# Large sample properties of full information estimators

Theorem 8. For a linear simultaneous equations model satisfying the assump­tions in Theorem 2 and with a nonsingular error covariance matrix, the following asymptotic properties of the 3SLS estimator hold: ‘ft(X 3SLS – §) ^ N(0, V3sls)

^3SLS – V2SLS
where

V3SLS = plim (Z'(X-1 ® PX)Z/T)-1

V2SLS = plim {1/T(Z'(I ® PX)Z)-1(Z'(X-1 ® PX)Z)(Z'(I ® Px)Z)-1}.

Theorem 9. Under regularity conditions which are satisfied by the classical simultaneous equations model

Vt(5fiml – 5) Л N {0, plim [(1/T)(d  log L/3535′)]-1}.

Furthermore, if there are no restrictions on X, then the asymptotic covariances of the 3SLS and FIML estimators coincide and thus, 3SLS and FIML are asymptot­ically equivalent. Finally, FIML and 3SLS are asymptotically efficient within the class of all consistent, uniformly asymptotically Gaussian estimators of 5.

Theorem 10. FIML and linearized FIML are asymptotically equivalent.

For the instrumental variable method, suppose the instrument matrix for Z in y = Z5 + u is Z, of the form

 ( Z Z11 Z12 " Z1G N> II Z21 Z22 " Z2G VZG1 ZG2 " ZGG

Recall that the data matrix Z is diagonal (Z1, Z2,…, ZG).

Following the partitioning of Z, into (Y,, X) we can further decompose each submatrix

Z as Zij = (Zj1, Zj2)

where Zij1 and Zij2 are parts of the instrument matrices for the right-hand side endogenous and exogenous variables, respectively, appearing in the jth equation with respective dimensions T x (G, – 1) and T x Kj.

Theorem 11. eous system y = Z5 + u, let XIV = (Z’Z) 1Z’y,

where Z is defined in the preceding paragraph. Suppose

1. Pr(|Z’ZM0) = 1

2. plim (Z’u/T) = 0

3. plim (Z’Z/T) is nonsingular and finite

4. plim (Z’Z/Т) = plim (Z'(X <8> I)Z/T).

Then XIV and X3SLS are asymptotically equivalent if and only if the following two conditions hold: