The asymptotic and finite sample distributions of OLS and simple IV in simultaneous equations

Abstract

In practice structural equations are often estimated by least-squares, thus neglecting any simultaneity. It is examined when this may be justifiable. Assuming data stationarity and existence of the first four moments of the disturbances the limiting distribution of the ordinary least-squares (OLS) estimator in a linear simultaneous equations model is derived. In simple static and dynamic models the asymptotic mean squared error of this inconsistent estimator is compared with that of consistent simple instrumental variable (IV) estimators and cases are depicted where—due to relative weakness of the instruments or mildness of the simultaneity—the inconsistent estimator is more precise. In addition, it is examined by simulation to what extent these first-order asymptotic findings are reflected in finite sample, taking into account non-existence of moments of the IV estimator. Dynamic visualization techniques enable to appreciate any differences in precision over a parameter space of a much higher dimension than just two, through colored animated image sequences (which are not very effective in print, but much more so in live-on-screen projection).