[Under classical assumptions, characterizations are given for two classes of instrumental variable estimators of an equation in a simultaneous system. IV estimators where all instruments are nonstochastic are expressed in terms of multinormal random vectors in exactly the same way as the 2SLS estimator of a just-identified equation. These estimators have no finite moments of positive integral order. The second class, consisting of IV estimators based on certain stochastic instruments, includes the OLS, 2SLS, and modified 2SLS estimators. The inadmissibility (under squared-error loss) of some estimators in this class is considered when the equation being estimated contains two endogenous variables.]

Roberto S. Mariano, Approximations to the Distribution Functions of the Ordinary Least-Squares and Two-Stage Least-Squares Estimators in the Case of Two Included Endogenous Variables, Econometrica, Vol. 41, No. 1 (Jan., 1973), pp. 67-77

[This paper deals with two single-equation estimators in a set of simultaneous linear stochastic equations--namely, ordinary least squares (OLS) and two-stage least squares (2SLS). Under the assumption that all predetermined variables in the model are exogenous, necessary and sufficient conditions are obtained for the existence of even moments of the above estimators. It is shown that for the general case with an arbitrary number of included endogenous variables, even moments of the 2SLS estimator are finite if and only if the order is less than K2 - G1 + 1. Furthermore, even moments of the OLS estimator exist if and only if the order is less than N - K1 - G1 + 1, where N is the sample size, G1 + 1 is the number of included endogenous variables, K1 and K2 respectively are the number of included and excluded exogenous variables in the equation to be estimated.]

This paper proposes the residual-based stochastic predictor as an alternative procedure for obtaining forecasts with a static nonlinear econometric model. This procedure modifies the usual Monte Carlo approach to stochastic simulations of the model in that calculated residuals over the sample period are used as proxies for disturbances instead of random draws from some assumed parametric distribution. In compar-ison with the Monte Carlo predictor, the residual-based should be less sensitive to distributional assumptions concerning disturbances in the system. It is also less demanding computationally. The large-sample asymptotic moments of the residual-based predictor are derived in this paper and compared with those of the Monte Carlo predictor. Both procedures are asymptotically unbiased. In terms of asymptotic mean squared prediction error (AMSPE), the Monte Carlo is efficient relative to the residual-based when the number of replications in the Monte Carlo simulations is large relative to sample size. This order of relative efficiency is reversed, however, when replication and sample sizes are similar. In any event, the amount by which the AMSPE of either predictor exceeds the lower bound for AMSPE is small as a percentage of the lower bound AMSPE when sample and replication sizes are at least of moderate magnitude. The paper also discusses the extension of the residual-based anld Monte Carlo procedures to the estimation of higher order moments and cumulative distribution functions of endogenous variables in the system.