Asymptotic Expansions of the Information Matrix Test Statistic

The information matrix (IM) test is shown to have a finite sample distribution which is poorly approximated by its asymptotic χ 2 distribution in models and sample sizes commonly encountered in applied econometric research. The quality of the χ 2 approximation depends upon the method chosen to compute the test. Failure to exploit restrictions on the covariance matrix of the test can lead to a test with appalling finite sample properties. Order O(n-1) approximations to the exact distribution of an efficient form of the IM test are reported. These are developed from asymptotic expansions of the Edgeworth and Cornish-Fisher types. They are compared with Monte Carlo estimates of the finite sample distribution of the test and are found to be superior to the usual χ 2 approximations in sample sizes of the magnitude found in applied micro-econometric work. The methods developed in the paper are applied to normal and exponential models and to normal regression models. Results are provided for the full IM test and for heteroskedasticity and nonnormality diagnostic tests which are special cases of the IM test. In general the quality of alternative approximations is sensitive to covariate design. However commonly used nonnormality tests are found to have distributions which, to order O(n-1), are invariant under changes in covariate design. This leads to simple design and parameter invariant size corrections for nonnormality tests.