# 99-082/4 (1999-11-02)

Luc Bauwens, CORE, Université Catholique de Louvain; Charles S. Bos, Erasmus University Rotterdam; Herman K. van Dijk, Econometric Institute, Erasmus University Rotterdam
Markov chain Monte Carlo; simulation; polar coordinates; GARCH; ill-behaved posterior; Value-at-Risk
JEL codes:
C11; C15; C63

Adaptive Polar Sampling (APS) is proposed as a Markov chain Monte Carlomethod for Bayesian analysis of models with ill-behaved posteriordistributions. In order to sample efficiently from such a distribution,a location-scale transformation and a transformation to polarcoordinates are used. After the transformation to polar coordinates, aMetropolis-Hastings algorithm is applied to sample directions and,conditionally on these, distances are generated by inverting the CDF.A sequential procedure is applied to update the location and scale.Tested on a set of canonical models that feature nearnon-identifiability, strong correlation, and bimodality, APS comparesfavourably with the standard Metropolis-Hastings sampler in terms ofparsimony and robustness. APS is applied within a Bayesian analysisof a GARCH-mixture model which is used for the evaluation of theValue-at-Risk of the return of the Dow Jones stock index.