I am estimating a linear model that contains endogenous
regressors (due to omitted variable problems). I first
applied a 2SLS procedure using a set of instruments but my
instruments seem to be week (low F in the first stage) and I
would like to estimate the same model with Limited
Information Maximum Likelihood.
My problem is the following: When I estimate the first stage
of the 2SLS, I exclude some exogenous regressors that are in
the main equation (I exclude them because of problems of
reverse causality in the first stage) and I am wondering
whether I can apply a LIML method and NOT use all the
exogenous variables from the main equation as explanatory
variables of the endogenous variables.
I know that ivreg2 does not allow to do it, so I was prepare
to compute the estimator "by hand" but when I look at the
matrix algebra of computing the LIML (using the formula
involving eigenvalue of a matrix of the data), I wonder
whether it can be used when some exogenous regressors are
excluded from the "first stage" (the derivation of this
formula is very intense in matrix algebra and quite obscure
to me).
Does anybody have an answer? Has anybody seen an application
of LIML where not all exogenous variables are used to
explain the endogenous variables?