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EXACT DISTRIBUTION THEORY IN STRUCTURAL ESTIMATION WITH AN IDENTITY

Published online by Cambridge University Press:  01 August 2009

Peter C.B. Phillips
Peter C.B. Phillips*
Affiliation:
Cowles Foundation, Yale University, University of Auckland, and, University of York
*
*Address correspondence to Peter C.B. Phillips, Department of Economics, Yale University, P.O. Box 208268, New Haven, CT 06520-8268, USA; e-mail: [email protected].
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Abstract

Some exact distribution theory is developed for structural equation models with and without identities. The theory includes LIML, IV, and OLS. We relate the new results to earlier studies in the literature, including the pioneering work of Bergstrom (1962). General IV exact distribution formulas for a structural equation model without an identity are shown to apply also to models with an identity by specializing along a certain asymptotic parameter sequence. Some of the new exact results are obtained by means of a uniform asymptotic expansion. An interesting consequence of the new theory is that the uniform asymptotic approximation provides the exact distribution of the OLS estimator in the model considered by Bergstrom (1962). This example appears to be the first instance in the statistical literature of a uniform approximation delivering an exact expression for a probability density.

Type
ARTICLES
Information
Econometric Theory , Volume 25 , Issue 4: Bergstrom Memorial Dedication Issue , August 2009 , pp. 958 - 984
Copyright
Copyright © Cambridge University Press 2009

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