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Folklore Theorems, Implicit Maps and New Unit Root Limit Theory

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Abstract

The delta method and continuous mapping theorem are among the most extensively used tools in asymptotic derivations in econometrics. Extensions of these methods are provided for sequences of functions, which are commonly encountered in applications, and where the usual methods sometimes fail. Important examples of failure arise in the use of simulation based estimation methods such as indirect inference. The paper explores the application of these methods to the indirect inference estimator (IIE) in first order autoregressive estimation. The IIE uses a binding function that is sample size dependent. Its limit theory relies on a sequence-based delta method in the stationary case and a sequence-based implicit continuous mapping theorem in unit root and local to unity cases. The new limit theory shows that the IIE achieves much more than bias correction. It changes the limit theory of the maximum likelihood estimator (MLE) when the autoregressive coefficient is in the locality of unity, reducing the bias and the variance of the MLE without affecting the limit theory of the MLE in the stationary case. Thus, in spite of the fact that the IIE is a continuously differentiable function of the MLE, the limit distribution of the IIE is not simply a scale multiple of the MLE but depends implicitly on the full binding function mapping. The unit root case therefore represents an important example of the failure of the delta method and shows the need for an implicit mapping extension of the continuous mapping theorem.

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File URL: http://cowles.econ.yale.edu/P/cd/d17b/d1781.pdf
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Paper provided by Cowles Foundation for Research in Economics, Yale University in its series Cowles Foundation Discussion Papers with number 1781.

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Length: 42 pages
Date of creation: Jan 2011
Date of revision:
Publication status: Published in Econometrica (January 2012): 80(1): 425-454
Handle: RePEc:cwl:cwldpp:1781

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Postal: Cowles Foundation, Yale University, Box 208281, New Haven, CT 06520-8281 USA

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Keywords: Binding function; Delta method; Exact bias; Implicit continuous maps; Indirect inference; Maximum likelihood;

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  1. Christian Gourieroux & Peter C. B. Phillips & Jun Yu, 2006. "Indirect Inference for Dynamic Panel Models," Cowles Foundation Discussion Papers 1550, Cowles Foundation for Research in Economics, Yale University.
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