My thanks to three referees and the co-editor for helpful comments. Some of the limit results in this paper were presented in the author's Durbin lecture at UCL in May 2009, at the Singapore SESG meetings in August 2009, and at the Nottingham Granger Memorial Conference in May 2010. The author acknowledges partial research support from the Kelly Fund and the NSF under Grants SES 06-47086 and SES 09-56687.
Folklore Theorems, Implicit Maps, and Indirect Inference
Article first published online: 10 JAN 2012
© 2012 The Econometric Society
Volume 80, Issue 1, pages 425–454, January 2012
How to Cite
Phillips, P. C. B. (2012), Folklore Theorems, Implicit Maps, and Indirect Inference. Econometrica, 80: 425–454. doi: 10.3982/ECTA9350
- Issue published online: 10 JAN 2012
- Article first published online: 10 JAN 2012
- Manuscript received June, 2010; final revision received June, 2011.
- Binding function;
- delta method;
- exact bias;
- implicit continuous maps;
- indirect inference;
- maximum likelihood
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.