The Hausman Test and Weak Instruments

We consider the following problem. There is a structural equation of interest that contains an explanatory variable that theory predicts is endogenous. There are one or more instrumental variables that credibly are exogenous with regard to this structural equation, but which have limited explanatory power for the endogenous variable. Further, there is one or more potentially strong instrument, which has much more explanatory power but which may not be exogenous. Hausman (1978) provided a test for the exogeneity of the second instrument when none of the instruments are weak. Here we focus on how the standard Hausman test does in the presence of weak instruments using the Staiger-Stock asymptotics. It is natural to conjecture that the standard version of the Hausman test would be invalid in the weak instruments case, which we con\u85rm. However, we provide a version of the Hausman test that is valid even in the presence of weak IV. We show that the situation we analyze occurs in several important economic examples. Our Monte Carlo experiments show that this procedure works relatively well in \u85nite samples. We should note that our test is not consistent, although we believe that it is impossible to construct a consistent test with weak instruments.