Series estimation for single-index models under constraints

This paper discusses a semiparametric single-index model. The link function is allowed to be unbounded and has unbounded support that fill the gap in the literature. The link function is treated as a point in an infinitely many dimensional function space which enables us to derive the estimates for the index parameter and the link function simultaneously. This approach is different from the profile method commonly used in the literature. The estimator is derived from an optimization with the constraint of an identification condition for the index parameter, which solves an important problem in the literature of single-index models. In addition, making use of a property of Hermite orthogonal polynomials, an explicit estimator for the index parameter is obtained. Asymptotic properties of the two estimators of the index parameter are established. Their efficiency is discussed in some special cases as well. The finite sample properties of the two estimators are demonstrated through an extensive Monte Carlo study and an empirical example.