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 heteroskedastic IV regression with many weak instruments. In particular, it is ¯rst shown that consistency and asymptotically normality can be obtained when estimating structural parameters using JIVE, even when errors exhibit heteroskedasticity of unkown form. This is not the case, however, with other well known IV estimators, such as LIML, Fuller's modi¯ed LIML, 2SLS, and B2SLS, which are shown to be inconsistent. Second, new covariance matrix estimators (and corresponding Wald test statistics) are proposed for JIVE, which are consistent even when instrument weakness is such that the rate of growth of the concentration parameter, rn is slower than the rate of growth of the the number of instruments, Kn and possibly much slower than the sample size, n, provided that (Kn)^.5 /rn, rn goes to 0 as n goes to infinity. The primary advantage of our tests, relative to those proposed previously in the literature, is that one can test general nonlinear hypotheses, as opposed to simple null hypotheses of the form H0: Beta=Beta star, where beta star is the value of beta under the null. We feel that this feature, taken together with the fact that the tests are robust to unconditional heteroskedasticity, is important from the perspective of empirical application, given that general linear and nonlinear hypotheses are often of interest to empirical researchers, and given that heteroskedasticity is prevalent, particularly in microeconomic datasets.