Estimation in nonlinear regression with Harris recurrent Markov chains

Abstract

In this paper, we study parametric nonlinear regression under the Harris recurrent Markov chain framework. We first consider the nonlinear least squares estimators of the parameters in the homoskedastic case, and establish asymptotic theory for the proposed estimators. Our results show that the convergence rates for the estimators rely not only on the properties of the nonlinear regression function, but also on the number of regenerations for the Harris recurrent Markov chain. Furthermore, we discuss the estimation of the parameter vector in a conditional volatility function, and apply our results to the nonlinear regression with $I(1)$ processes and derive an asymptotic distribution theory which is comparable to that obtained by Park and Phillips [Econometrica 69 (2001) 117-161]. Some numerical studies including simulation and empirical application are provided to examine the finite sample performance of the proposed approaches and results.Comment: Published at http://dx.doi.org/10.1214/15-AOS1379 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Local composite quantile regression smoothing for Harris recurrent Markov processes

“…An important feature of the Harris recurrence is to allow us to construct a split chain, which plays a critical role in the derivation of the asymptotic theory (c.f., Karlsen and Tjøstheim, 2001; Karlsen et al , 2007; and Li et al , 2016). With the help of the split chain technique, we can decompose the partial sum of functions of { X t } into blocks of i.i.d.…”

“…For the stationary and positive recurrent { X t }, β = 1. Myklebust et al (2012) and Li et al (2016) further provided an example for a general β-null recurrent Markov chain, where β could be any value between 0 and 1. The β-null recurrent process also has the so-called invariance property that if { X t } is β-null recurrent, then for a one-to-one transformation 𝒯(·), {𝒯( X t )} is still β-null recurrent (Teräsvirta et al , 2010).…”

“…This makes our methodology applicable to the case of heavy-tailed error distribution. It seems difficult to relax the condition that { e t } is independent of the regressor { X t }, which is a standard assumption in the literature (c.f., Karlsen et al , 2007; Gao et al , 2015; and Li et al , 2016). One possibility is to replace the mutual independence assumption by the condition that the compound process {( X t , e t )} is Harris recurrent, in which case the proofs of the asymptotic theorems in Appendix B need to be substantially modified.…”

“…The data (available at https://www.uktradeinfo.com/) were collected monthly from January 1996 to August 2013, and thus the sample size is n = 212. The same data set was also studied by Li et al (2016) by using the parametric nonlinear modelling framework and least squares estimation method. We next consider more flexible nonparametric nonlinear modelling, and estimate it by using the three nonparametric methods as in Example 5.1.…”

“…We plot the data { Y t } and { X t } in Figure 5.1. The empirical applications in both Gao et al (2013) and Li et al (2016) suggested that the real exchange rates { X t } follow the nonstationary threshold AR (1) process and belong to a 1/2-null recurrent Markov process.…”