Asymptotics for the Low‐frequency Ordinates of the Periodogram of a Long‐memory Time Series

Abstract

We consider the asymptotic distribution of the normalized periodogram ordinates Z(o+)/f(w,) ( j = 1,2, . . .) of a general long-memory time series. Here, I(. )is the periodogram based on a sample size n, f( .) is the spectral density and 0, = 27rj/n. We assume that n -+ m 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, limn E { I(oJ/)/ f ( w , ) } depends on both j and d and is typically greater than unity, implying a positive asymptotic relative bias in I(w,) as an estimator of f ( w , ) . Tapering is found to reduce this bias dramatically, except at frequency wi. The asymptotic distribution of I ( w~) / f ( w / ) for a Gaussian process is, in general, that of an unequally weighted linear combination of two independent x: 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 + m .

Semiparametric estimation in perturbed long memory series

“…The properties of several semiparametric estimators of d 0 depend on the adequacy of the approximation of the periodogram to the local specification of the spectral density. Hurvich and Beltrao (1993), Robinson (1995a) and Arteche and Velasco (2005) in an asymmetric long memory context, observed that the asymptotic relative bias of the periodogram produces the bias typically encountered in semiparametric estimates of the memory parameters. Deo and Hurvich (2001), Crato and Ray (2002) and Arteche (2004) detected that the bias is quite severe in perturbed long memory series if the added noise is not explicitly considered in the estimation.…”

Estimating fractional cointegration in the presence of polynomial trends

“…It is interesting to note that this expression for the expectation of the limiting distribution of n −2dx times the periodogram of {x t } n t=1 at frequency j is equivalent to the expression given in Theorem 1 of Hurvich and Beltrao (1993) for the limiting expectation of a normalized periodogram of a univariate series, where the normalization is by the spectral density at j .…”

Broadband Semiparametric Estimation of the Memory Parameter of a Long‐Memory Time Series Using Fractional Exponential Models

“…as f (ù j ) 1 2 ÷ 2 2 . These properties no longer hold when d T 0 (Kunsch, 1986;Hurvich and Beltrao, 1993;Robinson, 1995a;Deo, 1997), although the ®ction that they remain valid when d T 0 has helped to motivate both the GPH and GSE estimators.…”