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A COMPARISON OF ALTERNATIVE APPROACHES TO SUPREMUM-NORM GOODNESS-OF-FIT TESTS WITH ESTIMATED PARAMETERS

Published online by Cambridge University Press:  21 February 2013

Thomas Parker
Thomas Parker*
Affiliation:
University of Waterloo
*
*Address correspondence to Thomas Parker, Department of Economics, University of Waterloo, Hagey Hall of Humanities, Waterloo, Ontario, Canada: N2L 3G1; e-mail: tmparker@uwaterloo.ca.
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Abstract

Goodness-of-fit tests based on parametric empirical processes have nonstandard limiting distributions when the null hypothesis is composite — that is, when parameters of the null model are estimated. Several analytic solutions to this problem have been suggested, including the calculation of adjusted critical values for these nonstandard distributions and the transformation of the empirical process such that statistics based on the transformed process are asymptotically distribution-free. The approximation methods proposed by Durbin (1985, Journal of Applied Probability 22(1), 99–122) can be applied to conduct inference for tests based on supremum-norm statistics. The resulting tests have quite accurate size, a fact that has gone unrecognized in the econometrics literature. Some justification for this accuracy lies in the similar features that Durbin’s approximation methods share with the theory of extrema for Gaussian random fields and for Gauss-Markov processes. These adjustment techniques are also related to the transformation methodology proposed by Khmaladze (1981, Theory of Probability and Its Applications26(2), 240–257) through the score function of the parametric model. Simulation experiments suggest that in small samples, Durbin-style adjustments result in tests that have higher power than tests based on transformed processes, and in some cases they have higher power than parametric bootstrap procedures.

Type
ARTICLES
Information
Econometric Theory , Volume 29 , Issue 5 , October 2013 , pp. 969 - 1008
Copyright
Copyright © Cambridge University Press 2013 

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Footnotes

The author wishes to express great appreciation to Roger Koenker for countless helpful discussions and able direction. This research also benefited from the valuable comments of Juan Carlos Escanciano, Andreas Hagemann, Marie Hušková, and Kyungchul Song. The author also wishes to thank Peter C.B. Phillips, the co-editor, and referees at Econometric Theory for their helpful suggestions. Finally, the author wishes to thank the late James Durbin, who in many ways inspired this entire project.

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