Asymptotic expansions are made of the distributions of a class of semi-parametric estimators including the Maximum Empirical Likelihood (MEL) method and the Generalized Method of Moments (GMM) for the coefficients of a single structural equation in the linear simultaneous equations system. The expansions in terms of the sample size, when the non-centrality parameters increase proportionally, are carried out to the order of O (n^-2). Comparisons of the distributions of the MEL and GMM estimators are also made.