# Econometrica

## Journal Of The Econometric Society

### An International Society for the Advancement of Economic Theory in its Relation to Statistics and Mathematics

Edited by: Guido W. Imbens • Print ISSN: 0012-9682 • Online ISSN: 1468-0262

Econometrica: May, 1991, Volume 59, Issue 3

# Asymptotic Expansions of the Information Matrix Test Statistic

The information matrix $\mathrm{(IM)}$ test is shown to have a finite sample distribution which is poorly approximated by its asymptotic $\chi^2$ distribution in models and sample sizes commonly encountered in applied econometric research. The quality of the $\chi^2$ 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 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 $\chi^2$ 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 $\mathrm{IM}$ test and for heteroskedasticity and nonnormality diagnostic tests which are special cases of the $\mathrm{IM}$ 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.