Adaptive jump-preserving estimates in varying-coefficient models

Jump-Preserving Varying-Coefficient Models for Nonlinear Time Series

“…In this paper, we consider piecewise-smooth coefficient functions a(·) that can exhibit a finite set of discontinuities located at points {s q } Q q=1 , where the number Q of jumps, the jump locations s q , and the jump sizes d q of the coefficient functions are all unknown. Contrary to Zhao et al (2016), we assume that the conditional variance σ 2 (z) = E[σ 2 (X, Z)|Z = z] is not constant, but it is a continuous function of z in this section. The case with discontinuous σ 2 (z) will be investigated later in Section 4.…”

“…The semiparametric model (1) has been studied by Zhao et al (2016) for the independent and identically distributed data, and in the present setting, it includes many popular time-series models. When X i is a constant, the model is reduced to a nonpara-4 metric jump-preserving model in Gijbels et al (2007).…”

“…To the best of our knowledge, VCMs with discontinuities in coefficient functions have not been investigated before in heteroscedastic and time series setting. For independent and identically distributed data, Zhu et al (2014) and Zhao et al (2016) suggested methods for estimation of varying-coefficient models with discontinuities.…”

“…Although this has already been done by Zhao et al (2016) in the case of independently and identically sampled observations, we analyze this method in the context of heteroscedastic and dependent data and provide additional asymptotic results such as the uniform convergence rate of the coefficient estimates.…”

“…This paper is structured as follows. In Section 2, the VCM is introduced and the jumppreserving estimation procedure is introduced based on Gijbels et al (2007) and Zhao et al (2016). In Section 3, we establish the consistency and asymptotic normality of this estimator.…”