ASYMPTOTICS FOR THE LOW‐FREQUENCY ORDINATES OF THE PERIODOGRAM OF A LONG‐MEMORY TIME SERIES

Abstract

Abstract. We consider the asymptotic distribution of the normalized periodogram ordinates I(ω j )/f(ω j ) ( j = 1,2,…) of a general long‐memory time series. Here, I (ω;) is the periodogram based on a sample size n , f (ω) is the spectral density and ω j = 2π j/n. We assume that n →∝ with j held fixed, and so our focus is on low frequencies; these are the most important frequencies for the periodogram‐based estimation of the memory parameter d. Contrary to popular belief, the normalized periodogram ordinates obtained from a Gaussian process are asymptotically neither independent identically distributed nor exponentially distributed. In fact, lim n E{I(ω j )/f(ω j )} depends on both j and d and is typically greater than unity, implying a positive asymptotic relative bias in I(ω j ) as an estimator of f(ω j ). Tapering is found to reduce this bias dramatically, except at frequency ω 1 . The asymptotic distribution of I(ω j )/f(ω j ) for a Gaussian process is, in general, that of an unequally weighted linear combination of two independent X 2 1 random variables. The asymptotic mean of the log normalized periodogram depends on j and d and is not in general equal to the negative of Euler's constant, as is commonly assumed. Consequently, the regression estimator of d proposed by Geweke and Porter‐Hudak will be asymptotically biased if the number of frequencies used in the regression is held fixed as n →∝.