Skip to main content
SpringerLink
Log in

Bayesian Estimation of a Skew-Student-t Stochastic Volatility Model

  • Published:
Methodology and Computing in Applied Probability Aims and scope Submit manuscript

Abstract

In this paper we present a stochastic volatility (SV) model assuming that the return shock has a skew-Student-t distribution. This allows a parsimonious, flexible treatment of skewness and heavy tails in the conditional distribution of returns. An efficient Markov chain Monte Carlo (MCMC) algorithm is developed and used for parameter estimation and forecasting. The MCMC method exploits a skew-normal mixture representation of the error distribution with a gamma distribution as the mixing distribution. The developed methodology is applied to the NASDAQ daily index returns. Bayesian model selection criteria as well as out-of-sample forecasting in a value-at-risk (VaR) study reveal that the SV model based on skew-Student-t distribution provides significant improvement in model fit as well as prediction to the NASDAQ index data over the usual normal model.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Abanto-Valle CA, Bandyopadhyay D, Lachos VH, Enriquez I (2010) Robust Bayesian analysis of heavy-tailed stochastic volatility models using scale mixtures of normal distributions. Comput Stat Data Anal 54:2883–2898

    Article  MathSciNet  MATH  Google Scholar 

  • Abanto-Valle CA, Migon HS, Lachos VH (2011) Stochastic volatility in mean models with scale mixtures of normal distributions and correlated errors: a bayesian approach. J Stat Plan Infer 141:1875–1887

    Article  MathSciNet  MATH  Google Scholar 

  • Abanto-Valle CA, Migon HS, Lachos VH (2012) Stochastic volatility in mean models with heavy-tailed distributions. Braz J Probab Stat 26:402–422

    Article  MathSciNet  MATH  Google Scholar 

  • Ando T (2006) Bayesian inference for nonlinear and non-gaussian stochastic volatility model wit leverge effect. J Jpn Stat Soc 36:173–197

    Article  MATH  Google Scholar 

  • Ando T (2007) Bayesian predictive information criterion for the evaluation of hierarchical Bayesian and empirical Bayes models. Biometrika 94:443–458

    Article  MathSciNet  MATH  Google Scholar 

  • Azzalini A (1986) Further results on a class of distributions which include the normal ones. Statistica 46:199–208

    MathSciNet  MATH  Google Scholar 

  • Azzalini A (2005) The skew-normal distribution and related multivariate families. Scand J Stat 32:159–188

    Article  MathSciNet  MATH  Google Scholar 

  • Azzalini A, Capitanio A (2003) Distributions generated by perturbation of symmetry with emphasis on a multivariate skew-t distribution. J R Stat Soc: Ser B 65:367–389

    Article  MathSciNet  MATH  Google Scholar 

  • Bayes CL, Branco MD (2007) Bayesian inference for the skewness parameter of the scalar skew-normal distribution. Braz J Probab Stat 21:141–163

    MathSciNet  Google Scholar 

  • Branco MD, Dey DK (2001) A general class of multivariate skew-elliptical distributions. J Multivar Anal 79:99–113

    Article  MathSciNet  MATH  Google Scholar 

  • Cappuccio N, Lubian D, Raggi D (2004) Mcmc bayesian estimation of a skew-GED stochastic volatility model. Stud Nonlinear Dyn Econ 8:article 6

  • Cappuccio N, Lubian D, Raggi D (2006) Investigating asymmetry in U.S. stock market indexes: evidence from a stochastic volatility model. Appl Financ Econ 16:479–490

    Article  Google Scholar 

  • Carter CK, Kohn R (1994) On Gibbs sampling for state space models. Biometrika 81:541–553

    Article  MathSciNet  MATH  Google Scholar 

  • Chen Q, Gerlach R, Lub Z (2012) Bayesian value-at-risk and expected shortfall forecasting via the asymmetric laplace distribution. Comput Stat Data Anal 56:3498–3516

    Article  MATH  Google Scholar 

  • Chib S, Greenberg E (1995) Understanding the Metropolis-Hastings algorithm. Am Stat 49:327–335

    Google Scholar 

  • Chib S, Nardari F, Shepard N (2002) Markov Chain Monte Carlo methods for stochastic volatility models. J Econ 108:281–316

    Article  MATH  Google Scholar 

  • Christoffersen P (1998) Evaluating interval forecasts. Int Econ Rev 39:841–862

    Article  MathSciNet  Google Scholar 

  • Christoffersen P, Pelletier D (2004) Backtesting value-at-risk: a duration-based approach. J Financ Econ 2:84–108

    Google Scholar 

  • de Jong P, Shephard N (1995) The simulation smoother for time series models. Biometrika 82:339–350

    Article  MathSciNet  MATH  Google Scholar 

  • Delatola EI, Griffin JE (2011) Bayesian nonparametric modelling of the return distribution with stochastic volatility. Bayesian Anal 6:901–926

    Article  MathSciNet  Google Scholar 

  • Fonseca TCO, Ferreira MAR, Migon HS (2008) Objective Bayesian analysis for the student-t regression model. Biometrika 95:325–333

    Article  MathSciNet  MATH  Google Scholar 

  • Frühwirth-Schnater S (1994) Data augmentation and dynamic linear models. J Time Ser Anal 15:183–202

    Article  MathSciNet  Google Scholar 

  • Geweke J (1992) Evaluating the accuracy of sampling-based approaches to the calculation of posterior moments. In: Bernardo JM, Berger JO, Dawid AP, Smith AFM (eds) Bayesian statistics, vol 4. Oxford University Press, Oxford, pp 169–193

    Google Scholar 

  • Gneiting T, Raftery AE (2007) Strictly proper scoring rules, prediction and estimation. J Am Stat Assoc 6:901–926

    MathSciNet  Google Scholar 

  • Good IJ (1952) Rational decisions. J R Stat Soc, Ser B 14:107–114

    MathSciNet  Google Scholar 

  • Hansen BE (1994) Autoregressive conditional density estimation. J Financ 35:705–730

    MATH  Google Scholar 

  • Harvey C, Sidique A (1999) Autoregressive conditional skewness. J Financ Quant Anal 34:465–487

    Article  Google Scholar 

  • Harvey C, Sidique A (2000) Conditional skewness in asset pricing tests. J Financ 55:1263–1296

    Article  Google Scholar 

  • Henze N (1986) A probabilistic representation of the skew-normal distribution. Scand J Stat 13:271–275

    MathSciNet  MATH  Google Scholar 

  • Jacquier E, Polson N, Rossi P (2004) Bayesian analysis of stochastic volatility models with fat-tails and correlated errors. J Econ 122:185–212

    Article  MathSciNet  Google Scholar 

  • Jondeau E, Rockinger M (2003) Conditional volatility, skewness and kurtosis: existence, persistence and co-movements. J Econ Dyn Control 27:1699–1737

    Article  MathSciNet  MATH  Google Scholar 

  • Kim S, Shepard N, Chib S (1998) Stochastic volatility: likelihood inference and comparison with ARCH models. Rev Econ Stud 65:361–393

    Article  MATH  Google Scholar 

  • Koopman S (1993) Disturbance smoothers for State Space models. Biometrika 80:117–126

    Article  MathSciNet  MATH  Google Scholar 

  • Kupiec PH (1995) Techniques for verifying the accuracy of risk measurement models. J Deriv 3:73–84

    Article  Google Scholar 

  • Liesenfeld R, Jung RC (2000) Stochastic volatility models: conditional normality versus heavy-tailed distrutions. J Appl Econ 15:137–160

    Article  Google Scholar 

  • McAleer M, da Veiga B (2008) Single-index and portfolio models for forecasting value-at-risk thresholds. J Forecast 27:217–235

    Article  MathSciNet  Google Scholar 

  • Melino A, Turnbull SM (1990) Pricing foreign options with stochastic volatility. J Econ 45:239–265

    Article  Google Scholar 

  • Mittnik S, Paolella MS (2000) Conditional density and value-at-risk prediction of asian currency exchange rates. J Forecast 19:313–333

    Article  Google Scholar 

  • Nakajima J, Omori Y (2012) Stochastic volatility model with leverage and asymmetrically heavy-tailed error using gh skew student’s t-distribution. Comput Stat Data Anal 56:3690–3704

    Article  MathSciNet  MATH  Google Scholar 

  • Pemstein D, Quinn KV, Martin AD (2007) The Scythe statistical library: an open source C++ library for statistical computation. J Stat Softw VV(II):1–29

    Google Scholar 

  • Pitt M, Shephard N (1999) Filtering via simulation: auxiliary particle filter. J Am Stat Assoc 94:590–599

    Article  MathSciNet  MATH  Google Scholar 

  • Rosa GJM, Padovani CR, Gianola D (2003) Robust linear mixed models with Normal/Independent distributions and bayesian MCMC implementation. Biom J 45:573–590

    Article  MathSciNet  Google Scholar 

  • Shephard N, Pitt M (1997) Likelihood analysis of non-Gaussian measurements time series. Biometrika 84:653–667

    Article  MathSciNet  MATH  Google Scholar 

  • Taylor S (1982) Financial returns modelled by the product of two stochastic processes-a study of the daily sugar prices 1961–75. In: Anderson O (ed) Time series analysis: theory and practice, vol 1. North-Holland, Amsterdam, pp 203–226

    Google Scholar 

  • Taylor S (1986) Modeling financial time series. Wiley, Chichester

    Google Scholar 

  • Tierney L (1994) Markov chains for exploring posterior distributions (with discussion). Ann Stat 21:1701–1762

    Article  MathSciNet  Google Scholar 

  • Tsiotas G (2012) On generalised asymmetric stochastic volatility models. Comput Stat Data Anal 56:151–172

    Article  MathSciNet  MATH  Google Scholar 

  • Watanabe T, Omori Y (2004) A multi-move sampler for estimate non-Gaussian time series model: comments on Shepard and Pitt (1997). Biometrika 91:246–248

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Statistics, Federal University of Rio de Janeiro, CP 68530, Rio de Janeiro, CEP 21945-970, RJ, Brazil

    C. A. Abanto-Valle

  2. Department of Statistics, Campinas State University, CP 6065, Campinas, CEP 13083-859, SP, Brazil

    V. H. Lachos

  3. Department of Statistics, University of Connecticut, Storrs, CT, USA

    Dipak K. Dey

Authors
  1. C. A. Abanto-Valle
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. V. H. Lachos
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Dipak K. Dey
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Dipak K. Dey.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Abanto-Valle, C.A., Lachos, V.H. & Dey, D.K. Bayesian Estimation of a Skew-Student-t Stochastic Volatility Model. Methodol Comput Appl Probab 17, 721–738 (2015). https://doi.org/10.1007/s11009-013-9389-9

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11009-013-9389-9

Keywords

AMS 2000 Subject Classifications

Search

Navigation