This paper proposes a Bayesian approach to a vector autoregression with stochastic volatility, where the multiplicative evolution of the precision matrix is driven by a multivariate beta variate. Exact updating formulas are given to the nonlinear filtering of the precision matrix. Estimation of the autoregressive parameters requires numerical methods: an importance-sampling based approach is explained here.

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The long range dependence paradigm appears to be a suitable description of the data generating process for many observed economic time series. This is mainly due to the fact that it naturally characterizes time series displaying a high degree of persistence, in the form of a long lasting effect of unanticipated shocks, yet exhibiting mean reversion. Whereas linear long range dependent time series models have been extensively used in macroeconomics, empirical evidence from financial time series prompted the development of nonlinear long range dependent time series models, in particular models of changing volatility.We discuss empirical evidence of long range dependence as well as the theoretical issues, both for economics and econometrics, such evidence has stimulated.

In the empirical analysis of financial time series, multivariate GARCH models have been used in various forms.As it is typical for nonlinear models there is yet no unique framework available to uncover dynamic covariance relationships for vector return processes.We introduce a new concept of impulse response functions tracing the e#ects of independent shocks on volatility through time.The advocated methodology avoids typical orthogonalization and ordering problems.Theoretical properties of volatility impulse response functions are derived and compared with conditional moment profiles introduced by Gallant, Rossi and Tauchen(1LLM for semi-nonparametric models.In an empirical study of a bivariate foreign exchange rate series we use volatility impulse response functions to compare alternative parametric volatility specifications.It is shown that for shocks a#ecting foreign exchange rates in an asymmetric way, the di#erence between our methodology and conditional volatility profiles can be substantial. Keywords: Multivariate GARCH, impulse response, exchange rate, volatility JEL Classification: C22 1 Electrabel, R&D Energy Markets, Place de l'Universite1E B-1J: Louvain-la-Neuve, Belgium.email christian.haff . 2 Institut fur Statistik und Okonometrie, Wirtschaftswissenschaftliche Fakultat, Humboldt--Universitat zu Berlin, SpandauerStr.1 D-11u Berlin, Germany. e-mail helmut@wiwi.hu-berlin.de. This paper is a revised version of CORE discussion paper 9847.Both authors would like to thank Luc Bauwens, Jeroen Rombouts and Helmut Lutkepohl for helpful comments.Financial support by the Belgian Government and the Deutsche Forschungsgemeinschaft is gratefully acknowledged. This text presents research results of the Belgian Program on Interuniversity Poles of At...