# Estimation of simultaneous equation sample selection model

A two-stage estimation method can be easily generalized for the estimation of a simultaneous equation model. Consider the linear simultaneous equation y* = y*B + xC + u, which can be observed only if zY > e. For the estimation of structural parameters, consider the first structural equation y* = y(1) ри + x151 + ии where y*(1) consists of included endogenous variables on the right-hand side of the structural equation. The bias-corrected structural equation is уи = y(1p1 + x181 + o1e (—-фт|)-) + . The system implies the reduced form equations y* = хП + v.

Lee, Maddala, and Trost (1980) suggest the estimation of the reduced form parameters П by the Heckman two-stage method, and used the predicted y(1) to estimate the bias-corrected structural equation similar to Theil’s two-stage method for a conventional simultaneous equation model.

The structural parameters can also be estimated by a general minimum dis­tance procedure (Amemiya, 1979). Amemiya’s method is a systematic procedure for estimating structural parameters directly from estimated reduced form para­meters. Let J and J2 be the selection matrices such that y(l) = у*/и and хи = xJ2. As y* = y*Jipi + xJ251 + u = х(П/ири + J281) + v1, one has пи = П/ири + J281. Let П be the reduced form estimate from Heckman’s two-stage estimation. Amemiya’s minimum distance procedure is to estimate ри and 5и from the linear equation li = П/ipi + J2§ + Си, where Zi. = (| – пи) – (П – П)/ири is the disturbance, by

least squares or generalized least squares. The relative efficiency of an estimator
from this minimum distance approach depends on the relative efficiency of the reduced form parameter estimates (Amemiya, 1983). Lee (1981), Amemiya (1983) and Newey (1987) compare various two-stage IV estimation methods with Amemiya’s generalized minimum distance estimators. It was found that many two-stage IV estimators are special cases of Amemiya’s minimum distance estimators depending on appropriate reduced form estimates. Lee (1992a) shows that the minimized generalized sum of squares residuals from Amemiya’s gener­alized least-squares procedure also provides a test of overidentification restric­tions of a linear structural equation. However, because Amemiya’s approach relies on solving structural parameters from reduced form parameters, it cannot be generalized to the estimation of a nonlinear simultaneous equation system while many IV approaches can.