# Amemiya’s Least Squares and Generalized Least Squares Estimators

Amemiya (1978c, 1979) proposed a general method of obtaining the esti­mates of the structural parameters from given reduced-form parameter esti­mates in general Tobit-type models and derived the asymptotic distribution. Suppose that a structural equation and the corresponding reduced-form equations are given by

y = Yy + X,)? + u – (10.8.11)

[y> Y] = X[jr, П] + V,

where X, is a subset of X. Then the structural parameters у and fi are related to the reduced-form parameters n and П in the following way:

п = Пу + ЗР, (10.8.12)

where J is a known matrix consisting of only ones and zeros. If, for example, Xj constitutes the first Kx columns of X (K = Kx+ K2), then we have J = (I, 0)’, where I is the identity matrix of size Kx and 0 is the K2 X Kx matrix of zeros. It is assumed that n, y, and P are vectors and П and J are matrices of conformable sizes. Equation (10.8.12) holds for Heckman’s model and more general simultaneous equations Tobit models, as well as for the standard simultaneous equations model (see Section 7.3.6).

Now suppose certain estimates n and П of the reduced-form parameters are given. Then, using them, we can rewrite (10.8.12) as

n = fly + jp + (n — n) — (П — П)у. (10.8.13)

Amemiya proposed applying LS and GLS estimation to (10.8.13). From Amemiya’s result (Amemiya, 1978c), we can infer that Amemiya’s GLS applied to Heckman’s model yields more efficient estimates than Heckman’s simultaneous equations two-step estimator discussed earlier. Amemiya (1983b) showed the superiority of the Amemiya GLS estimator to the WLS version of the Lee-Maddala-Trost estimator in a general simultaneous equa­tions Tobit model.