On Bayesian Modelling of Fat Tails and Skewness

C. Fernández, M.F.J. Steel

Research output: Working paperDiscussion paperOther research output

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Abstract

We consider a Bayesian analysis of linear regression models that can account for skewed error distributions with fat tails.The latter two features are often observed characteristics of empirical data sets, and we will formally incorporate them in the inferential process.A general procedure for introducing skewness into symmetric distributions is first proposed.Even though this allows for a great deal of flexibility in distributional shape, tail behaviour is not affected.In addition, the impact on the existence of posterior moments in a regression model with unknown scale under commonly used improper priors is quite limited.Applying this skewness procedure to a Student-$t$ distribution, we generate a ``skewed Student'' distribution, which displays both flexible tails and possible skewness, each entirely controlled by a separate scalar parameter. The linear regression model with a skewed Student error term is the main focus of the paper: we first characterize existence of the posterior distribution and its moments, using standard improper priors and allowing for inference on skewness and tail parameters.For posterior inference with this model, a numerical procedure is suggested, using Gibbs sampling with data augmentation. The latter proves very easy to implement and renders the analysis of quite challenging problems a practical possibility.Two examples illustrate the use of this model in empirical data analysis.
Original languageEnglish
Place of PublicationTilburg
PublisherEconometrics
Number of pages31
Volume1996-58
Publication statusPublished - 1996

Publication series

NameCentER Discussion Paper
Volume1996-58

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Fat Tails
Bayesian Modeling
Skewness
Improper Prior
Linear Regression Model
Tail
Moment
Data Augmentation
Tail Behavior
t-distribution
Symmetric Distributions
Gibbs Sampling
Bayesian Analysis
Numerical Procedure
Error term
Posterior distribution
Regression Model
Data analysis
Flexibility
Scalar

Keywords

  • Bayesian statistics
  • linear regression model

Cite this

Fernández, C., & Steel, M. F. J. (1996). On Bayesian Modelling of Fat Tails and Skewness. (CentER Discussion Paper; Vol. 1996-58). Tilburg: Econometrics.
Fernández, C. ; Steel, M.F.J. / On Bayesian Modelling of Fat Tails and Skewness. Tilburg : Econometrics, 1996. (CentER Discussion Paper).
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Fernández, C & Steel, MFJ 1996 'On Bayesian Modelling of Fat Tails and Skewness' CentER Discussion Paper, vol. 1996-58, Econometrics, Tilburg.

On Bayesian Modelling of Fat Tails and Skewness. / Fernández, C.; Steel, M.F.J.

Tilburg : Econometrics, 1996. (CentER Discussion Paper; Vol. 1996-58).

Research output: Working paperDiscussion paperOther research output

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PY - 1996

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N2 - We consider a Bayesian analysis of linear regression models that can account for skewed error distributions with fat tails.The latter two features are often observed characteristics of empirical data sets, and we will formally incorporate them in the inferential process.A general procedure for introducing skewness into symmetric distributions is first proposed.Even though this allows for a great deal of flexibility in distributional shape, tail behaviour is not affected.In addition, the impact on the existence of posterior moments in a regression model with unknown scale under commonly used improper priors is quite limited.Applying this skewness procedure to a Student-$t$ distribution, we generate a ``skewed Student'' distribution, which displays both flexible tails and possible skewness, each entirely controlled by a separate scalar parameter. The linear regression model with a skewed Student error term is the main focus of the paper: we first characterize existence of the posterior distribution and its moments, using standard improper priors and allowing for inference on skewness and tail parameters.For posterior inference with this model, a numerical procedure is suggested, using Gibbs sampling with data augmentation. The latter proves very easy to implement and renders the analysis of quite challenging problems a practical possibility.Two examples illustrate the use of this model in empirical data analysis.

AB - We consider a Bayesian analysis of linear regression models that can account for skewed error distributions with fat tails.The latter two features are often observed characteristics of empirical data sets, and we will formally incorporate them in the inferential process.A general procedure for introducing skewness into symmetric distributions is first proposed.Even though this allows for a great deal of flexibility in distributional shape, tail behaviour is not affected.In addition, the impact on the existence of posterior moments in a regression model with unknown scale under commonly used improper priors is quite limited.Applying this skewness procedure to a Student-$t$ distribution, we generate a ``skewed Student'' distribution, which displays both flexible tails and possible skewness, each entirely controlled by a separate scalar parameter. The linear regression model with a skewed Student error term is the main focus of the paper: we first characterize existence of the posterior distribution and its moments, using standard improper priors and allowing for inference on skewness and tail parameters.For posterior inference with this model, a numerical procedure is suggested, using Gibbs sampling with data augmentation. The latter proves very easy to implement and renders the analysis of quite challenging problems a practical possibility.Two examples illustrate the use of this model in empirical data analysis.

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Fernández C, Steel MFJ. On Bayesian Modelling of Fat Tails and Skewness. Tilburg: Econometrics. 1996. (CentER Discussion Paper).