COWLES FOUNDATION FOR RESEARCH IN ECONOMICS
AT YALE UNIVERSITY

Box 208281
New Haven, CT 06520-8281

COWLES FOUNDATION DISCUSSION PAPER NO. 1378

Higher-order Improvements of the Parametric Bootstrap
for Long-memory Gaussian Processes

Donald W.K. Andrews and Offer Lieberman

August 2002

This paper determines coverage probability errors of both delta method and parametric bootstrap confidence intervals (CIs) for the covariance parameters of stationary long-memory Gaussian time series. CIs for the long-memory parameter d₀ 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 first estimator solves the first-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 n^1/2 rate.

Key words: Asymptotics, confidence intervals, delta method, Edgeworth expansion, Gaussian process, long memory, maximum likelihood estimator, parametric bootstrap, t statistic, Whittle likelihood

JEL Classification: C12, C13, C15