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MULTIMODALITY p**-FORMULA AND CONFIDENCE REGIONS

Published online by Cambridge University Press:  05 June 2017

Kees Jan Van Garderen  and
Fallaw Sowell
Kees Jan Van Garderen*
Affiliation:
University of Amsterdam
Fallaw Sowell
Affiliation:
Carnegie Mellon University
*
*Address correspondence to Kees Jan Van Garderen, Department of Economics and Econometrics, University of Amsterdam, Roetersstraat 11, P.O. Box 15867, 1001 NJ, Amsterdam, The Netherlands; e-mail: K.J.vanGarderen@uva.nl.
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Abstract

Barndorff-Nielsen’s celebrated p*-formula and variations thereof have amongst their various attractions the ability to approximate bimodal distributions. In this paper we show that in general this requires a crucial adjustment to the basic formula. The adjustment is based on a simple idea and straightforward to implement, yet delivers important improvements. It is based on recognizing that certain outcomes are theoretically impossible and the density of the MLE should then equal zero, rather than the positive density that a straight application of p* would suggest. This has implications for inference and we show how to use the new p**-formula to construct improved confidence regions. These can be disjoint as a consequence of the bimodality. The degree of bimodality depends heavily on the value of an approximate ancillary statistic and conditioning on the observed value of this statistic is therefore desirable. The p**-formula naturally delivers the relevant conditional distribution. We illustrate these results in small and large samples using a simple nonlinear regression model and errors in variables model where the measurement errors in dependent and explanatory variables are correlated and allow for weak proxies.

Type
ARTICLES
Information
Econometric Theory , Volume 34 , Issue 2 , April 2018 , pp. 416 - 446
Copyright
Copyright © Cambridge University Press 2017 

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Footnotes

We thank Grant Hillier, Peter Phillips, Richard Smith and participants at the Cambridge conference in Richard Smith’s honour, participants at the NESG conference, the Co-editors, and especially two referees for earlier comments that substantially improved the paper.

References

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