Estimation mainly for two classes of popular models, single-index and partially linear single-index models, is studied in this paper. Such models feature nonstationarity. Orthogonal series expansion is used to approximate the unknown integrable link functions in the models and a profile approach is used to derive the estimators. The findings include the dual rate of convergence of the estimators for the single-index models and a trio of convergence rates for the partially linear single-index models. A new central limit theorem is established for a plug-in estimator of the unknown link function. Meanwhile, a considerable extension to a class of partially nonlinear single-index models is discussed in Section 4. Monte Carlo simulation verifies these theoretical results. An empirical study furnishes an application of the proposed estimation procedures in practice.
Chaohua Dong. Jiti Gao. Dag Tjøstheim. "Estimation for single-index and partially linear single-index integrated models." Ann. Statist. 44 (1) 425 - 453, February 2016. https://doi.org/10.1214/15-AOS1372