## 5.2 Consistency of estimated solutions to the system of equations (3) and (16) 

Establishing consistency for the system of equations is more complicated since consistency requires not only conditions on estimators of known func- tions of the observables, ε$_{·}$, but also verification that the resulting estimators of φ satisfy the conditions of well-posedness in Theorem 5. 

The next Theorem gives conditions for a consistent plug-in nonparametric estimator for the model (7 − 9) that leads to the system (3) and thus (16) with continuously differentiable φ. 

Semiparametric generalized method of moments estimation of this model for a class of regression functions that includes functions in L$_{1}$ ( R $^{d}$) was proposed by Wang and Hsiao (2010); semiparametric estimation was also discussed for somewhat different classes of univariate parametric regression functions in Schennach (2007) and Zinde-Walsh (2009). Semiparametric esti- mation of polynomial regression is in Hausman, Newey, Ichimura and Powell (1991). 

Start by formally stating the assumptions. 

Assumption 4 (model) . In the model (7 − 9) in R d the moments of the errors E ( u$_{y}$ | z, u, u$_{x}$ ) = 0; E ( u$_{x}$ | z, v ) = 0; z is independent of u ; g is a generalized function in S $^{∗}$. 

This assumption implies that if the function g is a regular locally summable function it cannot grow at infinity at a rate faster than some polynomial rate. 

