Long Memory in Stochastic Volatility

Recent studies have suggested that stock markets' volatility has a type of long-range dependence that is not appropriately described by the usual Generalized Autoregressive Conditional Heteroskedastic (GARCH) and Exponential GARCH (EGARCH) models. In this paper, diierent models for describing this long-range dependence are examined and the properties of a Long-Memory Stochastic Volatility (LMSV) model, constructed by incorporating an Autoregressive Fractionally Integrated Moving Average (ARFIMA) process in a stochastic volatility scheme, are discussed. Strongly consistent estimators for the parameters of this LMSV model are obtained by maximizing the spectral likelihood. The distribution of the estimators is analyzed by means of a Monte Carlo study. The LMSV is applied to daily stock market returns providing an improved description of the volatility behavior. In order to assess the empirical relevance of this approach, tests for long-memory volatility are described and applied to an extensive set of stock market series, presenting substantial evidence for the existence of long-memory volatility. 1 The authors had the privilege of presenting this paper at the conference in honor of Professor Carl Christ held at the Johns Hopkins University in April 1995. We are thankful to Patrick Asea and participants at the conference, as well as to Francis Diebold, Thomas Epps, and Roger Pinkham for helpful comments and suggestions which led to improvements of this paper. Any remaining insuuciencies are our own.