Shirota, Shinichiro, Takayuki Hizu and Yasuhiro Omori

The daily return and the realized volatility are simultaneously modeled in the stochastic volatility model with leverage and long memory. The dependent variable in the stochastic volatility model is the logarithm of the squared return, and its error distribution is approximated by a mixture of normals. In addition, we incorporate the logarithm of the realized volatility into the measurement equation, assuming that the latent log volatility follows an Autoregressive Fractionally Integrated Moving Average (ARFIMA) process to describe its long memory property. Using a state space representation, we propose an ecient Bayesian estimation method implemented using Markov chain Monte Carlo method (MCMC). Model comparisons are performed based on the marginal likelihood, and the volatility forecasting performances are investigated using S&P500 stock index returns.