Bayesian Variations on the Frisch and Waugh Theme

The paper is devoted to discussing consequences of the so-called Frisch-Waugh Theorem to posterior inference and Bayesian model comparison. We adopt a generalised normal linear regression framework and weaken its assumptions in order to cover non-normal, jointly elliptical sampling distributions, autoregressive specifications, additional nuisance parameters and multi-equation SURE or VAR models. The main result is that inference based on the original full Bayesian model can be obtained using transformed data and reduced parameter spaces, provided the prior density for scale or precision parameters is appropriately modified.Bayesian inference, regression models, SURE models, VAR processes, data transformations

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

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

Bayesian Value-at-Risk for a Portfolio: Multi- and Univariate Approaches Using MSF-SBEKK Models

The s-period ahead Value-at-Risk (VaR) for a portfolio of dimension n is considered and its Bayesian analysis is discussed. The VaR assessment can be based either on the n-variate predictive distribution of future returns on individual assets, or on the univariate Bayesian model for the portfolio value (or the return on portfolio). In both cases Bayesian VaR takes into account parameter uncertainty and non-linear relationship between ordinary and logarithmic returns. In the case of a large portfolio, the applicability of the n-variate approach to Bayesian VaR depends on the form of the statistical model for asset prices. We use the n-variate type I MSF-SBEKK(1,1) volatility model proposed specially to cope with large n. We compare empirical results obtained using this multivariate approach and the much simpler univariate approach based on modelling volatility of the value of a given portfolio.Bayesian econometrics, risk analysis, multivariate GARCH processes, multivariate SV processes, hybrid SV-GARCH models

Bayesian Analysis for Hybrid MSF-SBEKK Models of Multivariate Volatility

The aim of this paper is to examine the empirical usefulness of two new MSF - Scalar BEKK(1,1) models of n-variate volatility. These models formally belong to the MSV class, but in fact are some hybrids of the simplest MGARCH and MSV specifications. Such hybrid structures have been proposed as feasible (yet non-trivial) tools for analyzing highly dimensional financial data (large n). This research shows Bayesian model comparison for two data sets with n = 2, since in bivariate cases we can obtain Bayes factors against many (even unparsimonious) MGARCH and MSV specifications. Also, for bivariate data, approximate posterior results (based on preliminary estimates of nuisance matrix parameters) are compared to the exact ones in both MSF-SBEKK models. Finally, approximate results are obtained for a large set of returns on equities (n = 34).Bayesian econometrics, Gibbs sampling, time-varying volatility, multivariate GARCH processes, multivariate SV processes

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.Bayesian inference; Exponential power distributions; Inference robustness; lq-norm; Symmetric multivariate distributions;

Posterior inference on long-run impulse responses

This paper describes a Bayesian analysis of impulse response functions. We show how many common priors imply that posterior densities for impulse responses at long horizons have no moments. Our results suggest that impulse responses should be assessed on the basis of their full posterior densities, and that standard estimates such as posterior means, variances or modes may be very misleading

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;

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;