# 12-026/4 (2012-03-23)

Lennart Hoogerheide, VU University Amsterdam; Anne Opschoor, Erasmus University Rotterdam; Herman K. van Dijk, Erasmus University Rotterdam, and VU University Amsterdam
mixture of Student-t distributions, importance sampling, Kullback-Leibler divergence, Expectation Maximization, Metropolis-Hastings algorithm, predictive likelihood, DCC GARCH, mixture GARCH, instrumental variables
JEL codes:
C11, C22, C26

This discussion paper was published in the Journal of Econometrics (2012). Vol. 171(2), 101-120.

A class of adaptive sampling methods is introduced for efficient posterior and predictive simulation. The proposed methods are robust in the sense that they can handle target distributions that exhibit non-elliptical shapes such as multimodality and skewness. The basic method makes use of sequences of importance weighted Expectation Maximization steps in order to efficiently construct a mixture of Student-t densities that approximates accurately the target distribution - typically a posterior distribution, of which we only require a kernel - in the sense that the Kullback-Leibler divergence between target and mixture is minimized. We label this approach Mixture of t by Importance Sampling and Expectation Maximization (MitISEM). The constructed mixture is used as a candidate density for quick and reliable application of either Importance Sampling (IS) or the Metropolis-Hastings (MH) method. We also introduce three extensions of the basic MitISEM approach. First, we propose a method for applying MitISEM in a sequential manner. Second, we introduce a permutation-augmented MitISEM approach. Third, we propose a partial MitISEM approach, which aims at approximating the joint distribution by estimating a product of marginal and conditional distributions. This division can substantially reduce the dimension of the approximation problem, which facilitates the application of adaptive importance sampling for posterior simulation in more complex models with larger numbers of parameters. Our results indicate that the proposed methods can substantially reduce the computational burden in econometric models like DCC or mixture GARCH models and a mixture instrumental variables model.