Weak Instruments: Diagnosis and Cures in Empirical Econometrics

What is the weak instruments (WI) problem and what causes it? Universal agreement does not exist on these questions. We define weak instruments by two features: (1) 2SLS is badly biased toward the OLS estimate and alternative “unbiased” estimators such as LIML may not solve the problem and (2) the standard (first order) asymptotic distribution does not give an accurate framework for inference. Thus, a researcher may estimate “bad results” and not be aware of the outcome. The cause of WI is often stated to be a low R or F statistic of the reduced form equation, in the most commonly occurring situation of one right hand side endogenous variable. We find the situation is more complex with an additional factor, the correlation between the stochastic disturbances of the structural equation and the reduced form, that needs to be taken into account. We discuss in this paper: a specification test of Hahn-Hausman (2002a) for WI, a caution against using “no moments” estimators such as LIML in the WI situation, suggestions for different estimators, an approach to inference of Frank Kleibergen (2002) for WI, and we end with a caution of how “small biases” can become “large biases” in the WI situation. We begin with the limited information structural model under the assumptions of e.g. Hausman (1983):