Higher-order improvements of the parametric bootstrap for long-memory Gaussian processesIntroductionModelParametric bootstrapAssumptionsPML assumptionsPWML assumptionsParameter valuesCondition CsCoverage probability errors of delta method CIsHigher-order improvements of the bootstrapMonte Carlo simulationsAcknowledgementsProofsEdgeworth expansion for the log-likelihood derivativesEdgeworth expansion for the Whittle log-likelihood derivativesProof of validity of the Edgeworth expansion for the log-likelihood derivativesLemmas used in the Proofs of Theorems 2 and 3Proofs of Theorems 2 and 3ReferencesJournal of Econometrics 133 (2006) 673–702Higher-order improvements of the parametricbootstrap for long-memory Gaussian processesDonald W.K. Andrewsa,, Offer Liebermanb, Vadim MarmercaCowles Foundation for Research in Economics, Yale University, P.O. Box 208281, New Haven,CT 06520-8281, USAbTechnion—Israel Institute of Technology and Cowles Foundation for Research in Economics,Yale University, USAcCowles Foundation for Research in Economics, Yale University, USAAvailable online 24 August 2005AbstractThis paper determines coverage probability errors of both delta method and parametricbootstrap confidence intervals (CIs) for the covariance parameters of stationary long-memoryGaussian time series. CIs for the long-memory parameter d0are included. The results establish thatthe bootstrap provides higher-order improvements over the delta method. Analogous results aregiven for tests. The CIs and tests are based on one or other of two approximate maximumlikelihood estimators. The first estimator solves the first-order conditions with respect to thecovariance parameters of a ‘‘plug-in’’ log-likelihood function that has the unknown mean replacedby the sample mean. The second estimator does likewise for a plug-in Whittle log-likelihood.The magnitudes of the coverage probability errors for one-sided bootstrap CIs for covarianceparameters for long-memory time series are shown to be essentially the same as they are with iiddata. This occurs even though the mean of the time series cannot be estimated at the usual n1=2rate.r 2005 Elsevier B.V. All rights reserved.JEL classification: C12; C13; C15Keywords: Asymptotics; Confidence intervals; Delta method; Edgeworth expansion; Gaussian process;Long memory; Maximum likelihood estimator; Parametric bootstrap; t statistic; Whittle likelihoodARTICLE IN PRESSwww.elsevier.com/locate/jeconom0304-4076/$ - see front matter r 2005 Elsevier B.V. All rights reserved.doi:10.1016/j.jeconom.2005.06.010Corresponding author.E-mail addresses: [email protected] (D.W.K. Andrews), [email protected](O. Lieberman), [email protected] (V. Marmer).1. IntroductionThis paper considers statistica l inference for a stationary long-memory Gaussiantime series with unknown mean m0and spectral density fy0that lies in a parametricfamily ffy: y 2 Y  RdimðyÞg. For this situation, Dahlhaus (1989) establishes theconsistency and asymptotic normality of a plug-in maximum likelihood estimator ofy0, which maximizes the likelihood function with the unknown mean m0replaced bya preliminar y consistent estimator, such as the sample mean. Dahlhaus showed thatthis estimator is asymptotically effici ent. His results allow one to constr uct ‘‘deltamethod’’ confidence intervals (CIs) and tests for elements of y0, including the long-memory parameter, d0, using an asymptotic normal approximation.1Fox andTaqqu (1986) provide similar results for the Whittle maximum likelihood estimatorof y0.In this paper, we establish the asymptotic order of magnitude of coverageprobability errors and null rejection rate errors of delta method CIs and testsconcerning elements of y0. We consider CIs and tests that are based on plug-inmaximum likelihood estimators that are defined in terms of the first-order conditions(FOCs) of a plug-in log-likelihood (PLL) function or a plug-in Whittle log-likelihood(PWLL) function. We refer to these estimators as PML and PWM L estimators. ThePLL and PWLL functions are the Gaussian log-likelihood and Gaussian Whittlelog-likelihood, respectively, with the sample mean plugged-in in place of theunknown mean.In addition, we introduce parametric bootstrap CIs and tests for elements of y0based on PML and PWML estimators and establish bounds on the asymptotic orderof magnitude of the coverage probability errors and null rejection rate errors of theseprocedures. We show that the bootstrap yields higher-order improvements over thedelta method in certain cases. To our knowledge there are no other results in theliterature, even first-order results, concerning the asymptotic properties of bootstrapmethods for long-memory processes.The results of the paper cover two- and one-sided delta method CIs and t tests.They cover symmetric two-sided and one-si ded parametric bootstrap CIs and tests.Both null-restricted and non-null-restricted parametric bootstrap tests are con -sidered. The former are preferred on theoretical grounds.The coverage probability errors of two- and one-sided delta method CIs forelements of y0are shown to be Oðn1Þ and Oðn1=2Þ, respectively, where n is thesample size. These errors are the same as for CIs in models for independent andidentically distributed (iid) observations. This occurs even though the mean m0cannot be estimated at the typical n1=2rate. Results for null rejection rate errors ofdelta method t tests are analogous.The coverage probability errors of symmetric tw o-sided and one-sided parametricbootstrap CIs are shown to be Oðn3=2lnðnÞÞ, and Oðn1lnðnÞÞ, respectively. Apartfrom the lnðnÞ term, the latter error is the same as for iid data. The error forARTICLE IN PRESS1Following the standard terminology in the bootstrap literature, we refer to asymptotic t tests and CIsbased on them using a normal approximation as ‘‘delta method’’ tests and CIs.D.W.K. Andrews et al. / Journal of Econometrics 133 (2006) 673–702674symmetric two-sided CIs is not as small as the error Oðn2Þ that has been establishedfor many CIs in iid contexts, see Hall (1988, 1992). This may be because our boundon the error is not sharp.The results show that symmetric two-sided and one-sided bootstrap CIs exhibithigher-order improvements in terms of coverage probabilities over their deltamethod counterparts of magnitude at least n1=2lnðnÞ.All of the bootstrap results just stated hold under a certain condition on