Abstract
Estimation in two classes of popular models, single-index models 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 function in the models and a prole approach is used to derive the estimators. The ndings include dual convergence rates of the estimators for the single-index models and a trio of convergence rates for the partially linear single-index models. More precisely, the estimators for single-index model converge along the direction of the true parameter vector at rate of n 1=4 , while at rate of n 3=4 along all directions orthogonal to the true parameter vector; on the other hand, the estimators of the index vector for the partially single-index model retain the dual convergence rates as in the single-index model but the estimators of the coecients in the linear part of the model possess rate n 1 .
Highlights
This paper considers the estimation of partially linear single-index models of the form yt = β0 xt + g(θ0 xt) + et, t = 1, · · ·, n, (1.1)
Most researchers only focus on the stationary covariate case so that their theoretical results are not applicable for practitioners who use partially linear single-index model to deal with nonstationary time series data
In view of (3.17), for θn estimated from the partially linear single-index model, we may still expect the results of Theorems 3.4 and 3.5 with θn,emp defined in the same way as before
Summary
In the last decade or so, nonlinear (nonparametric or semiparametric) and nonstationary time series models have been studied extensively and improved dramatically as witnessed by the literature, such as those based on the nonparametric kernel approach by Karlsen and Tjøstheim (2001); Karlsen et al (2007), Gao et al (2009a,b), Phillips (2009), Wang and Phillips (2009a,b), Gao (2012), Wang and Phillips (2012), Gao and Phillips (2013a,b) and Phillips et al (2013), among others. In the nonstationary context, Cai and Gao (2013) give a useful result about the convergence of the signal matrix that requires a truncation parameter to be relatively smaller than that used in the stationary time series case We shall adopt this result and in this paper the smoothness of the link function is required to be relatively higher. Guerre and Moon (2006) point out that their method may be used for the estimation of single-index models where the link function g(x) → ∞ when |x| → ∞ Both of them are quite different from the setting of this study. Convergence in probability and distribution are signified as →P and →D, respectively
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