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An exact, unified distributional characterization of statistics used to test linear hypotheses in simple regression models


  • Parker, Thomas


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.

Suggested Citation

  • Parker, Thomas, 2010. "An exact, unified distributional characterization of statistics used to test linear hypotheses in simple regression models," MPRA Paper 22841, University Library of Munich, Germany.
  • Handle: RePEc:pra:mprapa:22841

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    Test of linear restrictions; Generalized beta distribution; Small-sample probability distribution; Regression model;

    JEL classification:

    • C12 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Hypothesis Testing: General

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