Gibbs Sampling for VAR-ARCH Models in Finance

Wolfgang Polasek
University of Basel
wolfgang@iso.iso.unibas.ch

Abstract

ARCH models have become an important tool for modeling volatility in finance. Simple estimation procedures do not obey the parameter constraints and their finite sample distribution is not known. Recent advances in Bayesian inferences allow an exact derivation of the complete posterior distribution by stochastic simulation. Based on Gibbs sampling and Markov Chain Monte Carlo methods (MCMC) we show how the posterior distribution for a Bayesian ARCH model can be derived. It will be shown that the model can be quite easily extended to cope with further aspects of volatility, like outliers, mixture distributions or t-distributions.

The paper compares different estimation procedures for Bayesian ARCH models: The Gibbs-importance algorithm (also called independence chain) and the data augmentation procedure via random coefficient ARCH models. Furthermore, extensions are discussed where the ARCH coefficients follow a hierarchical prior distribution which imposes constraints in form of tightness on the lag distribution. The simulated posterior distribution also allows the simulation of the predictive distribution which can be used for forecasting. Finally, an example is discussed where the exchange rates and interest rates of daily time series between the Deutschmark, Yen and the Dollar are analyzed.

Keywords: Bayesian ARCH models (B-ARCH), volatility and tightness models, Gibbs sampling, Markov Chain Monte Carlo methods.