Asymptotic Expansions of the Distributions of Estimates in Simultaneous Equations for Alternative Parameter Sequences

Abstract

The distributions of the LIML and TSLS estimates of the coefficient of an endogenous variable in a single equation can be approximated by asymptotic expansions. This paper relates the expansions in terms of the noncentrality parameter and the sample size going to infinity, the noncentrality parameter going to infinity with the sample size held fixed, and the standard deviation of the disturbance going to zero (small-o). 1. INTRODUCriON RECENTLY, ASYMPTOTIC EXPANSIONS of the distributions of estimates of coefficients of a single equation in a system of simultaneous equations have been made by Anderson [1], Anderson and Sawa [2], Mariano [6 and 7], and Sargan and Mikhail [11]. The expansions have usually been carried out on the basis that the sample size increases and that the effect of the exogenous variables (the noncentrality parameter) increases along with the sample size. In this paper we consider the case of the covariance matrix of the disturbances known and alternatively the case of the sample size fixed. We relate these three cases to the approach of letting the disturbance decrease (the small-o- approach). The estimates treated are two-stage least squares (TSLS) and limited information maximum likelihood (LIML).