# Nonlinear Three-Stage Least Squares Estimator

As a natural extension of the class of the NL2S estimators—the version that minimizes (8.1.22), Jorgenson and Laffont (1974) defined the class of nonlin­ear three-stage least squares (NL3S) estimators as the value of a that mini­mizes

f'[2_1 ® W(W/W)-‘W,]f, (8.2.2)

A

where X is a consistent estimate of X. For example,

І-іІн&Жа)’, (8-2.3)

1 г-і

where a is the NL2S estimator obtained from each equation. This definiton of the NL3S estimators is analogous to the definition of the linear 3SLS as a generalization of the linear 2SLS. The consistency and the asymptotic nor­mality of the NL3S estimators defined in (8.2.2) and (8.2.3) have been proved by Jorgenson and Laffont (1974) and Gallant (1977).

The consistency of the NL2S and NL3S estimators of the parameters of model (8.2.1) can be proved with minimal assumptions on utt—namely, those stated after (8.2.1). This robustness makes the estimators attractive. Another important strength of the estimators is that they retain their consist­ency regardless of whether or not (8.2.1) yields a unique solution for y, and, in the case of multiple solutions, regardless of what additional mechanism chooses a unique solution. (MaCurdy, 1980, has discussed this point further.) However, in predicting the future value of the dependent variable, we must know the mechanism that yields a unique solution.

Amemiya (1977a) defined the class of the NL3S estimators more generally as the value of a that minimizes

f/A"1S(S/A-,Sr, S/A-I£ (8.2.4)

where A is a consistent estimate of Л and S is a matrix of constants with NT rows and with the rank of at least 2£L[ Kt. This definition is reduced to the Jorgenson-Laffont definition if S = diag(W, W,. . . , W). The asymptotic variance-covariance matrix of IT times the estimator is given by

V3 = plim TIG’A-‘SfS’A-‘Sr’S’A-‘G]"1. (8.2.5)

Its lower bound is equal to VB3 = lira 7′[£’G, A-l£’G]-1

which is attained when we choose S = EG. We call this estimator the BNL3S estimator (B for “best”).

We can also attain the lower bound (8.2.6) using the Jorgenson-Laifont definition, but that is possible if and only if the space spanned by the column vectors of W contains the union of the spaces spanned by the column vectors of EG і for / = 1, 2,. . . ,N. This necessitates including many columns in W, which is likely to increase the finite sample variance of the estimator although it has no effect asymptotically. This is a disadvantage of the Joigenson-LafFont definition compared to the Amemiya definition.

Noting that BNL3S is not practical, as was the case with BNL2S, Amemiya (1976a) suggested the following approximation:

Step 1. Compute &j, an SNL2S estimator of a„ /=1,2,. . . , N.

Step 2. Evaluate G, at a,—call it G,.

Step 3. T reat Clf as the dependent variables of a regression and search for the optimal set of independent variables W, that best predict 6,.

Step 4. Choose S = diagfP^!, P2G2,. . . , Рлгбд,}, where P, =

wxw/w. r’w;.

Applications of NL3S can be found in articles by Jorgenson and Lau (1975), who estimated a three-equation translog expenditure model; by Jorgenson and Lau (1978), who estimated a two-equation translog expenditure model; and by Haessel (1976), who estimated a system of demand equations, nonlin­ear only in parameters, by both NL2S and NL3S estimators.