Advanced Econometrics Takeshi Amemiya
Limited Information Model
7.3.1 Introduction
In this section we shall consider situations in which a researcher wishes to estimate only the parameters of one structural equation. Although these pa
rameters can, of course, be estimated simultaneously with the parameters of the remaining equations by FIML, we shall consider simpler estimators that do not require the estimation of the parameters of the other structural equations.
We shall assume for simplicity that a researcher wishes to estimate the parameters of the first structural equation
y^Y. y + Xj/H-UjC-Zje + u,), (7.3.1)
where we have omitted the subscript 1 from y, ft, and a. We shall not specify the remaining structural equations; instead, we shall merely specify the reduced form equations for Y,,
which is a subset of (7.1.2). The model defined by (7.3.1) and (7.3.2) can be regarded as a simplified simultaneous equations model in which simultaneity appears only in the first equation. We call this the limited information model In contrast, we call model (7.1.1) the full information model.
Assumptions 7.1.1 and 7.1.2 are still maintained. However, Assumption
7.1.3 need not be assumed if we assume (7.3.2). Assumption 7.1.1 implies that the rows of(Uj, V,) are i. i.d. with zero mean. Throughout this section we shall denote the variance-covariance matrix of each row of (и,, V,) by X. Partition Xas
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where о і is the variance of each element of u,. We assume that a is identifiable. It means that, if we partition П, as Ilj = (ГГП, П{ 0)' in such a way that X! is postmultiplied by Пп, then the rank of Пш is equal to , the number of elements of y.
