This paper proposes a class of models that jointly model returns and ex-post variance measures under a Markov switching framework. Both univariate and multivariate return versions of the model are introduced. Bayesian estimation can be conducted under a fixed dimension state space or an infinite one. The proposed models can be seen as nonlinear common factor models subject to Markov switching and are able to exploit the information content in both returns and ex-post volatility measures. Applications to U.S. equity returns and foreign exchange rates compare the proposed models to existing alternatives. The empirical results show that the joint models improve density forecasts for returns and point predictions of return variance. The joint Markov switching models can increase the precision of parameter estimates and sharpen the inference of the latent state variable.Keywords: infinite hidden Markov model, realized covariance, density forecast, MCMC * We thank Qiao Yang for comments. Maheu is grateful to the SSHRC for financial support.

Variance estimation is central to many questions in finance and economics. Until now ex post variance estimation has been based on infill asymptotic assumptions that exploit high-frequency data. This article offers a new exact finite sample approach to estimating ex post variance using Bayesian nonparametric methods. In contrast to the classical counterpart, the proposed method exploits pooling over high-frequency observations with similar variances. Bayesian nonparametric variance estimators under no noise, heteroskedastic and serially correlated microstructure noise are introduced and discussed. Monte Carlo simulation results show that the proposed approach can increase the accuracy of variance estimation. Applications to equity data and comparison with realized variance and realized kernel estimators are included.

This paper suggests a new approach to evaluate realized covariance (RCOV) estimators via their predictive power on return density. By jointly modeling returns and RCOV measures under a Bayesian framework, the predictive density of returns and ex-post covariance measures are bridged. The forecast performance of a covariance estimator can be assessed according to its improvement in return density forecasting. Empirical applications to equity data show that several RCOV estimators consistently perform better than others and emphasize the importance of RCOV selection in covariance modeling and forecasting.

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