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EXPANSIONS FOR THE DISTRIBUTION OF THE MAXIMUM LIKELIHOOD ESTIMATOR OF THE FRACTIONAL DIFFERENCE PARAMETER

Published online by Cambridge University Press:  08 June 2004

Offer Lieberman  and
Peter C.B. Phillips
Offer Lieberman
Affiliation:
Technion—Israel Institute of Technology
Peter C.B. Phillips
Affiliation:
Cowles Foundation for Research in Economics, Yale University, University of Auckland and University of York
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Abstract

The maximum likelihood estimator (MLE) of the fractional difference parameter in the Gaussian ARFIMA(0,d,0) model is well known to be asymptotically N(0,6/π2). This paper develops asymptotic expansions to the distribution of this statistic under the assumption of a known unit variance. The correction term for the density is shown to be independent of d, so that the MLE is second-order pivotal for d. This feature of the MLE is unusual, at least in time series contexts. Simulations show that the normal approximation is poor and that the expansions can make a significant improvement in accuracy provided the correction terms are computed without further asymptotic approximation.This paper was commenced and revised while Lieberman was visiting the Cowles Foundation during 2000–2002. Lieberman thanks the Cowles Foundation for support and hospitality during this visit. Phillips thanks the NSF for support under grants SBR 97-30295 and SES 0092509.

Information

Type
Research Article
Information
Econometric Theory , Volume 20 , Issue 3 , June 2004 , pp. 464 - 484
Copyright
© 2004 Cambridge University Press

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References

REFERENCES

Andrews, D.W.K. & O. Lieberman (2002a) Higher-Order Improvement of the Parametric Bootstrap for Long-Memory Gaussian Processes. Cowles Foundation discussion paper 1378, Yale University.
Andrews, D.W.K. & O. Lieberman (2002b) Valid Edgeworth Expansions for the Whittle Maximum Likelihood Estimator for Stationary Long-Memory Gaussian Time Series. Cowles Foundation discussion paper 1361, Yale University.
Barndorff-Nielsen, O.E. & D.R. Cox (1989) Asymptotic Expansions for Use in Statistics. Chapman and Hall.
Bhattacharya, R.N. & R.R. Rao (1976) Normal Approximation and Asymptotic Expansions. Wiley.
Chambers, J.M. (1967) On methods of asymptotic approximation for multivariate distributions. Biometrika 54, 367–383.Google Scholar
Chibisov, D.M. (1972) An asymptotic expansion for the distribution of a statistic admitting an asymptotic expansion. Theory of Probability and Its Applications 17, 620–630.Google Scholar
Dahlhaus, R. (1989) Efficient parameter estimation for self-similar processes. Annals of Statistics 17, 1749–1766.Google Scholar
Durbin, J. (1980) Approximation for densities of sufficient estimators. Biometrika 67, 311–333.Google Scholar
Fox, R. & M.S. Taqqu (1987) Central limit theorems for quadratic forms in random variables having long-range dependence. Probability Theory and Related Fields 74, 213–240.Google Scholar
Giraitis, L. & P.M. Robinson (2003) Edgeworth expansions for the semiparametric Whittle estimator of long memory. Annals of Statistics 31, 1325–1375.Google Scholar
Gradshteyn, I.S. & I.M. Ryzhik (1980) Table of Integrals, Series, and Products. Academic Press.
Lieberman, O., J. Rousseau, & D.M. Zucker (2000) Small-sample likelihood-based inference in the ARFIMA model. Econometric Theory 16, 231–248.Google Scholar
Lieberman, O., J. Rousseau, & D.M. Zucker (2003) Valid Edgeworth expansions for the maximum likelihood estimator of the parameter of a stationary, Gaussian, strongly dependent series. Annals of Statistics 31, 586–612.Google Scholar
Phillips, P.C.B. (1977) Approximations to some finite sample distributions associated with a first order stochastic difference equation. Econometrica 45, 463–485.Google Scholar
Searle, S.R. (1971) Linear Models. Wiley.
Silverman, B.W. (1986) Density Estimation for Statistics and Data Analysis. Chapman and Hall.
Taniguchi, M. (1984) Validity of Edgeworth expansions for statistics of time series. Journal of Time Series Analysis 5, 37–51.Google Scholar