  Overidentified two-stage least square (TSLS) is commonly adopted by applied
economists to address endogeneity. Though it potentially gives more efficient
or informative estimate, overidentification comes with a cost. The bias of TSLS
is severe when the number of instruments is large. Hence, Jackknife
Instrumental Variable Estimator (JIVE) has been proposed to reduce bias of
overidentified TSLS. A conventional heuristic rule to assess the performance of
TSLS and JIVE is approximate bias. This paper formalizes this concept and
applies the new definition of approximate bias to three classes of estimators
that bridge between OLS, TSLS and a variant of JIVE, namely, JIVE1. Three new
approximately unbiased estimators are proposed. They are called AUK, TSJI1 and
UOJIVE. Interestingly, a previously proposed approximately unbiased estimator
UIJIVE can be viewed as a special case of UOJIVE. While UIJIVE is approximately
unbiased asymptotically, UOJIVE is approximately unbiased even in finite
sample. Moreover, UOJIVE estimates parameters for both endogenous and control
variables whereas UIJIVE only estimates the parameter of the endogenous
variables. TSJI1 and UOJIVE are consistent and asymptotically normal under
fixed number of instruments. They are also consistent under many-instrument
asymptotics. This paper characterizes a series of moment existence conditions
to establish all asymptotic results. In addition, the new estimators
demonstrate good performances with simulated and empirical datasets.
