Hostname: page-component-6766d58669-nf276 Total loading time: 0 Render date: 2026-05-24T20:34:21.765Z Has data issue: false hasContentIssue false
  • English
  • Français

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.
Get access Rights & Permissions [Opens in a new window]

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.

Information

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

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

REFERENCES

Amari, S.-I. (1982) Differential geometry of curved exponential families-curvatures and information loss. The Annals of Statistics 10, 357–385.Google Scholar
Amari, S.-I. (1985) Differential Geometrical Methods in Statistics. Springer Lecture Notes in Statistics, vol. 28. Springer Verlag.Google Scholar
Ariza, C. & Van Garderen, K.J. (2010) Conditional Bimodality in a Structural Equations Model. University of Amsterdam Econometrics Discussion paper: 2009/12 (rev 2010).Google Scholar
Barndorff-Nielsen, O.E. (1980) Conditionality resolutions. Biometrika 67, 293–310.Google Scholar
Barndorff-Nielsen, O.E. (1983) On a formula for the distribution of the maximum likelihood estimator. Biometrika 70, 343–365.Google Scholar
Barndorff-Nielsen, O.E. & Cox, D.R. (1979) Edgeworth and saddlepoint approximations with statistical applications (with discussion). Journal of the Royal Statistical Society B 41, 279–312.Google Scholar
Barndorff-Nielsen, O.E. & Cox, D.R. (1989) Asymptotic Techniques for use in Statistics. Chapman and Hall.Google Scholar
Barndorff-Nielsen, O.E. & Cox, D.R. (1994) Inference and Asymptotics. Chapman and Hall.Google Scholar
Bergstrom, A.R. (1962) The exact sampling distributions of least squares and maximum likelihood estimators of the marginal propensity to consume. Econometrica 30, 480–490.Google Scholar
Basu, D. (1955) On statistics independent of a complete sufficient statistic. Sankhyā 15, 377–380.Google Scholar
Cox, D.R. (1980) Local ancillarity. Biometrika 67, 273–278.Google Scholar
Chesher, A. & Smith, R.J. (1997) Likelihood ratio specification tests. Econometrica 65, 627–646.Google Scholar
Daniels, H.E. (1954) Saddlepoint approximations in statistics. Annals of Mathematical Statistics 25, 631–650.Google Scholar
Daniels, H.E. (1980) Exact saddlepoint approximations. Biometrika 67, 59–63.Google Scholar
De Nadai, M. & Lewbel, A. (2016) Nonparametric errors in variables models with measurement errors on both sides of the equation. Journal of Econometrics 191, 19–32.Google Scholar
Durbin, J. (1954) Errors in variables. Revue de l’institut International de Statistique 22, 23–32.CrossRefGoogle Scholar
Durbin, J. (1980) Approximations for densities of sufficient estimators. Biometrika 67, 311–333.Google Scholar
Efron, B. (1975) Defining the curvature of a statistical problem (with applications to second order efficiency). The Annals of Statistics 3, 1189–1242.Google Scholar
Efron, B. (1978) The geometry of exponential families. The Annals of Statistics 6, 362–376.Google Scholar
Efron, B. & Hinkley, D.V. (1978) Assessing the accuracy of the maximum likelihood estimator: Observed versus expected Fisher information. Biometrika 65(3), 457–483.Google Scholar
Fisher, R.A. (1934) Two new properties of mathematical likelihood. Proceedings of the Royal Society of London, A 144, 285–307.Google Scholar
Forchini, G. (2006) On the bimodality of the exact distribution of the TSLS estimator. Econometric Theory 22, 932–946.CrossRefGoogle Scholar
Fraser, D.A.S. (1968) The Structure of Inference. Wiley.Google Scholar
Fraser, D.A.S. (1979) Probability and Statistics. Duxbury.Google Scholar
Hillier, G.H. (2006) Yet more on the exact properties of IV estimators. Econometric Theory 22, 913–931.Google Scholar
Hillier, G.H. & Armstrong, M. (1999) The density of the maximum likelihood estimator. Econometrica 67, 1459–1470.Google Scholar
Hinkley, D.V. (1980) Likelihood as approximate pivotal distribution. Biometrika 67, 287–292.Google Scholar
Juola, R.C. (1993) More on shortest confidence intervals. The American Statistician 47, 117–119.Google Scholar
Maddala, G.S. & Jeong, J. (1992) On the exact small sample distribution of the instrumental variable estimator. Econometrica 60, 181–183.CrossRefGoogle Scholar
Mavroeidis, S. & Van Garderen, K.J. (2006) Conditional Inference in the Cointegrated Vector Autoregressive Model. Mimeograph, Oxford & Amsterdam.Google Scholar
Nelson, C.R. & Startz, R. (1990) Some further results on the small sample properties of the instrumental variable estimator. Econometrica 58, 967–976.Google Scholar
Phillips, P.C.B. (2006) A remark on bimodality and weak instrumentation in structural equation estimation. Econometric Theory 22, 947–960.Google Scholar
Spady, R.H. (1991) Saddlepoint approximations for regression models. Biometrika 78, 879–889.Google Scholar
Stock, J.H., Wright, J.H., & Yogo, M. (2012) A survey of weak instruments and weak identification in generalized method of moments. Journal of Business & Economic Statistics 20, 518–529.Google Scholar
Van Garderen, K.J. (1997) Curved exponential models in econometrics. Econometric Theory 13, 771–790.Google Scholar
Woglom, G. (2001) More results on the exact small sample properties of the instrumental variable estimator. Econometrica 69, 1381–1389.Google Scholar