, N., Markov chain Monte Carlo methods for stochastic volatility models, Journal of econometrics, 108, 281-316, (2002) · Zbl 1099.62539
[16] Chow, S.T.B.; Chan, J.S.K., Scale mixtures distributions in statistical modelling, Australian & New Zealand journal of statistics, 50, 135-146, (2008)
[17] de Jong, P.; Shephard, N., The simulation smoother for time series models, Biometrika, 82, 339-350, (1995) · Zbl 0823.62072
[18] Fama, E., Portfolio analysis in a stable Paretian market, Management science, 11, 404-419, (1965) · Zbl 0129.11903
[19] Fernández, C.; Steel, M.F.J., Bayesian regression analysis with scale mixtures of normals, Econometric theory, 16, 80-101, (2000) · Zbl 0945.62031
[20] Frühwirth-Schnater, S., Data augmentation and dynamic linear models, Journal of time series analysis, 15, 183-202, (1994) · Zbl 0815.62065
[21] Geweke, J., Evaluating the accuracy of sampling-based approaches to the calculation of posterior moments, (), 169-193
[22] Gross, A.M., A Monte Carlo swindle for estimators of location, Journal of the royal statistical society, series C. applied statistics, 22, 347-353, (1973)
[23] Jacquier, E.; Polson, N.; Rossi, P., Bayesian analysis of stochastic volatility models, Journal of business and economic statistics, 12, 371-418, (1994)
[24] Jacquier, E.; Polson, N.; Rossi, P., Bayesian analysis of stochastic volatility models with fat-tails and correlated errors, Journal of econometrics, 122, 185-212, (2004) · Zbl 1328.91254
[25] Kim, S.; Shepard, N.; Chib, S., Stochastic volatility: likelihood inference and comparison with ARCH models, Review of economic studies, 65, 361-393, (1998) · Zbl 0910.90067
[26] Koopman, S.J.; Uspensky, E.H., The stochastic volatility in Mean model: empirical evidence from international stock markets, Journal of applied econometrics, 17, 667-689, (2002)
[27] Lachos, V.H., Ghosh, P., Arellano-Valle, R.B., 2009. Likelihood based inference for skew-normal/independent linear mixed models. Statistica Sinica (in press) · Zbl 1186.62071
[28] Lange, K.L.; Little, R.; Taylor, J., Robust statistical modeling using \(t\) distribution, Journal of the American statistical association, 84, 881-896, (1989)
[29] Lange, K.L.; Sinsheimer, J.S., Normal/independent distributions and their applications in robust regression, Journal of computational and graphical statistics, 2, 175-198, (1993)
[30] Liesenfeld, R.; Jung, R.C., Stochastic volatility models: conditional normality versus heavy-tailed distributions, Journal of applied econometrics, 15, 137-160, (2000)
[31] Little, R., Robust estimation of the Mean and covariance matrix from data with missing values, Applied statistics, 37, 23-38, (1988) · Zbl 0647.62040
[32] Madan, D.; Seneta, E., The variance gamma (v.g) model for share market return, Journal business, 63, 511-524, (1990)
[33] Mahieu, R.; Schotman, P.C., Am empirical application of stochastic volatility models, Journal of applied econometrics, 13, 333-360, (1998)
[34] Mandelbrot, B., The variation of certain speculative prices, Journal of business, 36, 314-419, (1963)
[35] Melino, A.; Turnbull, S.M., Pricing foreign options with stochastic volatility, Journal of econometrics, 45, 239-265, (1990)
[36] Morgenthaler, S.; Tukey, J., Configural polysampling: A route to practical robustness, (1991), Wiley New York · Zbl 0756.62016
[37] Nakajima, J.; Omori, Y., Leverage, heavy-tails and correlated jumps in stochastic volatility models, Computational statistics & data analysis, (2008) · Zbl 05687930
[38] Omori, Y.; Chib, S.; Shephard, N.; Nakajima, J., Stochastic volatility with leverage: fast likelihood inference, Journal of econometrics, 140, 425-449, (2007) · Zbl 1247.91207
[39] Omori, Y.; Watanabe, T., Block sampler and posterior mode estimation for asymmetric stochastic volatility models, Computational statistics & data analysis, 52, 2892-2910, (2008) · Zbl 05564678
[40] Pemstein, D.; Quinn, K.V.; Martin, A.D., The scythe statistical library: an open source C++ library for statistical computation, Journal of statistical software V, 1-29, (2007)
[41] Philippe, A., Simulation of right and left truncated gamma distributions by mixtures, Statistics and computing, 7, 173-181, (1997)
[42] Pitt, M.; Shephard, N., Filtering via simulation: auxiliary particle filter, Journal of the American statistical association, 94, 590-599, (1999) · Zbl 1072.62639
[43] Robert, C.P.; Titterington, D.M., Discussion on “bayesian measures of model complexity and fit”, Biometrical journal, 64, 573-590, (2002)
[44] Rosa, G.J.M.; Padovani, C.R.; Gianola, D., Robust linear mixed models with normal/independent distributions and Bayesian MCMC implementation, Biometrical journal, 45, 573-590, (2003)
[45] Shephard, N.; Pitt, M., Likelihood analysis of non-Gaussian measurements time series, Biometrika, 84, 653-667, (1997) · Zbl 0888.62095
[46] Shibata, M.; Watanabe, T., Bayesian analysis of a Markov switching stochastic volatility model, Journal of the Japan statistical society, 35, 205-219, (2005)
[47] So, M.; Lam, K.; Li, W., A stochastic volatility model with Markov switching, Journal of business and economic statistics, 15, 183-202, (1998)
[48] Spiegelhalter, D.J.; Best, N.G.; Carlin, B.P.; van der Linde, A., Bayesian measures of model complexity and fit, Journal of the royal statistical society, series B, 64, 621-622, (2002)
[49] Stone, M., Discussion on “bayesian measures of model complexity and fit”, Journal of the royal statistical society, series B, 64, 621, (2002)
[50] Tauchen, G.E.; Pitts, M., The price variability-volume relationships in speculative markets, Econometrica, 51, 485-506, (1983) · Zbl 0495.90026
[51] Taylor, S., Financial returns modelled by the product of two stochastic processes—a study of the daily sugar prices 1961-75, (), 203-226
[52] Taylor, S., Modeling financial time series, (1986), Wiley Chichester
[53] Tierney, L., Markov chains for exploring posterior distributions (with discussion), Annal of statistics, 21, 1701-1762, (1994) · Zbl 0829.62080
[54] Wang, J.; Genton, M., The multivariate skew-slash distribution, Journal of statistical planning and inference, 136, 209-220, (2006) · Zbl 1081.60013
[55] Watanabe, T.; Omori, Y., A multi-move sampler for estimate non-Gaussian time series model: comments on Shepard and pitt (1997), Biometrika, 91, 246-248, (2004) · Zbl 1132.62349
[56] Yu, J., On leverage in stochastic volatility model, Journal of econometrics, 127, 165-178, (2005) · Zbl 1335.91116
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