# What is an example to show Wald's test does not confirm significance at the same level for each variable?

Suppose that the true model is given by Y=0.3+X1+X2+X3. Assume we have 100 training examples where each covariate vector (X1, X2, X3) is randomly drawn from some distribution P and Y is generated according to the true model.

Consider three OSL models of Y against (i) X1, (ii) X2, (iii) X3 respectively. Is it true that if Wald’s test of significance confirms that X1 is significant at a level of 0.01 for (i), then it confirms that X2 and X3 are significant at a level of 0.01 for (ii) and (iii) respectively?

The Wald's test tends to be biased when the regression coefficient is large, therefore the above statement would be false. How would I go about fitting a model to provide a counter example against this statement. I was going to use gml in r but am having no such luck

• Hint: unless $X_1,X_2,X_3$ are exchangeable under $P,$ the apparent symmetry in the question is broken: that's how you can construct counterexamples to your conjecture. (Counterexamples exist even with exchangeable $P$ but might be a little harder to construct because they only occur at random rather than predictably.) – whuber Oct 17 '19 at 11:36