Limit Theorems For Nonlinear Cointegrating Regression

Qiying Wang

• Publisher: WSPC
• Year: 2015
• eBook ISBN: 9789814675642
• Topic: Biological Sciences
• Subtopic: Science General
• Index: Biological Sciences

This book provides the limit theorems that can be used in the development of nonlinear cointegrating regression. The topics include weak convergence to a local time process, weak convergence to a mixture of normal distributions and weak convergence to stochastic integrals. This book also investigates estimation and inference theory in nonlinear cointegrating regression.

The core context of this book comes from the author and his collaborator's current researches in past years, which is wide enough to cover the knowledge bases in nonlinear cointegrating regression. It may be used as a main reference book for future researchers.

This book provides the limit theorems that can be used in the development of nonlinear cointegrating regression. The topics include weak convergence to a local time process, weak convergence to a mixture of normal distributions and weak convergence to stochastic integrals. This book also investigates estimation and inference theory in nonlinear cointegrating regression.

The core context of this book comes from the author and his collaborator's current researches in past years, which is wide enough to cover the knowledge bases in nonlinear cointegrating regression. It may be used as a main reference book for future researchers.

Readership: Graduate students and researchers interested in nonlinear cointegrating regression.
Key Features:

• First of all, this book extends the classical martingale limit theorem. Unlike previous books, for a certain class of martingales, weak convergence to a mixture of normal distributions is established under the convergence in distribution for the conditional variance
• This extension partially removes a barrier in applications of the classical martingale limit theorem to non-parametric estimation and inference with non-stationarity
• This extension enhances the effectiveness of the classical martingale limit theorem in the investigation of asymptotics in statistics, econometrics and other fields
• Secondly, this book systemically introduces weak convergence to a local time process and weak convergence to stochastic integrals beyond martingale and semi-martingale structures (This kind of context is new to the field and is particularly useful in the framework of nonlinear cointegration)
• Finally, this book does not look for the most general theory from the view of probability, but provides enough details for those who are interested in nonlinear cointegrating regression

Chapter 1

Introduction

Consider a nonlinear regression model:

where f : R → R is an unknown function, x_t and u_t are regressors and regression errors, respectively. Let K(x) be a kernel function and h ≡ h_n → 0 be a bandwidth. The conventional kernel estimator of f (x) is given by

Let x₀ be fixed. We may split

as

The second term of (1.2) is easy to handle with and is negligible under mild conditions. The limit behavior of

is determined by the first term of (1.2), which follows from the asymptotics (jointly) of S_n1 and S_n2 defined by

If (x_t, y_t) is a stationary process, we may prove that

converges to a constant C under mild conditions on x_t and h. Moreover, a standard central limit theorem can be employed to show

where σ² > 0 is a constant. As a consequence,

converges to a normal distribution, an important result on the estimation and inference theory in nonlinear regression. There are extensive researches in the area. See, e.g., Härdle (1990), Fan and Gijbels (1996) and the references therein.

The situation becomes very different if (x_t, y_t) includes a non-stationary component. To illustrate, assume that

, where

is a sequence of i.i.d. N(0, 1) random variables. We further assume

and K(x) is a uniform kernel, i.e.,

. In this case, it is not true to say

. Instead, we may prove

where B = {B_t}_t≥0 is a Brownian motion and L_B(1, 0) is a local time of B. Furthermore, S_n2 forms a martingale, but limit behavior of S_n2 cannot be investigated by the classical martingale limit theorem that is stated in Theorem 3.2 of Hall and Heyde (1980). Indeed, as a martingale, the conditional variance

of S_n2 has the property:

by (1.4). Since the convergence in distribution in (1.5) for the conditional variance

cannot be extended to the convergence in probability, the classical martingale limit theorem is no longer useful to provide the asymptotics of S_n2.

When the regressor x_t is a nonstationary time series, in view of the obvious link to cointegration when f ...

Table of contents

• Cover Page
• Title Page
• Copyright
• Contents
• Preface
• 1. Introduction
• 2. Convergence to local time
• 3. Convergence to a mixture of normal distributions
• 4. Convergence to stochastic integrals
• 5. Nonlinear cointegrating regression
• Appendix A Concepts of stochastic processes
• Appendix B Metric space
• Appendix C Convergence of probability measure
• Bibliography
• Index