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' instruments, 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 StaigerStock asymptotics. It is natural to conjecture that the standard version of the Hausman test would be invalid in the weak instrument case, which we confirm. However, we provide a version of the Hausman test that is valid even in the presence of weak IV and illustrate how to implement the test in the presence of heteroskedasticity. We show that the situation we analyze occurs in several important economic examples. Our Monte Carlo experiments show that our procedure works relatively well in finite 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.
Bibliographical noteFunding Information:
We thank Takeshi Amemiya, an associate editor, two referees, and Joris Pinkse for helpful comments and suggestions, and Martin Weidner for proofreading. Hahn, Ham, and Moon acknowledge supports from the National Science Foundation . Any opinions, findings, conclusions, or recommendations in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.
All Science Journal Classification (ASJC) codes
- Economics and Econometrics