Power in Econometric Applications

This paper is concerned with the use of power properties of tests in econometric applications. Inverse power functions are defined. These functions are designed to yield summary measures of power that facilitate the interpretation of test results in practice. Simple approximations are introduced for the inverse power functions of Wald, likelihood ratio, Lagrange multiplier, and Hausman tests. These approximations readily convey the general qualitative features of the power of a test. Examples are provided to illustrate their usefulness in interpreting test results. A COMMON PROBLEM faced in applied econometrics is that of interpreting the results of a hypothesis test when the test fails to reject the null hypothesis. Most practitioners realize that just because a test fails to reject a hypothesis one cannot claim to accept it. Nevertheless, it is common for this to be ignored, since the practitioner is often in a position where he would like the outcome of the test to provide useful inferences whether or not the test rejects. The purpose of this paper is to introduce inverse power (IP) summary measures that enable the practitioner to avoid such errors and make valid inferences when a test fails to reject the null hypothesis. These summary measures are widely applicable, easy to use (especially in the common case of a test concerning a single restriction), and simple to compute. When a test rejects the null hypothesis, the implication is that the data are inconsistent with each parameter point in the null in the sense that the probabil- ity of type I error for each point is small, viz., a or less. Correspondingly, when a test fails to reject the null hypothesis an analogous statement is needed regarding the error probabilities for points in the alternative hypothesis. It is not the case that all points in the alternative are inconsistent with the data in the sense that their probability of type II error is small (a or less). It is possible, however, to determine the region S in the alternative parameter space that is inconsistent with the data in this sense. The IP function introduced below evaluated at