Parker, Thomas (2010): An exact, unified distributional characterization of statistics used to test linear hypotheses in simple regression models.

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Abstract
The Wald, likelihood ratio and Lagrange multiplier test statistics are commonly used to test linear restrictions in regression models. It is shown that for testing these restrictions in the classical regression model, the exact densities of these test statistics are special cases of the generalized beta distribution introduced by McDonald (1984); McDonald and Xu (1995a). This unified derivation provides a method by which one can derive small sample critical values for each test. These results may be indicative of the behavior of such test statistics in more general settings, and are useful in visualizing how each statistic changes with different parameter values in the simple regression model. For example, the results suggest that Wald tests may severely underreject the null hypothesis when the sample size is small or a large number of restrictions are tested.
Item Type:  MPRA Paper 

Original Title:  An exact, unified distributional characterization of statistics used to test linear hypotheses in simple regression models 
Language:  English 
Keywords:  Test of linear restrictions, Generalized beta distribution, Smallsample probability distribution, Regression model 
Subjects:  C  Mathematical and Quantitative Methods > C1  Econometric and Statistical Methods and Methodology: General > C12  Hypothesis Testing: General 
Item ID:  22841 
Depositing User:  Thomas Parker 
Date Deposited:  21 May 2010 19:28 
Last Modified:  28 Sep 2019 06:52 
References:  E. Berndt and N. E. Savin. Conflict among criteria for testing hypotheses in the multivariate linear regression model. Econometrica, 45(5):1263–1277, July 1977. A. Buse. The likelihood ratio, wald, and lagrange multiplier tests: An expository note. The American Statistician, 36(3):153–157, August 1982. R. Davidson and J. MacKinnon. Estimation and Inference in Econometrics. Oxford, 1993. R.F. Engle. Wald, likelihood ratio, and lagrange multiplier tests in econometrics. In Z. Griliches and M.D. Intriligator, editors, Handbook of Econometrics, volume II, pages 775–826. NorthHolland, 1984. J.B. McDonald. Some generalized functions for the size distribution of income. Econometrica, 52(3): 647–664, May 1984. J.B. McDonald and Y.J. Xu. A generalization of the beta distribution with applications. Journal of Econometrics, 69:133–152, 1995a. J.B. McDonald and Y.J. Xu. Errata — a generalization of the beta distribution with applications. Journal of Econometrics, 69:427–428, 1995b. 
URI:  https://mpra.ub.unimuenchen.de/id/eprint/22841 