Three-Stage Least Squares Estimator
In this section we shall again consider the full information model defined by
(7.1.1) . The 3SLS estimator of a in (7.1.5) can be defined as a special case of G2SLS applied to the same equation. The reduced form equation comparable to (7.3.22) is provided by
where X = I ® X, П = diagtdl,, J, ),(П2, J2), . . . , (Пдг, J*)], V = diagKVHOMV^O), . . . , (VN, 0)], and J, = (X’X)-‘X’X,.
To define 3SLS (proposed by Zellner and Theil, 1962), we need a consistent estimator of 2, which can be obtained as follows:
Step 1. Obtain the 2SLS estimator of a,-, /=1,2,. . . , N.
Step 2. Calculate fl, = у,- — Z,/*,^, /=1,2,. . . , N.
Step 3. Estimate аи by = T~lfiju,.
Next, inserting Z = Z, X = X, and = X ® I into (7.3.25), we have after some manipulation of Kronecker products (see Theorem 22 of Appendix 1).
<*3S = [Z'(2_1 © ® P)y (7.4.2)
= [Z'(i_1 © VZJr’Z’fX-1 ® I)y,
where Z = diag(Z[, Z^,. . . , ZN) and Z, = PZ,. The second formula of
(7.4.2) is similar to the instrumental variables representation of FIML given in
(7.2.12) . We can make use of this similarity and prove that 3SLS has the same asymptotic distribution as FIML. For exact distributions of 3SLS and FIML, see the survey articles mentioned in Section 7.3.5.