Testing for partial exogeneity with weak identification

We consider the following problem. A structural equation of interest contains two sets of explanatory variables which economic theory predicts may be endogenous. The researcher is interesting in testing the exogeneity of only one of them. Standard exogeneity tests are in general unreliable from the view point of size control to assess such a problem. We develop four alternative tests to address this issue in a convenient way. We provide a characterization of their distributions under both the null hypothesis (level) and the alternative hypothesis (power), with or without identification. We show that the usual 2 critical values are still applicable even when identification is weak. So, all proposed tests can be described as robust to weak instruments. We also show that test consistency may still hold even if the overall identification fails, provided partial identification is satisfied. We present a Monte Carlo experiment which confirms our theory. We illustrate our theory with the widely considered returns to education example. The results underscore: (1) how the use of standard tests to assess partial exogeneity hypotheses may be misleading, and (2) the relevance of using our procedures when checking for partial exogeneity.