Some Properties of a Modification of the Limited Information Estimator

THE SMALL SAMPLE PROPERTIES of the estimators of parameters in a single equation of a system of equations have been investigated in several ways. Some of the many Monte Carlo studies that have been conducted are reviewed in Johnston [8]. Nagar [14] used expansions in powers of terms whose order in probability was T1, where T is the sample size, to obtain approximations of the small-sample behavior of the fixed members of Theil's k-class estimators [18]. Basmann [4, 5], Kabe [9], Richardson [15], Sawa [17], and others, present exact distributions for the two-stage least squares estimators for certain models. Mariano and Sawa [13] give the exact distribution of the limited information estimator for the coefficient in a single equation containing two endogenous variables. The studies of the exact distributions show that the limited information estimator does not possess moments and that the first two moments of the two-stage least squares estimator exist only for certain levels of overidentification. Kadane [11] presents an approximation to the bias and mean square error of the fixed k-class estimator and the limited information estimator in terms of an expansion in o-, where o_2 iS (a multiple of) the variance of the residuals in the equation. Kadane's results for the fixed k-class agree with those of Nagar [14]. Recently asymptotic expansions of tie distribution function of the estimators of a single equation have appeared (e.g., see Anderson [1], and Sargan and Mikhail [16]). We present a modification of the limited information estimator and demonstrate that the modified estimator possesses finite moments and that one member of the class has bias of order T2. The estimator is a member of the k-class estimators originally introduced by Theil [18]. We also introduce a modification of the Nagar [14] "fixed" k-class estimators to ensure finite moments. The modified estimators have the same limiting distribution as the unmodified estimators, and the approximations presented by Nagar hold for the first two moments of the modified fixed k-class estimator. Restricting the modified fixed k-class estimator and the modified limited information estimator to have the same, but arbitrary, bias we show that to order T2 the modified limited information estimator dominates the fixed k-class 1 Journal Paper number J-8374 of the Iowa Agriculture and Home Economics Experiment Station, Ames, Iowa; Project 2039. This research was partly supported by the United States Bureau of the Census through Joint Statistical Agreements 74-1 and 75-1.