Large sample properties of limited information estimators
We now consider the large sample asymptotic behavior of the limited information estimators described in Section 2. Because of space limitation, theorems are stated without proof. We start with a theorem for limited information instrumental variable estimators in general.
Thus, first consider the IV estimator defined in (6.5): XIV = (WZ) 2W'y1, where the instrument matrix W is of the same dimension as Z = (Y1, Xa) such that
(1) plim W ' u1/T = 0 (6.25)
(2) plim W' W/T = Qw, positive definite and finite (6.26)
(3) plim W'Z/T = M', nonsingular (6.27)
(4) W'щ/ VT converges in distribution to N(0, о2Qw). (6.28)
Note that (6.25) together with (6.27) implies that XIV is consistent. Also, (6.25) is a consequence of (6.28) and hence (6.25) would be unnecessary if (6.28) is assumed.
(6.26) is not necessarily satisfied by the original model; for example, when X contains a trending variable. For cases like these, the theorems we discuss presently will require further modifications. Both (6.26) and (6.27) will need minimally the assumption that (XX/ T) has a finite positive definite limit as T approaches infinity. (6.28) depends on a multivariate version (e. g. Lindeberg-Feller) of the central limit theorem for independent but nonidentically distributed random vectors.
Assuming that the instrument matrix W satisfies the three properties (6.26)- (6.28), we have
1. Vt(XIV - 5) A N(0, o2(MQw1M')-1) = N[0, о2plim(Z'?WZ/T)-1]
2. A consistent estimator of о2 is (y1 - ZX IV)'( y1 - ZX IV)/T.
Applying the above result to two-stage least squares, we get
Theorem 2. Vi (X2SLS - 5) A N(0, o2(RQ-1R')-1) = N[0, о2 plim (Z'PXZ/T)-1], where R = plim (Z'X/T) and Q = plim (XX/T), under the assumptions that
1. R exists, is finite, and has full row rank,
2. Q exists and is finite and positive definite,
3. Xu1/(T1/2) ^ N(0, o2Q).
Corollary 1. A consistent estimator of о2 is (y1 - zX2SLS)' (y1 - ZX2SLS)/T.
For the fc-class estimator, we have already indicated that plim k = 1 is a sufficient condition for consistency. A sharper result derives from the two previous theorems.
Theorem 3. If k is such that plim T 1/2(k - 1) = 0, then T1/2(X2SLS - 5) and T 1/2(X(k) - 5) are asymptotically equivalent.
For the LIML estimator, we have the following result as a consequence of the preceding theorem.
Theorem 4. If the data matrix X is such that (X'X/T) has a finite positive definite limit, then, under assumptions A1, A2, and A4' in Section 1, LIML is consistent and asymptotically equivalent to 2SLS; that is ft (5 LIML - 5) ^ N(0, a2(RQ-1R')-1) where R and Q are as defined in Theorem 2.
Under the conditions of the theorem, it can be shown that plim ft € = 0 where € is the smallest characteristic root of | W - XS | = 0, where W and S are the second moment residual matrices defined in (6.15). Thus, Theorem 4 now follows directly from Theorem 3 and the fact that LIML is equivalent to k-class with k equal to 1 + €.
Note that ft € 0 is a consequence of the stronger result that T€ converges
in distribution to a central chi-squared distribution with degrees of freedom equal to the number of overidentifying restrictions.
If the structural errors ut are normally distributed, then LIML is asymptotically efficient among all consistent and uniformly asymptotic normal (CUAN) estimators of в and у based on the "limited information" model discussed in the preceding section. By Theorem 4, under normality assumptions, 2SLS shares with LIML this property of being asymptotically efficient within this class of CUAN estimators of в and y.
Theorem 5. Let X LIVE be the IV estimator of 5 based on the instrument matrix
WL = (XП1, Xj), where } is any consistent estimator of Па. Then XLIVE is asymptotically equivalent to X 2SLS.
Theorem 6. Consider the class of IV estimators of the first equation where instruments are of the form W = XF, where F is either stochastic or nonstochastic and of size K x (K1 + G1 - 1) and of full rank. Within this class, two-stage least squares is asymptotically efficient.
Theorem 7. For the modified 2SLS estimator XM2SLS, where the first stage regressor matrix is H instead of X,
1. XM2SLS is exactly equivalent to the IV estimator of 5 using the instrument matrix (PHY1, X1) if X1 is contained in the column space of H.
2. XM2SLS is consistent if and only if the column space of H contains the column space of X1.