Estimators for the long-memory parameter in LARCH models, and fractional Brownian motion

This paper investigates several strategies for consistently estimating the so-called Hurst parameter H responsible for the long-memory correlations in a linear class of ARCH time series, known as LARCH(∞) models, as well as in the continuous-time Gaussian stochastic process known as fractional Brownian motion (fBm). A LARCH model’s parameter is estimated using a conditional maximum likelihood method, which is proved to have good stability properties. A local Whittle estimator is also discussed. The article further proposes a specially designed conditional maximum likelihood method for estimating the H which is closer in spirit to one based on discrete observations of fBm. In keeping with the popular financial interpretation of ARCH models, all estimators are based only on observation of the “returns” of the model, not on their “volatilities”.