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COPYRIGHT 2003 American Statistical Association
This note compares a Bayesian Markov chain Monte Carlo approach implemented by Watanabe with a maximum likelihood ML approach based on an efficient importance sampling procedure to estimate dynamic bivariate mixture models. In these models, stock price volatility and trading volume am jointly directed by the unobservable number of price-relevant information arrivals, which is specified as a serially correlated random variable. It is shown that die efficient importance sampling technique is extremely accurate and that it produces results that differ significantly from those reported by Watanabe.
KEY WORDS: Bayesian posterior means; Efficient importance sampling; Latent variable; Markov chain Monte Carlo; Maximum likelihood.
1. INTRODUCTION
Watanabe (2000) performed a Bayesian analysis of dynamic bivariate mixture models for the Nikkei 225 stock index futures. Under these models, which were introduced by Tauchen and Pitts (1983) and modified by Andersen (1996), stock price volatility and trading volume are jointly determined by the unobservable flow of price-relevant information arrivals. In particular, the high persistence in stock price volatility typically found under autoregressive conditional heteroscedastic (ARCH) and stochastic volatility (SV) models is accounted for by serial correlation of the latent information arrival process. But because this process enters the models nonlinearly, the likelihood function and Bayesian posterior densities depend on high-dimensional interdependent integrals, whose evaluation requires the application of Monte Carlo (MC) integration techniques.
Watanabe (2000) used a Markov chain Monte Carlo (MCMC) integration technique, initially proposed by Jacquier, Polson, and Rossi (1994) in the context of univariate SV models. Specifically, he applied a variant of the Metropolis-Hasting acceptance-rejection algorithm proposed by Tierney (1994) (see also Chib and Greenberg 1995 for details) incorporating a multimove sampler proposed by Shephard and Pitt (1997). Details of this nontrivial algorithm were provided in appendix A of Watanabe's article. Although Watanabe (2000) did not report computing times, the MCMC algorithm is computer-intensive (he used 18,000 draws for each integral evaluation), and assessing its convergence is delicate in the presence of highly correlated variables. Note, in particular, that his results pass a convergence diagnostics (CD) test proposed by Geweke (1992).
Watanabe's Bayesian results for the bivariate mixture models are distinctly at odds with results found in the literature for other datasets. Specifically, his estimates of volatility persistence are very close to those found for returns alone under univariate models. In contrast, the generalized method of moments (GMM) estimates obtained by Andersen (1996) for the U.S. stock market unequivocally indicate that persistence drops significantly under bivariate specifications. This finding is confirmed by the study of Liesenfeld (1998) for German stocks. Liesenfeld (1998) computed maximum likelihood (ML) estimates using the accelerated Gaussian importance sampling (AGIS) MC integration technique proposed by Danielsson and Richard (1993). This technique has since been generalized into an efficient importance sampling (EIS) procedure by Richard and Zhang (1996, 1997) and Richard (1998).
In this present article, we rely on the EIS procedure to reestimate the Tauchen-Pitts model using Watanabe's data. Our ML estimates and posterior means all differ significantly from Watanabe's results and are fully consistent with the earlier findings of Andersen (1996) and Liesenfeld (1998). We show that EIS likelihood estimates are numerically accurate even though they are based on as few as 50 MC draws. We also demonstrate that Watanabe's results are the consequence of an apparent lack of convergence of the implemented MCMC algorithm in a single dimension of the parameter space. After discussions between the editors, Shephard, Watanabe, and ourselves, it now appears that the problem with Watanabe's implementation of the multimove sampler originates from a typographical error in the work of Shephard and Pitt (1997). (A corrigendum to this article is soon to appear in Biometrika.) In the meantime, Watanabe has also corrected his own implementation and recently informed us that he now obtains results that are very similar to ours (see Watanabe and Omori 2002 for further details). Hence the focus of this article is to illustrate how EIS contributed to detecting a problem of an MCMC implementation that passed standard convergence tests. But EIS is more than just a procedure to check MCMC results--it is a powerful MC integration technique on its own that should be considered as a potential alternative to MCMC.
The remainder of the article is organized as follows. Section 2 briefly reviews the bivariate mixture model of Tauchen and Pitts (1983), and Section 3 outlines the EIS procedure. Section 4 compares the MCMC results of Watanabe (2000) for the Tauchen-Pitts model with those obtained by EIS. Tentative explanations for the observed differences in these results are discussed. Section 5 summarizes our findings and concludes.
2. THE TAUCHEN-PITTS MODEL
Watanabe (2000) analyzed both the bivariate model of Tauchen and Pitts (1983) and a modified version thereof, proposed by Andersen (1996). Because the salient features of both models are similar, here we discuss only the Tauchen and Pitts version, which is given...
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