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(1) 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 and to perform well numerically. 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 observtions of fBm. In keeping with the popular …nancial interpretation of ARCH models, all estimators are based only on observation of the “returns”of the model, not on their “volatilities”. 1 Introduction Long-memory behavior is one of the most important empirical properties exhibited by …nancial time series, such as asset returns and exchange rates. It is well known that, for the most part, the values of such a time series rt, t 2 N are uncorrelated but not independent, with most of dependency “hidden”within some nonlinear functions of rt, such as r t or jrtj. Historically, this has been modeled by conditional variance (volatility) models, such as the models traditionally included in the so-called (G)ARCH framework (see Gourieroux (1997) [20] and also Ghysels, Harvey, Renault (1996) [21]). However, typically, these models possess the so-called short memory property, and more speci…cally, exponential decay in autocorrelations of the respective nonlinear function of rt, such as r t . A symptomatic situation is found in Dan Nelson’s well-known convergence results of ARCH/GARCH models to stochastic volatility models (see Nelson (1990) [34]). The linear autoregressive conditional heteroscedasticity model (LARCH), …rst introduced in Robinson (1991) [35], has long been considered a very convenient vehicle for long-memory modeling. Its name is probably due to Giraitis, Robinson and Surgailis (2000) [17]. This model can be described as rt = t"t; 2 t = @a+ 1 X j=1 bjrt j 1A2 ; t 2 Z (1) Key words and phrases: ARCH, time series, fractional Brownian motion, maximum likelihood estimator, long memory Whittle estimator, moving average MSC 2000 subject classi…cations: primary 62M09; secondary 60G18, 62M10, 91B84. yDept. Statistics, Purdue University 150 N. University St., West Lafayette, IN 47907-2067, USA [email protected] [email protected] zDepto de Estadística –CIMFAV, Universidad de Valparaíso, 1091 Av. Gran Bretaña, Playa Ancha, Valparaíso, Chile [email protected]