Robust bayesian inference in empirical regression models

Broadening the stochastic assumptions on the error terms of regression models was prompted by the analysis of linear multivariate t models in Zellner (1976). We consider a possible non-linear regression model under any multivariate elliptical data density, and examine Bayesian posterior and productive results. The latter are shown to be robust with respect to the specific choice of a sampling density within this elliptical class. In particular, sufficient conditions for such model robustness are that we single out a precision factor T2 on which we can specify an improper prior density. Apart from the posterior distribution of this nuisance parameter T 2, the entire analysis will then be completely unaffected by departures from Normality. Similar results hold in finite mixtures of such elliptical densities, which can be used to average out specification uncertainty

Bayesian econometrics:conjugate analysis and rejection sampling using mathematica

Mathematica is a powerful "system for doing mathematics by computer" which runs on personal computers (Macs and MS-DOS machines), workstations and mainframes. Here we show how Bayesian methods can be implemented in Mathematica. One of the drawbacks of Bayesian techniques is that they are computation-intensive, and every computation is a little different. Since Mathematica is so flexible, it can easily be adapted to solving a number of different Bayesian estimation problems. We illustrate the use of Mathematica functions (i) in a traditional conjugate analysis of the linear regression model and (ii) in a completely nonstandard model -where rejection sampling is used to sample from the posterior

A decision theoretic analysis of the unit root hypothesis using mixtures of elliptical models

This paper develops a formal decision theoretic approach to testing for a unit root in economic time series. The approach is empirically implemented by specifying a loss function based on predictive variances; models are chosen so as to minimize expected loss. In addition, the paper broadens the class of likelihood functions traditionally considered in the Bayesian unit root literature by: i) Allowing for departures from normality via the specification of a likelihood based on general elliptical densities; ii) allowing for structural breaks to occur; iii) allowing for moving average errors; and iv) using mixtures of various submodels to create a very flexible overall likelihood. Empirical results indicate that, while the posterior probability of trend-stationarity is quite high for most of the series considered, the unit root model is often selected in the decision theoretic analysis

Robust Bayesian inference in Iq-Spherical models

The class of multivariate lq-spherical distributions is introduced and defined through their isodensity surfaces. We prove that, under a Jeffreys' type improper prior on the scale parameter, posterior inference on the location parameters is the same for all lq-spherical sampling models with common q. This gives us perfect inference robustness with respect to any departures from the reference case of independent sampling from the exponential power distribution

Rejection sampling in demand systems

We illustrate the method of rejection sampling in a Bayesian application of a new approach toı estimating Demand Systems. This approach, suggested by Varian (1990), is based on a generalization of Afriat's (1967) efficiency index. Rejection sampling is applied to the prior-to-posterior mapping enabling us to obtain posterior results in a nonstandard model

Bayesian marginal equivalence of elliptical regression models.

The use of proper prior densities in regression models with multivariate non-Normal elliptical error distributions is examined when the scale matrix is known up to a precision factor T, treated as a nuisance parameter. Marginally equivalent models preserve the convenient predictive and posterior results on the parameter of interest B obtained in the reference case of the Normal model and its conditionally natural conjugate gamma prior. Prior densities inducing this property are derived for two special cases of non-Normal elliptical densities representing very different patterns of tail behavior. In a linear framework, so-called semi-conjugate prior structures are defined as leading to marginal equivalence to a Normal data density with a fully natural conjugate prior.Multivariate elliptical data densities; Proper priors; Model robustness; Student t density;

Mixtures of g-priors for bayesian model averaging with economic applications

We examine the issue of variable selection in linear regression have a potentially large amount of possible covariates and economic theory offers insufficient guidance on how to select the Model Averaging presents uncertainty. Our main interest here is the effect of the prior on the results, such as posterior inclusion probabilities of regressors and predictive performance. We combine a Binomial-Beta prior on model size with a g addition, we assign a hyperprior to g, as the choice impact on the results. For the prior of Beta shrinkage priors, which covers most choices in the recent literature. We propose a benchmark Beta prior, inspired by earlier findings with fixed g, and show it leads to selection. Inference is conducted through a Markov chain Monte Carlo sampler over model space and g. We examine the performance of the various priors in the context of simulated and real data. For the latter, we consider two important appl economics, namely cross-country growth regression and returns to schooling. Recommendations to applied users are provided.Consistency, Model uncertainty, Posterior odds, Prediction, Robustness

Bayesian inference with stochastic volatility models using continuous superpositions of non-Gaussian Ornstein-Uhlenbeck processes

This paper discusses Bayesian inference for stochastic volatility models based on continuous superpositions of Ornstein-Uhlenbeck processes. These processes represent an alternative to the previously considered discrete superpositions. An interesting class of continuous superpositions is defined by a Gamma mixing distribution which can define long memory processes. We develop efficient Markov chain Monte Carlo methods which allow the estimation of such models with leverage effects. This model is compared with a two-component superposition on the daily Standard and Poor's 500 index from 1980 to 2000.Leverage effect; Levy process; Long memory; Markov chain Monte Carlo; Stock price

Robust bayesian inference in empirical regression models.

Broadening the stochastic assumptions on the error terms of regression models was prompted by the analysis of linear multivariate t models in Zellner (1976). We consider a possible non-linear regression model under any multivariate elliptical data density, and examine Bayesian posterior and productive results. The latter are shown to be robust with respect to the specific choice of a sampling density within this elliptical class. In particular, sufficient conditions for such model robustness are that we single out a precision factor T2 on which we can specify an improper prior density. Apart from the posterior distribution of this nuisance parameter T 2, the entire analysis will then be completely unaffected by departures from Normality. Similar results hold in finite mixtures of such elliptical densities, which can be used to average out specification uncertainty.Multivariate elliptical data densities; Model robustness; Improper priors; Finite mixtures;

Posterior moments of scale parameters in elliptical regression models.

In the general multivariate elliptical class of data densities we define a scalar precision parameter r through a normalization of the scale matrix V. Using the improper prior on r which preserves the results under Normality for all other parameters and prediction, we consider the posterior moments of r. For the subclass of scale mixtures of Normals we derive the Bayesian counterpart to a sampling theory result concerning uniformly minimum variance unbiased estimation of 7. 2 • If the sampling variance exists, we single out the common variance factor i' as the scalar multiplying V in this sampling variance. Moments of i' are examined for various elliptical subclasses and a sampling theory result regarding its unbiased estimation is mirrored.Multivariate elliptical data densities; Bayesian analysis; Unbiased estimation;