# zbMATH — the first resource for mathematics

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.

##### MSC:
 91G70 Statistical methods; risk measures 62F35 Robustness and adaptive procedures (parametric inference) 62F15 Bayesian inference 91B70 Stochastic models in economics
Scythe
Full Text:
##### References:
 [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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.