Another Look at Single-Index Models Based on Series Estimation

In this paper, a semiparametric single-index model is investigated. The link function is allowed to be unbounded and has unbounded support that answers a pen ding issue in the literature. Meanwhile, 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 derive d from an optimization with the constraint of identification condition for index parameter, which is a natural way but ignored in the literature. In addition, making use of a property of Hermite orthogonal polynomials, an explicit estimator for the index parameter is obtained. Asymptotic properties for 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 estimates are demonstrated through an extensive Monte Carlo study and an empirical example.