On extended partially linear single-index models

Abstract

Aiming to explore the relation between the response y and the stochastic explanatory vector variable X beyond the linear approximation, we consider the single-index model, which is a well-known approach in multidimensional cases. Specifically, we extend the partially linear single-index model to take the form y =β T 0 X + φ(θ T 0 X ) + ε, where ε is a random variable with Εε=0 and var(ε)=σ 2 , unknown, β 0 and θ 0 are unknown parametric vectors and φ(.) is an unknown real function. The model is also applicable to nonlinear time series analysis. In this paper, we propose a procedure to estimate the model and prove some related asymptotic results. Both simulated and real data are used to illustrate the results.