Tests for Independence between Categorical Variables

I prove the numerical equivalence between Pearson's independence test statistic for categorical variables and the Lagrange Multiplier and overidentifying restrictions test statistics in several popular linear and non-linear regression models. I also show that its asymptotically equivalent Likelihood Ratio test is numerically identical in the non-linear regression models, and that the heteroskedasticity-robust Wald test statistics in the multivariate linear probability model and GMM coincide with the Wald test statistic in the conditional multinomial model. Finally, I show that all these equivalences also apply to serial independence tests in discrete Markov chains