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Symmetry, Regression Design, and Sampling Distributions

Published online by Cambridge University Press:  11 February 2009

Andrew Chesher  and
Simon Peters
Andrew Chesher
Affiliation:
University of Bristol
Simon Peters
Affiliation:
University of Bristol
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Abstract

When values of regressors are symmetrically disposed, many M-estimators in a wide class of models have a reflection property, namely, that as the signs of the coefficients on regressors are reversed, their estimators' sampling distribution is reflected about the origin. When the coefficients are zero, sign reversal can have no effect. So in this case, the sampling distribution of regression coefficient estimators is symmetric about zero, the estimators are median unbiased and, when moments exist, the estimators are exactly uncorrelated with estimators of other parameters. The result is unusual in that it does not require response variates to have symmetric conditional distributions. It demonstrates the potential importance of covariate design in determining the distributions of estimators, and it is useful in designing and interpreting Monte Carlo experiments. The result is illustrated by a Monte Carlo experiment in which maximum likelihood and symmetrically censored least-squares estimators are calculated for small samples from a censored normal linear regression, Tobit, model.

Type
Articles
Information
Econometric Theory , Volume 10 , Issue 1 , March 1994 , pp. 116 - 129
Copyright
Copyright © Cambridge University Press 1994

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References

1.Andrews, D.W.K.A note on the unbiasedness of feasible GLS, quasimaximum likelihood, robust, adaptive and spectral estimators of the linear model. Econometrica 54 (1986): 687–698.CrossRefGoogle Scholar
2.Box, G.E.P. & Watson, G.S.. Robustness to non-normality of regression tests. Biometrika 49 (1962): 93–106.CrossRefGoogle Scholar
3.Chesher, A.D.A reflection property of M estimators. Discussion Paper No. 90/282, Department of Economics, University of Bristol, 1990.Google Scholar
4.Chesher, A.D., Peters, S. & Spady, R.. Approximations to the distributions of heterogeneity tests in the censored normal linear regression model. Discussion Paper No. 89/240, Department of Economics, University of Bristol, 1989.Google Scholar
5.Cox, D.R.Regression models and life tables. Journal of the Royal Statistical Society, Series B 34 (1972): 187–220.Google Scholar
6.Cryer, J.D., Nankervis, J.C. & Savin, N.E.. Mirror image and invariant distributions in ARMA models. Econometric Theory 5 (1989): 36–52.Google Scholar
7.Huber, P.J.Robust Statistical Procedures. Philadelphia: Society for Industrial and Applied Mathematics, 1977.Google Scholar
8.Kakwani, N.C.The unbiasedness of Zellner's seemingly unrelated regression equation estimators. Journal of the American Statistical Association 62 (1967): 141–142.Google Scholar
9.Moolgavkar, S.H. & Venzon, D.J.. Confidence regions for parameters of the proportionate hazard model: A simulation study. Scandinavian Journal of Statistics 14 (1987): 43–56.Google Scholar
10.Powell, J.L.Symmetrically trimmed least squares estimation for Tobit models. Econometrica 54 (1986): 1435–1460.Google Scholar
11.Press, W.H., Flannery, B.P., Teukolsky, S.A. & Vetterling, W.T.. Numerical Recipes: The Art of Scientific Computing. Cambridge: Cambridge University Press, 1986.Google Scholar
12.Weisberg, S. Comment on “Some large sample tests for non–normality in the linear regression model” by H. White and G.M. MacDonald. Journal of the American Statistical Association 75 (1980): 28–31.CrossRefGoogle Scholar
13.Wichman, B.A. & Hill, I.D.. Algorithm AS183: An efficient portable pseudo–random number generator. Applied Statistics 31 (1982): 188–190.Google Scholar