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

**Series:** 1998-09, Preprints

- MSC:
- 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