Bayesian modeling of autoregressive partial linear models with scale mixture of normal errors

Abstract

Normality and independence of error terms are typical assumptions for partial linear models. However, these assumptions may be unrealistic in many fields, such as economics, finance and biostatistics. In this paper, a Bayesian analysis for partial linear model with first-order autoregressive errors belonging to the class of the scale mixtures of normal distributions is studied in detail. The proposed model provides a useful generalization of the symmetrical linear regression model with independent errors, since the distribution of the error term covers both correlated and thick-tailed distributions, and has a convenient hierarchical representation allowing easy implementation of a Markov chain Monte Carlo scheme. In order to examine the robustness of the model against outlying and influential observations, a Bayesian case deletion influence diagnostics based on the Kullback–Leibler (K–L) divergence is presented. The proposed method is applied to monthly and daily returns of two Chilean companies.

Keywords:

• autoregressive time series model
• Gibbs sampler
• influential observations
• partial linear models
• scale mixtures of normal distributions

Acknowledgements

We thank the Editor, an Associate Editor and two referees whose constructive comments led to a much improved presentation. The first author would like to thank the support from ECOS-CONICYT C10E03 and DIUC No. 213.014.021-1.0 from the Universidad de Concepción. L.M. Castro acknowledges funding support by Grant FONDECYT 1130233 from the Chilean government. The research of V.H. Lachos was supported by Grant 3305054/2011-2 from Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq-Brazil) and Grant 2011/17400-6 from Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP-Brazil). Ronaldo Dias acknowledges the support from CNPq-Brazil