Advanced Econometrics Takeshi Amemiya
Asymptotic Distribution of the Limited Information Maximum Likelihood Estimator and the Two-Stage Least Squares Estimator
The LIML and 2SLS estimators of a have the same asymptotic distribution. In this subsection we shall derive it without assuming the normality of the observations.
We shall derive the asymptotic distribution of 2SLS. From (7.3.4) we have
yfT(a2s - a) = (Г-*ZJ PZ^-'r-^Z; Pu,.
The limit distribution of VT(d2s — a) is derived by showing that plim T~lZ PZ, exists and is nonsingular and that the limit distribution of T~i/2Z[ Pu, is normal.
First, consider the probability limit of T~lZ PZ,. Substitute (ХП, + V,, X,) for Zx in T~ lZ PZ,. Then any term involving У, converges to 0 in probability. For example,
plim T~lX[ X(X'X)-lX'Vl = plim T~lX X(T~lX, XrlT-lX, Vl
= plim Г-’Х; X(plim T-'X'X)-1 plim Г^Х'У, = 0.
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The second equality follows from Theorem 3.2.6, and the third equality follows from plim r^’X'V, = 0, which can be proved using Theorem 3.2.1. Therefore
Furthermore, A is nonsingular because rank (П10) = У,, which is assumed for identifiability.
Next, consider the limit distribution of T’_1/2Z,1Pu1. From Theorem 3.2.7 we have
-Lz'Pu rn'iX,"L >/T|_ x; J *’
where=means that both sides of it have the same limit distribution. But, using Theorem 3.5.4, we can show
■ iV(0, crfA). (7.3.8)
Thus, from Eqs. (7.3.5) through (7.3.8) and by using Theorem 3.2.7 again, we conclude
mchs - a) -> ЩО, aA"1). (7.3.9)
To prove that the LIML estimator of a has the same asymptotic distribution as 2SLS, we shall first prove
plim л/Г(А — 1) = 0. (7.3.10)
We note
х~1ТТш'
which follows from the identity
ifW'1/2W, W'l/2i; <J'W<5 rfn
where г/ = W U2S, and from Theorems 5 and 10 of Appendix 1. Because
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we have
_ u;[X(X'Xr'X' - X,(Xi X,)-'Xi]u, uJMu,
Therefore the desired result (7.3.10) follows from noting, for example, plim r-^uJXtX'Xr'X'u, = 0. From (7.3.3) we have
yff(dL-a) = [T-1 ZJPZ, - (A - Dr^ZJMZ,]-1 (7.3.13)
X [T-^ZJPu, - (A - OT^ZJMu,].
But (7.3.10) implies that both (A — l)r_1ZJMZ, and (A — )T~mZJMu, converge to 0 in probability. Therefore, by Theorem 3.2.7,
VT(dL - a) = >/f(Z; PZ, Г *Z{ Pu,, (7.3.14)
which implies that the LIML estimator of a has the same asymptotic distribution as 2SLS.
