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Folklore theorems, implicit maps, and indirect inference

Folklore theorems, implicit maps, and indirect inference
Folklore theorems, implicit maps, and indirect inference
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 that 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 (partial) 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.
binding function, delta method, exact bias, implicit continuous maps, indirect inference, maximum likelihood
0012-9682
425-454
Phillips, Peter C.B.
f67573a4-fc30-484c-ad74-4bbc797d7243
Phillips, Peter C.B.
f67573a4-fc30-484c-ad74-4bbc797d7243

Phillips, Peter C.B. (2012) Folklore theorems, implicit maps, and indirect inference. Econometrica, 80 (1), 425-454. (doi:10.3982/ECTA9350).

Record type: Article

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 that 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 (partial) 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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More information

e-pub ahead of print date: 10 January 2012
Published date: January 2012
Keywords: binding function, delta method, exact bias, implicit continuous maps, indirect inference, maximum likelihood
Organisations: Social Sciences

Identifiers

Local EPrints ID: 358335
URI: https://eprints.soton.ac.uk/id/eprint/358335
ISSN: 0012-9682
PURE UUID: 9fe85ffb-3d62-4d0e-952c-09db95a8b878
ORCID for Peter C.B. Phillips: ORCID iD orcid.org/0000-0003-2341-0451

Catalogue record

Date deposited: 03 Oct 2013 13:45
Last modified: 15 Oct 2019 00:39

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