Volatility Processes and Volatility Forecast with Long Memory

Abstract

We introduce a new family of processes that include the long memory (power law) in the volatility correlation. This is achieved by measuring the historical volatility on a set of increasing time horizons and by computing the resulting effective volatility by a sum with power law weights. In the limit where only one component is included, the models are equivalent to GARCH(1,1) and I-GARCH(1). The models have 2 parameters (integrated processes) or 4 parameters (mean reverting processes). Volatility forecast in the context of quadratic processes is discussed, in particular as a mean to estimate process parameters. Using hourly data, the empirical properties of the new models are compared to existing models (GARCH, FIGARCH, ...), in particular log-likelihood estimates and volatility forecast errors. These studies show the advantage of the long memory processes as they give a good description of the empirical data from 1 hour to 1 month, with the same parameters.

Keywords: long memory volatility processes, ARCH model, volatility forecast

JEL Classification: C22