Theoretical Economics Letters, 2012, 2, 408-411

http://dx.doi.org/10.4236/tel.2012.24075 Published Online October 2012 (http://www.SciRP.org/journal/tel)

Further Results on Convergence for Nonlinear

Transformations of Fractionally Integrated Time Series

Chien-Ho Wang

Department of Economics, National Taipei University, New Taipei City, Chinese Taipei

Email: wangchi3@mail.ntpu.edu.tw

Received May 6, 2012; revised June 4, 2012; accepted July 2, 2012

ABSTRACT

This paper presents some new results for the nonlinear tran sformations of the fractional integration process. Specifically,

this paper reviews the weight fractional integration process with the Hurst parameter, 32> >56d, and investigates

the asymptotics of asymptotically homogeneous functional transformations of weight fractional integration process.

These new results improve upon the earlier research of Tyurin and Phillips [1].

Keywords: Long Memory; Fractional Brownian Motion; Tanaka Formula; Nonlinearity

1. Introduction

Since the breakthrough papers of Park and Phillips [2, 3 ],

the research on the nonlin ear cointegration has generated

a lot of interest in recent years. In traditional research on

cointegration, econometricians always adopt linear models.

Using a linear cointegration model, econometricians can

derive large sample properties easily. However, these

settings have a serious drawback: there are many non-

linear relationships between dependent variables and

independent v ariables in the cointegration mo del. Thus, it

is a subjective process to set cointegration as a linear

form in advance. A nonlinear regression model may

improve this problem in a cointegration system. Although

nonlinear regressions have obvious merit for cointe-

gration models, it is difficult to derive the asymptotics

for their estimated parameters and test statistics. Park and

Phillips [2, 3] were the first to use local time to obtain

asymptotics under nonlinear transformations of the I(1)

process. Pötscher [4] and de Jong and Wang [5] later

extended to these results to more flexible assumptions.

The asymptotics of nonlinear transformations for non-

stationary time series consistently concentrated on the I(1)

process in early nonlinear cointegration research. Tyurin

and Phillips [1] extended their method to the nonsta-

tionary I(d) process. Jeganathan [6] investigated the asy-

mptotics of nonlinear transformations for generalized

fractional stable motions. Although they presented some

new results for the nonlinear transformations of the non-

stationary fractionally integrated process, they only con-

centrated on integrable functions.

This paper uses a weight nonstationary fractionally

integrated process instead of the standard nonstationary

fractionally integrated process. This paper extends the

results of Tyurin and Phillips [1] to asymptotically

homogeneous functions. Specifically, this paper uses the

fractional Brownian motion Tanaka formula to obtain the

asymptotics of nonlinear transformations for the non-

stationary fractional integration process. The results of

this paper address the shortcomings of Tyurin and Phillips

[1].

2. Assumptions and Basic Results

Consider the following fractional integration processes:

1=

d

t

Lx t

(1)

where t

is an

2

... 0,iid

and 32> >56d. t

is called a nonstationary fractionally integrated processes.

In addition to the definition of nonstationary fractionally

integrated processes, This paper uses the following ad-

ditional assumptions.

x

Assumption 1. For some

>2>2max1,2qp H,

<

q

k

E

and

2<

t

E

,where

is the Hurst

exponent, =12,and 2Hd p

Assumpti on 2.

1)

2

=1 =

nH

t

t

VarxnM n

, where

n is a

slowly varying function.

2) The distribution of k

, , is abso-

lutely continuous with respect to the Lebesgue measure

and has characteristic function for which

=0, 1,2,k

=it k

tEe

=0

limttt

for some >0

.

Based on these assumptions, we can obtain the frac-

tional central limit theorem for the nonstationary I(d) p ro-

cesses.

Theorem 1. Consider the process defined by

C

opyright © 2012 SciRes. TEL