Realized Volatility and Modeling Stock Returns as a Mixture of Normal Random Variables: the GARCH-Skew-t Model

This paper provides a new empirical guidance for modeling a skewed and fat-tailed error distribution underlying the traditional GARCH models for equity returns based on empirical findings on Realized Volatility (RV), constructed from the summation of higher-frequency squared (demeaned) returns. Based on an 80-year sample of U.S. daily stock market returns, I find that the distribution of monthly RV conditional on past returns is approximately the inverted-chi-square. I also find that monthly market returns, conditional on RV and past returns are normally distributed with RV in both mean and variance ) , ( RV RV N β μ + . These empirical findings serve as building blocks for the distributional assumption underlying the GARCH-skew-t model, which is derived as a mixture distribution of normal and inverted-chi-square in Kim & McCulloch (2007). Thus, this paper provides a new empirical support for the GARCH-skew-t modeling of equity returns. I show that the implied GARCH-skew-t model adequately describes the pattern of U.S. stock market returns. Moreover, the GARCH-skew-t accurately represents the three important stylized facts of stock market returns: volatility clustering, fat-tails and negative skewness. JEL Codes: C10, C22, G10