The Asymptotic Bias and Mean-Squared Error of Double K-Class Estimators When the Disturbances are Small

Abstract

Two STRUCTURAL ESTIMATION TECHNIQUES for the parameters of a single equation in a system of simultaneous linear equations were available at the time of the Cowles Commission work (Koopmans [13], Hood and Koopmans [8]): ordinary least squares (OLS) and limited-information maximum likelihood (LIML). Later Theil [19, 20, 21] and Basinann [2] developed two-stage least squares (2SLS). Theil [20, 21] invented the k-class, which includes OLS (k 0), 2SLS (k = 1), and LIML (k = 2, the minimum root of a certain determinental equation (Anderson and Rubin [1])). Because the k-class contains these three leading contenders among single-equation estimators, it has been a convenient class of estimators for theoretical study. Theil [21, (353-354)] also proposed the h-class, which contains OLS and 2SLS buLt not LIML. Nagar [14] proposed the double kclass, a two-parameter family which embraces both the hand k-classes. The double k-class allows considerable flexibility with regard to the choice of an estimation teclhique, as well as providing a useful sumllmary statement of previoLusly proposed single-equnation estimators. Comparisons among competing estimators are greatly facilitated if the estimators can all be regarded as special instances of a general procedure. This paper provides small-disturbance asymptotic moment expressions for the general case of the double k-class estimator. These expressions can be applied to make comparisons among any double k-class estimators. Previous results describing the behavior of special cases of the double k-class estimators have been derived by various methods, including large-sample asymptotic theory, exact finite-sample theory, Monte Carlo studies, and small-disturbance asymptotic theory. Kadane [10] gave a review of these methods and their results as applied to the (single) k-class. He also derived the small-disturbance asymptotic bias and mean-squared error of the k-class estimators. The only analytic results on the entire double k-class previously available are those of Nagar [14], who derived expressions for the large-sample asymptotic bias to order (lIT) and the asymptotic mean-squared error to order (I/T2) for all double k-class estimators with parameters of the form k, 1 + c1/T, k2 = 1 +a2/T for constant a, and a2. The method of analysis used in this paper to investigate the double k-class estimators is based upon the small-disturbance asymptotic approach introduced