Series Estimation of Semilinear Models

This paper discusses estimation of the semilinear model E[y | x, z] = x′β + g(z) using series approximations to the unknown function g(z) under much weaker conditions than heretofore given in the literature. In particular, we allow for z being multidimensional and to have a discrete distribution, features often present in applications. In addition, the smoothness conditions are quite weak: it will suffice for √n consistency of β̂ that the modulus of continuity of g(z) and E[x | z] be higher than one-fourth the dimension of z and that the number of terms be chosen appropriately.