Estimation of semi-varying coefficient models with nonstationary regressors

Abstract

We study a semi-varying coefficient model where the regressors are generated by the multivariate unit root I(1) processes. The influence of the explanatory vectors on the response variable satisfies the semiparametric partially linear structure with the nonlinear component being functional coefficients. A semiparametric estimation methodology with the first-stage local polynomial smoothing is applied to estimate both the constant coefficients in the linear component and the functional coefficients in the nonlinear component. The asymptotic distribution theory for the proposed semiparametric estimators is established under some mild conditions, from which both the parametric and nonparametric estimators are shown to enjoy the well-known superconsistency property. Furthermore, a simulation study is conducted to investigate the finite sample performance of the developed methodology and results.

Uniform Consistency of Nonstationary Kernel-Weighted Sample Covariances for Nonparametric Regression

“…where f Z (•) is the density function of Z t . Similar to the argument in the proof of Proposition A.1 in Li et al (2014), it is easy to show that…”

“…Much of the existing literature on the limit theory of Q n (•) for the random design case imposes a martingale difference structure on e t , which excludes the possibility of correlation between X t and e t (c.f., Cai et al, 2009;Li et al, 2014). However, for consistency with the framework of Section 2, we follow the same structure as Assumption 1 to generate the unit root process X t and the stationary process e t , thereby allowing for possible correlation between X t and e t .…”

Adaptive estimation for varying coefficient models with nonstationary covariates

“…In addition, it will be interesting to consider the hypothesis tests for the functional coefficients with the adaptive estimators. Finally, the idea of the adaptive estimation might also be extended to many other semiparametric models with non stationary regressors, such as semivarying coefficient models (Li et al 2017), single-index and partially linear single-index integrated models (Dong, Gao, and Tjøstheim 2016), and varying coefficient partially non linear models (Zhou and Lin 2018).…”

“…Nonstationarity is a very important empirical feature in many economic and financial time series. Over the past decade, there has been great interests in nonparametric and semiparametric models with non stationary covariates, existing literature includes Cai, Li, and Park (2009), Chan and Wang (2015), Chen, Fang, and Li (2015), Chen, Gao, and Li (2012), Dong, Gao, and Tjøstheim (2016), Gu and Liang (2014), Gao and Phillips (2013), Juhl and Xiao (2005), Karlsen, Myklebust, and Tjøstheim (2007), Karlsen and Tjostheim (2001), Liang, Lin, and Hsiao (2015), Li et al (2017), Sun, Cai, and Li (2013), Sun and Li (2011), Wang (2014), Wang (2015), Wang and Phillips (2009a), Wang and Phillips (2009b), Wang and Phillips (2016), Xiao (2009), Zhou and Lin (2018). As we know, compared with nonparametric regression model, semiparametric regression models have the advantage of attenuating the problem of "curse of dimensionality. "…”

Kernel-based Inference in Time-Varying Coefficient Cointegrating Regression

“…Notwithstanding this body of work many practical implementations reveal that parametric linear cointegration models are often rejected by the data even when there is evident co-movement among the trending series. Acknowledgement of this weakness has led to the recent development of econometric methodology for treating various nonlinear and nonparametric cointegrating models (Park and Phillips, 2001;Karlsen, Myklebust and Tjøstheim, 2007;Cai, Li and Park, 2009;Wang and Phillips, 2009a,b;Xiao, 2009;Gao and Phillips, 2013;Li et al, 2017;Phillips, Li and Gao, 2017). For the important case of multivariate integrated covariates, much of this nonparametric research on nonlinear cointegration excludes possible co-movement among the regressors and the presence of deterministic drift.…”