Testing unconfoundedness in regression models with normally distributed regressors

by Gaffke, N.; Steyer, R.; von Davier, A.


Preprint series: 98-09, Preprints

62F05 Asymptotic properties of tests
62J05 Linear regression


Abstract: The regression of a real valued response variable Y on two multi-dimensional regressor variables X and W is considered where Y, X, and W follow a joint multivariate normal distribution. The null hypothesis of unconfoundedness of the regression E(Y | X) w.r.t. the potential confounder W is to be tested on the basis of n i.i.d. multivariate normal observations. The paper focusses on the large sample Wald test statistic which is known to be asymptotically chi-square distributed under the null hypothesis, provided that the Jacobian of the restriction function describing the null hypothesis has full rank. However, there are points in the null hypothesis which do not meet this assumption. In fact, it turns out that the standard result on the asymptotic distribution of the Wald statistic is not true at those `singular points\' of the null hypothesis. The question arises whether or not the Wald test is (asymptotically) conservative at those points. Results are presented which indicate that this question can be answered

Keywords: Nonlinear hypothesis, Wald test, asymptotically normal estimator, asymptotic distribution

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