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Vector Autoregression (VAR) has been a standard empirical tool used in macroeconomics and finance. In this paper we discuss how to compare alternative VAR models after they are estimated by Bayesian MCMC methods. In particular we apply a robust version of deviance information criterion (RDIC) recently developed in Li, Zeng, and Yu (2014b) to determine the best candidate model. RDIC is a better information criterion than the widely used deviance information criterion (DIC) when latent variables are involved in candidate models. Empirical analysis using US data shows that the optimal model selected by RDIC can be different from that by DIC.

In this chapter we develop and implement a method for maximum simulated likelihood estimation of the continuous time stochastic volatility model with the constant elasticity of volatility. The approach does not require observations on option prices, nor volatility. To integrate out latent volatility from the joint density of return and volatility, a modified efficient importance sampling technique is used after the continuous time model is approximated using the Euler–Maruyama scheme. The Monte Carlo studies show that the method works well and the empirical applications illustrate usefulness of the method. Empirical results provide strong evidence against the Heston model.