Do we keep the instrument in the OLS, if Wu-Hausman test (using that instrument) fails to reject that OLS is consistent? If two instrument variables (for example $Z_1$ and $Z_2$) are both exogenous and relevant. And we found Wu-Hausman test fails to reject the $H_0$: Both IV regression and OLS are consistent. When we want to report the OLS result:
We have two options: $$(1) Y = X_1 + X_2 + ... + Z_1 + Z_2 + e $$ 
$$(2) Y = X_1 + X_2 + ... + e $$
Is there any compulsory reasons that $Z_1 + Z_2$ need to be in the equation?
Thanks very much!
 A: The null hypothesis of the Wu-Hausman test is that both estimators are consistent. When you use statistical tests, you either reject the null, or you fail to reject the null, but you cannot accept the null. Therefore, you cannot use the result of this test to support the claim that the OLS is consistent. 
An example for this is that, if you use weak instruments and the precision of your IV estimator is poor, you will frequently end up with a Wu-Hausman test that fails to reject the null hypothesis. This happens not because the point estimates are close, but because the standard error of the IV estimator is large. 
Look at both the OLS and the IV estimates, and in particular, look at their precisions. If they are imprecise, that might explain why the test does not reject the null. If both are very precise (and close to each other), then you have no reason not to report it, because it is an interesting result. In general, my advice would be to report both the OLS (without including $Z_1$ and $Z_2$ among the regressors) and the IV estimate results (as well as the result of the Wu-Hausman test). 
