Asymptotic Properties of the LSE in a Regression Model with Long-Memory Stationary Errors

Adapting to Unknown Disturbance Autocorrelation in Regression with Long Memory

“…If u t is, for example, an instantaneous nonlinear function of a Gaussian long memory process v t , then by extension of results for the sample mean of Rosenblatt (1961) or Taqqu (1975), one expects the limit distributions of OLS and GLS to be non-normal in case u t has Hermite rank greater than 1 (that is, its expansion in Hermite polynomials of v t contains no linear component). Even when OLS and GLS are asymptotically normal, they have rate of convergence which is not only affected by trending behaviour in deterministic (such as polynomial-in-t) x t but, unlike our GLS estimates with stochastic x t , is also adversely affected by long memory u t when the limiting spectral distribution function of the normalized x t has a jump at frequency zero (as in the polynomial-in-t case); see Yajima (1988Yajima ( , 1991. Dahlhaus (1995) studied a class of weighted estimates deriving from Adenstedt's (1974) treatment of the simple location model, studying to what extent they achieve the asymptotic Gauss-Markov bound in the presence of long (and negative) memory u t .…”

“…Initial study of the implications of long memory u t in (1.1) focussed on ordinary least squares (OLS) estimates of β. In case of deterministic, such as polynomial-in-t regressors, Yajima (1988Yajima ( , 1991 found that OLS estimates can still be asymptotically normal, even if u t is non-Gaussian but a linear process. However their asymptotic variance, indeed their rate of convergence, is adversely affected by the long memory in u t .…”

Confidence intervals for long memory regressions

“…The explicit computations presented here allow us to account for the effects of a finite sample size in confidence intervals and should have other uses. The performance of our method improves on the OLS methods of Yajima (1988of Yajima ( , 1991 and bypasses the computationally demanding WLS methods of Robinson and Hidalgo (1997). Our method merely requires an estimate of the long memory parameters.…”

“…Ordinary least squares (OLS) estimators, though subefficient to WLS estimators in long memory settings, often perform well and were studied in Yajima (1988), who established their asymptotic normality and derived their explicit asymptotic variance. Yajima (1991) and Kleiber (2001) present efficiency relations between WLS and OLS estimators in long memory error settings.…”

11 Autocorrelation-robust inference

“…However he discussed studentization only under (3.2), which can apply only when H = 1/2. Yajima (1988Yajima ( , 1991 discussed the structure of the asymptotic variance of jJ for polynomial-in-t and other Zt, and Robinson (1994b) explicitly justified a suitable studentization in the polynomial-in-t case. Dahlhaus (1992) established the asymptotic normality of a studentized GLSE in case of Gaussian X¡.…”