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Testing unconfoundedness in regression models with normally distributed regressors

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

Series: 1998-09, Preprints

62F05 Asymptotic properties of tests
62J05 Linear regression

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

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