Limited Information Maximum Likelihood Estimator
The LIML estimator is obtained by maximizing the joint density of y! and Y! under the normality assumption with respect to а, П, and X without any constraint. Anderson and Rubin (1949) proposed this estimator (without the particular normalization on Г we have adopted here) and obtained an explicit formula for the LIML estimator:4
and m, = і – x, (x; x, r1 x;.
7.3.2 Two-Stage Least Squares Estimator
The two-stage least squares (2SLS) estimator of a, proposed by Theil (1953),5 is defined by
d2S = (Z’1PZ1)-1Z’1Pyl,
where P = X(X’X)-1X.