Large Sample Tests for Comparing Regression Coefficients in Models With Normally Distributed Variables

The analysis of regression coefficients is an important issue in different scientific areas, mostly because conclusions about the relationship between variables, like causal interpretations, are drawn based on these coefficients. This paper focuses on the description of the null hypothesis of invariance of regression coefficients for multidimensional stochastic regressors. In this study, it is assumed that the variables have a joint normal distribution with unknown expectation and unknown positive definite covariance matrix. In this context, it is shown that the null hypothesis contains special parameter points, called singular and stationary parameter points, that influence the distribution of the commonly used test statistics under the null hypothesis. Three large sample statistics—the Wald test, the likelihood ratio test, and the efficient score test—are compared when testing this nonlinear null hypothesis. The results of a simulation study are presented. The goal of the simulations is to check the distributions of the three statistics for finite sample sizes and at a stationary point of the null hypothesis. Another aim is to compare the empirical values of the three statistics to one another, for different parameter constellations. It is shown that all three statistics present deviations from the expected chi-squared distribution at this special parameter point. However, any of the three statistical tests might be used for testing the hypothesis of the invariance of the regression coefficients since they remain asymptotically conservative at the stationary points of the null hypothesis.