This paper analyzes the conditions under which consistent estimation can be achieved in instrumental variables (IV) regression when the available instruments are weak and the number of instruments, Kn, goes to infinity with the sample size. We show that consistent estimation depends importantly on the strength of the instruments as measured by rn, the rate of growth of the so-called concentration parameter, and also on Kn. In particular, when Kn→∞, the concentration parameter can grow, even if each individual instrument is only weakly correlated with the endogenous explanatory variables, and consistency of certain estimators can be established under weaker conditions than have previously been assumed in the literature. Hence, the use of many weak instruments may actually improve the performance of certain point estimators. More specifically, we find that the limited information maximum likelihood (LIML) estimator and the bias-corrected two-stage least squares (B2SLS) estimator are consistent when , while the two-stage least squares (2SLS) estimator is consistent only if Kn/rn→0 as n→∞. These consistency results suggest that LIML and B2SLS are more robust to instrument weakness than 2SLS.

We introduce a class of strategies that generalizes examples constructed in two-player games under imperfect private monitoring. A sequential equilibrium is belief-free if, after every private history, each player's continuation strategy is optimal independently of his belief about his opponents' private histories. We provide a simple and sharp characterization of equilibrium payoffs using those strategies. While such strategies support a large set of payoffs, they are not rich enough to generate a folk theorem in most games besides the prisoner's dilemma, even when noise vanishes.