Theoretical Economics Letters, 2012, 2, 408-411
http://dx.doi.org/10.4236/tel.[phone] 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
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
H
is the Hurst
exponent, =12,and 2Hd p


Assumpti on 2.
1)



2
=1 =
nH
t
t
VarxnM n
, where


M
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

C.-H. WANG 409

=1 ,
d
tt
xL



where 32> >56d.

t

satisfies and
Assumption (1). Then the process

=0
t
E 


 
1/2, dH
nr
nx nBr 
  (2)
where


. is Gauss sign and


0,1r.


H
Br is a
fractional Brownian motion with Hurst exponent defined
by stochastic integral.

 
12
1
()=d ,
12
rH
H
Brrs Ws
H

 

 (3)
where W(.) is a Brownian Motion on .
[0,1]


n 

is a
slowly varying function with
 
=nMn 
konurie7]
o obtain the asymptotics of the transformed frac-
tio
.
Proof of Theorem 1:
See Aom and Goroux [ 
T
nally integrated series, it is necessary to use the local
time

,Lts, which is generally defined as quadratic
variatte in the stochastic process literature. Qua-
dratic variation is infinite when we use fractional Brow-
nian motion instead of Brownian motion. To prevent this
problem, this paper adopts the fractional Brownian
motion Tanaka formula from Coutin, Nualart and Tudor
[8].
ion fini
 
   
0
=0
sgnd,
HH
r
HH
Bt sBs
Bt sBtLts



where as and

sgn=1,0, 1z>0,=0,<0z
1> >H13.
H
is a Hurst exponent.
Remark ula) Let be
lo 1Occupation time form. (f cally integrable and 32> >56H. Then
 
n
 
0d=, d.
H
f
Bt tfsLtss


 (4)
for all e set t
If w
 
=1 <
f
xxs
ula. u
, we can obtain frac-
tional local time form
 
0
0
1
,=1 <d
lim 2
t
u
Ltsx sur
u

 (5)
3. Asymptotically Homogeneous Functions
of Park and Phillips [2] considered the transformation
asymptotically homogeneous functions for I(1) process.
 


=,,Tx FxRx

   (6)
where

F
x is locally integrable function and


...R is
a reminder. This paper defines the notion of an asymp-
totically homogeneous function following de Jong and
Wang [4]:
Definition 1. A function is called asymptotically
ho

.T>K
mogeneous if for all 0 and some function


.
F
,
 
1d=0.
lim K
KTxFxx




 
 (7)
If
 (8)
and
 
1asTx Fx
 

 
1Tx Gx

 for a locally integrable
ion funct


.G, then


.T is asymptotically homo-
geneous. ition to Definition 1,

Tx must satisfy
monotonic regular. Use the followingma to prove
the asymptotics:
Lemma 1. Und
In add Lem
er Assumption, for any
(9)
where “” denotes weak convergence in
0K,


1
1/2
nd

1
nIn

 
0
=1 d,
tH
t
xx IBrxr 




,DKK
on (i.e. the space of functions that are continuous


0,1
except for a finite number of discontinuities) and
H
is
a Hurst parameter.
Proof of Lemma 1: because slowly varying function, From Jeganathan [6],


n, will not affect our proof, we set 


=1n 
.
ise in Pointw
x
, the result follows from Remf
Tyurin and Phlips [1], and therefore it suffices to show
stochastic equicontinuity of
ark 3.5 o
il



12
=1 t txx.
By the Skorokhod representation, we can assume that
1nd
nIn






12 as
d
0,1 0
sup H
rrn
nxBr
.
Then for n large enough,




12
sup d nx
0,1H rrn Br



almost surely, implying that for large enough n






12 12
1
:< <=1
sup supndd 
tt
xKxxxx t
nInxxInxx




 



 
 
12
1
=1
1
0
2
sup
2d
sup
=1,d3 sup sup
ndt
xK t
H
xK
x
x
xK sK
nIxn xx
1,
I
xBrx
LssLs


r

















(10)
where the equality follows from the occupation times
formula (see Tyurin and Phillips [1]) and because


1, sup sK Ls
 is a well-defined random variable. The
above chain of inequalities establishes stochastic equi-
continuity of

C.-H. WANG 410


12
1
=1
ndt
t
nInxx


,
 






11
112
12 1
=1
/1
12 11
/=1
12
=
=
=()
1d,
nn
nd
dtt
t
n
K
dt
Kt
dt
SS
nnTnxInx
nnTnx
Ij nxjj









 

 



 



K
(13)
w hich completes the proof.
Theorem 2. Suppose Assumption holds. Also assume

that


.T is asymptotically homogeneous. In additio
as n, sume that

.
F
is continuous and


.T is monotone
regular. Then, for 13> 0

, 32 56d and
1/2>d0



11d
d


 
,

/1
1/211/21
2/
12
=1
=( )()
((1)),
K
dd
nK
ndt
t
Sn Tnjn
I
jnx jdj







 


 

 (14)



1
1/2
0
=1
d
=()(1,)d.
n
tH
t
nnTnxFBr
FsLs s








r
(11)
Proof of Theorem 2:
To simplify our proof, we assume. Because


  
1/1
12 12
3/
1
0
=
1dd,
K
dd
nK
H
Sn Tnj
I
jBr jrj






 



 (15)

=1n 


12
1=1
pdtp
tn
nxO



su , it now sufat
fo fices to show th r any 0K,
 






11
12 12
4
12 112
=1
=()()1,d,
K
dd
nK
K
dd
K
Sn TnjLj
nTnsLs








 


 







 
1
1/21
=1
1
0
d
=1,d.
n
d
t
d
HH
K
K
nnT
FBrI BrKr
FsLs s


 





(12)
Now, by Lemma 1,
,dj
s

(16)
12
d
tt
nxInxK




 




55
1
0
== 1,d
=d.
K
nK
HH
SS FsLss
F
BrIBr Kr




 (17)
We will show that


 
12
1
=1
1
0
d
ndt
t
nI

H
nxx
IBrx r



.
By the Skorokhod representation theorem, we can
assume without loss of generality that
01, =0
lims u p
lim jnj n
nSS
 
 
almost surely for . By the monotone regular
condition, we can act as if is monotone without
loss of generality. For
=1, ,4j

.T
12n
SS

 we then have (See the
Equation [18] below) and as 0

, the last term
disappears because of continuity of

.
F
, the second
inequality follows from monotonicity of


.T, and the
third by our definition of a n as ymptotically homogeneous
function. To show that 0lims
lim  


 
1
12
1
=1 0
d
ndtH
t
nInxxIBrxr

 

=0
.
as
n
c
Now for all 23
=0
nn
SS

upn
almost surely, note that (See the Equation [19] below)

>0

, let
 









1/1 12
12 112
12
msup msup
nt
nn
SS
 
 /=1
1/1 12
12 1 1/212
/=1
1d
li li
11d limsup
n
Kd
dd
t
Kt
n
Kd
dddt
K
nt
nnTnx TnjIjnxjj
nnTnjTnjIjnxjj












 











  
   
11
/1 12 1/212 12
/
/1
/
11d
limsup
1d=d,
Kdd dd
K
n
KK
KK
nTnjnTnjFjFjj
FjFjjFxFxx
 




 

 





 


(18)
 



  
 

 
1/1 1
12
1212 1
/ 0
=1
1/1
12 12
/
1
112 121
11
2
22d=1
n
Kd
dd tH
Kt
K
dd
nK
KK
dd
n n
KK
nTnjnIjnxjIjBrj
cnTn jdj
cnTnxFxdxcFxxo









 

 


 


 
 










ddrj 
(19)

C.-H. WANG 411
almost surely under our assumptions and by the definition
of . For
n
c34nn SS


 we have
 
 



 



34
1/
12 12
/
1
1
0
/
1
1
0
||
1d1,d
d1,.
sup
nn
K
dd
K
H
K
H
xK
SS
nTnj
I
1/
12 12 d
K
dd
jBrjrLj j
IxBrxrLx










 












(20)
By the earlier argument,
nT njj


 

 

 
1
12 12
1>0 d< ,
supsup K
dd
K
nnTnj






 

j


and therefore it suffices to show that as
(21)
0

,



1
1
0d1,
sup H
xK IxBrxr Lx



 
0. (22)
ormula, the above expressi
satisfies
By the occupation times fon

  

 
1
1
(1, )d1,
sup
=1,1,d sup
x
x
xK
LssLx
LsLxs











continuity of on
,1, 1,0as0
sup sup
xK sxxLsLx 

 


by uniform
x
xK
x 

(23)

1,.L


,
K
K. Finally, for
45n
SS

, we have







1
12 12
1
12 12
1,d
lim
1, d
sup lim
=0
Kdd
K
n
Kdd
K
n
sK
nTnsHsLs
LsnTn sFss





 



 







s

by the defiptotically homogeneous fu nc-
tion, which completes th e proof.
Theorem (2) expands the transformations asymptoti-
cally homogeneous functions to scaled nonstationary I(d)
processes. This new result can be used to obtain the asy-
mptotics of the nonlinear fractional cointegration.
4. Acknowledgements
The author thanks financial support from National Science
NCES
[2] J. Y. Park and P. C. B. Phillips, “Asymptotics for Nonlin-
ear Transforme Series,” Econo-
metric Theory. 269-298.
(24)
nition of an asym
of Council of Taiwan under grant NSC 93-2415-H-305-
010 and Chun-Chieh Huang, Chu-Chien Lee and Shiwei
Wang for excellent research assistants.
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