Directory UMM :Data Elmu:jurnal:J-a:Journal of Econometrics:Vol99.Issue1.Nov2000:

Journal of Econometrics 99 (2000) 107}137

Gaussian tests for seasonal unit roots
based on Cauchy estimation and recursive
mean adjustments
Dong Wan Shin*, Beong Soo So
Ewha Womans University, Department of Statistics, DeaHyunDong, SeoDaeMoonGu, Seoul,
120-750 South Korea
Received 25 March 1997; received in revised form 13 March 2000; accepted 14 March 2000

Abstract
We propose tests for seasonal unit roots whose limiting null distributions are always
standard normal regardless of the period of seasonality and types of mean adjustments.
The seasonal models of Dickey, Hasza and Fuller (1984. Journal of American Statistical
Association 79, 355}367) (DHF) and Hylleberg, Engle, Granger and Yoo (1990. Journal
of Econometrics 44, 215}238) (HEGY) are considered. For estimating parameters related
to the seasonal unit roots, regressor signs are used as instrumental variables while
recursive sample means are used for adjusting the seasonal means. In addition to
normality of the limiting null distributions, in seasonal mean models, the recursive mean
adjustment provides the new tests with locally higher powers than those of the existing
tests of DHF and HEGY based on the ordinary least-squares estimators. If data have

a strong linear time trend, the recursive mean adjustment is a source of both power gains
of some tests for local alternatives and power losses of all tests for other alternatives.
Limiting normality allow evaluation of p-values and testing joint signi"cance of subsets
of seasonal unit roots. ( 2000 Elsevier Science S.A. All rights reserved.
JEL classixcation: C12; C22
Keywords: Instrumental variable; Normal tests; Sign

* Corresponding author. Fax: #82-2-3277-2614.
E-mail address: shindw@mm.ewha.ac.kr (D.W. Shin).

0304-4076/00/$ - see front matter ( 2000 Elsevier Science S.A. All rights reserved.
PII: S 0 3 0 4 - 4 0 7 6 ( 0 0 ) 0 0 0 3 2 - 4

108

D.W. Shin, B.S. So / Journal of Econometrics 99 (2000) 107}137

1. Introduction
Since the pioneering works of Fuller (1976) and Dickey and Fuller (1979),
a great deal of attention has been paid to testing for unit roots in economic time

series. Economic time series, usually available in monthly or quarterly forms,
reveal various kinds of seasonal patterns. As to the issue of random walk
versus trend stationarity for yearly or nonseasonal economic time series (Fuller,
1996; Stock, 1994; Hamilton, 1994; and references therein), a major question is
whether the main variation of the seasonal time series is a consequence of the
nonstationary stochastic seasonality due to the seasonal unit roots or a consequence of a deterministic seasonal trend with stationary stochastic seasonality.
Dickey et al. (1984) (DHF in the sequel) consider a model that is null
stationary if seasonally di!erenced. Under their model, the time series has all
nonstationary stochastic seasonalities of di!erent frequencies corresponding to
the unit root and all the other seasonal unit roots. Their model considers only
seasonal means. Extensions to models containing both seasonal means and
a time trend are made by Ahn and Cho (1993) and Cho et al. (1995). Hylleberg et
al. (1990) (HEGY in the sequel) and Beaulieu and Miron (1993) point out that
seasonal time series can have nonstationary stochastic seasonalities of the
frequencies only corresponding to speci"c seasonal unit roots on the unit circle.
They propose tests which can identify the speci"c frequencies corresponding to
the signi"cant seasonal unit roots. Smith and Taylor (1998a) make extensions
which include seasonal means and seasonal trends. Franses (1994), Ghysels et al.
(1996) and Boswijk and Franses (1996) develop tests for seasonal unit roots for
periodic seasonal autoregressive (AR) models in which values of the AR coe$cients di!er seasonally.

A special feature of unit root tests for seasonal autoregression is that the
limiting null distributions of the unit root tests depend not only on the form of
mean adjustments but also on the period of seasonality. At least "ve di!erent
mean functions may be considered: no mean; a simple mean; seasonal means;
a time trend; seasonal means with a time trend. The limiting null distributions of
the above-mentioned tests are not normal and are complicated functions of
standard Brownian motions. Moreover, the limiting null distributions get more
involved because they depend also on the period of seasonality. Hence, in order
to have complete test procedures for various mean functions and various periods
of seasonality, a large bulk of distributional results and tables of the corresponding percentage points are required. Another special feature is that the usual
ordinary least squares estimators (OLSEs) adopted by almost all of the above
authors su!er from large downward biases. The biases are, in large part, due to
estimation of the parameters of the mean functions. One serious consequence of
the large downward biases of the OLSEs is poor power of unit root tests.
In this paper, we develop tests for seasonal unit roots whose limiting null
distributions are always normal irrespective of the type of mean function and the

D.W. Shin, B.S. So / Journal of Econometrics 99 (2000) 107}137

109

period of seasonality. Both classes of the models of DHF and HEGY are
considered. Applications to the periodic models of Franses (1994), Ghysels et al.
(1996) and Boswijk and Franses (1996) are obvious. In our new method, the
Cauchy estimator developed by So and Shin (1999a) in which the signs of the
regressor variables are used as instrumental variables for estimating the coe$cients of seasonal AR models is utilized. The Cauchy estimators allow us to
construct asymptotically normal tests for seasonal unit roots. The means are
adjusted by &recursive detrending' developed by So and Shin (1999b) and Shin and
So (1999) in which observations at any time are adjusted for the mean by using
observations up to that time point. The recursive mean adjustment provides us
with less-biased AR parameter estimates. By adopting the recursive mean adjustment, in addition to the smaller biases, we attain limiting normality of the test
statistics. The resulting tests for seasonal mean models are shown to be locally
more powerful than the existing tests based on the OLSEs. If the mean function
consists of a strong linear time trend, then the recursive mean adjustment provides
power gains of some tests for some cases of alternative hypotheses close to the null
hypotheses but causes power degradation for the other cases.
Limiting null normality of the proposed tests for any seasonality and for any
mean function implies several merits. First, the normal tests can be used for a much
wider class of models than the usual nonnormal tests which require di!erent tables
of percentage points for each model and for each period of seasonality. Second, we

can compute the p-values of the test statistics. Since only percentage points of
speci"c probabilities, say, 1%, 2.5%, 5%, and 10%, are usually available for the
usual OLSE-based tests, the p-values of the OLSE-based tests cannot be easily
computed. Availability of the p-values for our tests is a clear advantage over the
OLSE-based tests because the p-values give us detailed information about the
signi"cances of the unit root hypotheses. Third, tests for joint signi"cance of
a subset of seasonal unit roots are possible using the chi-square distributions.
The remainder of this paper is organized as follows. In Section 2, the recursive
mean adjustment is introduced. In Section 3, based on Cauchy estimation, tests
for seasonal unit roots for the model of DHF are developed. In Section 4, tests
for seasonal unit roots for the model of HEGY are developed. In Section 5,
extensions to higher-order AR models are made. In Section 6, "nite sample
properties of the proposed tests are investigated through Monte-Carlo experiments. In the appendix, proofs of the theoretical results are provided.

2. Recursive mean adjustments
We present our new mean adjustment scheme through a seasonal unit root
AR model
y "k #u ,
t
t

t

(2.1)

110

D.W. Shin, B.S. So / Journal of Econometrics 99 (2000) 107}137

u "ou #e ,
t
t~d
t

(2.2)

where y , t"1,2, n, are observations, k is a mean function, o is the AR
t
t
parameter of interest, and d is a positive integer representing the period of
seasonality. The errors e are independent and identically distributed with zero

t
mean and "nite variance p2. We are interested in testing for the seasonal unit
root hypothesis H : o"1. For the mean function k , we may consider a simple
0
t
mean, seasonal means, a simple mean with a trend, seasonal means with a trend,
seasonal means and seasonal trends which are given by
k "b ,
t
0

(2.3)

d
k"+ bd ,
(2.4)
t
i it
i/1
k "b #b t,

(2.5)
t
0
q
d
k " + b d #b t,
(2.6)
t
i it
q
i/1
d
k " + (b #b t)d ,
(2.7)
t
i
qi it
i/1
respectively, where d are seasonal indicator variables such that d "1 if
it

it
i,t (mod d); d "0 otherwise. Among (2.3)}(2.7), the models with seasonal
it
means (2.4) and the seasonal means with a trend (2.6) are frequently encountered
in practice. If a time series reveals a cycling pattern without time trend, k in (2.4)
t
would be a plausible choice. If a time series reveals a cycling pattern with a linear
trend, then one might consider (2.6). One can also choose one of (2.3)}(2.7) using
the standard model selection criteria such as AIC and BIC.
The traditional methods of DHF, HEGY, and many others are based on the
mean-adjusted observations y !k8 , where the mean function k8 is constructed
t
t
t
from the OLSEs of the parameters in k which uses all the observations
t
My ,2, y N. For example, the mean-adjusted estimator of DHF is
1
n
o8"+ (y !y6 )(y !y6 )/+ (y !y6 )2, where y6 "n~1+n y . However, several

t/1 t
t
t~d
t~d
authors such as Tanaka (1984), Shaman and Stine (1988), Cruddas et al. (1989),
and So and Shin (1999b) point out that this mean adjustment scheme induces
downward biases for estimates of o. They report that the OLSE has nonnegligible bias especially when the AR root is close to unity or the number of
nuisance parameters is large. For example, when d"1 and k "b , Shaman
t
0
and Stine (1988) show that, for DoD(1,
E(o8!o)"!n~1(1#3o)#o(n~1).
Also, for o"1, E(o8!1)"!5n~1#o(n~1).

D.W. Shin, B.S. So / Journal of Econometrics 99 (2000) 107}137

111

We brie#y discuss how the downward bias a!ects powers of the existing tests
based on the OLSE. When DoD(1, the convergence rate of o8 is Jn and the

n~1-order bias is asymptotically negligible. On the other hand, when o"1, the
convergence rate of o8 is n and the n~1-order bias is nonnegligible. Thus the null
distribution of n(o8!1) is more strongly a!ected than the alternative distribution of Jn(o8!o) by the n~1-order bias. Suppose that we have an estimator
oH with smaller null bias than o8. Assume that, due to the smaller bias, the 5th
null percentile gH of n(oH!1) is closer to zero by, say, 4 than g8"!13.7, the 5th
null percentile of n(o8!1) when n"100 and o"1. This is the case of recursively
mean-adjusted OLSE. Now gH"!9.7. Assume further that, when n"100 and
o(1, the distributions of Jn(o8!o) and Jn(oH!o) are N(0, 1!o2) and
N(0, i2(1!o2)), respectively. Usually i'1 because reduced-bias estimation
in#ates variances of estimators. Powers of level 5% tests n(o8!1) and n(oH!1)
at o(1 are p8 "P[n(o8!1)(!13.7 D o]"U[10(0.863!o)/J1!o2] and pH"
P[n(oH!1)(!9.7 D o]"U[10(0.903!o)/iJ1!o2], respectively, where
U is the normal distribution function. Therefore, if 0.863!0.04/
(i!1)(o(1, then pH'p8 . If 0.863(o(1, pH'p8 regardless of values of
i'1. If i is close to one as is the usual case, then pH'p8 for a wide range of o. If,
for example, i"1.1, then the range is 0.463(o(1. This example shows
clearly that we can achieve considerable power improvement of unit root tests
against stationary alternatives by reducing biases of estimators. We should,
however, admit that the situation is reversed for explosive alternatives o'1 in
which n(oH!1) becomes less powerful than n(o8!1) due to the reduced bias.
Cruddas et al. (1989) consider a situation in which several AR(1) processes
have a common AR coe$cient and di!erent means. This model is identical to
model (2.1)}(2.2) with mean function (2.4). They apply the method of restricted
maximum likelihood (REML) estimation and conduct Monte-Carlo simulations showing that the REML estimator of the AR coe$cient is substantially
less-biased than the maximum likelihood estimator.
In our seasonal models, we now have two sources of large bias for the OLSE:
the existence of unit roots and the large number of nuisance parameters for the
seasonal mean functions. We achieve bias reduction by adjusting the mean
through the recursive adjustment of So and Shin (1999b) in which k is estimated
t
by the OLSE based on the observations My ; s"1,2, tN. Now, the mean
s
functions k of (2.3)}(2.7) are estimated by
t
k( "bK (t),
(2.8)
t
0
d
k( " + bK (t)d ,
(2.9)
t
i it
i/1
k( "bK (t)#bK (t)t,
(2.10)
t
0
q
d
k( " + bK (t)d #bK (t)t,
(2.11)
t
i it
q
i/1

112

D.W. Shin, B.S. So / Journal of Econometrics 99 (2000) 107}137

d
k( " + MbK (t)#bK (t)tNd ,
(2.12)
t
i
qi
it
i/1
where, for (2.8)}(2.12),
bK (t)"t~1+t y , the sample mean of My , j"1, 2,2, tN,
j/1 j
j
0
bK (t)"the sample mean of My
, s"0, 1, 2,2, s N,
i
i`ds
t
(bK (t), bK (t))"coe$cients of (1, j) in the regression of y on (1, j), for
0
q
j
j"1, 2,2, t,
(bK (t), bK (t))"coe$cients of d in the regression of y on d ,2, d , j, for
i
q
ij,j
j
1j
dj
j"1, 2,2, t,
(bK (t), bK (t))"coe$cients of (1, s) in the regression of y
on (1, s), for
i
qi
i`ds
s"0, 1, 2,2, s ,
t
respectively, and s is the largest integer such that i#ds )t.
t
t
We call y !k( the &recursive mean adjustment' because the parameters b , b ,
t
t
0 i
b , b are recursively estimated. The recursively adjusted observations y !k(
q qi
t
t
will be used in Sections 3}5 for estimating the AR coe$cients. Kianifard and
Swallow (1996) discuss applications of slightly di!erent versions of the recursive
residuals for various linear models.
In (2.3)}(2.7), we described the most widely used mean functions. Extensions
to general mean functions are obvious. For model (2.1)}(2.2), let the mean
function be k "x@ b where x is a sequence of known deterministic k-vectors
t
t
t
and b is a k-vector of unknown parameters. Now, the AR coe$cient o is
estimated by an ordinary least squares (OLS) "tting to the recursively demeaned
model
y !x@ bK (t!d)"oMy !x@ bK (t!d)N#e( ,
t~d
t
t~d
t~d
t
where

A

B

~1 t
t
bK (t)" + x x@
+ xy
j j
j j
j/1
j/1
is the OLSE of b based on Mx , y , j"1,2, tN. This scheme should provide us
j j
with a less-biased estimator of o because, with k(
"x@ bK (t!d), the regressor
t~d
t~d
(y !k(
) is uncorrelated with the error term e , implying negligible correlat~d
t~d
t
tion between (y !k(
) and e( . On the other hand, if x@ bK (n) is used to
t~d
t~d
t~d
t
estimate k
as in the classical approaches, then the regressor is correlated with
t~d
the error term e due to the correlation between bK (n) and e , implying nonneglit
t
gible correlation between (y !k(
) and e( . This, in turn, induces large biases
t~d
t~d
t
in estimates of o. The biases are more severe for models (2.4), (2.6), and (2.7)
because many extra nuisance parameters are estimated. The biases also become
more severe as o gets close to one because the correlation between bK (n) and
e gets stronger as o approaches one.
t

D.W. Shin, B.S. So / Journal of Econometrics 99 (2000) 107}137

113

3. Model of Dickey}Hasza}Fuller
We "rst consider testing seasonal unit root H : o"1 for model (2.1)}(2.2).
0
Given the recursive estimate k( , the parameter o is estimated from the meant
adjusted model
y !k(
"o(y !k(
)#e( .
(3.1)
t
t~d
t~d
t~d
t
The seasonal unit root o is estimated by using the sign of (y !k(
) as an
t~d
t~d
instrumental variable. Now, our estimator of o is

N

n
n
D,
(3.2)
o( " + [sign(y !k(
)](y !k(
) + Dy !k(
c
t~d
t~d
t
t~d
t~d
t~d
t/d`1
t/d`1
where sign(x)"1 if x'0 and sign(x)"!1 if x)0. The estimator (3.2) may be
called the Cauchy estimator because Cauchy (1836) "rst introduced such an
estimator in a simple linear regression model. This strategy is reviewed in Kotz
and Johnson (1983) in the context of interpolation. The Cauchy estimator o( is
#
also the weighted least-squares estimator for a conditionally heteroscedastic
model y "oy #e , E(e D y , y ,2)"0, and E(e2 D y , y ,2)"
t t~1 t~2
t
t~d
t
t t~1 t~2
pDy D.
t~d
Our test for the seasonal unit root is the pivotal statistic of (o( !1) de"ned by
#
n
D(o( !1),
(3.3)
q( "(o( !1)/se(o( )"(p( Jn)~1 + Dy !k(
t~d
t~d #
#
#
#
t/d`1
Dy !k(
D)~1 is the standard error of o( and p( 2 is
where se(o( )"p( Jn(+n
t/d`1 t~d
t~d
#
#
a consistent estimator of p2. For p( 2, we use the OLS variance estimator. Observe
that, under H : o"1 and y "y #e ,
0
t
t~d
t
n
n
+ Dy !k(
D(o( !1)" + sign(y !k(
)e
(3.4)
t~d
t~d #
t~d
t~d t
t/d`1
t/d`1
is a martingale because, due to the recursive nature of k(
, sign(y !k(
) is
t~d
t~d
t~d
independent of e and hence E[sign(y !k(
)e D y , j(t]"0. Now, by the
t
t~d
t~d t j
martingale central limit theorem (Brown, 1971; Fuller, 1996, p. 235), we get
asymptotic normality of the statistic q( .
#
Theorem 1. Consider model (2.1)}(2.2). Assume that o"1 and k "0. Then, for
t
any d and any p(y ,2, y )-measurable mean adjustment k( , we have q( NN(0, 1),
1
t
t
#
where N denotes convergence in distribution and p(y ,2, y ) is the p-xeld
1
t
generated by My ,2, y N.
1
t
Note that, in (3.1), in addition to y , y is also adjusted by
t~d
t
k(
"x@ bK (t!d) instead of x@ bK (t!d). The main reason for adjusting y by
t
t
t~d
t~d
k(
is to have y !k(
!o(y !k(
)"e and hence retain the martingale
t~d
t
t~d
t~d
t~d
t

114

D.W. Shin, B.S. So / Journal of Econometrics 99 (2000) 107}137

structure of (3.4) under o"1. However, since y is adjusted by k(
, the
t
t~d
remaining part k !k
is not adjusted. If the mean function k contains only
t
t~d
t
seasonal means, then k !k "0 and our scheme of adjusting y by k(
cret
t~d
t
t~d
ates no problem. If k contains a time trend, then k !k
does not vanish,
t
t
t~d
whose e!ect on power performances of test statistics is investigated in Section
6.4. In order to retain the advantage of the limiting normality of the test
statistics, we adjust y by k(
.
t
t~d
Theorem 1 states that limiting normality of the pivotal statistic q( holds for all
#
positive integers d and for all o3[!1, 1]. This allows us to construct a simple
level (1!a) con"dence interval o( $z se(o( ) for o, where z is the a-percentile
#
a@2
#
a
of the standard normal distribution. The asymptotic con"dence interval has
coverage probability (1!a) for all o. On the other hand, the OLSE o( cannot be
0
directly used to construct an asymptotic con"dence interval of the form
o( $z se(o( ) because limiting distribution of (o( !o)/se(o( ) is not normal
0
a@2
0
0
0
when o"1 although it is normal when DoD(1. Moreover, if mean is adjusted,
the "nite sample distribution of Jn(o( !o) for o close to one is substantially
0
skewed. By inverting the "nite sample distributions of the OLSEs numerically,
Andrews (1993) constructs con"dence intervals of o for d"1. In comparison
with Andrews intervals, our con"dence intervals are much simpler and valid for
any positive integer d.
As for the tests based on the OLSE, our test q( also has asymptotically
#
nonzero power against local alternatives of the form o"1!a/n, a'0. Under
such alternatives, the limiting distribution of q( can easily be shown to be of the
#
form Z!a; where Z is a standard normal random variable and ; is a
a
a
strictly positive random variable. Thus, limiting power of size-a test q( is strictly
#
greater than a because lim
P[q( )!z ]"P[Z!a ; )!z ]'
n?=
#
a
a
a
P[Z)!z ]"a.
a
The sign(y ) is also used by Burridge and Guerre (1996) in a di!erent context.
t
They observe that the number of sign changes of an integrated process is smaller
than that of a stationary process. They propose to use the number of sign changes as
a nonparametric test for a unit root. On the other hand, our method is based on the
parametric estimation which uses the sign(y ) as an instrumental variable.
t~1
4. Model of Hylleberg}Engle}Granger}Yoo
4.1. Models and estimation
We present our method through the quarterly model of HEGY. General
d-period model is analyzed after the quarterly model. Consider a fourth order
AR model
t(B)y "e ,
t
t

(4.1)

D.W. Shin, B.S. So / Journal of Econometrics 99 (2000) 107}137

115

where B is the back shift operator such that By "y
and t(B) is a fourtht
t~1
order polynomial of B. Assume, without loss of generality, that the constant is 1.
HEGY observe the identity
t(B)"!n B(1#B#B2#B3)!n (!B)(1!B#B2!B3)
0
2
! (n B!n )(!B)(1!B2)#(1!B4)
(4.2)
1
3
where n , n , n , and n are determined from t(B) by comparing both sides of
0 1 2
3
the identity (4.2). Expressions for n are given in HEGY.
i
The parameters n determine cycling patterns of the observations y . If all
i
t
n are zero, t(B)"(1!B4) and y satis"es (1!B4)y "e . Note that the
i
t
t
t
equation (1!B4)"(1!B)(1#B)(1#B2) has the four seasonal unit roots
eBiC2n*@4, i2"!1, at three seasonal frequencies h "2ni/4, i"0, 1, 2. The AR
i
polynomial t(B) has factors (1!B), (1#B), (1#B2) if and only if n "0,
0
n "0, n "n "0, respectively. If t(B) has only a factor (1!B), then
2
1
3
y reveals a #uctuation of a random walk. If t(B) has only a factor (1#B), then
t
y reveals a #uctuation oscillating two times per year between a random walk
t
= , say, and its negative mirror image != . If t(B) has only a factor (1#B2),
t
t
then y reveals a #uctuation oscillating once per year between a random walk
t
and its negative mirror image. When t(B) has seasonal unit roots at more than
one frequency, y reveals a combined #uctuation pattern corresponding to
t
di!erent seasonal unit roots. See HEGY for other characteristics of y according
t
to the seasonal unit roots.
Using (4.2), we reparametrize (4.1) into
z "n y
#n y
#n y
#n y
#e ,
t
0 0,t~1
1 1,t~1
2 2,t~1
3 3,t~1
t
where

(4.3)

z "(1!B4)y ,
t
t
y "(1#B#B2#B3)y ,
y "!(1!B2)By ,
(4.4)
0t
t
1t
t
y "!(1!B#B2!B3)y , y "(1!B2)y .
2t
t
3t
t
In (4.3)}(4.4) and in the sequel, we use separating comma &,' in subscripts
wherever requiring clarity and do not use comma elsewhere. Therefore, for
example, both y and y are used to denote the same quantity and both q( and
0t
0,t
i#
q( are used to denote the same quantity.
i,#
One advantage of (4.3) over (2.1)}(2.2) is that we can identify signi"cant
seasonal frequencies of the observations y by checking n "0. HEGY apply
t
i
OLS-"tting to (4.3). Their test for signi"cance of n is q( "n( /se(n( ), where
i
i0
i0
i0
n( is the OLSE of n and se(n( ) is the standard error of n( . The distribution of
i0
i
i0
i0
q( is almost independent of each other because the regressors y
,y
,
i0
0,t~1 1,t~1
y
, y
are almost orthogonal. However, its limiting null distribution
2,t~1 3,t~1
depends not only on d and i but also on the type of mean adjustment. In

116

D.W. Shin, B.S. So / Journal of Econometrics 99 (2000) 107}137

addition, the estimators n( are highly biased when seasonal means or trends are
i0
adjusted. In order to overcome these di$culties, we apply recursive adjustment and
Cauchy estimation for estimating n . We "rst extend the de"nition of sign(y ).
i
it
Dexnition 1. Let sign (y )"y /Dy D, i"0, 2; sign (y )"y /(y2 #y2 )1@2, i"1, 3,
3t
i it
it it
i it
it 1t
where it is understood that 0/0"0.
Note that if we de"ne the sign of a complex number z by sgn(z)"z/DzD, the
complex number sign (y )#i sign (y ) is the same as sgn(y #iy ). One
1 1t
3 3t
1t
3t
reason for using the complex signs, sign (y ) and sign (y ), for i"1, 3 corre1 1t
3 3t
sponding to the complex root i instead of the usual sign(y ) and sign(y ) is that
1t
3t
the complex signs enable us to establish joint normality of our tests. Another
reason is that the tests based on the complex signs have higher power than the
tests based on the usual sign.
A mean-adjusted model is (4.3) with y replaced by y8 , where y8 is the same as
it
it
it
y except that y8 is constructed from (4.4) using the recursively mean-adjusted
it
it
observations y !k( instead of y . Note that z is y !y . De"ne
t
t
t
t
t
t~d
v "sign (y8 ), t"1,2, n, i"0, 1, 2, 3,
it
i it
and use them as instrumental variables. Let
X "(y8 , 2,y8
)@,
i
i,d
i,n~1
and let

V "(v ,2, v
)@, i"0,2, 3,
i
i,d
i,n~1

X"(X DX D X D X ),
V"(V D V D V D V ), Z"(z
, , z )@.
0 1 2 3
0 1 2 3
d`1 2 n
Then our Cauchy estimator of n"(n , n , n , n )@ is given by
0 1 2 3
n( "(n( , n( , n( , n( )@"(V @X)~1(V @Z).
#
0# 1# 2# 3#
The covariance matrix is estimated by
var( (n( )"p( 2(V @X)~1(V @V) (X @V)~1,
#
where p( 2 is the OLSE of p2. Signi"cance of n is tested by the pivotal statistics
i
q( "n( /se(n( ), i"0, 1, 2, 3,
i#
i#
i#
where se(n( )"p( c , c is the (i#1, i#1) element of (V@X )~1(V@V)(X@V )~1.
i#
ii ii
We next discuss extensions to general d. Beaulieu and Miron (1993), Franses
(1992), and Taylor (1998) extend the tests of HEGY to the cases d"12 and
d"6. Several other extensions have appeared in Franses and Hobijn (1997)
and Smith and Taylor (1998b, c). Let d be a positive integer. The extended
version of (4.3)}(4.4) is
d~1
z"+ ny
#e ,
t
i i,t~1
t
i/0

D.W. Shin, B.S. So / Journal of Econometrics 99 (2000) 107}137

117

A B
A B

d
2i
y " + cos
jn Bj~1y ,
i"0,2, r,
i,t
t
d
j/1
d
2i
y
" + sin
jn Bj~1y , i"1,2, r ,
(4.5)
i`r,t
t
#
d
j/1
where r"[d/2], the integer part of [d/2], and r "[(d!1)/2]. Now, the
#
complex signs in De"nition 1 are extended so that sign (y )"y /Dy D;
0 0t
0t 0t
sign (y )"y /Dy D if d is even; sign (y )#i sign (y
)"(y #iy
)/
r rt
rt rt
i i,t
i`r i`r,t
i,t
i`r,t
Dy #iy
D, i"1,2, r . Extensions of the tests q( are obvious. The limiting
i,t
i`r,t
#
i#
null distributions of the test statistics are given below.
Theorem 2. Consider (4.5). For any d and any p(y ,2, y )-measurable mean
1
t
adjustment k( ,
t
(i) if n "0, then q( NN(0, 1); if d is even and n "0, then q( NN(0, 1),
0
0#
r
r,#
(ii) if n "n "0, then (q9 , q(
)NN (0, I ), i"1,2, r , where N (0, I ) is
i
i`r
ic i`r,c
2
2
#
2
2
the standard bivariate normal distribution.
4.2. Hypothesis testing
For testing n "0, we use the normal test q( , i"0,2, d!1. For testing the
i
i#
joint hypothesis (n , n )"(0,0) which states signi"cance of the seasonal unit
i i`r
root of frequency 2ni/d, we may use QK "q( 2 #q( 2 , whose limiting null
i`r,#
i,#
i#
distribution is s2 by Theorem 2, where s2 is a chi-square random variable with
2
k
k degrees of freedom. However, as observed by HEGY, if explosive processes are
excluded, the alternative hypothesis is (n (0, n O0) and therefore is partly
i
i`r
one sided. Hence, we can improve the power of the test QK by restricting n to
i#
i
n )0. Then the restricted estimator of n is n8 "n( if n( )0; n8 "0 if
i
i
i#
i#
i#
i#
n( '0. Also, the restricted version of QK is QI "q82 #q( 2 . The limiting null
i`r,#
i,#
i#
i#
i#
distribution of QI is a chi-bar square distribution which is a convex combinai#
P(n( '0)"2~1 and
tion of s2 and s2 distributions with weights w "lim
2
1
n?=
i#
1
w "lim
P(n( )0)"2~1, respectively. That is, when n "n "0,
2
n?=
i#
i
i`r
(4.6)
QI Nw s2 #w s2 .
2 2
i#
1 1
In (4.6) and in the sequel, +l w s2 denotes a random variable with distribution
k/0 k k
function F(x)"+l w P[s2)x], s2 "0, if +l w "1, w *0, and l is
k/0 k
k
0
k
k/0 k
a positive integer. Now, the joint hypothesis (n , n )"(0, 0) is tested by the
i i`r
statistic QI whose percentage points are available from (4.6) and is given in
i#
Table 1.
Our approach allows us to construct joint tests for seasonal unit roots of
di!erent frequencies. Let I be a subset of M0, 1,2, rN. Let n "Mn N; n "Mn ,
(0)
0
(i)
i
n N, i"1,2, r ; n "Mn N if d is even. Suppose we want to test the hypothei`r
# (r)
r
sis of joint signi"cance of the frequencies corresponding to the seasonal unit

118

D.W. Shin, B.S. So / Journal of Econometrics 99 (2000) 107}137

Table 1
Percentage points of the chi-bar square distribution 0.5s2 #0.5s2
1
2
p"0.5P[s2 )x]#0.5P[s2 )x]
1
2
p
x

0.010
0.006

0.025
0.036

0.050
0.014

0.100
0.049

0.500
0.867

0.900
3.808

0.950
5.138

0.975
6.483

0.990
8.273

roots eBiC2n*@d for all i3I. Now, the joint signi"cance is represented by the null
hypothesis H : n "M0N for all i3I. The joint hypothesis H can be tested
0 (i)
0
using the statistics QK "+ QK or QI "+ QI , where QI "q82 and, if d is
0#
I#
i|I i#
I#
i|I i#
0#
even, QI "q82 . The limiting null distributions of the tests QK and QI follow
r,#
I#
I#
r,#
from Theorem 2 and are given below.
Theorem 3. Consider (4.5). Under H : n "M0N for all i3I, for any d and any
0 (i)
p(y ,2, y )-measurable mean adjustment k( ,
1
t
t
(i) q( NN(0, 1) for all i3I and the limiting distributions are independent,
i#
(ii) QK Ns2 and QI N+ l (l)2~ls2 ,
l{`k
k/0 k
2l
I#
I#
where l is the number of elements in I. Also, l@"l if (d is even, 0NI, and rNI) or (d is
odd, 0NI); l@"l!1 if (d is even and only one of 0 or r is in I) or (d is odd and
03I); l@"l!2 if d is even and 0, r3I.
A simple example consists of testing the joint hypothesis n "n "n "
0
1
2
n "0 for the quarterly model. Now I"M0, 1, 2N and l"3. Thus, the limiting
3
null distribution of the statistic QI "q82 #q82 #q82 #q( 2 is the chi-bar
3#
2#
1#
0#
I#
square distribution (s2 #3s2 #3s2 #s2 )/8 by Theorem 3(ii). In general, the
4
3
2
1
limiting null distributions of the one-sided tests QI are chi-bar square distribuI#
tions and are more complicated than the limiting null distributions of the
two-sided chi-square tests QK . However, the one-sided tests QI
are more
I#
I#
powerful than the two-sided tests QK . Percentage points of the chi-bar square
I#
distributions and the p-values of the test statistics can be easily computed using
the chi-square distributions.

5. Extensions to higher-order autoregressive models
A higher-order extension of the model of DHF is
z "(o!1)y #h z #2#h z #e ,
(5.1)
t
t~d
1 t~1
p t~p
t
where z "y !y . We assume that the characteristic equation
t
t
t~d
h(B)"1!h B!2!h Bp has all roots outside the unit circle. For estima1
p
ting the stationary parameters h ,2, h , Cauchy estimation would give us less
1
p

D.W. Shin, B.S. So / Journal of Econometrics 99 (2000) 107}137

119

e$cient estimators. Thus, we use the e$cient OLSE hK "(hK ,2, hK )@ which is the
1
p
vector of regression coe$cients of z ,2, z
in the regression of z on
t~1
t~p
t
y !k(
, z , , z , t"p#d#1,2, n. Now, the nonstationary parat~d
t~d t~1 2 t~p
meter (o!1) is estimated by applying Cauchy estimation to the model
z8 "(o!1)(y !k(
)#e(
(5.2)
t
t~d
t~d
t
of the residuals z8 "z !hK z !2!hK z . Thus the Cauchy estimator is
t
t
1 t~1
p t~p
o( "1#+sign(y !k(
)z8 /+Dy !k(
D. The q-statistic q( for the unit
#
t~d
t~d t
t~d
t~d
#
root is de"ned similarly as in (3.3).
Model (4.5) of HEGY can also be extended to a higher-order model
d~1
z"+ ny
#h z #2#h z #e .
(5.3)
t
i i,t~1
1 t~1
p t~p
t
i/0
First, the parameters n ,2, n
,h , , h are estimated by OLS regression of
0
d~1 1 2 p
z on y8
,2, y8
, z ,2, z , t"p#d#1,2, n. Let hK ,2, hK be
t
0,t~1
d~1,t~1 t~1
t~p
1
p
the estimators of h ,2, h . Then the nonstationary parameters n ,2, n
are
1
p
0
d~1
estimated by applying Cauchy estimation to the residual model
d~1
z !hK z !2!hK z " + n y8
#e( .
(5.4)
t
1 t~1
p t~p
i i,t~1
t
i/0
The test statistics q( , i"0,2, d!1, (QK , QI ), i"1,2, r , are computed in
i#
i# i#
#
the same manner as in Section 4 using z8 "z !hK z !2!hK z
instead
t
t
1 t~1
p t~p
of z .
t
Since the OLSE hK is Jn-consistent by Chan and Wei (1988), the limiting null
distributions of the test statistics are the same as those with known h.
Theorem 4. Consider (5.1). Assume that all the characteristic roots of h(B) lie
outside the unit circle and assume o"1. Then, for any d and any p(y ,2, y )1
t
measurable mean adjustment k( , q( NN(0,1).
t #
Theorem 5. Consider (5.3). Assume that all the characteristic roots of h(B) lie
outside the unit circle. Then, for any d and any p(y ,2, y )-measurable mean
1
t
adjustment k( , the limiting null distributions of the test statistics q( , QK , QI ,
t
i# i# i#
QK , QI are the same as those in Theorems 2 and 3.
I# I#
Instead of the two-stage "tting of the residual regression (5.2), one can
construct a one-stage instrumental variable estimator for (5.1) by using sign
(y !k(
) as an instrument for y !k(
and Mz , j"1,2, pN as instrut~d
t~d
t~d
t~d
t~j
ments for themselves. Similarly we can construct another instrumental variable
estimator for (5.3) using Msign (y8
), i"0,2, d!1; z , j"1,2, pN as ini i,t~1
t~j
strumental variables. The limiting null distributions of the resulting test statistics are the same as those in Theorems 4 and 5.

120

D.W. Shin, B.S. So / Journal of Econometrics 99 (2000) 107}137

Table 2
Empirical sizes (%) of q( for testing H : o"1 against H : o(1 in model y !k "o(y !
#
0
1
t
t
t~d
k )#e !
t~d
t
d"4

d"12

Adjustment (n)

24

48

120

240

480

48

120

240

480

No adjustment
Simple mean
Seasonal means
Simple mean and trend
Seasonal means and trend

4.3
4.4
2.6
3.7
2.4

4.6
4.9
3.8
4.4
3.6

4.8
4.6
5.1
5.0
4.4

5.3
5.2
5.0
5.3
5.2

4.7
4.8
4.7
5.2
5.2

3.3
2.8
1.1
3.1
1.0

4.2
4.4
3.4
4.1
3.4

4.8
4.9
4.2
5.1
4.6

5.0
5.0
4.6
4.7
4.6

!Nominal level "5%, number of samples "10,000, critical value "!1.645.

6. Monte-Carlo study
6.1. Simple models
We "rst consider the DHF model (2.1)}(2.2) and investigate "nite sample sizes
of the tests q( . Observations y , t"1,2, n are generated from (2.1)}(2.2) using
#
t
k "0; o"1; n"24, 48, 120, 240, 480; d"4, 12, where e are standard nort
t
mal errors simulated by RNNOA, a FORTRAN subroutine of IMSL (1989).
The initial values y , y , y , y are all set to zero. In Table 2, the empirical
0 ~1 ~2 ~3
sizes of the tests q( with various mean adjustments are given. Each number in
#
Table 2 represents percentage of 10,000 independent test statistics q( smaller
#
than !1.645, the 5th left percentile of the standard normal distribution. For the
quarterly case (d"4), the sizes are all between 4.4% and 5.2% and are close to
the nominal level 5% when n is greater than 120 (30 yr). However, when n"48
or 24, tests adjusted for seasonal means (2.9) and seasonal means with a trend
(2.11) are somewhat undersized. For the monthly case (d"12), for n*240, sizes
are all between 4.2% and 5.1%. When n"48 (4 yr), all the tests are downsized.
Generally, we can say that if we have data for more than 20 yr, we have sizes
between 4% and 5.2% for all of the "ve mean adjustments and for d"4, 12.
We next compare the empirical powers of our tests q( with those of the
#
OLSE-based tests q( . We consider two adjustments of seasonal means (2.9) and
0
seasonal means with a trend (2.11). Data y are generated from model (2.1)}(2.2)
t
with k "0, o"0.95, 0.9, 0.8, 0.7 and n"48, 120, 240, 480. Number of replit
cations is 10,000. Table 3 presents the percentages of the test statistics smaller
than the left 5th percentiles of the null distributions of the corresponding test
statistics. The 5th percentile for q( is !1.645 for all cases. The 5th percentiles for
#
q( di!er for each (n, d, k( ) and are available in DHF (1984, Table 7) for the
0
t

D.W. Shin, B.S. So / Journal of Econometrics 99 (2000) 107}137

121

Table 3
Empirical powers (%) of q( and q( for testing H : o"1 against H : o(1 in model
#
0
0
1
y !k "o(y !k )#e !
t
t
t~d
t~d
t
Seasonal means

Seasonal means and trend

q(
0
d"4

q(
#
d"4

q(
0
d"12

q(
#
d"12

q(
0
d"4

q(
#
d"4

q(
0
d"12

q(
#
d"12

0.95
0.90
0.80
0.70

7.9
10.1
15.5
25.3

7.7
11.6
22.0
34.0

6.7
8.7
13.1
16.3

1.9
2.3
4.2
5.9

7.7
9.8
14.9
24.1

6.5
8.7
15.7
25.0

6.7
8.7
13.7
17.5

1.6
2.0
3.0
4.5

120
120
120
120

0.95
0.90
0.80
0.70

10.0
19.1
57.3
92.1

16.7
34.0
68.8
91.1

10.0
15.8
29.7
51.1

9.5
18.3
40.3
61.6

9.4
18.1
52.1
88.5

13.4
25.4
56.8
82.3

10.0
15.7
29.7
51.3

8.2
15.8
35.6
56.4

240
240
240
240

0.95
0.90
0.80
0.70

18.2
55.3
99.6
100.0

35.0
71.6
98.7
100.0

15.5
29.7
74.8
97.9

24.5
50.3
88.8
98.9

17.6
50.3
99.0
100.0

27.4
57.8
95.3
99.8

15.7
29.8
74.1
97.8

21.6
47.0
84.9
98.1

480
480
480
480

0.95
0.90
0.80
0.70

53.8
99.2
100.0
100.0

73.0
98.6
100.0
100.0

28.8
72.9
100.0
100.0

55.1
91.9
100.0
100.0

47.8
98.4
100.0
100.0

59.2
95.8
100.0
100.0

29.1
72.9
100.0
100.0

51.4
88.4
100.0
100.0

n

o

48
48
48
48

!Nominal level "5%; number of samples "10,000; critical value of q( "!1.645. Critical values
c
of q( for n"(48,120,240,480) are (!4.18,!4.10,!4.06,!4.05) for (d"4, k( " seasonal means);
o
t
(!4.39,!4.34,!4.29,!4.27) for (d"4, k( " seasonal means and trend); (!6.20,!5.86,
t
!5.83,!5.82) for (d"12, k( " seasonal means); (!6.02,!6.01,!5.98,!5.95) for
t
(d"12, k( " seasonal means and trend).
t

seasonal means k in (2.4). For a model with seasonal means and a trend (2.6), we
t
use the empirical 5th percentiles of 10,000 values of q( simulated with o"1. For
0
almost all cases considered here, we see that our tests q( have higher powers than
#
q( . Better powers of q( over q( are more evident for d"12 than for d"4, which
0
#
0
is a consequence of reduced-bias estimation of o using the recursive mean
adjustments (2.8)}(2.12).
We next consider the quarterly model (4.3) of HEGY. We compare our tests
q( , q( , and QI with the corresponding tests q( , q( , and QK of HEGY based
0# 2#
1#
00 20
10
on the OLSE. The statistics (q( , q( ), (q( , q( ), and (QI , QK ) test for signi"00 0#
20 2#
10 1#
cance of n , n , and (n , n ), respectively. Data y are generated from model
0 1
1 3
t
(4.3) with n "0,$0.02,$0.04,$0.08,$0.12, i"0, 1, 2, 3 and n"120. The
i
test statistics are adjusted for the seasonal means k of (2.4). The nominal level is
t
5%. The 5% critical values of q( and q( are !1.645 and that of QI is 5.138
0#
2#
1#

122

D.W. Shin, B.S. So / Journal of Econometrics 99 (2000) 107}137

Table 4
Empirical size (%) and power (%) of the tests for model (4.3) adjusted for means!
n
0

n
2

n
1

n
3

q(
00

q(
0#

q(
20

q(
2#

QK

0
0
0
0
0
0
0
0
0
0
0
0
!0.02
!0.02
!0.02
!0.02
!0.02
!0.02
!0.02
!0.02
!0.02
!0.02
!0.02
!0.02
0
0
0
0
0
0
0
0
0
0
0
!0.04
!0.04
!0.04
!0.04
!0.04
!0.04
!0.04
!0.04
!0.04
!0.04
!0.04

0
0
0
0
0
0
!0.02
!0.02
!0.02
!0.02
!0.02
!0.02
0
0
0
0
0
0
!0.02
!0.02
!0.02
!0.02
!0.02
!0.02
0
0
0
0
0
!0.04
!0.04
!0.04
!0.04
!0.04
!0.04
0
0
0
0
0
0
!0.04
!0.04
!0.04
!0.04
!0.04

0
0
0
!0.02
!0.02
!0.02
0
0
0
!0.02
!0.02
!0.02
0
0
0
!0.02
!0.02
!0.02
0
0
0
!0.02
!0.02
!0.02
0
0
!0.04
!0.04
!0.04
0
0
0
!0.04
!0.04
!0.04
0
0
0
!0.04
!0.04
!0.04
0
0
0
!0.04
!0.04

0
0.02
!0.02
0
0.02
!0.02
0
0.02
!0.02
0
0.02
!0.02
0
0.02
!0.02
0
0.02
!0.02
0
0.02
!0.02
0
0.02
!0.02
0.04
!0.04
0
0.04
!0.04
0
0.04
!0.04
0
0.04
!0.04
0
0.04
!0.04
0
0.04
!0.04
0
0.04
!0.04
0
0.04

4.3
4.0
4.6
4.5
3.9
4.2
4.1
4.1
4.8
4.5
4.2
4.2
6.7
6.1
7.2
6.7
6.3
7.1
6.6
6.3
6.5
6.6
6.0
6.5
3.8
4.2
4.1
4.2
4.4
4.1
4.5
4.2
4.2
4.4
4.3
10.4
8.9
10.9
11.0
9.2
10.4
10.6
9.5
10.6
10.6
10.1

4.4
4.7
4.6
4.6
4.5
4.7
4.9
4.5
4.7
4.8
4.6
5.0
9.3
9.5
10.3
10.7
10.2
10.4
10.4
10.1
9.7
10.2
10.3
10.1
4.7
4.3
4.4
4.8
4.5
4.3
4.9
4.7
4.6
4.8
4.6
16.8
15.9
15.9
17.4
16.4
17.1
17.1
17.5
16.9
17.1
17.4

4.8
4.6
4.6
4.7
4.6
4.6
7.1
7.3
6.3
7.2
7.1
6.6
4.3
4.8
4.2
4.8
4.7
4.5
7.0
7.2
6.9
7.0
6.7
7.1
4.4
3.9
4.2
4.5
4.4
11.1
11.1
9.7
11.4
12.1
10.9
4.7
4.7
3.9
4.7
4.8
4.4
11.4
11.6
9.7
11.2
11.9

4.4
4.7
4.6
4.8
4.5
5.2
9.8
10.1
9.4
10.1
10.5
9.7
4.9
4.8
4.7
4.4
4.7
4.8
10.6
9.8
10.3
10.5
9.7
10.6
4.4
4.5
4.5
4.4
4.9
16.8
16.9
16.3
17.3
18.1
17.4
4.3
4.4
4.3
4.7
4.8
4.3
17.3
16.7
16.7
17.9
17.4

5.0
7.6
7.0
7.2
8.7
8.3
5.5
7.0
6.7
7.3
8.6
8.5
5.2
7.4
6.6
7.1
8.6
8.5
5.4
7.5
7.2
7.3
8.5
9.4
20.2
20.5
9.9
15.6
15.3
5.2
21.3
20.5
9.7
16.1
15.1
5.2
21.0
22.1
9.8
15.6
16.3
5.5
22.5
21.8
10.2
16.6

10

QI

1#

4.3
9.8
9.8
8.1
12.3
11.6
4.7
10.0
9.7
8.1
12.6
12.1
4.1
10.4
10.1
7.9
12.2
11.7
4.2
10.3
10.2
8.1
12.0
11.9
26.7
26.5
10.8
20.8
20.1
4.5
34.2
32.1
12.4
25.9
24.2
4.5
32.2
34.6
12.5
25.2
26.6
4.7
34.8
34.6
12.9
26.0

D.W. Shin, B.S. So / Journal of Econometrics 99 (2000) 107}137

123

Table 4 (continued)
n
0

n
2

n
1

n
3

q(
00

q(
0#

q(
20

q(
2#

QK

!0.04
0
0
0
0
0
0
0
0
0
0
0
!0.08
!0.08
!0.08
!0.08
!0.08
!0.08
!0.08
!0.08
!0.08
!0.08
!0.08
!0.08
0
0
0
0
0
0
0
0
0
0
0
!0.12
!0.12
!0.12
!0.12
!0.12
!0.12
!0.12
!0.12
!0.12
!0.12
!0.12
!0.12

!0.04
0
0
0
0
0
!0.08
!0.08
!0.08
!0.08
!0.08
!0.08
0
0
0
0
0
0
!0.08
!0.08
!0.08
!0.08
!0.08
!0.08
0
0
0
0
0
!0.12
!0.12
!0.12
!0.12
!0.12
!0.12
0
0
0
0
0
0
!0.12
!0.12
!0.12
!0.12
!0.12
!0.12

!0.04
0
0
!0.08
!0.08
!0.08
0
0
0
!0.08
!0.08
!0.08
0
0
0
!0.08
!0.08
!0.08
0
0
0
!0.08
!0.08
!0.08
0
0
!0.12
!0.12
!0.12
0
0
0
!0.12
!0.12
!0.12
0
0
0
!0.12
!0.12
!0.12
0
0
0
!0.12
!0.12
!0.12

!0.04
0.08
!0.08
0
0.08
!0.08
0
0.08
!0.08
0
0.08
!0.08
0
0.08
!0.08
0
0.08
!0.08
0
0.08
!0.08
0
0.08
!0.08
0.12
!0.12
0
0.12
!0.12
0
0.12
!0.12
0
0.12
!0.12
0
0.12
!0.12
0
0.12
!0.12
0
0.12
!0.12
0
0.12
!0.12

11.4
4.0
4.6
4.3
4.2
4.2
4.5
3.7
4.2
4.3
3.8
4.0
26.6
22.4
28.9
29.1
24.9
31.4
28.7
24.2
30.3
31.1
26.4
34.6
4.1
4.1
4.4
4.0
4.4
4.4
4.3
3.8
3.9
4.3
4.6
54.7
45.9
59.0
60.6
51.7
66.0
59.7
51.5
64.4
66.7
56.7
73.1

17.9
4.7
4.2
4.9
4.8
4.1
4.8
5.1
4.4
4.5
4.5
3.9
33.8
34.2
33.4
36.6
35.3
37.0
36.6
36.2
34.9
38.6
36.9
39.7
5.0
4.0
4.5
4.9
4.4
4.7
5.3
4.2
4.7
5.1
4.3
55.2
54.7
48.9
59.9
55.4
61.7
58.4
58.5
55.3
64.0
60.5
66.1

10.8
4.3
4.3
4.5
4.9
4.7
29.4
29.9
24.2
31.7
33.3
26.3
4.8
4.3
4.3
4.6
4.7
4.0
30.6
32.0
25.6
32.4
35.6
27.9
4.3
4.5
4.8
4.6
4.3
56.7
61.2
47.3
61.2
68.0
52.9
4.6
4.1
4.0
4.2
4.6
4.5
61.2
66.5
53.0
67.9
74.8
59.0

17.3
4.8
5.1
4.1
4.5
4.8
35.4
33.5
34.3
34.7
37.2
35.3
4.5
4.3
4.8
4.9
4.2
4.8
37.5
34.9
36.3
37.9
39.5
37.4
4.1
6.0
4.6
4.2
4.8
55.0
49.2
54.9
59.2
61.4
56.3
4.7
3.8
5.3
4.2
4.4
4.7
58.7
55.2
59.1
62.8
66.6
60.0

17.0
68.2
68.3
18.1
44.5
44.5
5.0
74.6
68.8
19.9
51.1
46.3
4.7
68.1
73.9
18.9
45.3
50.9
4.9
74.8
74.7
20.3
51.5
51.3
92.5
92.3
34.5
79.7
80.1
5.4
95.9
92.4
39.1
86.7
80.6
5.0
92.1
96.1
39.2
81.4
86.5
5.4
95.9
96.3
41.9
88.4
88.3

10

!Nominal level "5%; number of samples "10,000; period of seasonality "4; n"120.

QI

1#

27.0
79.9
79.9
23.0
56.2
55.7
4.7
84.3
80.3
25.2
60.0
57.3
4.3
79.5
83.7
23.9
56.4
60.7
4.1
84.5
84.6
25.8
61.0
59.8
95.6
95.8
39.4
81.5
81.6
4.6
97.7
95.6
42.5
86.4
82.4
4.6
95.6
97.8
43.4
82.9
86.0
4.3
97.7
97.9
44.7
87.7
87.0

124

D.W. Shin, B.S. So / Journal of Econometrics 99 (2000) 107}137

from Table 1. The 5% critical values of q( , q( , and QK
are
00 20
10
!2.944,!2.918, 6.603 which are interpolated values from HEGY (1990,
Tables 1a and b).
In Table 4, percentages of rejected test statistics out of 10,000 independent
tests of n"120 are reported. Numbers below q( and q( are sizes if they
i#
i0
correspond to n "0 and are powers if they correspond to n O0, i"0, 2.
i
i
Similarly, numbers below QI
and QK
are sizes if they correspond to
1#
10
n "n "0 and are powers otherwise. Even though, the tests are slightly
1
3
undersized, the empirical sizes of all the tests are close to the nominal 5% level
for all n considered here. When Dn D)0.08, our tests q( , q( , and QI are more
i
i
0# 2#
1#
powerful than the corresponding OLSE-based tests q( , q( , and QK . When
00 20
10
Dn D"0.12, our tests are not more powerful than the OLSE-based tests.
i
6.2. Tests based on the OLSE and recursive mean adjustment
Power advantage of the proposed Cauchy tests over the existing OLSE-based
tests is mainly due to the recursive mean adjustment. Thus, the recursive mean
adjustment would also improve powers of the existing tests. We compare power
performance of the proposed tests (q( , q( , q( , QK ), the existing OLSE-based
# 0# 2# 1#
tests (q( , q( , q( , QK ), and the tests (q( , q( , q( , QK ) based on the OLSE and
0 00 20 10
03 003 203 103
the recursive mean adjustment. The tests are all adjusted for seasonal means
(2.4). The simulation schemes are similar to those for Tables 3 and 4. The
statistics (q( , q( , q( , QK ) are the same as (q( , q( , q( , QK ) except that the
03 003 203 103
0 00 20 10
recursively adjusted observations are used instead of the observations adjusted
by global seasonal means.
In Table 5, rejection frequencies of the tests q( , q( , q( for the DHF model (3.1)
03 0 #
are reported. The results are based on 10,000 independent replications. As
a critical value of q( , we use the empirical left 5th percentile from 10,000 values
03
of q( simulated under o"1. We see that in terms of power performance, q( is
03
03
best, q( is intermediate, and q( is worst. The Cauchy test q( gains power thanks
#
0
#
to the recursive mean adjustment but loses power owing to lower e$ciency of
the Cauchy estimator than the OLSE especially for o not close to one. Thus,
power performance of q( is in the middle of those of q( and q( .
#
0
03
In Table 6, rejection frequencies of the tests for the HEGY model (4.3) are
reported. The results are based on 10,000 replications. As critical values of
(q( , q( ), we use the left 5th empirical percentiles (or right 5th percentile for
003 203
QK ) of 10,000 values of (q( , q( ) simulated under n "n "n "n "0.
103
003 203
0
1
2
3
Power performances of the tests are similar to those in Table 5. The tests
(q( , q( , QK ) are more powerful than (q( , q( , QK ) which are in turn more
003 203 103
0# 2# 1#
powerful than (q( , q( , QK ).
00 20 10
The Monte-Carlo results in Tables 5 and 6 indicate that the recursive mean
adjustment substantially improves powers of tests for seasonal unit roots.
A detailed study on the tests for seasonal unit roots based on the OLSEs and

D.W. Shin, B.S. So / Journal of Econometrics 99 (2000) 107}137

125

Table 5
Empirical powers (%) of q( and q( for testing H : o"1 against H : o(1 in model y "oy #e .
#
0
0
1
t
t~d
t
Test statistics are adjusted for seasonal means!

n

o

q(
03
d"4

q(
0
d"4

q(
#
d"4

q(
03
d"12

q(
0
d"12

q(
#
d"12

120
120
120
120

0.95
0.90
0.80
0.70

23.8
50.4
91.8
99.8

10.0
19.1
57.3
92.1

16.7
34.0
68.8
91.1

16.1
33.9
69.0
90.7

10.0
15.8
29.7
51.1

9.5
18.3
40.3
61.6

!Nominal level "5%; number of samples "10,000.

recursive mean adjustments would be a good topic for a future research.
However, it would require a large number of null distributions and their
percentage points for each combination of parameters of seasonal models. On
the other hand, our tests based on the Cauchy estimator and the recursive mean
adjustment have reasonable power properties and still retain the advantage of
standard normal asymptotics for any combination of the parameters of seasonal
models.
6.3. Higher-order models
We now investigate "nite sample properties of the test statistics for higherorder models. We "rst consider a model
(1!hB)(1!oBd)y "e ,
t
t
which is equivalent to the DHF model (5.1) with p"1. We compare the test
statistics q( and q( adjusted for seasonal means. Parameters are set to
#
0
d"4, 12; n"120; h"$0.5; o"1, 0.95, 0.9, 0.8, 0.7; and 10,000 replications
are used. The nominal level is set to 5%. The test statistic q( is computed from
#
(5.2) with p"1 by the Cauchy method. The test statistic q( is computed from
0
OLS-"tting to model (5.1) with p"1. In Table 7, we present rejection percentages of the test statistics. We see that the size of our test q( is close to the nominal
#
level 5% and that of q( is slightly smaller than 5%. The power of q( looks greater
0
#
than that of q( for o"0.95, 0.9, 0.8. The seemingly higher power of q( is partly
0
#
due to the better performance of the recursive mean adjustment and partly due
to the higher size of q( than that of q( . Thus power advantage of q( over q( would
#
0
#
0
not be as good as in Table 7. Size and power properties of the test statistics for
h"!0.5 are similar to those for h"0.5.
We next consider a higher-order HEGY model of (5.3) with p"1;
n"120, 240; n , n , n "0,!0.08; n "0; h"$0.5; and 10,000 replica0 1 2
3
tions. We compute Cauchy tests (q( , q( , QK ) from (5.4) with p"1 and
0# 2# 1#

126

D.W. Shin, B.S. So / Journal of Econometrics 99 (2000) 107}137

Table 6
Empirical size (%) and power (%) of the tests for model (4.3) adjusted for seasonal means!
100]
n
n
2
1

n
0

n
3

q(

003

q(
00

q(

0#

q(
203

q(

20

q(
2#

QK

103

QK
10

QI

1#

0
0
0
0
0

0
0
0
0
0

0
0
!8
!8
!8

8
!8
0
8
!8

4.7
4.4
4.9
5.0
4.4

4.0
4.6
4.3
4.2
4.2

4.7
4.2
4.9
4.8
4.1

4.6
5.2
4.7
5.0
5.0

4.3
4.3
4.5
4.9
4.7

4.8
5.1
4.1
4.5
4.8

83.8
84.3
26.9
64.5
63.7

68.2
68.3
18.1
44.5
44.5

79.9
79.9
23.0
56.2
55.7

0
0
0
0
0
0

!8
!8
!8
!8
!8
!8

0
0
0
!8
!8
!8

0
8
!8
0
8
!8

4.8
4.5
4.6
4.5
4.8
4.1

4.5
3.7
4.2
4.3
3.8
4.0

4.8
5.1
4.4
4.5
4.5
3.9

50.5
51.5
48.1
53.4
53.8
49.9

29.4
29.9
24.2
31.7
33.3
26.3

35.4
33.5
34.3
34.7
37.2
35.3

5.3
87.5
84.5
28.7
67.3
65.2

5.0
74.6
68.8
19.9
51.1
46.3

4.7
84.3
80.3
25.2
60.0
57.3

!8
!8
!8
!8
!8
!8

0
0
0
0
0
0

0
0
0
!8
!8
!8

0
8
!8
0
8
!8

47.6
46.6
49.9
51.7
47.7
52.5

26.6
22.4
28.9
29.1
24.9
31.4

33.8
34.2
33.4
36.6
35.3
37.0

5.1
4.6
4.8
5.1
4.5
4.7

4.8
4.3
4.3
4.6
4.7
4.0

4.5
4.3
4.8
4.9
4.2
4.8

4.9
83.9
86.9
27.7
64.5
68.5

4.7
68.1
73.9
18.9
45.3
50.9

4.3
79.5
83.7
23.9
56.4
60.7

!8
!8
!8
!8
!8
!8

!8
!8
!8
!8
!8
!8

0
0
0
!8
!8
!8

0
8
!8
0
8
!8

50.7
47.8
51.8
52.9
50.5
55.1

28.7
24.2
30.3
31.1
26.4
34.6

36.6
36.2
34.9
38.6
36.9
39.7

53.2
53.3
50.1
54.4
56.6
51.6

30.6
32.0
25.6
32.4
35.6
27.9

37.5
34.9
36.3
37.9
39.5
37.4

5.1
88.0
87.8
28.8
69.0
68.4

4.9
74.8
74.7
20.3
51.5
51.3

4.1
84.5
84.6
25.8
61.0
59.8

!Nominal level "5%; number of samples "10,000; period of seasonality "4; n"120.

compute (q( , q( , QK ) from OLS-"tting to model (5.3) with p"1. All tests are
00 20 10
adjusted for seasonal means. In Table 8, we report rejection frequencies of the
test statistics. The nominal level is 5%. When n"120, the OLSE-based tests
q(
and q(
are slightly under-sized; the test q(
is slightly over-sized for
00
20
0#
h"!0.5; and the test q( is slightly over-sized for h"0.5. When n"240, the
2#
sizes of our tests, as well as the sizes of the OLSE-based tests, get closer to the
nominal level within 1% for almost all cases. The seemingly higher powers