Abstract

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 pending 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 derived from an optimization with the constraint of identification condition for index parameter, which is a natural way but ignored in the literature.

Highlights

  • Single-index models have been studied extensively in the econometrics and statistics literature in the past thirty years or so and cover many classic parametric models by using a general function form g0 (x θ0)

  • One important class of estimation methods is based on using a nonparametric kernel method, (e.g. Ichimura (1993), Hardle, W. and Hall, P. and Ichimura, H. (1993), Carroll et al (1997), Chapter 2 of Gao (2007) and Lee (2015), for example)

  • Since establishing the corresponding theory for this cross-sectional dependence setting involves much more techniques than what will be involved in this paper, this paper focuses on the i.i.d. setting, and the CSD case will be left for future research

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Summary

Introduction

Single-index models have been studied extensively in the econometrics and statistics literature in the past thirty years or so and cover many classic parametric models by using a general function form g0 (x θ0). Building on the above literature and imposing similar boundedness conditions to those in Xia (2006), Cai et al (2015) recently have introduced variable selection procedure to the single-index model by employing the kernel based on method. One is that we are able to have a closed-form estimate for θ0; another is that, like a parametric model, we need not impose the compactness on the parameter space where θ0 belongs to These two improvements being offered by this paper cannot be achieved in the existing literature where θ0 is estimated based on the assumption of the compactness of the parameter space, see, for example, Dong et al (2016).

Estimation Methods
Extremum Estimation Method
Closed-Form Estimation Method
Estimation of the link function
Monte Carlo Study
Hypothesis Test
Bias and RMSE
Empirical Study
Conclusion
N ZN EN σe2 N2

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