Full Information Maximum Likelihood Estimator

In this section we shall define the maximum likelihood estimator of the parameters of model (7.1.1) obtained by assuming the normality of U, and we shall derive its asymptotic properties without assuming normality. We attach the term full information (FIML) to distinguish it from the limited information maximum likelihood (LIML) estimator, which we shall discuss later.

The logarithmic likelihood function under the normality assumption is given by

NT T

log L = – ~ log (2к) + Tlog ІІГІІ – – log |2| (7.2.1)

– j tr 2-‘(УГ – XB)'(YT – XB),

where ІІГІІ denotes the absolute value of the determinant of Г. We define the FIML estimators as the values of 2, Г, and В that maximize log L subject to the normalization on Г and the zero restrictions on Г and B. We shall derive its properties without assuming normality; once we have defined FIML, we shall treat it simply as an extremum estimator that maximizes the function given in

(7.2.1) . We shall prove the consistency and asymptotic normality of FIML under Assumptions 7.1.1, 7.1.2, and 7.1.3; in addition we shall assume the identifiability condition (7.1.8) and TS N+ K.

Differentiating log L with respect to 2 using the rules of matrix differentia­tion given in Theorem 21 of Appendix 1 and equating the derivative to 0, we obtain

2= Г-1(УГ — XB)’ (YT – XB). (7.2.2)

Inserting (7.2.2) into (7.2.1) yields the concentrated log likelihood function

log L* = – j log |(Y – XBT_1)'(Y – ХВГ-‘)|, (7.2.3)

where we have omitted terms that do not depend on the unknown parameters. The condition T Ш N + К is needed because without it the determinant can be made equal to 0 for some choice of В and Г. Thus the FIML estimators of the unspecified elements of В and Г can be defined as those values that minimize

ST = T~l( – XBT’O’CY – ХВГ-1)|. (7.2.4)

Inserting Y = ХВ0Го1 + V, where B0 and Г0 denote the true values, into

(7.2.4) and taking the probability limit, we obtain

р1іт5г = |Л0 + (В0Го1-ВГ-1),А(В0Го1-ВГ-1)|, (7.2.5)

where A = lim Г_1Х’Х. Moreover, the convergence is clearly uniform in В and Г in the neighborhood of B„ and Г0 in the sense defined in Chapter 4. Inasmuch as plim ST is minimized uniquely at В = B0 and Г = Г0 because of

image498

the identifiability condition, the FIML estimators of the unspecified elements of Г and В are consistent by Theorem 4.1.2. It follows from (7.2.2) that the FIML estimator of 2 is also consistent.

The asymptotic normality of the FIML estimators of Г and В can be proved by using Theorem 4.1.3, which gives the conditions for the asymptotic nor­mality of a general extremum estimator. However, here we shall prove it by representing FIML in the form of an instrumental variables estimator. The instrumental variables interpretation of FIML is originally attributable to Durbin (1963) and is discussed by Hausman (1975).

Differentiating (7.2.1) with respect to the unspecified elements of В (cf. Theorem 21, Appendix 1) and equating the derivatives to 0, we obtain

Х'(УГ-ХВ)2-‘4 0, (7.2.6)

where = means that only those elements of the left-hand side of (7.2.6) that correspond to the unspecified elements of В are set equal to 0. The zth column of the left-hand side of (7.2.6) is X’ (YT — ХВ)<т’, where a‘ is the zth column of 2-1. Note that this is the derivative of log L with respect to the zth column of B. But, because only the Kt elements of the zth column of В are nonzero,

(7.2.6) is equivalent to

X<(YT – XB)<r’ = 0, /—1,2……….. iV. (7.2.7)

We can combine the N equations of (7.2.7) as

-Xj 0 • • 0-

"Уі — Z, a, ■

0 X’2

(X-1 ® I)

y2 ~~ 22a2

-o Xir

"Улг ~~ ZjvOttf

Differentiating (7.2.1) with respect to the unspecified elements of Г and equating the derivatives to 0, we obtain

Т(Г T1 “ Y'(Yr – XB)2-‘ = 0, (7.2.9)

where = means that only those elements of the left-hand side of (7.2.9) that correspond to the unspecified elements of Г are set equal to 0. Solving (7.2.2) for 71 and inserting it into (7.2.9) yield

Y'(YT – XB)2-‘ * 0, (7.2.10)

we can rewrite (7.2.10) as

where Y( consists of the nonstochastic part of those vectors of Y that appear in the right-hand sid£ of the zth structural_equation.

Defining Z( = (Y,, X,) and Z = diag(Z, ,Ъг,………….. ZN), we can combine

(7.2.8) and (7.2.11) into a single equation

a = [Z'(X_1 © I)Z]-lZ'(X~l © I)y. (7.2.12)

The FIML estimator of a, denoted a, is a solution of£7.2.12), where X is replaced by the right-hand side of (7.2.2). Because both Z and X in the right – hand side of (7.2.12) depend on a, (7.2.12) defines a implicitly. Nevertheless, this representation is useful for deriving the asymptotic distribution of FIML as well as for comparing it with the 3SLS estimator (see Section 7.4).

Equation (7.2.12), with X replaced by the right-hand side of (7.2.2), can be used as an iterative algorithm for calculating a. Evaluate Z and X in the right-hand side of (7.2.12) using an initial estimate of a, thus obtaining a new estimate of a by (7.2.12). Insert the new estimate into the right-hand side, and so on. However, Chow (1968) found a similar algorithm inferior to the more standard Newton-Raphson iteration (cf. Section 4.4.1).

The asymptotic normality of a, follows easily from (7.2.12). Let Z and X be Z and X evaluated at a, respectively. Then we have from (7.2.12)

4f{6t -a) = [T~lZ'(i-1 ® I © I)u. (7.2.13)

Because a and X are consistent estimators, we can prove, by a straightforward application of Theorems 3.2.7, 3.5.4, and 3.5.5, that under Assumptions

7.1.1, 7.1.2, and 7.1.3

4f(a-a)-^N{0, [lim r-1Z'(X-1®I)Z]-1}. (7.2.14)

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