Weak Instruments and Spurious Regressions

This paper develops an alternative asymptotic approach to Staiger-Stock's (1997) local to zero IV estimation of a structural parameter when instruments are weakly related to the endogenous variables. Rather than treating the limiting coefficients of the instruments in the first-stage regression as being in the neighborhood of Zero, we treat the limiting coefficients as random variables that are centered around zero when instruments are weak. This is done by transforming the first-stage regression into a spurious, higher order regression between weakly related processes. Our methodology exploits the well-known property that in higher order regressions, the rank condition on the moment matrix of instruments and regressors is no longer necessary to obtain standard distributions for estimators and test statistics. Since there is a failure of this rank condition in Staiger-Stock's framework, it seems reasonable to translate the problem to a framework where only the order condition is required. It turns out that our IV estimator has properties similar to the split sample IV estimator of Angrist-Krueger (1995), even though they are based on entirely different first-stage regressions. Unlike the local to zero IV estimator which is biased toward OLS and has a nonpivotal distribution, our estimator is biased toward zero, its distribution is pivotal, and converges at a faster rate of (i.e. it is hyperconsistent relative to the local to zero IV estimator). This implies that many of the tests that were shown to have nonpivotal distributions when instruments are weak, now have pivotal distributions when constructed from our IV estimator