Averaged Periodogram Spectral Estimation with Long‐memory Conditional Heteroscedasticity

Abstract

The empirical relevance of long‐memory conditional heteroscedasticity has emerged in a variety of studies of long time series of high frequency financial measurements. A reassessment of the applicability of existing semiparametric frequency domain tools for the analysis of time dependence and long‐run behaviour of time series is therefore warranted. To that end, in this paper the averaged periodogram statistic is analysed in the framework of a generalized linear process with long‐memory conditional heteroscedastic innovations according to a model specification first proposed by Robinson (Testing for strong serial correlation and dynamic conditional heteroscedasticity in multiple regression. J. Economet. 47 (1991), 67–84). It is shown that the averaged periodogram estimate of the spectral density of a short‐memory process remains asymptotically normal with unchanged asymptotic variance under mild moment conditions, and that for strongly dependent processes Robinson's averaged periodogram estimate of long memory (Semiparametric analysis of long memory time series. Ann. Stat. 22 (1994), 515–39) remains consistent.