Triangular simultaneous equations model under structural misspecification

Triangular simultaneous equation models are commonly used in econometric analysis to analyse endogeneity problems caused, among others, by individual choice or market equilibrium. Empirical researchers usually specify the simultaneous equation models in an ad hoc linear form; without testing the validity of such specification. In this paper, approximation properties of a linear fit for structural function in a triangular system of simultaneous equations are explored. I demonstrate that, linear fit can be interpreted as the best linear prediction to the underlying structural function in a weighted mean squared (WMSE) error sense. Furthermore, it is shown that with the endogenous variable being a continuous treatment variable, under misspecification, the pseudo-parameter that defines the returns to treatment intensity is weighted average of the Marginal Treatment Effects (MTE) of Heckman and Vytlacil (2001). Misspecification robust asymptotic variance formulas for estimators of pseudo-true parameters are also derived. The approximation properties are further investigated with Monte-Carlo experiments.