β-Spline Estimation in a Semiparametric Regression Model with Nonlinear Time Series Errors
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Copyright: © 2020 Jinhong You, Gemai Chen and Xian Zhou. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
We study the estimation problems for a partly linear regression model with a nonlinear time series error structure. The model consists of a parametric linear component for the regression coefficients and a nonparametric nonlinear component. The random errors are unobservable and modeled by a first-order Markov bilinear process. Based on a B-spline series approximation of the nonlinear component function, we propose a semiparametric ordinary least squares estimator and a semiparametric generalized least squares estimator of the regression coefficients, a least squares estimator of the autoregression parameter for the errors, and a B-spline series estimator of the nonparametric component function. The asymptotic properties of these estimators are investigated and their asymptotic distributions are derived. We also provide a consistent estimator for the asymptotic covariance matrix of the semiparametric generalized least squares estimator of the regression coefficients. Our results can be used to make asymptotically efficient statistical inferences. In addition, a small simulation is conducted to evaluate the performance of the proposed estimators, which shows that the semiparametric generalized least squares estimator of the regression coefficients is more efficient than the semiparametric ordinary least squares estimator.
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- Semiparametric regression model
- first-order Markovian bilinear process
- β-splines series estimation
- semiparametric least squares estimators
- asymptotic normality