Estimation and Testing Using Jackknife IV in Heteroskedastic Regressions with Many Weak Instruments

This paper develops Wald-type tests for general (possibly nonlinear) restrictions in the context of a weakly-identified heteroskedastic IV regression. In particular, it is first shown that, in a framework with many weak instruments, consistency and asymptotic normality can be obtained when estimating structural parameters using JIVE, even if disturbances exhibit heteroskedasticity of unknown form. This is not the case, however, with other well-known IV estimators, such as LIML, Fuller's modified LIML, 2SLS, and B2SLS, which are shown to be inconsistent in general under heteroskedasticity. We also introduce new covariance matrix estimators for JIVE, which are consistent even when instrument weakness is such that the rate of growth of the concentration parameter, r(n), is slower than that of the number of instruments, K(n), and possibly much slower than the sample size n, provided that K(n)^0.5/r(n) goes to zero as n approaches infinity. Wald test statistics are then constructed using these covariance matrix estimators, and the resulting statistics are shown to have limiting chi-square distributions under the null hypothesis. A primary advantage of our approach is that, relative to other testing frameworks which have previously been proposed in the weak instruments literature, our framework allows one to test hypotheses more general than simple point null hypotheses. We feel that this feature, taken together with the fact that our tests are robust to heteroskedasticity of unknown form, is important from the perspective of empirical application, given that testing general linear and nonlinear restrictions are often of interest to empirical researchere, and given that heteroskedasticity is prevalent, particularly in microeconomic datasets