Metropolis-Hastings Algorithms for DSGE Models

This chapter talks about the most widely used method to generate draws from posterior distributions of a DSGE model: the random walk MH (RWMH) algorithm. The DSGE model likelihood function in combination with the prior distribution leads to a posterior distribution that has a fairly regular elliptical shape. In turn, the draws from a simple RWMH algorithm can be used to obtain an accurate numerical approximation of posterior moments. However, in many other applications, particularly those involving medium- and large-scale DSGE models, the posterior distributions could be very non-elliptical. Irregularly shaped posterior distributions are often caused by identification problems or misspecification. In lieu of the difficulties caused by irregularly shaped posterior surfaces, the chapter reviews various alternative MH samplers, which use alternative proposal distributions.

Bayesian Inference of a Multivariate Regression Model

We explore Bayesian inference of a multivariate linear regression model with use of a flexible prior for the covariance structure. The commonly adopted Bayesian setup involves the conjugate prior, multivariate normal distribution for the regression coefficients and inverse Wishart specification for the covariance matrix. Here we depart from this approach and propose a novel Bayesian estimator for the covariance. A multivariate normal prior for the unique elements of the matrix logarithm of the covariance matrix is considered. Such structure allows for a richer class of prior distributions for the covariance, with respect to strength of beliefs in prior location hyperparameters, as well as the added ability, to model potential correlation amongst the covariance structure. The posterior moments of all relevant parameters of interest are calculated based upon numerical results via a Markov chain Monte Carlo procedure. The Metropolis-Hastings-within-Gibbs algorithm is invoked to account for the construction of a proposal density that closely matches the shape of the target posterior distribution. As an application of the proposed technique, we investigate a multiple regression based upon the 1980 High School and Beyond Survey.

BAYESIAN REGRESSION ANALYSIS WITH SCALE MIXTURES OF NORMALS

This paper considers a Bayesian analysis of the linear regression model under independent sampling from general scale mixtures of normals. Using a common reference prior, we investigate the validity of Bayesian inference and the existence of posterior moments of the regression and scale parameters. We find that whereas existence of the posterior distribution does not depend on the choice of the design matrix or the mixing distribution, both of them can crucially intervene in the existence of posterior moments. We identify some useful characteristics that allow for an easy verification of the existence of a wide range of moments. In addition, we provide full characterizations under sampling from finite mixtures of normals, Pearson VII, or certain modulated normal distributions. For empirical applications, a numerical implementation based on the Gibbs sampler is recommended.