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Higher-order improvements of the parametric bootstrap



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ARTICLE IN PRESS Journal of Econometrics 133 2006 673 702 www elsevier com locate jeconom Higher order improvements of the parametric bootstrap for long memory Gaussian processes Donald W K Andrewsa Offer Liebermanb Vadim Marmerc a Cowles Foundation for Research in Economics Yale University P O Box 208281 New Haven CT 06520 8281 USA b Technion Israel Institute of Technology and Cowles Foundation for Research in Economics Yale University USA c Cowles Foundation for Research in Economics Yale University USA Available online 24 August 2005 Abstract This paper determines coverage probability errors of both delta method and parametric bootstrap con dence intervals CIs for the covariance parameters of stationary long memory Gaussian time series CIs for the long memory parameter d 0 are included The results establish that the bootstrap provides higher order improvements over the delta method Analogous results are given for tests The CIs and tests are based on one or other of two approximate maximum likelihood estimators The rst estimator solves the rst order conditions with respect to the covariance parameters of a plug in log likelihood function that has the unknown mean replaced by 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 covariance parameters for long memory time series are shown to be essentially the same as they are with iid data This occurs even though the mean of the time series cannot be estimated at the usual n1 2 rate r 2005 Elsevier B V All rights reserved JEL classification C12 C13 C15 Keywords Asymptotics Con dence intervals Delta method Edgeworth expansion Gaussian process Long memory Maximum likelihood estimator Parametric bootstrap t statistic Whittle likelihood Corresponding author E mail addresses donald andrews yale edu D W K Andrews offerL technion ac il O Lieberman vadim marmer yale edu V Marmer 0304 4076 see front matter r



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