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On Monte Carlo methods for Bayesian multivariate regression models with heavy-tailed errors

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  • Roy, Vivekananda
  • Hobert, James P.

Abstract

We consider Bayesian analysis of data from multivariate linear regression models whose errors have a distribution that is a scale mixture of normals. Such models are used to analyze data on financial returns, which are notoriously heavy-tailed. Let [pi] denote the intractable posterior density that results when this regression model is combined with the standard non-informative prior on the unknown regression coefficients and scale matrix of the errors. Roughly speaking, the posterior is proper if and only if n>=d+k, where n is the sample size, d is the dimension of the response, and k is number of covariates. We provide a method of making exact draws from [pi] in the special case where n=d+k, and we study Markov chain Monte Carlo (MCMC) algorithms that can be used to explore [pi] when n>d+k. In particular, we show how the Haar PX-DA technology studied in Hobert and Marchev (2008) [11] can be used to improve upon Liu's (1996) [7] data augmentation (DA) algorithm. Indeed, the new algorithm that we introduce is theoretically superior to the DA algorithm, yet equivalent to DA in terms of computational complexity. Moreover, we analyze the convergence rates of these MCMC algorithms in the important special case where the regression errors have a Student's t distribution. We prove that, under conditions on n, d, k, and the degrees of freedom of the t distribution, both algorithms converge at a geometric rate. These convergence rate results are important from a practical standpoint because geometric ergodicity guarantees the existence of central limit theorems which are essential for the calculation of valid asymptotic standard errors for MCMC based estimates.

Suggested Citation

  • Roy, Vivekananda & Hobert, James P., 2010. "On Monte Carlo methods for Bayesian multivariate regression models with heavy-tailed errors," Journal of Multivariate Analysis, Elsevier, vol. 101(5), pages 1190-1202, May.
  • Handle: RePEc:eee:jmvana:v:101:y:2010:i:5:p:1190-1202
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    References listed on IDEAS

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    1. repec:bla:jfinan:v:44:y:1989:i:4:p:889-908 is not listed on IDEAS
    2. Zhou, Guofu, 1993. "Asset-Pricing Tests under Alternative Distributions," Journal of Finance, American Finance Association, vol. 48(5), pages 1927-1942, December.
    3. Jones, Galin L. & Haran, Murali & Caffo, Brian S. & Neath, Ronald, 2006. "Fixed-Width Output Analysis for Markov Chain Monte Carlo," Journal of the American Statistical Association, American Statistical Association, vol. 101, pages 1537-1547, December.
    4. Marchev, Dobrin & Hobert, James P., 2004. "Geometric Ergodicity of van Dyk and Meng's Algorithm for the Multivariate Student's t Model," Journal of the American Statistical Association, American Statistical Association, vol. 99, pages 228-238, January.
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    Cited by:

    1. Jung, Yeun Ji & Hobert, James P., 2014. "Spectral properties of MCMC algorithms for Bayesian linear regression with generalized hyperbolic errors," Statistics & Probability Letters, Elsevier, vol. 95(C), pages 92-100.
    2. Qin, Qian & Hobert, James P., 2018. "Trace-class Monte Carlo Markov chains for Bayesian multivariate linear regression with non-Gaussian errors," Journal of Multivariate Analysis, Elsevier, vol. 166(C), pages 335-345.
    3. Li, Haoxiang & Qin, Qian & Jones, Galin L., 2024. "Convergence analysis of data augmentation algorithms for Bayesian robust multivariate linear regression with incomplete data," Journal of Multivariate Analysis, Elsevier, vol. 202(C).
    4. Choi, Hee Min & Hobert, James P., 2013. "Analysis of MCMC algorithms for Bayesian linear regression with Laplace errors," Journal of Multivariate Analysis, Elsevier, vol. 117(C), pages 32-40.

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