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ADMISSIBLE SIGNIFICANCE TESTS IN SIMULTANEOUS EQUATION MODELS

Published online by Cambridge University Press:  21 April 2017

Theodore W. Anderson
Theodore W. Anderson*
Affiliation:
Stanford University
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Abstract

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Consider testing the null hypothesis that a single structural equation has specified coefficients. The alternative hypothesis is that the relevant part of the reduced form matrix has proper rank, that is, that the equation is identified. The usual linear model with normal disturbances is invariant with respect to linear transformations of the endogenous and of the exogenous variables. When the disturbance covariance matrix is known, it can be set to the identity, and the invariance of the endogenous variables is with respect to orthogonal transformations. The likelihood ratio test is invariant with respect to these transformations and is the best invariant test. Furthermore it is admissible in the class of all tests. Any other test has lower power and/or higher significance level. In particular, this likelihood ratio test dominates a test based on the Two-Stage Least Squares estimator.

Type
T.W. ANDERSON MEMORIAL
Information
Econometric Theory , Volume 33 , Issue 3 , June 2017 , pp. 534 - 550
Copyright
Copyright © Cambridge University Press 2017 

Footnotes

The author (deceased, September 17, 2016) thanked Naoto Kunitomo, Graduate School of Economics, University of Tokyo for his generous assistance. An early version of this paper was presented to the James Durbin Seminar sponsored by the London School of Economics and University College, London, on October 29, 2009. This version was presented to the Haavelmo Centennial Symposium, Oslo, on December 14, 2011. Minor editorial changes made by the Editor, December 12, 2016.

References

REFERENCES

Abramowitz, M. & Stegun, I.A. (1972) Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover Books on Mathematics, Dover Publications.Google Scholar
Anderson, T.W. (1976) Estimation of linear functional relationships: Approximate distributions and connections with simultaneous equations in econometrics. Journal of the Royal Statististical Society Series B 38(1), 1–36. With a discussion by Sprent, P., Copas, J.B., Bartlett, M.S., Chan, N.N., Solari, Mary E., Hendry, David F., Walker, A.M., Andersen, E.B., Bibby, John, Farebrother, R.W., Lyttkens, Ejnar, Maxwell, A.E., O’Brien, R.J., Phillips, P.C.B., Williams, D.A., Williams, E.J. and Patefield, W.M. and a reply by the author.Google Scholar
Anderson, T.W. (1984) The 1982 Wald Memorial Lectures: Estimating linear statistical relationships. Annals of Statistics 12(1), 1–45. URL 10.1214/aos/1176346390.CrossRefGoogle Scholar
Anderson, T.W. (2003) An Introduction to Multivariate Statistical Analysis, 3rd ed. Wiley Series in Probability and Statistics. Wiley.Google Scholar
Anderson, T.W. & Girshick, M.A. (1944) Some extensions of the Wishart distribution. Annals of Mathematical Statistics 15, 345–357 (Correction: 1964, 35, 923).CrossRefGoogle Scholar
Anderson, T.W. & Kunitomo, N. (2007) On Likelihood Ratio Tests of Structural Coefficients: Anderson–Rubin (1949) Revisited. URL http://www.cirje.e.u-tokyo.ac.jp/research/dp/2007/2007cf499.pdf. Discussion Paper CIRJE-F-499.Google Scholar
Anderson, T.W. & Kunitomo, N. (2009) Testing a hypothesis about a structural coefficient in a simultaneous equation model for known covariance matrix (In honor of Raul P. Mentz). Estadistica 61, 7–16.Google Scholar
Anderson, T.W. & Rubin, H. (1949) Estimation of the parameters of a single equation in a complete system of stochastic equations. Annals of Mathematical Statistics 20, 46–63.CrossRefGoogle Scholar
Anderson, T.W., Stein, C., & Zaman, A. (1985) Best invariant estimation of a direction parameter. Annals of Statistics 13 (2), 526–533. URL 10.1214/aos/1176349536.CrossRefGoogle Scholar
Creasy, M.A. (1956) Confidence limits for the gradient in the linear functional relationship. Journal of the Royal Statistical Society Series B 18, 65–69.Google Scholar
Lehmann, E.L. (1986) Testing Statistical Hypotheses, 2nd ed. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. Wiley.CrossRefGoogle Scholar
Moreira, M.J. (2003) A conditional likelihood ratio test for structural models. Econometrica 71(4), 1027–1048. URL 10.1111/1468-0262.00438.CrossRefGoogle Scholar
Zaman, A. (1996) Statistical Foundations for Econometric Techniques. Economic Theory, Econometrics, and Mathematical Economics. Academic Press Inc.Google Scholar