Conflict Among Testing Procedures in a Linear Regression Model.

摘要

Savin and Berndt and Savin have shown that an inequality relation exists between different test statistics used for testing hypotheses of the form r-R Beta = 0. They found that the value of the likelihood ratio test statistic (LR = 21og lambda), the Wald test statistic (W), and the Lagrange multiplier test statistic (LM) are always such that (1) W greater than or = LR greater than or = LM. This result has been generalized by Breusch who showed that the only necessary assumption for this inequality to hold is, that the disturbances follow a distribution which allows maximum-likelihood estimation. However, neither Breusch nor any of the authors before him were able to conclude anything about the power of the different tests. In this paper it will be shown that for finite but large samples a similar inequality relation to (1) exists between the powers of the three tests. The Wald test is uniformly more powerful than either of the other two tests, and the likelihood ratio test is more powerful than the Lagrange multiplier test for very large samples and for moderate-to-large differences between the null hypothesis and the true value of the tested parameters. The assumption of a scalar covariance matrix is made to simplify the exposition. The results can probably be generalized to hold for any disturbance vector which allows maximum-likelihood estimation.