Volatility processes and volatility forecast with long memory

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 volatilities on a set of increasing time horizons and by computing the resulting effective volatility by a sum with power law weights. The processes have 2 parameters (linear processes) or 4 parameters (affine processes). In the limit where only one component is included, the processes are equivalent to GARCH(1,1) and I-GARCH(1). Volatility forecast is discussed in the context of processes with quadratic equations, in particular as a mean to estimate process parameters. Using hourly data, the empirical properties of the new processes are compared to existing processes (GARCH, I-GARCH, FIGARCH, ...), in particular log-likelihood estimates and volatility forecast errors. This study covers time horizons ranging from 1 hour to 1 month. We also study the variation of the estimated parameters with respect to changing sample by introducing a natural coordinate invariant distance. The long memory processes show a small but systematic quantitative improvement with respect to the standard GARCH(1,1) process. Yet, the main advantage of the new long memory processes is that they give a good description of the empirical data from 1 hour to 1 month, with the same parameters. Their other advantages is that they are efficient to evaluate numerically, that they behave well with respect to the cut-off (i.e. the largest time horizon included in the process) and that they can be extended along several directions. JEL: C22