Finite Sample Properties of Estimators
Let us first consider the available results dealing with the general case where there are no further restrictions on the linear system nor on the equation being estimated apart from the condition of identifiability and the classical Gaussian assumptions. The characterization in (6.29)-(6.31) is the starting point of the analysis.
The first group of results here deals with the existence and nonexistence of moments of various estimators and is best summarized in the following theorem.
Theorem 12. In an identified equation in a linear simultaneous system satisfying assumptions A1-A4′, absolute moments of positive order for indicated estimators of coefficients in this structural equation are finite up to (and including) order
1. K2 – G1 for 2SLS and 3SLS;
2. h – K1 – G1, for modified 2SLS (the class of stochastic IV estimators) where h is the number of linearly independent first-stage regressors;
3. T – K1 – G1, for the k-class estimators with k nonstochastic and 0 < k < 1;
4. 0, for the k-class with nonstochastic k exceeding 1;
5. 0, for instrumental variable estimators with nonstochastic instruments; and
6. 0, for LIML and FIML.
Because of the distributional equivalence results discussed in the preceding section, (2) follows from (1). As a corollary to (1), 2SLS in the just identified case will have no finite absolute moments of positive order; as a consequence (5) also follows from (1).
References to specific authors who derived these results are given in Mariano (1982) and Phillips (1983).
From Theorem 12, we see that the LIML and FIML estimators (as well as 2SLS if the degree of over-identification is less than two) are inadmissible under a strictly quadratic loss function. Of course, admissibility comparisons may change under alternative loss functions with finite risks for the estimators. Such alternatives could be based on probability concentration around the true parameter value or other loss functions that increase at a slower rate than quadratic.
Beyond these moment existence results, various authors have derived closed – form expressions for the exact moments and probability density functions of these estimators – see Phillips (1983) for specific references. The expressions are rather complicated, in terms of zonal polynomials.
Another class of results in the general case deals with asymptotic expansions for the estimators themselves – namely, expressions of the type
7 = £ a, + Op(N-(r+1)/2), j=0
where a, = Op(N-/2) and a0 is the probability limit of 7 as N ^ «>. See Rothenberg (1984) for an extensive review. For the k-class estimator, such an expansion can be obtained by noting that we can write S(k) – в = H (I + A)-1n, where H is nonstochastic, A = Op(N~1/2) and n = Op(1) and then applying an infinite series expansion for the matrix inverse. One of the earliest papers on this in the econometric literature is Nagar (1959), where expansions of this type are formally obtained for the k-class estimator when k is of the form 1 + c/T where c is nonstochastic.
Given an asymptotic expansion like
S(k) – в = F0 + F—1/2 + F-1 + Op(T -3/2) = F + Op(T -3/2),
we can then proceed to define the "asymptotic moments" of (0^ – в) to be the moments of the stochastic approximation up to a certain order; for example, the moments of F. Thus, asymptotic moments determined this way relate directly to the moments of an approximation to the estimator itself and will not necessarily be approximations to the exact moments of 0W. For example, this will not hold for LIML and other estimators for which moments of low positive order do not exist. In cases where moments do exist, Sargan (1974, 1976) gives conditions under which these asymptotic moments are valid approximations to the moments of the estimator.
For a more concrete discussion, let us assume now that the equation being estimated contains only two endogenous variables. The early work on the analytical study of finite sample properties of SEM estimators dealt with this case. The equation now takes the following form:
y1 = у2в + Xay + щ, (6.32)
where в is scalar and y2 is T x 1. The characterization of the limited information estimators in (6.29)-(6.31) can be reduced further to canonical form for a better understanding of how parameter configurations affect the statistical behavior of the estimators (for example, see Mariano, 1977 and 1982). One reduction to canonical form simplifies in the case of two included endogenous variables to
0(щ – в = (о/ю^Т^р2 (0*) – X), (6.33)
where о2 = variance of the structural disturbance ut1, ю2 = variance of yt2, p = correlation coefficient between yt2 and ut1, X = р//1 – р2, and
S*k) = (xy + lu’v)/(y’y + kv’v). (6.34)
The vectors x, y, u, and v are mutually independent multivariate normal (the first two are K2 x 1 and the last two are (T – K) x 1) with unit variances, mean vector equal to zero for u and v and equal to ax = (0,…, 0, pX)’ and ay = (0,…, 0, p)’ for x and y, with p2 = [(E(y2))'(Px – Px1)(E(y2))]/®2.
The last quantity, p2, has been called the "concentration parameter" in the literature. This derives from the fact that as this parameter increases indefinitely, with sample size staying fixed, for k nonstochastic as well as for LIML, 0(k) converges in probability to the true parameter value в; Basmann (1961) and Mariano (1975). Also, in large sample asymptotics with the usual assumption that XX/T tends to a finite, positive definite limit, the variance in the limiting normal distribution of VT (02SLS – в) is inversely proportional to the limit of p2/ T. (The asymptotic variance is (о2/ю2) (lim(p2/T ))-1.)
One can then use (6.34) to derive exact expressions for moments and probability distributions of the k-class estimator – e. g. see Mariano (1982) and Phillips (1983) for a more detailed survey. The critical parameters are the correlation between error and endogenous regressor, the concentration parameter, the degree of over-identification, the difference between sample size and number of exogenous variables, and the size of structural error variance relative to the variance of the endogenous regressor.