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NONLINEAR COINTEGRATING REGRESSION UNDER WEAK IDENTIFICATION

Published online by Cambridge University Press:  25 November 2011

Xiaoxia Shi  and
Peter C.B. Phillips
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Abstract

An asymptotic theory is developed for a weakly identified cointegrating regression model in which the regressor is a nonlinear transformation of an integrated process. Weak identification arises from the presence of a loading coefficient for the nonlinear function that may be close to zero. In that case, standard nonlinear cointegrating limit theory does not provide good approximations to the finite-sample distributions of nonlinear least squares estimators, resulting in potentially misleading inference. A new local limit theory is developed that approximates the finite-sample distributions of the estimators uniformly well irrespective of the strength of the identification. An important technical component of this theory involves new results showing the uniform weak convergence of sample covariances involving nonlinear functions to mixed normal and stochastic integral limits. Based on these asymptotics, we construct confidence intervals for the loading coefficient and the nonlinear transformation parameter and show that these confidence intervals have correct asymptotic size. As in other cases of nonlinear estimation with integrated processes and unlike stationary process asymptotics, the properties of the nonlinear transformations affect the asymptotics and, in particular, give rise to parameter dependent rates of convergence and differences between the limit results for integrable and asymptotically homogeneous functions.

Type
Research Article
Information
Econometric Theory , Volume 28 , Issue 3 , June 2012 , pp. 509 - 547
Copyright
Copyright © Cambridge University Press 2011

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Footnotes

Our thanks to two referees and the co-editor, Pentti Saikkonen, for helpful comments on the original version. The paper originated in a 2008 Yale take-home examination. The first complete draft was circulated in December 2009. Xiaoxia Shi gratefully acknowledges support from the Cowles Foundation via a Carl Arvid Anderson Fellowship at Yale University. Peter Phillips thanks the NSF for support under grant SES 06-47086 and 09-56687.

References

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