## The Annals of Statistics

### 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.

#### Article information

Source
Ann. Statist., Volume 44, Number 5 (2016), 1957-1987.

Dates
Revised: August 2015
First available in Project Euclid: 12 September 2016

https://projecteuclid.org/euclid.aos/1473685265

Digital Object Identifier
doi:10.1214/15-AOS1379

Mathematical Reviews number (MathSciNet)
MR3546440

Zentralblatt MATH identifier
1349.62380

Subjects
Primary: 62F12: Asymptotic properties of estimators
Secondary: 62M05: Markov processes: estimation

#### Citation

Li, Degui; Tjøstheim, Dag; Gao, Jiti. Estimation in nonlinear regression with Harris recurrent Markov chains. Ann. Statist. 44 (2016), no. 5, 1957--1987. doi:10.1214/15-AOS1379. https://projecteuclid.org/euclid.aos/1473685265

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#### Supplemental materials

• Supplement to “Estimation in nonlinear regression with Harris recurrent Markov chains”. We provide some additional simulation studies, the detailed proofs of the main results in Section 3, the proofs of Lemmas A.1 and A.2 and Theorem 4.1.