Description
This article intends to notify that asymptotic distributions of the nonlinear unit root test statistics must
be rigorously treated if deterministic components are included in the estimated regression. The simple
inductive argument of replacing the standard Brownian motion by either the demeaned or demeaned
and detrended ones usually adopted in the literature is invalid. New results on the asymptotic distributions
of the t-ratio to test the null of the unit root against the nonlinear exponential smooth transition
autoregressive (ESTAR) with deterministic components are provided.
Unit root test against ESTAR with deterministic components
Tsai-Yin Lin
a, *
, Chih-Hsien Lo
b
a
Department of Finance, National Kaohsiung First University of Science and Technology, Kaohsiung City, Taiwan
b
Department of Finance, National Kaohsiung University of Applied Sciences, Kaohsiung City, Taiwan
a r t i c l e i n f o
Article history:
Received 17 February 2014
Accepted 29 April 2014
Available online 9 March 2015
JEL classi?cation:
C12
C32
Keywords:
Deterministic components
Exponential smooth transition
autoregressive
Unit root test
a b s t r a c t
This article intends to notify that asymptotic distributions of the nonlinear unit root test statistics must
be rigorously treated if deterministic components are included in the estimated regression. The simple
inductive argument of replacing the standard Brownian motion by either the demeaned or demeaned
and detrended ones usually adopted in the literature is invalid. New results on the asymptotic distri-
butions of the t-ratio to test the null of the unit root against the nonlinear exponential smooth transition
autoregressive (ESTAR) with deterministic components are provided.
© 2015 College of Management, National Cheng Kung University. Production and hosting by Elsevier
Taiwan LLC. All rights reserved.
1. Introduction
Dissatisfaction with the linear ARMA framework, within which
unit root tests are applied, has recently prompted econometricians
to consider nonlinear alternatives to develop unit root tests.
Kapetanios, Shin, and Snell (2003) popularized an ADF type statistic
to test the null of unit root against the stationary exponential
smooth transition autoregressive (ESTAR) alternative, which has
been found in international monetary economics as an effective
way to describe real exchange rates dynamics. In contrast, Rothe
and Sibbertsen (2006) developed Phillips-Perron type unit root
tests against the ESTAR alternative, in which error terms are
allowed to be strong mixing.
Both Kapetanios et al. (2003) and Rothe and Sibbertsen (2006)
began with the leading case without deterministic components to
derive asymptotic distributions of their test statistics. However,
they claim that in the presence of deterministic components,
through a bypass remark mimicking the arguments used in linear
unit root tests, the asymptotic distribution follows only by replac-
ing the standard Brownian motion with either the demeaned mo-
tion or the demeaned and detrended motion. We argue in this
article that this is not the case, since what actually should be
demeaned or detrended is the cubic of lag series. Asymptotic dis-
tributions of the nonlinear unit root test statistics must be rigor-
ously considered if deterministic components are included in the
estimated regression. Newresults on the asymptotic distribution of
the t-ratio proposed in Kapetanios et al. (2003) and Rothe and
Sibbertsen (2006) are provided when the estimated regression
includes deterministic components.
The rest of this article is organized as follows. Section 2 de-
scribes the basic ESTAR model, assumptions, and the test statistics
of the ESTAR unit root test. In section 3, new results on the
asymptotic distributions with deterministic components included
in the estimated regression are proven. Section 4 then concludes.
It should be noted that in this paper, kXk
p
¼ ðEðjXj
p
ÞÞ
1
p
denotes
the L
p
norm, “/
p
” ð“÷!
a:s:
”Þ denotes convergence in probability
(almost surely), W(r) refers to a standard Brownian motion de?ned
onr2½0; 1?; “0” and represents weak convergence in distribution.
2. The model, assumption and test statistics
Consider a univariate ESTAR of order 1 model, ESTAR(1), which
is described in Kapetanios et al. (2003) and Rothe and Sibbertsen
(2006):
y
t
¼ d
0
kt
j þ ry
tÀ1
þgy
tÀ1
Gðy
tÀ1
;q Þ þ ?
t
; (1)
Gðy
tÀ1
; qÞ ¼ 1 Àexp
_
Àqy
2
tÀ1
_
; (2)
* Corresponding author. Department of Finance, National Kaohsiung First Uni-
versity of Science and Technology, Number 1, University Road, Yanchao District,
Kaohsiung City 824, Taiwan.
E-mail address: [email protected] (T.-Y. Lin).
Peer review under responsibility of College of Management, National Cheng
Kung University.
HOSTED BY
Contents lists available at ScienceDirect
Asia Paci?c Management Review
j ournal homepage: www. el sevi er. com/ l ocat e/ apmrvhttp://dx.doi.org/10.1016/j.apmrv.2014.12.006
1029-3132/© 2015 College of Management, National Cheng Kung University. Production and hosting by Elsevier Taiwan LLC. All rights reserved.
Asia Paci?c Management Review 20 (2015) 44e47
where d
kt
is the deterministic component, r and g are unknown
scalar parameters, and is a vector parameter conformable with d
kt
.
We distinguish three kinds of deterministic components,
d
1t
¼ f?g; d
2t
¼ ð1Þ; d
3t
¼ ð1; tÞ
0
: The transition function G(y
tÀ1
;q )
is a scaling parameter which controls the shift of regimes and is
assumed to be of exponential form. The parameter q 0 de-
termines the speed of transition between regimes. Kapetanios et al.
(2003) and Rothe and Sibbertsen (2006) both impose r ¼ 1
and0 2, {?
t
} is a zero mean, strong
mixing sequence with mixing coef?cients a
m
of size Àpb/(pÀb)
andsup
i1
k?
i
k
p
¼ C
This article intends to notify that asymptotic distributions of the nonlinear unit root test statistics must
be rigorously treated if deterministic components are included in the estimated regression. The simple
inductive argument of replacing the standard Brownian motion by either the demeaned or demeaned
and detrended ones usually adopted in the literature is invalid. New results on the asymptotic distributions
of the t-ratio to test the null of the unit root against the nonlinear exponential smooth transition
autoregressive (ESTAR) with deterministic components are provided.
Unit root test against ESTAR with deterministic components
Tsai-Yin Lin
a, *
, Chih-Hsien Lo
b
a
Department of Finance, National Kaohsiung First University of Science and Technology, Kaohsiung City, Taiwan
b
Department of Finance, National Kaohsiung University of Applied Sciences, Kaohsiung City, Taiwan
a r t i c l e i n f o
Article history:
Received 17 February 2014
Accepted 29 April 2014
Available online 9 March 2015
JEL classi?cation:
C12
C32
Keywords:
Deterministic components
Exponential smooth transition
autoregressive
Unit root test
a b s t r a c t
This article intends to notify that asymptotic distributions of the nonlinear unit root test statistics must
be rigorously treated if deterministic components are included in the estimated regression. The simple
inductive argument of replacing the standard Brownian motion by either the demeaned or demeaned
and detrended ones usually adopted in the literature is invalid. New results on the asymptotic distri-
butions of the t-ratio to test the null of the unit root against the nonlinear exponential smooth transition
autoregressive (ESTAR) with deterministic components are provided.
© 2015 College of Management, National Cheng Kung University. Production and hosting by Elsevier
Taiwan LLC. All rights reserved.
1. Introduction
Dissatisfaction with the linear ARMA framework, within which
unit root tests are applied, has recently prompted econometricians
to consider nonlinear alternatives to develop unit root tests.
Kapetanios, Shin, and Snell (2003) popularized an ADF type statistic
to test the null of unit root against the stationary exponential
smooth transition autoregressive (ESTAR) alternative, which has
been found in international monetary economics as an effective
way to describe real exchange rates dynamics. In contrast, Rothe
and Sibbertsen (2006) developed Phillips-Perron type unit root
tests against the ESTAR alternative, in which error terms are
allowed to be strong mixing.
Both Kapetanios et al. (2003) and Rothe and Sibbertsen (2006)
began with the leading case without deterministic components to
derive asymptotic distributions of their test statistics. However,
they claim that in the presence of deterministic components,
through a bypass remark mimicking the arguments used in linear
unit root tests, the asymptotic distribution follows only by replac-
ing the standard Brownian motion with either the demeaned mo-
tion or the demeaned and detrended motion. We argue in this
article that this is not the case, since what actually should be
demeaned or detrended is the cubic of lag series. Asymptotic dis-
tributions of the nonlinear unit root test statistics must be rigor-
ously considered if deterministic components are included in the
estimated regression. Newresults on the asymptotic distribution of
the t-ratio proposed in Kapetanios et al. (2003) and Rothe and
Sibbertsen (2006) are provided when the estimated regression
includes deterministic components.
The rest of this article is organized as follows. Section 2 de-
scribes the basic ESTAR model, assumptions, and the test statistics
of the ESTAR unit root test. In section 3, new results on the
asymptotic distributions with deterministic components included
in the estimated regression are proven. Section 4 then concludes.
It should be noted that in this paper, kXk
p
¼ ðEðjXj
p
ÞÞ
1
p
denotes
the L
p
norm, “/
p
” ð“÷!
a:s:
”Þ denotes convergence in probability
(almost surely), W(r) refers to a standard Brownian motion de?ned
onr2½0; 1?; “0” and represents weak convergence in distribution.
2. The model, assumption and test statistics
Consider a univariate ESTAR of order 1 model, ESTAR(1), which
is described in Kapetanios et al. (2003) and Rothe and Sibbertsen
(2006):
y
t
¼ d
0
kt
j þ ry
tÀ1
þgy
tÀ1
Gðy
tÀ1
;q Þ þ ?
t
; (1)
Gðy
tÀ1
; qÞ ¼ 1 Àexp
_
Àqy
2
tÀ1
_
; (2)
* Corresponding author. Department of Finance, National Kaohsiung First Uni-
versity of Science and Technology, Number 1, University Road, Yanchao District,
Kaohsiung City 824, Taiwan.
E-mail address: [email protected] (T.-Y. Lin).
Peer review under responsibility of College of Management, National Cheng
Kung University.
HOSTED BY
Contents lists available at ScienceDirect
Asia Paci?c Management Review
j ournal homepage: www. el sevi er. com/ l ocat e/ apmrvhttp://dx.doi.org/10.1016/j.apmrv.2014.12.006
1029-3132/© 2015 College of Management, National Cheng Kung University. Production and hosting by Elsevier Taiwan LLC. All rights reserved.
Asia Paci?c Management Review 20 (2015) 44e47
where d
kt
is the deterministic component, r and g are unknown
scalar parameters, and is a vector parameter conformable with d
kt
.
We distinguish three kinds of deterministic components,
d
1t
¼ f?g; d
2t
¼ ð1Þ; d
3t
¼ ð1; tÞ
0
: The transition function G(y
tÀ1
;q )
is a scaling parameter which controls the shift of regimes and is
assumed to be of exponential form. The parameter q 0 de-
termines the speed of transition between regimes. Kapetanios et al.
(2003) and Rothe and Sibbertsen (2006) both impose r ¼ 1
and0 2, {?
t
} is a zero mean, strong
mixing sequence with mixing coef?cients a
m
of size Àpb/(pÀb)
andsup
i1
k?
i
k
p
¼ C