Long and short memory conditional heteroskedasticity in estimating the memory parameter of levels

Semiparametric estimates of long memory seem useful in the analysis of long financial time series because they are consistent under much broader condi tions than parametric estimates. However, recent large sample theory for se miparametric estimates forbids conditional heteroskedasticity, We show that a leading semiparametric estimate, the Gaussian or local Whittle one, can be consistent and have the same limiting distribution under conditional het eroskedasticity as under the conditional homoskedasticity assumed by Robins on (1995, Annals of Statistics 23, 1630-61), Indeed, noting that long memor y has been observed in the squares of financial time series, we allow, unde r regularity conditions, for conditional heteroskedasticity of the general form introduced by Robinson (1991, Journal of Econometrics 47, 67-84), whic h may include long memory behavior for the squares, such as the fractional noise and autoregressive fractionally integrated moving average form, and a lso standard short memory ARCH and GARCH specifications.