This paper derives the limiting distributions of
alternative jackknife instrumental variables (JIV)
estimators and gives formulas for accompanying
consistent standard errors in the presence of
heteroskedasticity and many instruments. The
asymptotic framework includes the many instrument
sequence of Bekker (1994, Econometrica 62,
657–681) and the many weak instrument sequence of
Chao and Swanson (2005, Econometrica 73,
1673–1691). We show that JIV estimators are
asymptotically normal and that standard errors are
consistent provided that as
n→∞, where
Kn and
rn denote,
respectively, the number of instruments and the
concentration parameter. This is in contrast to the
asymptotic behavior of such classical instrumental
variables estimators as limited information maximum
likelihood, bias-corrected two-stage least squares,
and two-stage least squares, all of which are
inconsistent in the presence of heteroskedasticity,
unless
Kn/rn→0.
We also show that the rate of convergence and the
form of the asymptotic covariance matrix of the JIV
estimators will in general depend on the strength of
the instruments as measured by the relative orders
of magnitude of rn and
Kn.