Manajemen | Fakultas Ekonomi Universitas Maritim Raja Ali Haji 073500102288618874
Journal of Business & Economic Statistics
ISSN: 0735-0015 (Print) 1537-2707 (Online) Journal homepage: http://www.tandfonline.com/loi/ubes20
On Unit-Root Tests When the Alternative Is a
Trend-Break Stationary Process
Amit Sen
To cite this article: Amit Sen (2003) On Unit-Root Tests When the Alternative Is a TrendBreak Stationary Process, Journal of Business & Economic Statistics, 21:1, 174-184, DOI:
10.1198/073500102288618874
To link to this article: http://dx.doi.org/10.1198/073500102288618874
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Date: 13 January 2016, At: 00:39
On Unit-Root Tests When the Alternative
Is a Trend-Break Stationary Process
Amit Sen
Department of Economics & Human Resources, Xavier University, Cincinnati, OH
45207-3212 (sen@xavier.edu)
Downloaded by [Universitas Maritim Raja Ali Haji] at 00:39 13 January 2016
Minimum t statistics to test for a unit-root are available when the form of break under the alternative
evolves according to the crash, changing growth, and mixed models. It is shown that serious power
distortions occur if the form of break is misspecied, and thus the practitioner should use the mixed
model as the appropriate alternative in empirical applications. The mixed model may reveal useful
information regarding the location and form of break. The maximum F statistic for the joint null of a
unit-root and no breaks is shown to have greater and less erratic power compared to the minimum t
statistic. Stronger evidence against the unit-root is found for the Nelson–Plosser series and U.S. Postwar
quarterly real gross national product.
KEY WORDS: Break-date; Form of break; Maximum F statistic; Minimum t statistic.
1.
INTRODUCTION
Perron (1989) demonstrated that the conventional Dickey
and Fuller (1979) t statistic (tDF ) accepts the null hypothesis of a unit-root far too often when the true data-generating
process is in fact trend stationary with a break in the intercept or the slope of the trend function. Three different characterizations of the trend-break alternative were considered:
(a) the crash model, which allows for a break in the intercept;
(b) the changing growth model, which allows for a break in
the slope with the two segments joined at the break-date; and
(c) the mixed model, which allows for a simultaneous break
in the intercept and the slope. To devise unit-root tests that
have power against the trend-break stationary alternative, Perron (1989) proposed the following methodology: First specify the location of the break-date (Tb ), and then estimate a
regression that nests the random walk null and the trend-break
stationary alternative of choice. Under the null hypothesis,
Perron derived the limiting distribution of the t statistic on the
i
4Tb 5, where
rst lag of the dependent variable, denoted by tDF
i D A, B, or C corresponds to the crash, the changing growth,
or the mixed model, respectively. The limiting distribution of
i
tDF
4Tb 5 depends on the location and form of break under the
alternative hypothesis.
The assumption that the location of break is known a priori was criticized by a number of studies, most notably by
Christiano (1992). Also, see Banerjee, Lumsdaine, and Stock
(1992), Zivot and Andrews (1992), and Perron and Vogelsang
(1992). Christiano (1992) argued that the choice of the breakdate is in most cases correlated with the data; for example,
the practitioner may determine the location of the break-date
by visually inspecting a plot of the time series. And because
Perron’s (1989) methodology does not account for this ‘pretest
examination of data,’ the unit-root null will be rejected too
often. A number of studies proposed extensions for unit-root
tests that do not require the practitioner to prespecify the
location of break; see Zivot and Andrews (1992), Banerjee
et al. (1992), Perron and Vogelsang (1992), Perron (1997),
and Vogelsang and Perron (1998). The strategy adopted by
these studies is to apply Perron’s (1989) methodology for each
possible break-date in the sample, which yields a sequence
of t statistics. Based on this sequence, numerous minimum
t statistics can be constructed by choosing the t statistic, based
on some algorithm, that maximizes evidence against the null
hypothesis. For example, one may simply use the minimum of
min
4i51 i D A, B, C.
the sequence of t statistics, denoted by tDF
With the availability of the minimum t statistics, the practitioner no longer needs to prespecify the location of break.
However, one must still specify the form of break under the
alternative hypothesis. If one assumes that the location of
break is unknown, it is most likely that the form of break will
be unknown as well. It is argued here that selection of the form
of break is also correlated with the data and so one must proceed with the break specication according to the most general
mixed model. In addition, one may expect power distortions
if the form of break is misspecied, that is, if one imposes
the crash (changing growth) model when in fact the changing
growth (crash) or the mixed model is appropriate. Consider,
for example, the use of the crash model specication under
the alternative hypothesis when the break occurs according to
the changing growth or the mixed model. The crash model
min
4A5 is designed to maximize evidence
minimum t statistic tDF
against the null hypothesis in favor of this particular alternative. If the true data-generating process is given by the mixed
min
4A5 can be expected to be lower
model, the power of tDF
min
than that of tDF 4C5, especially if the magnitude of the slope
min
4A5 may be
break is relatively large. Even lower power of tDF
expected if the alternative evolves according to the changing
growth model. On the other hand, if the crash hypothesis is
the correct specication, then its use will yield superior power
compared to the mixed model statistics. In practice, however,
one is uncertain about the form of break. Because one would
like to guard against distortions in power owing to misspecication of the form of break, it is recommended that the practitioner use the mixed model specication under the alternative
hypothesis.
The rst objective is to assess the performance of minimum
t statistics when the form of break is misspecied. To this
end, simulation evidence is provided (a) on the performance
174
© 2003 American Statistical Association
Journal of Business & Economic Statistics
January 2003, Vol. 21, No. 1
DOI 10.1198/073500102288618874
Downloaded by [Universitas Maritim Raja Ali Haji] at 00:39 13 January 2016
Sen: Unit-Root Tests
of the crash model statistics when the break occurs according
to either the changing growth or the mixed model; (b) on the
performance of the changing growth model statistics when the
break occurs according to either the crash or the mixed model;
and (c) to assess the loss in power of the mixed model statistics when the break occurs according to either the crash or the
changing growth model. It is found that the loss in power is
quite small if the mixed model specication is used when in
fact the break occurs according to the crash model. The loss
in power is more substantial if the break occurs according to
the changing growth model. However, serious distortions can
occur if the crash (changing growth) model is used when the
appropriate model is either the changing growth (crash) model
or the mixed model. Therefore, the results indicate that one
should use the form of the break specied under the mixed
model, unless prior information about the nature of the break
suggests using either the crash model or the changing growth
model. In most cases, it is expected that information on the
form of break will be accompanied with information on the
location of break, in which case Perron’s (1989) tests should
be used. Also studied are the power properties the supWald (or
equivalently the maximum F statistic, denoted by FTmax ) proposed by Murray (1998) and Murray and Zivot (1998) for the
joint null hypothesis of a unit-root and no break in the intercept and slope of the trend function. Simulation results show
that the power of FTmax is less erratic and can be greater than
the mixed model minimum t statistics.
Second, the arguments are illustrated within the context
of an empirical example that has received much attention in
the literature. Numerous studies have used the Nelson and
Plosser (1982) data and postwar quarterly real gross national
product (GNP) to test for the presence of a unit-root. Perron (1989) specied the great crash of 1929 as the breakdate for all Nelson–Plosser series and the oil price shock of
1973 for quarterly real GNP. Conditional on these break-dates,
the crash model was specied for all Nelson–Plosser series
except common stock prices and real wages, for which the
mixed model was used, and the changing growth model was
used for quarterly real GNP. Whereas subsequent studies by
Zivot and Andrews (1992), Perron (1997), Nunes, Newbold,
and Kuan (1997), and Murray and Nelson (2000) allowed for
an unknown break-date, they retained the model specication
proposed by Perron (1989). Unlike in these studies, empirical
evidence is presented here for the presence of a unit-root in all
series when the alternative allows for a simultaneous break in
the intercept and slope. It is found that the unit-root null can be
rejected for all series with the exception of the GNP deator,
consumer prices, velocity, and interest rate series. The results
are robust to possible misspecication of the form of break
and therefore reveal useful information regarding the location
and form of break. For instance, the empirical evidence of
Zivot and Andrews (1992) indicates that real per capita GNP
is characterized as a stationary process with break in the intercept occurring in 1929, and both money stock and quarterly
real GNP contain a unit-root. By using the mixed model as
the appropriate alternative, evidence is strengthened against a
unit-root in real per capita GNP, and evidence is uncovered
against the unit-root for money stock and quarterly real GNP.
The break-date is estimated for real per capita GNP in 1938,
175
for money stock in 1930, and for quarterly real GNP in 1964,
fourth quarter. The estimated break-dates for these series are
different from the estimated break-dates reported by Zivot and
Andrews (1992).
The outline of this article is as follows. Section 2 discusses
the minimum t statistics and the maximum F statistic that
have been proposed in the literature. In Section 3, simulation
evidence is presented that enables one to study the nite sample size and power properties of all statistics when the form
of break may be misspecied. Empirical evidence for the Nelson and Plosser (1982) data and U.S. postwar quarterly real
GNP is presented in Section 4. Concluding comments appear
in Section 5.
2.
TEST STATISTICS AND METHODOLOGY
We begin with a brief discussion of the null and alternative hypotheses of interest, the class of minimum t statistics,
and the maximum F statistic. The discussion follows the analysis by Zivot and Andrews (1992). Consider the time series
8yt 9TtD1 , where T is the available sample size. The three different characterizations of the alternative hypothesis, originally
discussed by Perron (1989), are given by
Model (A)2 yt D Œ0 C Œ1 DUt 4Tbc 5 C Œ2 t C et 1
(1)
yt D Œ0 C Œ2 t C Œ3 DTt 4Tbc 5 C et 1
(2)
Model (B)2
Model (C)2
yt D Œ0 C Œ1 DUt 4Tbc 5 C Œ2 t C Œ3 DTt 4Tbc 5 C et 1
(3)
where Tbc is the correct break-date, DUt 4Tbc 5 D 14t>Tbc 5 ,
DTt 4Tbc 5 D 4t ƒ Tbc 514t>Tbc 5 , and 14t>Tbc 5 is an indicator function that takes on the value 0 if t µ Tbc and 1 if t > Tbc .
For the asymptotic results, it is assumed that the break-date
is a constant fraction of the sample size, that is, Tbc D ‹c T
with the correct break-fraction ‹c 2 401 15. It is assumed that
A4L5et D B4L5t , and t is a sequence of iid 401‘ 2 5 random
variables, A(L) and B(L) are polynomials in the lag operator of order p C 1 and q, respectively, with all roots outside
the unit circle. Model (A) is referred to as the crash model
because it allows for a break in the intercept alone, Model (B)
is referred to as the changing growth model because it allows
for a break in the slope with the two segments joined at the
break-date, and Model (C) is referred to as the mixed model
because it allows for a simultaneous break in the intercept and
the slope of the trend function.
The data-generating process given in (1)–(3), known as the
additive outlier (AO) model, incorporates a break in the trend
function that occurs instantaneously. This article considers the
innovation outlier (IO) model, wherein the change in the trend
function evolves in the same manner as any other shock; see
section 4.2 in the work by Perron (1989). That is,
Model (A)2 yt D Œ0 C Œ2 t C –4L56Œ1 DUt 4Tbc 5 C t 71 (4)
Model (B)2 yt D Œ0 C Œ2 t C –4L56Œ3 DTt 4Tbc 5 C t 71 (5)
Model (C)2 yt D Œ0 C Œ2 t C –4L5
6Œ1 DUt 4Tbc 5 C Œ3 DTt 4Tbc 5 C t 71
(6)
where –4L5 D A4L5ƒ1 B4L5. The impact of the break is different in the AO and IO models. Consider, for example, the
176
Journal of Business & Economic Statistics, January 2003
crash model specication of the break. In the AO characterization, the magnitude of the break is Œ1 . However, in the IO
characterization the immediate impact of the break is equal to
Œ1 , but the long-run impact is equal to –415Œ1 .
Under the null hypothesis, the data-generating process contains a unit-root, that is,
ü
yt D Œ C ytƒ1 C – 4L5t 1
(7)
Downloaded by [Universitas Maritim Raja Ali Haji] at 00:39 13 January 2016
where – ü 4L5 D Aü 4L5ƒ1 B4L51 A4L5 D 41 ƒ L5Aü 4L5, and
D 1. To test for a unit-root in the IO framework, the following methodology has been suggested. Specify an interval
å D 6‹0 1 1 ƒ ‹0 7 401 15 that is believed to contain the true
break-fraction. For each possible ‹ 2 å, estimate the following
regression that nests the null and the appropriate alternative:
O A0 C Œ
O A1 DUt 46‹T 75 C Œ
O A2 t C O A ytƒ1
yt D Œ
C
k
X
cO jA ãytƒj C eO tA 1
(8)
cO jB ãytƒj C eO tB 1
(9)
jD1
O B0 C Œ
O B2 t C Œ
O B3 DTt 46‹T 75 C O B ytƒ1
yt D Œ
C
k
X
jD1
O C0 C Œ
O C1 DUt 46‹T 75 C Œ
O C2 t C Œ
O C3 DTt 46‹T 75
yt D Œ
C O C ytƒ1 C
k
X
cO jC ãytƒj C eO tC 1
(10)
jD1
where [.] is the smallest integer function. The additional k
regressors 8ãytƒj 9kjD1 are included in the regression to eliminate the correlation in the disturbance term. Typically, the
value of the lag-truncation parameter (k) is unknown, and so
a data-dependent method for choosing the appropriate value
of k is used; see Perron and Vogelsang (1992), Hall (1992,
1994), Perron (1997), and Ng and Perron (1995). We use Perron and Vogelsang’s (1992) data-dependent method k4t-sig)
for selecting the lag-truncation parameter, which is described
in what follows. Specify an upper bound k-max for the
lag-truncation parameter. For each break-date 6‹T 7, the chosen value of the lag-truncation parameter (k ü ) is determined
according to the following general to specic procedure: The
last lag in an autoregression of order k ü is signicant, but
the last lag in an autoregression of order greater than k ü is
insignicant. The signicance of the coefcient is assessed
using the 10% critical values based on a standard normal distribution. Regressions (8)–(10) are estimated for the breakdates 86‹ 0 T 71 6‹0 T 7 C 11 : : : 1 T ƒ 6‹0 T 79, and the sequence
T ƒ6‹ T 7
i
4Tb 59Tb D6‹00 T 7
of t statistics for H0 2 D 1, denoted by 8tDF
(i D A, B, C) is calculated. Based on this sequence, a number
of minimum t statistics can be obtained by using an algorithm to choose the appropriate break-date that maximizes evidence against Equation (7); see Perron and Vogelsang (1992),
Banerjee et al. (1992), Perron (1997), and Vogelsang and
Perron (1998).
We consider two particular algorithms for choosing the
break-date. The rst statistic, originally proposed by Perron
and Vogelsang (1992) and Zivot and Andrews (1992), is
obtained by choosing the break-date that maximizes evidence
against the unit-root null, that is,
i
min
tDF
4Tb 5
4i5 D MinTb 286‹ 0 T 716‹ 0 T 7C11 : : : 1T ƒ6‹0 T 79 tDF
(11)
min
4i5 can be
for i D A, B, C. Perron (1997) demonstrated that tDF
calculated with å D 401 15 so that no trimming of the sample
is necessary. The second statistic, proposed by Banerjee et al.
(1992) and Vogelsang and Perron (1998), is dened as
O 51
tODF 4i5 D tDF 4‹
i
(12)
O (i D A, B, C) is the break-date that maximizes the
where ‹
i
Wald statistic, FTi 46‹T 75, corresponding to H0 2 Œ1 D 0 for the
crash model (i D A), H0 2 Œ3 D 0 for the changing growth
model (i D B), and H0 2 Œ1 D Œ3 D 0 for the mixed model
(i D C).
min
4i5 and tODF 4i5 (i D A, B, C),
The minimum t statistics, tDF
do not require specication of the location of break. However,
the practitioner does need to specify the form of break under
the alternative hypothesis. Once the break-date is treated as
unknown, the practitioner will in most cases be unaware of the
form of break. Because the crash and changing growth models
are special cases of the mixed model, the practitioner should
specify the mixed model as the appropriate alternative. Some
loss in power can be expected from using the statistics from
the mixed model when in fact the break occurs according to
the crash or changing growth model. However, the behavior
of the statistics from the crash (changing growth) model when
the break occurs according to the changing growth (crash) or
mixed model is uncertain. The next section presents simulation evidence regarding the power properties of the minimum
t statistics when the form of break is misspecied.
Next, we briey describe the supWald statistic proposed
by Murray (1998) and Murray and Zivot (1998) for the joint
null hypothesis of a unit-root and no break in the intercept
and slope of the trend function. As before, it is assumed that
the break-fraction lies in å D 6‹0 1 1 ƒ ‹0 7. For each possible break-date Tb 2 86‹ 0 T 71 6‹ 0 T 7 C 11 : : : 1 T ƒ 6‹0 T 79, the F
statistic, denoted by FT 4Tb 5 from regression (10) that corresponds to the joint null hypothesis H 0J 2 D 11 Œ1 D 0, Œ3 D 0
is calculated as
FT 4Tb 5 D
iƒ1
h ¡P
¢
T
0 ƒ1
O b 5 ƒ r5
O b 5 ƒ r5 R
4R Œ4T
R0
4RŒ4T
tD1 xt 4Tb 5xt 4Tb 5
q‘O 2 4Tb 5
1
(13)
O b 5 is the ordinary least squares (OLS) estimator
where Œ4T
of Œ D 4Œ0 1 Œ1 1 Œ2 1 Œ3 1 1 c1 1 : : : 1 ck 50 , xt 4Tb 5 D 411 DUt 4Tb 5,
t1 DTt 4Tb 51 ytƒ1 1 ãytƒ1 1 : : : 1 ãytƒk 50 , r D 401 01 150 , Pq is the
number of restrictions, ‘O 2 4Tb 5 D 4T ƒ 5 ƒ k5ƒ1 TtD1 4yt ƒ
O b 552 , and R is dened so that RΠD r corresponds
xt 4Tb 50 Œ4T
to the restrictions imposed on the parameter vector Πby the
joint null H0J . Specically,
3
2
0 1 0 0 0 00k
R D 40 0 0 1 0 00k 5 0
0 0 0 0 1 00k
Sen: Unit-Root Tests
177
Table 1. Critical Values for FTmax With å D [ ‹0 , 1 - ‹0 ]
1%
Downloaded by [Universitas Maritim Raja Ali Haji] at 00:39 13 January 2016
5%
10%
‹0 D 015
T D 50
k D0
k(t-sig)
1301238
1500101
1106071
1302783
1005840
1202464
904661
1008523
T D 100
k D0
k(t-sig)
1200157
1300842
1008744
1108875
1000248
1007809
900628
907820
T D 150
k D0
k(t-sig)
1108505
1208437
1007593
1102872
909333
1003951
809723
904935
T D 200
k D0
k(t-sig)
1106450
1206811
1004785
1100691
906730
1003038
808708
903692
1001691
904376
806958
T Dˆ
1009288
‹0 D 010
T D 50
k D0
k(t-sig)
1301646
1500177
1106795
1303184
1006061
1202799
905237
1009242
T D 100
k D0
k(t-sig)
1201731
1300842
1009391
1108875
1000970
1008409
901282
908127
T D 150
k D0
k(t-sig)
1108765
1208440
1008124
1102896
909692
1004511
900420
905204
T D 200
k D0
k(t-sig)
1107058
1207662
1005348
1101565
907310
1003460
809164
903941
1009841
1002152
904931
807353
T Dˆ
‹0 D 005
T D 50
k D0
k(t-sig)
1302127
1501077
1107261
1303226
1006409
1202862
905796
1009674
T D 100
k D0
k(t-sig)
1201958
1300842
1009940
1109215
1001425
1008752
901871
908674
T D 150
k D0
k(t-sig)
1109328
1208440
1008546
1102896
1000083
1004922
900895
905456
T D 200
k D0
k(t-sig)
1107236
1207662
1005892
1101565
907763
1004089
809535
904368
1100364
1002414
905427
807946
T Dˆ
T ƒ6‹ T 7
The sequence of F statistics 8FT 4Tb 59Tb D6‹00 T 7 is used to calculate the maximum F statistic for H 0J as
FTmax D MaxTb 286‹ 0 T 716‹ 0 T 7C11 : : : 1T ƒ6‹0 T 79 FT 4Tb 50
(14)
The limiting distribution of FTmax under H0J is easily obtained
from theorem 3.2F by Murray and Zivot (1998). Note that in
this article the F statistic is used for H0J rather than the Wald
statistic considered by Murray and Zivot (1998). Therefore,
the critical values of the supWald statistic and FTmax will differ
by a multiplicative constant equal to q D 3. Asymptotic critical values for the supWald statistic (without any trimming of
the sample) are given by table 6 in Murray and Zivot (1998).
Asymptotic and nite sample critical values for the limiting
null distribution of FTmax are given in Table 1 when the trimming parameter ‹0 D 0151 0101 005.
3.
2.5%
FINITE SAMPLE SIZE AND POWER
In this section, we present nite sample size and power
results for the minimum t statistics and the FTmax statistic. The
differences in the power of FTmax and the minimum t statistics
min
4C5 and tODF 4C5, are noted.
from the mixed model, namely, tDF
We follow the experimental design of Vogelsang and Perron
(1998) and generate data according to
61 ƒ 4 C 5L C L2 7yt D 41 C –L56Œ1 DUtc C Œ3 DTtc C et 71
t D 11 21 : : : 1 T 0
(15)
We set D 1 for the size simulations and D 008 for
the power simulations. The sample size for all simulations
is T D 100, the correct break-date is Tbc D 50, DUtc D
DUt 4Tbc 5, DTtc D DTt 4Tbc 5, et ¹ iid N(0,1), and y0 D
yƒ1 D 0. The following combinations of (1 –) are used:
8401 053 4061 053 4ƒ061 053 401 0553 401 ƒ0559. For the size simulations, set Œ1 D Œ3 D 0. For the power simulations, we consider the data-generating process under (a) the crash model
with a break in the intercept for Œ1 D 811 21 41 61 89; (b) the
changing growth model with a break in the slope for Œ3 D
8011 021 031 041 059; and (c) the mixed model with a simultaneous break in the intercept and slope for Œ1 D 811 21 41 69 and
Œ3 D 8011 021 031 049. For each parameter combination 2,000
replications were generated. We calculate the size and power
of following statistics at the 5% signicance level using the
min
4i5 and OtDF 4i5
appropriate nite sample critical values: tDF
178
Journal of Business & Economic Statistics, January 2003
min
( i ) , tODF ( i ) for i D A, B, C, and FTmax
Table 2. Finite Sample Size of tDF
–
min
(A)
tDF
tODF (A)
min
(B)
tDF
Ot (B)
DF
min
(C)
tDF
tODF (C)
FTmax
.0
.6
ƒ.6
.0
.0
.0
.0
.0
.5
ƒ.5
.0420
.0550
.0405
.0640
.3160
.0405
.0505
.0390
.0585
.2815
.0495
.0605
.0530
.0785
.3180
.0530
.0615
.0545
.0790
.3080
.0585
.0580
.0540
.0845
.4320
.0535
.0545
.0520
.0800
.4030
.0525
.0695
.0510
.0865
.4125
Downloaded by [Universitas Maritim Raja Ali Haji] at 00:39 13 January 2016
NOTE:
5% nominal size, DGP ( D 1): yt D ( C )yt ƒ1 ƒ yt ƒ2 C (1 C –L) e t .
for i D A, B, C, and FTmax . We use ‹0 D 015 for calculating OtDF 4i5 and FTmax . The lag-truncation parameter (k) in the
regressions (8)–(10) is chosen according to the k4t-sig5 procedure mentioned in Section 2 with k-max D 5. Sen (2001)
presented additional simulation evidence in which the lagtruncation parameter is set equal to its correct value; see his
tables 1–3. The nite sample size for all statistics is presented
in Table 2, and the power of all statistics under the crash
model, the changing growth model, and the mixed model is
given in Tables 3, 4, and 5, respectively. The following interesting features emerge from the simulations:
1. The exact size of all statistics is fairly close to the nominal size (see Table 1), except when there is a negative moving
average component, that is, (1 –5 D 401 ƒ055.
2. When the break occurs according to the crash model
min
4A5 and tODF 4A5 have the largest power. There is
(Table 3), tDF
very little loss in power from using the mixed-model statistics,
min
tDF
4C5, OtDF 4C5, and FTmax . The power of the changing growth
min
4B5 and tODF 4B5, is close to zero except
model statistics, tDF
for small intercept breaks (Œ1 ).
3. When the break occurs according to the changing growth
min
4B5 and OtDF 4B5 have the largest power.
model (Table 4), tDF
There is some loss in power from using the mixed model
statistics, but this loss in power diminishes with the size of
the slope break magnitude. In all cases, the power of the crash
min
4A5 and OtDF 4A5, is close to zero.
model statistics, tDF
4. When the break occurs according to the mixed model
(Table 5), the power of the mixed-model statistics, in general,
increases with the size of the intercept break (Œ1 ) and slope
break (Œ3 ) magnitudes. For a xed Œ3 , the power of the crash
model statistics increases with Œ1 , and the power of the changing growth model statistics decreases with Œ1 . For a xed Œ1 ,
min
( i ) , tODF ( i ) for i D A, B, C, and FTmax : True DGP is Crash Model
Table 3. Finite Sample Power of tDF
Œ1
min
tDF
(A)
tODF (A)
min
t DF
(C)
tODF (C)
FTmax
1
2
4
6
8
02380
03660
09825
100000
100000
02440
03800
09815
100000
100000
D 081 D 001 – D 00
.0740
.0725
.0010
.0010
.0000
.0000
.0000
.0000
.0000
.0000
01995
02305
07875
09970
100000
01885
02270
07860
09945
100000
01560
03640
09850
100000
100000
1
2
4
6
8
08030
08800
09985
100000
100000
07655
08690
09985
100000
100000
D 081 D 061 – D 00
.5695
.5510
.1965
.1955
.0060
.0075
.0000
.0000
.0000
.0000
07620
08200
09825
100000
100000
07150
07920
09790
100000
100000
06875
07870
09915
100000
100000
1
2
4
6
8
00780
02100
09685
100000
100000
00825
02100
09545
100000
100000
D 081 D ƒ061 – D 00
.0205
.0215
.0005
.0005
.0000
.0000
.0000
.0000
.0000
.0000
00670
00585
04515
09530
100000
00640
00575
04355
09305
09975
00770
03440
09940
100000
100000
1
2
4
6
8
02245
03710
09820
100000
100000
02185
03655
09810
100000
100000
D 081 D 001 – D 05
.0750
.0730
.0015
.0010
.0000
.0000
.0000
.0000
.0000
.0000
02080
02280
07795
09975
100000
01965
02160
07790
09970
100000
01740
03755
09850
100000
100000
1
2
4
6
8
07265
07215
09820
100000
100000
06935
06775
09810
100000
100000
D 081 D 001 – D ƒ05
.4345
.4145
.0770
.0755
.0000
.0000
.0000
.0000
.0000
.0000
07580
06840
08685
09965
100000
07065
06275
08525
09960
100000
07325
07060
09830
09995
100000
NOTE:
min
tDF
(B)
tODF (B)
5% nominal size, crash model DGP: yt D ( C )yt ƒ1 ƒ ytƒ2 C (1 C – L)[Œ1 DU tc C e t ].
Sen: Unit-Root Tests
179
Downloaded by [Universitas Maritim Raja Ali Haji] at 00:39 13 January 2016
min
( i ) , tODF ( i ) for i D A, B, C, and FTmax : True DGP is
Table 4. Finite Sample Power of tDF
Changing Growth Model
Œ3
min
tDF
(A)
tODF (A)
min
tDF
(B)
.1
.2
.3
.4
.5
.0000
.0000
.0000
.0000
.0000
.0000
.0000
.0000
.0000
.0000
.1
.2
.3
.4
.5
.1010
.0000
.0000
.0000
.0000
.1
.2
.3
.4
.5
min
tDF
(C)
tODF (C)
FTmax
D 081 D 001 – D 00
.3110
.3340
.5150
.5400
.7470
.7680
.9365
.9445
.9915
.9930
.1870
.2120
.3153
.5190
.7385
.1720
.1720
.2390
.3475
.4495
.1440
.2350
.4400
.7655
.9450
.1085
.0000
.0000
.0000
.0000
D 081 D 061 – D 00
.8225
.8315
.8625
.8765
.9225
.9305
.9675
.9725
.9880
.9930
.7215
.7030
.7430
.7845
.8190
.7040
.6740
.6815
.7020
.7000
.6265
.6360
.7165
.7925
.8790
.0000
.0000
.0000
.0000
.0000
.0000
.0000
.0000
.0000
.0000
D 081 D ƒ061 – D 00
.1985
.2105
.4055
.4215
.7940
.8000
.9800
.9780
.9990
.9955
.0680
.1085
.3220
.6755
.9165
.0590
.0825
.2105
.4380
.6150
.0775
.2350
.7140
.9600
.9870
.1
.2
.3
.4
.5
.0010
.0000
.0000
.0000
.0000
.0010
.0000
.0000
.0000
.0000
D 081 D 001 – D 05
.3155
.3415
.5375
.5625
.7450
.7660
.9360
.9430
.9910
.9910
.1860
.2220
.3160
.5040
.7230
.1725
.1840
.2340
.3310
.4610
.1555
.2570
.4495
.7200
.9045
.1
.2
.3
.4
.5
.0100
.0000
.0000
.0000
.0000
.0095
.0000
.0000
.0000
.0000
D 081 D 001 – D ƒ05
.7600
.7700
.8125
.8205
.8940
.9015
.9565
.9630
.9920
.9925
.7210
.7110
.7000
.7460
.8095
.7215
.7050
.6840
.7025
.7035
.7060
.7000
.7280
.8150
.9245
NOTE:
tODF (B)
5% nominal size, changing growth model DGP: yt D ( C )yt ƒ1 ƒ yt ƒ2 C (1 C –L)[Œ3 DTtc C e t ].
min
( i ) , OtDF ( i ) for i D A, B, C, and FTmax : True DGP is Mixed Model
Table 5. Finite Sample Power of tDF
Œ1
Œ3
min
tDF
(A)
tODF (A)
tODF (B)
min
tDF
(C)
1
2
4
6
.1
.0005
.0015
.2545
.9920
.0005
.0015
.2615
.9805
– D 00
.3345
.0845
.0000
.0000
02115
01770
06180
09930
01900
01640
06175
09915
02100
04350
09890
100000
1
2
4
6
.2
.0000
.0000
.0110
.5835
.0000
.0000
.0115
.5720
.6450
.5835
.0380
.0000
.6720
.6130
.0455
.0000
03100
02735
05065
09645
02335
01525
04995
09605
03650
06625
09960
100000
1
2
4
6
.3
.0000
.0000
.0005
.1360
.0000
.0000
.0005
.1350
.8705
.9480
.4920
.0250
.8820
.9545
.5290
.0320
04750
05885
04235
09245
03325
02540
03610
09190
06075
08525
09995
100000
1
2
4
6
.4
.0000
.0000
.0005
.0330
.0000
.0000
.0000
.0350
.9740
.9950
.9600
.3915
.9770
.9970
.9695
.4430
06530
08010
05945
08350
03930
02950
02545
08285
08595
09660
09985
100000
1
2
4
6
.1
.1025
.0635
.3505
.9905
.1110
.0690
.3610
.9905
– D 00
.7945
.5185
.0890
.0045
07480
08035
09720
100000
07295
07875
09630
100000
07015
08150
09905
100000
1
2
4
6
.2
.0000
.0005
.0025
.3135
.0000
.0010
.0030
.3505
.9015
.8040
.2775
.0590
07695
08000
09650
09985
07240
07645
09585
09985
07375
08625
09960
100000
min
tDF
(B)
D 11 D 001
.3070
.0760
.0000
.0000
D 11 D 061
.7845
.4970
.0825
.0040
.8875
.7855
.2525
.0500
tODF (C)
FTmax
(continued)
180
Journal of Business & Economic Statistics, January 2003
Downloaded by [Universitas Maritim Raja Ali Haji] at 00:39 13 January 2016
Table 5 (continued)
Œ1
Œ3
min
tDF
(A)
tODF (A)
min
tDF
(B)
tODF (B)
min
tDF
(C)
1
2
4
6
.3
.0000
.0000
.0005
.0110
.0000
.0000
.0005
.0150
.9565
.9470
.5760
.2205
.9610
.9550
.6085
.2375
08005
08390
09655
09995
07140
07605
09510
09995
07980
09205
09985
100000
1
2
4
6
.4
.0000
.0000
.0000
.0005
.0000
.0000
.0000
.0005
.9885
.9900
.8835
.5025
.9895
.9915
.9065
.5360
08500
08935
09685
09990
07155
07540
09415
09990
08860
09615
09995
100000
1
2
4
6
.1
.0010
.0005
.2660
.9910
.0010
.0000
.2365
.9675
– D 00
.2935
.1545
.0000
.0000
01265
00740
02205
08315
01105
00545
02155
08190
01465
04850
09950
100000
1
2
4
6
.2
.0000
.0000
.0130
.6400
.0000
.0000
.0075
.5860
.6715
.8605
.2205
.0035
.6930
.8745
.2525
.0040
02700
04295
01395
06130
02025
01530
01050
05965
04265
07825
09980
100000
1
2
4
6
.3
.0000
.0000
.0000
.1585
.0000
.0000
.0000
.1205
.9145
.9765
.9515
.2765
.9245
.9795
.9580
.3170
05390
07645
05210
03380
03500
02760
00460
03245
08140
09455
09995
100000
1
2
4
6
.4
.0000
.0000
.0000
.0075
.0000
.0000
.0000
.0050
.9865
.9980
.9995
.9555
.9870
.9980
.9945
.9535
08080
09215
09295
04935
04850
03870
00625
01620
09625
09930
100000
100000
1
2
4
6
.1
.0000
.0010
.2775
.9920
.0010
.0010
.2920
.9890
– D 05
.3530
.1240
.0005
.0000
02220
02065
06005
09805
02010
01875
05900
09805
02405
04995
09860
100000
1
2
4
6
.2
.0000
.0000
.0180
.6055
.0000
.0000
.0165
.6355
.6875
.6685
.0500
.0005
.7145
.6965
.0620
.0015
03720
03495
04805
09530
02855
01865
04640
09505
04170
06750
09940
100000
1
2
4
6
.3
.0000
.0000
.0025
.1820
.0000
.0000
.0015
.1965
.8910
.9635
.6265
.0530
.8995
.9700
.6620
.0690
04930
06435
04140
08820
03300
02905
03190
08810
06260
08640
09965
100000
1
2
4
6
.4
.0000
.0000
.0000
.0430
.0000
.0000
.0000
.0470
.9735
.9915
.9795
.5100
.9785
.9920
.9845
.5610
06630
08160
06635
07755
04255
03515
02335
07605
07200
08395
09405
100000
1
2
4
6
.1
.0075
.0015
.1755
.9780
.0085
.0015
.1950
.9810
D 11 D 001 – D ƒ05
.7275
.7385
.4005
.4110
.0030
.0030
.0000
.0000
07410
07120
07960
09890
07325
06880
07680
09890
07265
07785
09845
100000
1
2
4
6
.2
.0000
.0000
.0030
.3810
.0000
.0000
.0025
.4205
.8565
.7580
.0770
.0000
.8660
.7755
.0840
.0000
07565
07450
07500
09860
07405
07125
06945
09855
07575
08375
09935
100000
1
2
4
6
.3
.0000
.0000
.0000
.0650
.0000
.0000
.0000
.0745
.9365
.9635
.4025
.0035
.9445
.9705
.4405
.0045
07760
08380
07205
09640
07330
07425
06240
09640
07950
09250
09975
100000
1
2
4
6
.4
.0000
.0000
.0000
.0075
.0000
.0000
.0000
.0080
.9840
.9925
.9030
.1200
.9855
.9940
.9175
.1450
08250
09000
07665
09210
07555
07680
05975
09185
08925
09680
09995
100000
NOTE:
D 11 D ƒ061
.2710
.1365
.0000
.0000
D 11 D 001
.3285
.1120
.0005
.0000
5% nominal size, mixed model DGP: yt D ( C )yt ƒ1 ƒ yt ƒ2 C (1 C – L)[Œ 1 DU tc C Œ3 DTtc C e t ].
tODF (C)
FTmax
Downloaded by [Universitas Maritim Raja Ali Haji] at 00:39 13 January 2016
Sen: Unit-Root Tests
the power of the crash model statistics decreases with Œ3 and
the power of the changing growth statistics increases with Œ3 .
min
min
4B5 and tODF 4B5] can fail
4A5 and OtDF 4A5 [tDF
Therefore, tDF
to reject the unit-root null if the true data-generating process
is given by the mixed model with a relatively large Œ3 (Œ1 ).
5. When the break occurs according to mixed model, the
min
4C5 and OtDF 4C5 is sometimes erratic in that
behavior of tDF
their power does not always increase with the size of Œ1 and
min
4C5 and
Œ3 , except when 41 –5 D 401 065. The power of tDF
OtDF 4C5 tends to fall slightly with the size of Œ3 when Œ1 is
min
4C5 and OtDF 4C5 is even more erratic
large. The behavior of tDF
when Œ3 is xed and Œ1 increases. On the other hand, FTmax
has the desirable property in that its power always increases
with the size of Œ1 and Œ3 .
6. In all cases, the power of FTmax is greater than the power
min
4C5 and tODF 4C5 except when the magnitude of intercept
of tDF
break or slope break is relatively small. In these cases, the use
of FTmax does not result in large loss of power.
Given that the power of FTmax is larger than the mixed-model
min
4C5 and OtDF 4C5], it would to be interesting to
statistics [tDF
ascertain how much of the power of FTmax can be attributed to
the break in the trend, that is, Œ1 6D0 or Œ3 6D 0 or both. The
size of all statistics under the unit-root null hypothesis when
there is a break in the intercept and/or slope was calculated.
The results, available on request, indicate that the size of all
statistics is close to the nominal size when the break magnitudes are small. However, the size of FTmax increases with
the break magnitudes. In all cases, the size of FTmax is slightly
min
4C5 and tODF 4C5, and this difference
larger than the size of tDF
in size increases slightly with the break magnitudes.
Simulation evidence presented by Vogelsang and Perron
(1998) illustrated that all statistics considered within the IO
model framework retain their size and power properties if the
break in the trend function occurs according to the AO model.
While the break in the trend is abrupt and instantaneous within
the AO framework, the break in the trend is more gradual
within the IO framework. In unreported simulations, we calculated the nite sample size and power of all IO statistics
when the break occurs according to the AO model. Our results
indicate that, in general, the size and power properties of all
statistics remain the same whether the break occurs according
to the AO model or the IO model. However, in most cases, the
min
4C5, tODF 4C5, and FTmax is larger when the break
power of tDF
occurs according to the AO model compared to their power
when the break occurs according to the IO model. The only
exception occurs in the changing growth model simulation,
where the power of OtDF 4C5 levels off with an increase in the
size of the slope break. The AO model simulation results are
available from the author on request.
4.
EMPIRICAL RESULTS FOR NELSON
AND PLOSSER DATA
In this section, we revaluate the evidence regarding the presence of a unit-root in all Nelson and Plosser (1982) series,
except unemployment, and postwar U.S. quarterly real GNP.
The natural logarithms of all series were analyzed, except for
the interest rate series, which was analyzed in level form.
181
Empirical evidence for the Nelson–Plosser data can be found
in Zivot and Andrews (1992), Perron (1997), Nunes et al.
(1997), Murray and Zivot (1998), and Murray and Nelson
(2000). Although these studies do not prespecify the location
of break, they retain Perron’s (1989) characterization of the
form of break under the alternative hypothesis. In particular,
Perron (1989) specied the changing growth model for quarterly real GNP, the mixed model for real wages and common
stock prices, and the crash model for all remaining series.
However, Perron’s (1989) specication of the form of break
is based on his selection of the great crash of 1929 for all
Nelson–Plosser series and the oil price shock of 1973 for quarterly real GNP series as appropriate break-dates. We argue that
once the location of break is treated as unknown, the form of
break should also be treated as unknown. Therefore, one must
proceed with the most general characterization of the break
under the alternative hypothesis, namely, a simultaneous break
in the intercept and slope.
Empirical evidence for all Nelson and Plosser (1982) series
and quarterly real GNP when the mixed model is used as the
appropriate alternative is presented. The k4t-sig5 method of
Perron and Vogelsang (1992) is used to determine the appropriate order of the lag-truncation parameter in regression (10).
Following Zivot and Andrews (1992), k-max D 8 is used for
the Nelson–Plosser (1982) series and k-max D 12 for quarmin
4C5,
terly real GNP. Table 6 reports the calculated statistics tDF
OtDF 4C5, and FTmax for each series, and the corresponding estimin
bb 4OtDF 5, and
bb 4tDF
51 T
mated break-dates which are denoted by T
max
b
Tb 4FT 5, respectively. The results for regression (10) corresponding to the different estimated break-dates are presented
in Table 7. [The results for common stock prices and real
wages given in table 6 of Zivot and Andrews (1992) were
found to be incorrect. The estimated regression results for real
wages with Tb D 1940 and k D 8 yield the results reported
by Zivot and Andrews (1992), but the t statistic on the ãytƒ8
was found to be 1.2277, which is less than 1.6. Although
the estimated regression results for common stock prices with
Tb D 1936 and k D 1 yield the results reported by Zivot and
Andrews (1992), it was found that with k D 3 the t statistic
on ãytƒ3 is 1.96, which is greater than 1.6. Therefore, in both
cases the k4t-sig5 procedure for choosing the lag-truncation
parameter was not applied correctly.]
min
min
bb 4tDF
4C5 and T
5. If the
First, consider the results for tDF
min
4C5 given by Zivot
asymptotic critical values are used for tDF
and Andrews (1992), the unit-root null is rejected at the 1%
signicance level for real GNP, nominal GNP, and industrial production; at the 2.5% signicance level for real per
capita GNP, common stock prices, and real wages; at the 5%
signicance level for employment, nominal wages, and quarterly real GNP; and at the 10% signicance level for money
stock. The unit-root null for the GNP deator, consumer
prices, velocity, and interest rate series were not rejected. Of
the series for which the unit-root null is rejected, the estimin
bb 4tDF
5 does not coincide with the great
mated break-date T
crash (1929) for real per capita GNP, money stock, common
stock prices, and real wages. In addition, the estimated breakdate for quarterly real GNP does not coincide with the oil
price shock of 1973. The evidence against the unit-root null
182
Journal of Business & Economic Statistics, January 2003
Table 6. Empirical Results for the Nelson–Plosser (1982) Data and Quarterly Real GNP
Series
Downloaded by [Universitas Maritim Raja Ali Haji] at 00:39 13 January 2016
Real GNP
Nominal GNP
Real per capita GNP
Industrial production
Employment
GNP de‘ator
Consumer prices
Nominal wages
Money stock
Velocity
Interest rate
Common stock prices
Real wages
Quarterly real GNP
min
tDF
(C)
ƒ506577(a1c)
ƒ601741(a1b)
ƒ502983(b1 d)
ƒ508192(a1c)
ƒ501995(c10)
ƒ401723
ƒ305120
ƒ502147(c10)
ƒ409709(d1 0)
ƒ309737
ƒ108785
ƒ505152(b1 d)
ƒ504509(b1 d)
ƒ501135(c10)
b (t min )
T
b DF
tODF (C)
b (tO )
T
b DF
FTmax
Tb (FTmax )
1929
1929
1938
1929
1929
1929
1893
1929
1930
1929
1963
1939
1940
1964.IV
ƒ506577(b1b)
ƒ601741(a1a)
ƒ502983(c1 c)
ƒ508192(a1b)
ƒ501995(c1 d)
ƒ401723
ƒ201086
ƒ306338
ƒ409709(d1d)
ƒ309737
ƒ103131
ƒ505015(b1c)
ƒ504509(b1c)
ƒ501135(d1d)
1929
1929
1938
1929
1929
1920
1869
1920
1930
1929
1964
1936
1940
1964.IV
1009804(b1c)
1300122(a1 b)
1002524(b1d)
1200478(a1 b)
1002154(c1d)
805103
803915
909920(c1d)
807065
505704
908964(c1d)
1002942(b1d)
1103206(a1 c)
807649
1929
1929
1938
1929
1929
1920
1863
1920
1930
1929
1964
1939
1940
1964.IV
P ü
O 1 DU t (Tb ) C Œ
O 0 CŒ
O 2t C Œ
O 3 DTt (Tb ) C y
O t ƒ1 C kD1 cO j ãyt ƒj C eO t . The small letters in parentheses
NOTE: Mixed-model regression: yt D Œ
j
that appear as superscript indicate the signi’cance of these statistics. The ’rst letter in the parentheses indicates signi’cance with
respect to the asymptotic critical values, and the second letter indicates signi’cance with respect to the appropriate ’nite sample
critical values. a, b, c, and d indicate signi’cance at the 1%, 2.5%, 5%, and 10% signi’cance level, and . appears if the statistic is not
signi’cant at the 10% signi’cance level. The asymptotic and ’nite sample critical values of t min (C) are taken from table 4 in Zivot and
DF
Andrews (1992) and table 1 in Perron (1997), respectively. The asymptotic and ’nite sample critical values for OtDF (C ) were obtained
from table 3 in Vogelsang and Perron (1998). The asymptotic and ’nite sample critical values for FTmax were taken from Table 1 in this
article. ‹ 0 D 005 was used.
Table 7. Estimated Regressions for the Nelson–Plosser (1982) Data and Quarterly Real GNP
Series
Real GNP
Tb
kü
1929
8
O0
Œ
306921
(506850)
Nominal GNP
1929
8
504047
(602240)
Real per capita GNP
1938
2
302484
(503017)
Industrial production
1929
8
01339
(403925)
Employment
1929
8
501709
(502386)
GNP De‘ator
1929
5
06601
(402638)
1920
5
02720
(204134)
Consumer prices
1893
5
03813
(303150)
1869
0
03109
(307658)
Nominal wages
1929
7
201348
(502841)
1920
7
09733
(304607)
Money stock
1930
8
04831
(504325)
Velocity
1929
1
03540
(306305)
Interest rate
1963
3
04049
(108000)
1964
0
02403
(102593)
Common stock prices 1939
1
05320
(502391)
1936
3
05743
(502399)
Real wages
1940
3
108163
(504785)
Quarterly real GNP
1964.IV 12 103289
(501524)
O1
Œ
O2
Œ
O3
Œ
ƒ01793
00241
00045
(ƒ402164)
(401459)
(100059)
ƒ02799
00224
00115
(ƒ405282)
(300520)
(107811)
01326
00010
00072
(400687)
(07412)
(301672)
ƒ03239
00331
00012
(ƒ501854)
(506933)
(09001)
ƒ00747
ƒ00015
00098
(ƒ308435)
(405430) (ƒ106187)
ƒ00899
00061
00010
(301706)
(306491)
(100358)
ƒ00997
ƒ00026
00060
(402269)
(400137) (ƒ200290)
ƒ00205
ƒ00009
00031
(ƒ101960) (ƒ08578)
(202153)
ƒ00077
ƒ00193
00204
(ƒ02035) (ƒ208023)
(209934)
ƒ01640
ƒ00004
00177
(ƒ308359)
(404476) (ƒ01847)
ƒ01516
ƒ00133
00211
(ƒ403287)
(404541) (ƒ300775)
ƒ01053
ƒ00023
00212
(ƒ304871)
(406864) (ƒ109341)
ƒ00473
ƒ00041
00060
(ƒ107643) (ƒ303186)
(306149)
ƒ00231
ƒ00016
01594
(ƒ00987) (ƒ07641)
(208927)
ƒ01287
ƒ00004
02157
(ƒ05393) (ƒ01989)
(304392)
ƒ01942
00078
00261
(ƒ205958)
(407365)
(406067)
ƒ02783
00090
00252
(ƒ306409)
(408941)
(409949)
00842
00086
00047
(403846)
(503509)
(303869)
ƒ00001
00171
00015
(401128)
(408642) (ƒ107395)
O
02352
(ƒ506577)
05057
(ƒ601741)
05490
(ƒ502983)
03005
(ƒ508192)
04914
(ƒ501995)
07831
(ƒ401723)
09042
(ƒ206100)
08921
(ƒ305120)
09528
(ƒ201086)
06581
(ƒ502147)
08357
(ƒ306338)
06893
(ƒ409709)
07635
(ƒ309737)
09047
(ƒ108785)
09421
(ƒ103131)
06038
(ƒ505152)
05702
(ƒ505015)
03892
(ƒ504509)
08218
(ƒ501135)
O
S(e)
.0503
.0680
.0522
.0876
.0289
.0435
.0418
.0361
.0520
.0542
.0535
.0419
.0636
.2481
.2475
.1397
.1392
.0307
.0086
P ü
O 1 DU t (Tb ) C Œ
O 0 CŒ
O 2t C Œ
O 3 DTt (Tb ) C y
O t ƒ1 C kD1 cO j ãyt ƒj C eO t . Tb is the chosen break-date,
NOTE: Mixed-model regression: yt D Œ
j
k ü is the value of the lag-truncation parameter chosen according to the k (t -sig) procedure of Perron and Vogelsang (1992) with k O tests
max D 8 for all series except quarterly real GNP, for which k -max D 12. The t statistics appear in parentheses. The t statistic for
O is the estimated standard deviation of the regression error.
the hypothesis that D 1. S(e)
Downloaded by [Universitas Maritim Raja Ali Haji] at 00:39 13 January 2016
Sen: Unit-Root Tests
is weakened if the nite sample critical values given by Permin
4C5 with lag-truncation parameter
ron (1997) are used for tDF
selected using the k4t-sig5 method. With the nite sample critical values, the unit-root null for employment, nominal wages,
money stock, and quarterly real GNP cannot be rejected.
Because a break is not allowed under the unit-root null
hypothesis, these results should be compared to the results
of Zivot and Andrews (1992). The unit-root null hypothesis is rejected here for money stock, real wages, and quarterly real GNP in addition to all series for which Zivot and
Andrews (1992) rejected the unit-root null. Further, the present
evidence is stronger for real per capita GNP. Unlike that of
Zivot and Andrews (1992), the break-date estimated here for
real per capita GNP is 1938, for money stock is 1930, for
common stock prices is 1939, and for quarterly real GNP is
1964, fourth quarter. The estimated break-date coincides with
the great crash of 1929 for real GNP, nominal GNP, industrial production, employment, and nominal wages. In cases for
which the unit-root null is rejected, one would like to determine the signicance of the trend function coefcients Œ0 ,
min
bb 4tDF
5. Given that the location
Œ1 , Œ2 , and Œ3 evaluated at T
min
bb 4tDF 5, the asymptotic distribution
of break is estimated by T
of t statistics for H0 2 Œi D 0, denoted by tŒO i , (i D 01 11 21 3)
may not be well approximated by the standard normal distribution. In unreported simulations, the empirical distribution of
tŒO i (i D 01 11 21 3) corresponding to the estimated regression at
bb 4t min 5 was examined. The results indicate that the empiriT
DF
cal distributions of tŒO i (i D 01 21 3) are symmetric, have mean
close to zero, but have a slightly larger variance compared to
the standard normal distribution. On the other hand, the empirical distribution of tŒO 1 is bimodal with a much larger variance
compared to the standard normal distribution. [It was found
that the empirical distributions of tŒO i (for i D 01 11 21 3) evalubb 4OtDF 5 and T
bb 4FTmax 5 exhibit similar behavior.] Results
ated at T
pertaining to the empirical distribution of tŒO i (i D 01 11 21 3) are
available on request. As a rst approximation, the signicance
of the intercept coefcient (Œ0 ), the trend coefcient (Œ2 ), and
the slope break coefcient (Œ3 ) using the critical values from
a standard normal distribution are assessed. [However, the signicance of the intercept break coefcient (Œ1 ) using the standard normal critical values is not determined.] The estimated
coefcients and their respective t statistics are presented in
Table 7. It was found that tŒO 0 and tŒO 2 are signicant for all
series, with the sole exception of Œ2 for real per capita GNP,
and tŒO 3 is signicant for nominal GNP, real per capita GNP,
money stock, common stock prices, real wages, and quarterly
real GNP.
Second, the results for tODF 4C5 are qualitatively similar to
min
4C5. The critical values for OtDF 4C5, both asympthose of tODF
totic and nite sample with lag-truncation parameter chosen
according to the k4t-sig5 method, can be found in Vogelsang
and Perron (1998). Based on the asymptotic critical values, the
unit-root null is rejected at the 1% signicance level for nominal GNP and industrial production; at the 2.5% signicance
level for real GNP, common stock prices, and real wages; at
the 5% signicance level for real per capita GNP and employment; and at the 10% signicance level for money stock and
quarterly real GNP. The unit-root null cannot be rejected for
GNP deator, consumer prices, nominal wages, velocity, and
183
interest rate. The results for all series are in agreement with
min
4C5, with the exception of
the results corresponding to tDF
nominal wages. The estimated break-date Tbb 4tODF 5 is the same
min
bb 4tDF
5 for all series for which the unit-root is rejected,
as T
except common stock prices, for which the estimated breakdate is 1936. The use of nite sample critical values weakens
the evidence against the unit-root null somewhat.
Finally, consider the results with FTmax . The critical values
for FTmax , both asymptotic and nite sample, are obtained from
Table 1. Based on the asymptotic critical values, the joint null
hypothesis can be rejected at the 1% signicance level f
ISSN: 0735-0015 (Print) 1537-2707 (Online) Journal homepage: http://www.tandfonline.com/loi/ubes20
On Unit-Root Tests When the Alternative Is a
Trend-Break Stationary Process
Amit Sen
To cite this article: Amit Sen (2003) On Unit-Root Tests When the Alternative Is a TrendBreak Stationary Process, Journal of Business & Economic Statistics, 21:1, 174-184, DOI:
10.1198/073500102288618874
To link to this article: http://dx.doi.org/10.1198/073500102288618874
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Date: 13 January 2016, At: 00:39
On Unit-Root Tests When the Alternative
Is a Trend-Break Stationary Process
Amit Sen
Department of Economics & Human Resources, Xavier University, Cincinnati, OH
45207-3212 (sen@xavier.edu)
Downloaded by [Universitas Maritim Raja Ali Haji] at 00:39 13 January 2016
Minimum t statistics to test for a unit-root are available when the form of break under the alternative
evolves according to the crash, changing growth, and mixed models. It is shown that serious power
distortions occur if the form of break is misspecied, and thus the practitioner should use the mixed
model as the appropriate alternative in empirical applications. The mixed model may reveal useful
information regarding the location and form of break. The maximum F statistic for the joint null of a
unit-root and no breaks is shown to have greater and less erratic power compared to the minimum t
statistic. Stronger evidence against the unit-root is found for the Nelson–Plosser series and U.S. Postwar
quarterly real gross national product.
KEY WORDS: Break-date; Form of break; Maximum F statistic; Minimum t statistic.
1.
INTRODUCTION
Perron (1989) demonstrated that the conventional Dickey
and Fuller (1979) t statistic (tDF ) accepts the null hypothesis of a unit-root far too often when the true data-generating
process is in fact trend stationary with a break in the intercept or the slope of the trend function. Three different characterizations of the trend-break alternative were considered:
(a) the crash model, which allows for a break in the intercept;
(b) the changing growth model, which allows for a break in
the slope with the two segments joined at the break-date; and
(c) the mixed model, which allows for a simultaneous break
in the intercept and the slope. To devise unit-root tests that
have power against the trend-break stationary alternative, Perron (1989) proposed the following methodology: First specify the location of the break-date (Tb ), and then estimate a
regression that nests the random walk null and the trend-break
stationary alternative of choice. Under the null hypothesis,
Perron derived the limiting distribution of the t statistic on the
i
4Tb 5, where
rst lag of the dependent variable, denoted by tDF
i D A, B, or C corresponds to the crash, the changing growth,
or the mixed model, respectively. The limiting distribution of
i
tDF
4Tb 5 depends on the location and form of break under the
alternative hypothesis.
The assumption that the location of break is known a priori was criticized by a number of studies, most notably by
Christiano (1992). Also, see Banerjee, Lumsdaine, and Stock
(1992), Zivot and Andrews (1992), and Perron and Vogelsang
(1992). Christiano (1992) argued that the choice of the breakdate is in most cases correlated with the data; for example,
the practitioner may determine the location of the break-date
by visually inspecting a plot of the time series. And because
Perron’s (1989) methodology does not account for this ‘pretest
examination of data,’ the unit-root null will be rejected too
often. A number of studies proposed extensions for unit-root
tests that do not require the practitioner to prespecify the
location of break; see Zivot and Andrews (1992), Banerjee
et al. (1992), Perron and Vogelsang (1992), Perron (1997),
and Vogelsang and Perron (1998). The strategy adopted by
these studies is to apply Perron’s (1989) methodology for each
possible break-date in the sample, which yields a sequence
of t statistics. Based on this sequence, numerous minimum
t statistics can be constructed by choosing the t statistic, based
on some algorithm, that maximizes evidence against the null
hypothesis. For example, one may simply use the minimum of
min
4i51 i D A, B, C.
the sequence of t statistics, denoted by tDF
With the availability of the minimum t statistics, the practitioner no longer needs to prespecify the location of break.
However, one must still specify the form of break under the
alternative hypothesis. If one assumes that the location of
break is unknown, it is most likely that the form of break will
be unknown as well. It is argued here that selection of the form
of break is also correlated with the data and so one must proceed with the break specication according to the most general
mixed model. In addition, one may expect power distortions
if the form of break is misspecied, that is, if one imposes
the crash (changing growth) model when in fact the changing
growth (crash) or the mixed model is appropriate. Consider,
for example, the use of the crash model specication under
the alternative hypothesis when the break occurs according to
the changing growth or the mixed model. The crash model
min
4A5 is designed to maximize evidence
minimum t statistic tDF
against the null hypothesis in favor of this particular alternative. If the true data-generating process is given by the mixed
min
4A5 can be expected to be lower
model, the power of tDF
min
than that of tDF 4C5, especially if the magnitude of the slope
min
4A5 may be
break is relatively large. Even lower power of tDF
expected if the alternative evolves according to the changing
growth model. On the other hand, if the crash hypothesis is
the correct specication, then its use will yield superior power
compared to the mixed model statistics. In practice, however,
one is uncertain about the form of break. Because one would
like to guard against distortions in power owing to misspecication of the form of break, it is recommended that the practitioner use the mixed model specication under the alternative
hypothesis.
The rst objective is to assess the performance of minimum
t statistics when the form of break is misspecied. To this
end, simulation evidence is provided (a) on the performance
174
© 2003 American Statistical Association
Journal of Business & Economic Statistics
January 2003, Vol. 21, No. 1
DOI 10.1198/073500102288618874
Downloaded by [Universitas Maritim Raja Ali Haji] at 00:39 13 January 2016
Sen: Unit-Root Tests
of the crash model statistics when the break occurs according
to either the changing growth or the mixed model; (b) on the
performance of the changing growth model statistics when the
break occurs according to either the crash or the mixed model;
and (c) to assess the loss in power of the mixed model statistics when the break occurs according to either the crash or the
changing growth model. It is found that the loss in power is
quite small if the mixed model specication is used when in
fact the break occurs according to the crash model. The loss
in power is more substantial if the break occurs according to
the changing growth model. However, serious distortions can
occur if the crash (changing growth) model is used when the
appropriate model is either the changing growth (crash) model
or the mixed model. Therefore, the results indicate that one
should use the form of the break specied under the mixed
model, unless prior information about the nature of the break
suggests using either the crash model or the changing growth
model. In most cases, it is expected that information on the
form of break will be accompanied with information on the
location of break, in which case Perron’s (1989) tests should
be used. Also studied are the power properties the supWald (or
equivalently the maximum F statistic, denoted by FTmax ) proposed by Murray (1998) and Murray and Zivot (1998) for the
joint null hypothesis of a unit-root and no break in the intercept and slope of the trend function. Simulation results show
that the power of FTmax is less erratic and can be greater than
the mixed model minimum t statistics.
Second, the arguments are illustrated within the context
of an empirical example that has received much attention in
the literature. Numerous studies have used the Nelson and
Plosser (1982) data and postwar quarterly real gross national
product (GNP) to test for the presence of a unit-root. Perron (1989) specied the great crash of 1929 as the breakdate for all Nelson–Plosser series and the oil price shock of
1973 for quarterly real GNP. Conditional on these break-dates,
the crash model was specied for all Nelson–Plosser series
except common stock prices and real wages, for which the
mixed model was used, and the changing growth model was
used for quarterly real GNP. Whereas subsequent studies by
Zivot and Andrews (1992), Perron (1997), Nunes, Newbold,
and Kuan (1997), and Murray and Nelson (2000) allowed for
an unknown break-date, they retained the model specication
proposed by Perron (1989). Unlike in these studies, empirical
evidence is presented here for the presence of a unit-root in all
series when the alternative allows for a simultaneous break in
the intercept and slope. It is found that the unit-root null can be
rejected for all series with the exception of the GNP deator,
consumer prices, velocity, and interest rate series. The results
are robust to possible misspecication of the form of break
and therefore reveal useful information regarding the location
and form of break. For instance, the empirical evidence of
Zivot and Andrews (1992) indicates that real per capita GNP
is characterized as a stationary process with break in the intercept occurring in 1929, and both money stock and quarterly
real GNP contain a unit-root. By using the mixed model as
the appropriate alternative, evidence is strengthened against a
unit-root in real per capita GNP, and evidence is uncovered
against the unit-root for money stock and quarterly real GNP.
The break-date is estimated for real per capita GNP in 1938,
175
for money stock in 1930, and for quarterly real GNP in 1964,
fourth quarter. The estimated break-dates for these series are
different from the estimated break-dates reported by Zivot and
Andrews (1992).
The outline of this article is as follows. Section 2 discusses
the minimum t statistics and the maximum F statistic that
have been proposed in the literature. In Section 3, simulation
evidence is presented that enables one to study the nite sample size and power properties of all statistics when the form
of break may be misspecied. Empirical evidence for the Nelson and Plosser (1982) data and U.S. postwar quarterly real
GNP is presented in Section 4. Concluding comments appear
in Section 5.
2.
TEST STATISTICS AND METHODOLOGY
We begin with a brief discussion of the null and alternative hypotheses of interest, the class of minimum t statistics,
and the maximum F statistic. The discussion follows the analysis by Zivot and Andrews (1992). Consider the time series
8yt 9TtD1 , where T is the available sample size. The three different characterizations of the alternative hypothesis, originally
discussed by Perron (1989), are given by
Model (A)2 yt D Œ0 C Œ1 DUt 4Tbc 5 C Œ2 t C et 1
(1)
yt D Œ0 C Œ2 t C Œ3 DTt 4Tbc 5 C et 1
(2)
Model (B)2
Model (C)2
yt D Œ0 C Œ1 DUt 4Tbc 5 C Œ2 t C Œ3 DTt 4Tbc 5 C et 1
(3)
where Tbc is the correct break-date, DUt 4Tbc 5 D 14t>Tbc 5 ,
DTt 4Tbc 5 D 4t ƒ Tbc 514t>Tbc 5 , and 14t>Tbc 5 is an indicator function that takes on the value 0 if t µ Tbc and 1 if t > Tbc .
For the asymptotic results, it is assumed that the break-date
is a constant fraction of the sample size, that is, Tbc D ‹c T
with the correct break-fraction ‹c 2 401 15. It is assumed that
A4L5et D B4L5t , and t is a sequence of iid 401‘ 2 5 random
variables, A(L) and B(L) are polynomials in the lag operator of order p C 1 and q, respectively, with all roots outside
the unit circle. Model (A) is referred to as the crash model
because it allows for a break in the intercept alone, Model (B)
is referred to as the changing growth model because it allows
for a break in the slope with the two segments joined at the
break-date, and Model (C) is referred to as the mixed model
because it allows for a simultaneous break in the intercept and
the slope of the trend function.
The data-generating process given in (1)–(3), known as the
additive outlier (AO) model, incorporates a break in the trend
function that occurs instantaneously. This article considers the
innovation outlier (IO) model, wherein the change in the trend
function evolves in the same manner as any other shock; see
section 4.2 in the work by Perron (1989). That is,
Model (A)2 yt D Œ0 C Œ2 t C –4L56Œ1 DUt 4Tbc 5 C t 71 (4)
Model (B)2 yt D Œ0 C Œ2 t C –4L56Œ3 DTt 4Tbc 5 C t 71 (5)
Model (C)2 yt D Œ0 C Œ2 t C –4L5
6Œ1 DUt 4Tbc 5 C Œ3 DTt 4Tbc 5 C t 71
(6)
where –4L5 D A4L5ƒ1 B4L5. The impact of the break is different in the AO and IO models. Consider, for example, the
176
Journal of Business & Economic Statistics, January 2003
crash model specication of the break. In the AO characterization, the magnitude of the break is Œ1 . However, in the IO
characterization the immediate impact of the break is equal to
Œ1 , but the long-run impact is equal to –415Œ1 .
Under the null hypothesis, the data-generating process contains a unit-root, that is,
ü
yt D Œ C ytƒ1 C – 4L5t 1
(7)
Downloaded by [Universitas Maritim Raja Ali Haji] at 00:39 13 January 2016
where – ü 4L5 D Aü 4L5ƒ1 B4L51 A4L5 D 41 ƒ L5Aü 4L5, and
D 1. To test for a unit-root in the IO framework, the following methodology has been suggested. Specify an interval
å D 6‹0 1 1 ƒ ‹0 7 401 15 that is believed to contain the true
break-fraction. For each possible ‹ 2 å, estimate the following
regression that nests the null and the appropriate alternative:
O A0 C Œ
O A1 DUt 46‹T 75 C Œ
O A2 t C O A ytƒ1
yt D Œ
C
k
X
cO jA ãytƒj C eO tA 1
(8)
cO jB ãytƒj C eO tB 1
(9)
jD1
O B0 C Œ
O B2 t C Œ
O B3 DTt 46‹T 75 C O B ytƒ1
yt D Œ
C
k
X
jD1
O C0 C Œ
O C1 DUt 46‹T 75 C Œ
O C2 t C Œ
O C3 DTt 46‹T 75
yt D Œ
C O C ytƒ1 C
k
X
cO jC ãytƒj C eO tC 1
(10)
jD1
where [.] is the smallest integer function. The additional k
regressors 8ãytƒj 9kjD1 are included in the regression to eliminate the correlation in the disturbance term. Typically, the
value of the lag-truncation parameter (k) is unknown, and so
a data-dependent method for choosing the appropriate value
of k is used; see Perron and Vogelsang (1992), Hall (1992,
1994), Perron (1997), and Ng and Perron (1995). We use Perron and Vogelsang’s (1992) data-dependent method k4t-sig)
for selecting the lag-truncation parameter, which is described
in what follows. Specify an upper bound k-max for the
lag-truncation parameter. For each break-date 6‹T 7, the chosen value of the lag-truncation parameter (k ü ) is determined
according to the following general to specic procedure: The
last lag in an autoregression of order k ü is signicant, but
the last lag in an autoregression of order greater than k ü is
insignicant. The signicance of the coefcient is assessed
using the 10% critical values based on a standard normal distribution. Regressions (8)–(10) are estimated for the breakdates 86‹ 0 T 71 6‹0 T 7 C 11 : : : 1 T ƒ 6‹0 T 79, and the sequence
T ƒ6‹ T 7
i
4Tb 59Tb D6‹00 T 7
of t statistics for H0 2 D 1, denoted by 8tDF
(i D A, B, C) is calculated. Based on this sequence, a number
of minimum t statistics can be obtained by using an algorithm to choose the appropriate break-date that maximizes evidence against Equation (7); see Perron and Vogelsang (1992),
Banerjee et al. (1992), Perron (1997), and Vogelsang and
Perron (1998).
We consider two particular algorithms for choosing the
break-date. The rst statistic, originally proposed by Perron
and Vogelsang (1992) and Zivot and Andrews (1992), is
obtained by choosing the break-date that maximizes evidence
against the unit-root null, that is,
i
min
tDF
4Tb 5
4i5 D MinTb 286‹ 0 T 716‹ 0 T 7C11 : : : 1T ƒ6‹0 T 79 tDF
(11)
min
4i5 can be
for i D A, B, C. Perron (1997) demonstrated that tDF
calculated with å D 401 15 so that no trimming of the sample
is necessary. The second statistic, proposed by Banerjee et al.
(1992) and Vogelsang and Perron (1998), is dened as
O 51
tODF 4i5 D tDF 4‹
i
(12)
O (i D A, B, C) is the break-date that maximizes the
where ‹
i
Wald statistic, FTi 46‹T 75, corresponding to H0 2 Œ1 D 0 for the
crash model (i D A), H0 2 Œ3 D 0 for the changing growth
model (i D B), and H0 2 Œ1 D Œ3 D 0 for the mixed model
(i D C).
min
4i5 and tODF 4i5 (i D A, B, C),
The minimum t statistics, tDF
do not require specication of the location of break. However,
the practitioner does need to specify the form of break under
the alternative hypothesis. Once the break-date is treated as
unknown, the practitioner will in most cases be unaware of the
form of break. Because the crash and changing growth models
are special cases of the mixed model, the practitioner should
specify the mixed model as the appropriate alternative. Some
loss in power can be expected from using the statistics from
the mixed model when in fact the break occurs according to
the crash or changing growth model. However, the behavior
of the statistics from the crash (changing growth) model when
the break occurs according to the changing growth (crash) or
mixed model is uncertain. The next section presents simulation evidence regarding the power properties of the minimum
t statistics when the form of break is misspecied.
Next, we briey describe the supWald statistic proposed
by Murray (1998) and Murray and Zivot (1998) for the joint
null hypothesis of a unit-root and no break in the intercept
and slope of the trend function. As before, it is assumed that
the break-fraction lies in å D 6‹0 1 1 ƒ ‹0 7. For each possible break-date Tb 2 86‹ 0 T 71 6‹ 0 T 7 C 11 : : : 1 T ƒ 6‹0 T 79, the F
statistic, denoted by FT 4Tb 5 from regression (10) that corresponds to the joint null hypothesis H 0J 2 D 11 Œ1 D 0, Œ3 D 0
is calculated as
FT 4Tb 5 D
iƒ1
h ¡P
¢
T
0 ƒ1
O b 5 ƒ r5
O b 5 ƒ r5 R
4R Œ4T
R0
4RŒ4T
tD1 xt 4Tb 5xt 4Tb 5
q‘O 2 4Tb 5
1
(13)
O b 5 is the ordinary least squares (OLS) estimator
where Œ4T
of Œ D 4Œ0 1 Œ1 1 Œ2 1 Œ3 1 1 c1 1 : : : 1 ck 50 , xt 4Tb 5 D 411 DUt 4Tb 5,
t1 DTt 4Tb 51 ytƒ1 1 ãytƒ1 1 : : : 1 ãytƒk 50 , r D 401 01 150 , Pq is the
number of restrictions, ‘O 2 4Tb 5 D 4T ƒ 5 ƒ k5ƒ1 TtD1 4yt ƒ
O b 552 , and R is dened so that RΠD r corresponds
xt 4Tb 50 Œ4T
to the restrictions imposed on the parameter vector Πby the
joint null H0J . Specically,
3
2
0 1 0 0 0 00k
R D 40 0 0 1 0 00k 5 0
0 0 0 0 1 00k
Sen: Unit-Root Tests
177
Table 1. Critical Values for FTmax With å D [ ‹0 , 1 - ‹0 ]
1%
Downloaded by [Universitas Maritim Raja Ali Haji] at 00:39 13 January 2016
5%
10%
‹0 D 015
T D 50
k D0
k(t-sig)
1301238
1500101
1106071
1302783
1005840
1202464
904661
1008523
T D 100
k D0
k(t-sig)
1200157
1300842
1008744
1108875
1000248
1007809
900628
907820
T D 150
k D0
k(t-sig)
1108505
1208437
1007593
1102872
909333
1003951
809723
904935
T D 200
k D0
k(t-sig)
1106450
1206811
1004785
1100691
906730
1003038
808708
903692
1001691
904376
806958
T Dˆ
1009288
‹0 D 010
T D 50
k D0
k(t-sig)
1301646
1500177
1106795
1303184
1006061
1202799
905237
1009242
T D 100
k D0
k(t-sig)
1201731
1300842
1009391
1108875
1000970
1008409
901282
908127
T D 150
k D0
k(t-sig)
1108765
1208440
1008124
1102896
909692
1004511
900420
905204
T D 200
k D0
k(t-sig)
1107058
1207662
1005348
1101565
907310
1003460
809164
903941
1009841
1002152
904931
807353
T Dˆ
‹0 D 005
T D 50
k D0
k(t-sig)
1302127
1501077
1107261
1303226
1006409
1202862
905796
1009674
T D 100
k D0
k(t-sig)
1201958
1300842
1009940
1109215
1001425
1008752
901871
908674
T D 150
k D0
k(t-sig)
1109328
1208440
1008546
1102896
1000083
1004922
900895
905456
T D 200
k D0
k(t-sig)
1107236
1207662
1005892
1101565
907763
1004089
809535
904368
1100364
1002414
905427
807946
T Dˆ
T ƒ6‹ T 7
The sequence of F statistics 8FT 4Tb 59Tb D6‹00 T 7 is used to calculate the maximum F statistic for H 0J as
FTmax D MaxTb 286‹ 0 T 716‹ 0 T 7C11 : : : 1T ƒ6‹0 T 79 FT 4Tb 50
(14)
The limiting distribution of FTmax under H0J is easily obtained
from theorem 3.2F by Murray and Zivot (1998). Note that in
this article the F statistic is used for H0J rather than the Wald
statistic considered by Murray and Zivot (1998). Therefore,
the critical values of the supWald statistic and FTmax will differ
by a multiplicative constant equal to q D 3. Asymptotic critical values for the supWald statistic (without any trimming of
the sample) are given by table 6 in Murray and Zivot (1998).
Asymptotic and nite sample critical values for the limiting
null distribution of FTmax are given in Table 1 when the trimming parameter ‹0 D 0151 0101 005.
3.
2.5%
FINITE SAMPLE SIZE AND POWER
In this section, we present nite sample size and power
results for the minimum t statistics and the FTmax statistic. The
differences in the power of FTmax and the minimum t statistics
min
4C5 and tODF 4C5, are noted.
from the mixed model, namely, tDF
We follow the experimental design of Vogelsang and Perron
(1998) and generate data according to
61 ƒ 4 C 5L C L2 7yt D 41 C –L56Œ1 DUtc C Œ3 DTtc C et 71
t D 11 21 : : : 1 T 0
(15)
We set D 1 for the size simulations and D 008 for
the power simulations. The sample size for all simulations
is T D 100, the correct break-date is Tbc D 50, DUtc D
DUt 4Tbc 5, DTtc D DTt 4Tbc 5, et ¹ iid N(0,1), and y0 D
yƒ1 D 0. The following combinations of (1 –) are used:
8401 053 4061 053 4ƒ061 053 401 0553 401 ƒ0559. For the size simulations, set Œ1 D Œ3 D 0. For the power simulations, we consider the data-generating process under (a) the crash model
with a break in the intercept for Œ1 D 811 21 41 61 89; (b) the
changing growth model with a break in the slope for Œ3 D
8011 021 031 041 059; and (c) the mixed model with a simultaneous break in the intercept and slope for Œ1 D 811 21 41 69 and
Œ3 D 8011 021 031 049. For each parameter combination 2,000
replications were generated. We calculate the size and power
of following statistics at the 5% signicance level using the
min
4i5 and OtDF 4i5
appropriate nite sample critical values: tDF
178
Journal of Business & Economic Statistics, January 2003
min
( i ) , tODF ( i ) for i D A, B, C, and FTmax
Table 2. Finite Sample Size of tDF
–
min
(A)
tDF
tODF (A)
min
(B)
tDF
Ot (B)
DF
min
(C)
tDF
tODF (C)
FTmax
.0
.6
ƒ.6
.0
.0
.0
.0
.0
.5
ƒ.5
.0420
.0550
.0405
.0640
.3160
.0405
.0505
.0390
.0585
.2815
.0495
.0605
.0530
.0785
.3180
.0530
.0615
.0545
.0790
.3080
.0585
.0580
.0540
.0845
.4320
.0535
.0545
.0520
.0800
.4030
.0525
.0695
.0510
.0865
.4125
Downloaded by [Universitas Maritim Raja Ali Haji] at 00:39 13 January 2016
NOTE:
5% nominal size, DGP ( D 1): yt D ( C )yt ƒ1 ƒ yt ƒ2 C (1 C –L) e t .
for i D A, B, C, and FTmax . We use ‹0 D 015 for calculating OtDF 4i5 and FTmax . The lag-truncation parameter (k) in the
regressions (8)–(10) is chosen according to the k4t-sig5 procedure mentioned in Section 2 with k-max D 5. Sen (2001)
presented additional simulation evidence in which the lagtruncation parameter is set equal to its correct value; see his
tables 1–3. The nite sample size for all statistics is presented
in Table 2, and the power of all statistics under the crash
model, the changing growth model, and the mixed model is
given in Tables 3, 4, and 5, respectively. The following interesting features emerge from the simulations:
1. The exact size of all statistics is fairly close to the nominal size (see Table 1), except when there is a negative moving
average component, that is, (1 –5 D 401 ƒ055.
2. When the break occurs according to the crash model
min
4A5 and tODF 4A5 have the largest power. There is
(Table 3), tDF
very little loss in power from using the mixed-model statistics,
min
tDF
4C5, OtDF 4C5, and FTmax . The power of the changing growth
min
4B5 and tODF 4B5, is close to zero except
model statistics, tDF
for small intercept breaks (Œ1 ).
3. When the break occurs according to the changing growth
min
4B5 and OtDF 4B5 have the largest power.
model (Table 4), tDF
There is some loss in power from using the mixed model
statistics, but this loss in power diminishes with the size of
the slope break magnitude. In all cases, the power of the crash
min
4A5 and OtDF 4A5, is close to zero.
model statistics, tDF
4. When the break occurs according to the mixed model
(Table 5), the power of the mixed-model statistics, in general,
increases with the size of the intercept break (Œ1 ) and slope
break (Œ3 ) magnitudes. For a xed Œ3 , the power of the crash
model statistics increases with Œ1 , and the power of the changing growth model statistics decreases with Œ1 . For a xed Œ1 ,
min
( i ) , tODF ( i ) for i D A, B, C, and FTmax : True DGP is Crash Model
Table 3. Finite Sample Power of tDF
Œ1
min
tDF
(A)
tODF (A)
min
t DF
(C)
tODF (C)
FTmax
1
2
4
6
8
02380
03660
09825
100000
100000
02440
03800
09815
100000
100000
D 081 D 001 – D 00
.0740
.0725
.0010
.0010
.0000
.0000
.0000
.0000
.0000
.0000
01995
02305
07875
09970
100000
01885
02270
07860
09945
100000
01560
03640
09850
100000
100000
1
2
4
6
8
08030
08800
09985
100000
100000
07655
08690
09985
100000
100000
D 081 D 061 – D 00
.5695
.5510
.1965
.1955
.0060
.0075
.0000
.0000
.0000
.0000
07620
08200
09825
100000
100000
07150
07920
09790
100000
100000
06875
07870
09915
100000
100000
1
2
4
6
8
00780
02100
09685
100000
100000
00825
02100
09545
100000
100000
D 081 D ƒ061 – D 00
.0205
.0215
.0005
.0005
.0000
.0000
.0000
.0000
.0000
.0000
00670
00585
04515
09530
100000
00640
00575
04355
09305
09975
00770
03440
09940
100000
100000
1
2
4
6
8
02245
03710
09820
100000
100000
02185
03655
09810
100000
100000
D 081 D 001 – D 05
.0750
.0730
.0015
.0010
.0000
.0000
.0000
.0000
.0000
.0000
02080
02280
07795
09975
100000
01965
02160
07790
09970
100000
01740
03755
09850
100000
100000
1
2
4
6
8
07265
07215
09820
100000
100000
06935
06775
09810
100000
100000
D 081 D 001 – D ƒ05
.4345
.4145
.0770
.0755
.0000
.0000
.0000
.0000
.0000
.0000
07580
06840
08685
09965
100000
07065
06275
08525
09960
100000
07325
07060
09830
09995
100000
NOTE:
min
tDF
(B)
tODF (B)
5% nominal size, crash model DGP: yt D ( C )yt ƒ1 ƒ ytƒ2 C (1 C – L)[Œ1 DU tc C e t ].
Sen: Unit-Root Tests
179
Downloaded by [Universitas Maritim Raja Ali Haji] at 00:39 13 January 2016
min
( i ) , tODF ( i ) for i D A, B, C, and FTmax : True DGP is
Table 4. Finite Sample Power of tDF
Changing Growth Model
Œ3
min
tDF
(A)
tODF (A)
min
tDF
(B)
.1
.2
.3
.4
.5
.0000
.0000
.0000
.0000
.0000
.0000
.0000
.0000
.0000
.0000
.1
.2
.3
.4
.5
.1010
.0000
.0000
.0000
.0000
.1
.2
.3
.4
.5
min
tDF
(C)
tODF (C)
FTmax
D 081 D 001 – D 00
.3110
.3340
.5150
.5400
.7470
.7680
.9365
.9445
.9915
.9930
.1870
.2120
.3153
.5190
.7385
.1720
.1720
.2390
.3475
.4495
.1440
.2350
.4400
.7655
.9450
.1085
.0000
.0000
.0000
.0000
D 081 D 061 – D 00
.8225
.8315
.8625
.8765
.9225
.9305
.9675
.9725
.9880
.9930
.7215
.7030
.7430
.7845
.8190
.7040
.6740
.6815
.7020
.7000
.6265
.6360
.7165
.7925
.8790
.0000
.0000
.0000
.0000
.0000
.0000
.0000
.0000
.0000
.0000
D 081 D ƒ061 – D 00
.1985
.2105
.4055
.4215
.7940
.8000
.9800
.9780
.9990
.9955
.0680
.1085
.3220
.6755
.9165
.0590
.0825
.2105
.4380
.6150
.0775
.2350
.7140
.9600
.9870
.1
.2
.3
.4
.5
.0010
.0000
.0000
.0000
.0000
.0010
.0000
.0000
.0000
.0000
D 081 D 001 – D 05
.3155
.3415
.5375
.5625
.7450
.7660
.9360
.9430
.9910
.9910
.1860
.2220
.3160
.5040
.7230
.1725
.1840
.2340
.3310
.4610
.1555
.2570
.4495
.7200
.9045
.1
.2
.3
.4
.5
.0100
.0000
.0000
.0000
.0000
.0095
.0000
.0000
.0000
.0000
D 081 D 001 – D ƒ05
.7600
.7700
.8125
.8205
.8940
.9015
.9565
.9630
.9920
.9925
.7210
.7110
.7000
.7460
.8095
.7215
.7050
.6840
.7025
.7035
.7060
.7000
.7280
.8150
.9245
NOTE:
tODF (B)
5% nominal size, changing growth model DGP: yt D ( C )yt ƒ1 ƒ yt ƒ2 C (1 C –L)[Œ3 DTtc C e t ].
min
( i ) , OtDF ( i ) for i D A, B, C, and FTmax : True DGP is Mixed Model
Table 5. Finite Sample Power of tDF
Œ1
Œ3
min
tDF
(A)
tODF (A)
tODF (B)
min
tDF
(C)
1
2
4
6
.1
.0005
.0015
.2545
.9920
.0005
.0015
.2615
.9805
– D 00
.3345
.0845
.0000
.0000
02115
01770
06180
09930
01900
01640
06175
09915
02100
04350
09890
100000
1
2
4
6
.2
.0000
.0000
.0110
.5835
.0000
.0000
.0115
.5720
.6450
.5835
.0380
.0000
.6720
.6130
.0455
.0000
03100
02735
05065
09645
02335
01525
04995
09605
03650
06625
09960
100000
1
2
4
6
.3
.0000
.0000
.0005
.1360
.0000
.0000
.0005
.1350
.8705
.9480
.4920
.0250
.8820
.9545
.5290
.0320
04750
05885
04235
09245
03325
02540
03610
09190
06075
08525
09995
100000
1
2
4
6
.4
.0000
.0000
.0005
.0330
.0000
.0000
.0000
.0350
.9740
.9950
.9600
.3915
.9770
.9970
.9695
.4430
06530
08010
05945
08350
03930
02950
02545
08285
08595
09660
09985
100000
1
2
4
6
.1
.1025
.0635
.3505
.9905
.1110
.0690
.3610
.9905
– D 00
.7945
.5185
.0890
.0045
07480
08035
09720
100000
07295
07875
09630
100000
07015
08150
09905
100000
1
2
4
6
.2
.0000
.0005
.0025
.3135
.0000
.0010
.0030
.3505
.9015
.8040
.2775
.0590
07695
08000
09650
09985
07240
07645
09585
09985
07375
08625
09960
100000
min
tDF
(B)
D 11 D 001
.3070
.0760
.0000
.0000
D 11 D 061
.7845
.4970
.0825
.0040
.8875
.7855
.2525
.0500
tODF (C)
FTmax
(continued)
180
Journal of Business & Economic Statistics, January 2003
Downloaded by [Universitas Maritim Raja Ali Haji] at 00:39 13 January 2016
Table 5 (continued)
Œ1
Œ3
min
tDF
(A)
tODF (A)
min
tDF
(B)
tODF (B)
min
tDF
(C)
1
2
4
6
.3
.0000
.0000
.0005
.0110
.0000
.0000
.0005
.0150
.9565
.9470
.5760
.2205
.9610
.9550
.6085
.2375
08005
08390
09655
09995
07140
07605
09510
09995
07980
09205
09985
100000
1
2
4
6
.4
.0000
.0000
.0000
.0005
.0000
.0000
.0000
.0005
.9885
.9900
.8835
.5025
.9895
.9915
.9065
.5360
08500
08935
09685
09990
07155
07540
09415
09990
08860
09615
09995
100000
1
2
4
6
.1
.0010
.0005
.2660
.9910
.0010
.0000
.2365
.9675
– D 00
.2935
.1545
.0000
.0000
01265
00740
02205
08315
01105
00545
02155
08190
01465
04850
09950
100000
1
2
4
6
.2
.0000
.0000
.0130
.6400
.0000
.0000
.0075
.5860
.6715
.8605
.2205
.0035
.6930
.8745
.2525
.0040
02700
04295
01395
06130
02025
01530
01050
05965
04265
07825
09980
100000
1
2
4
6
.3
.0000
.0000
.0000
.1585
.0000
.0000
.0000
.1205
.9145
.9765
.9515
.2765
.9245
.9795
.9580
.3170
05390
07645
05210
03380
03500
02760
00460
03245
08140
09455
09995
100000
1
2
4
6
.4
.0000
.0000
.0000
.0075
.0000
.0000
.0000
.0050
.9865
.9980
.9995
.9555
.9870
.9980
.9945
.9535
08080
09215
09295
04935
04850
03870
00625
01620
09625
09930
100000
100000
1
2
4
6
.1
.0000
.0010
.2775
.9920
.0010
.0010
.2920
.9890
– D 05
.3530
.1240
.0005
.0000
02220
02065
06005
09805
02010
01875
05900
09805
02405
04995
09860
100000
1
2
4
6
.2
.0000
.0000
.0180
.6055
.0000
.0000
.0165
.6355
.6875
.6685
.0500
.0005
.7145
.6965
.0620
.0015
03720
03495
04805
09530
02855
01865
04640
09505
04170
06750
09940
100000
1
2
4
6
.3
.0000
.0000
.0025
.1820
.0000
.0000
.0015
.1965
.8910
.9635
.6265
.0530
.8995
.9700
.6620
.0690
04930
06435
04140
08820
03300
02905
03190
08810
06260
08640
09965
100000
1
2
4
6
.4
.0000
.0000
.0000
.0430
.0000
.0000
.0000
.0470
.9735
.9915
.9795
.5100
.9785
.9920
.9845
.5610
06630
08160
06635
07755
04255
03515
02335
07605
07200
08395
09405
100000
1
2
4
6
.1
.0075
.0015
.1755
.9780
.0085
.0015
.1950
.9810
D 11 D 001 – D ƒ05
.7275
.7385
.4005
.4110
.0030
.0030
.0000
.0000
07410
07120
07960
09890
07325
06880
07680
09890
07265
07785
09845
100000
1
2
4
6
.2
.0000
.0000
.0030
.3810
.0000
.0000
.0025
.4205
.8565
.7580
.0770
.0000
.8660
.7755
.0840
.0000
07565
07450
07500
09860
07405
07125
06945
09855
07575
08375
09935
100000
1
2
4
6
.3
.0000
.0000
.0000
.0650
.0000
.0000
.0000
.0745
.9365
.9635
.4025
.0035
.9445
.9705
.4405
.0045
07760
08380
07205
09640
07330
07425
06240
09640
07950
09250
09975
100000
1
2
4
6
.4
.0000
.0000
.0000
.0075
.0000
.0000
.0000
.0080
.9840
.9925
.9030
.1200
.9855
.9940
.9175
.1450
08250
09000
07665
09210
07555
07680
05975
09185
08925
09680
09995
100000
NOTE:
D 11 D ƒ061
.2710
.1365
.0000
.0000
D 11 D 001
.3285
.1120
.0005
.0000
5% nominal size, mixed model DGP: yt D ( C )yt ƒ1 ƒ yt ƒ2 C (1 C – L)[Œ 1 DU tc C Œ3 DTtc C e t ].
tODF (C)
FTmax
Downloaded by [Universitas Maritim Raja Ali Haji] at 00:39 13 January 2016
Sen: Unit-Root Tests
the power of the crash model statistics decreases with Œ3 and
the power of the changing growth statistics increases with Œ3 .
min
min
4B5 and tODF 4B5] can fail
4A5 and OtDF 4A5 [tDF
Therefore, tDF
to reject the unit-root null if the true data-generating process
is given by the mixed model with a relatively large Œ3 (Œ1 ).
5. When the break occurs according to mixed model, the
min
4C5 and OtDF 4C5 is sometimes erratic in that
behavior of tDF
their power does not always increase with the size of Œ1 and
min
4C5 and
Œ3 , except when 41 –5 D 401 065. The power of tDF
OtDF 4C5 tends to fall slightly with the size of Œ3 when Œ1 is
min
4C5 and OtDF 4C5 is even more erratic
large. The behavior of tDF
when Œ3 is xed and Œ1 increases. On the other hand, FTmax
has the desirable property in that its power always increases
with the size of Œ1 and Œ3 .
6. In all cases, the power of FTmax is greater than the power
min
4C5 and tODF 4C5 except when the magnitude of intercept
of tDF
break or slope break is relatively small. In these cases, the use
of FTmax does not result in large loss of power.
Given that the power of FTmax is larger than the mixed-model
min
4C5 and OtDF 4C5], it would to be interesting to
statistics [tDF
ascertain how much of the power of FTmax can be attributed to
the break in the trend, that is, Œ1 6D0 or Œ3 6D 0 or both. The
size of all statistics under the unit-root null hypothesis when
there is a break in the intercept and/or slope was calculated.
The results, available on request, indicate that the size of all
statistics is close to the nominal size when the break magnitudes are small. However, the size of FTmax increases with
the break magnitudes. In all cases, the size of FTmax is slightly
min
4C5 and tODF 4C5, and this difference
larger than the size of tDF
in size increases slightly with the break magnitudes.
Simulation evidence presented by Vogelsang and Perron
(1998) illustrated that all statistics considered within the IO
model framework retain their size and power properties if the
break in the trend function occurs according to the AO model.
While the break in the trend is abrupt and instantaneous within
the AO framework, the break in the trend is more gradual
within the IO framework. In unreported simulations, we calculated the nite sample size and power of all IO statistics
when the break occurs according to the AO model. Our results
indicate that, in general, the size and power properties of all
statistics remain the same whether the break occurs according
to the AO model or the IO model. However, in most cases, the
min
4C5, tODF 4C5, and FTmax is larger when the break
power of tDF
occurs according to the AO model compared to their power
when the break occurs according to the IO model. The only
exception occurs in the changing growth model simulation,
where the power of OtDF 4C5 levels off with an increase in the
size of the slope break. The AO model simulation results are
available from the author on request.
4.
EMPIRICAL RESULTS FOR NELSON
AND PLOSSER DATA
In this section, we revaluate the evidence regarding the presence of a unit-root in all Nelson and Plosser (1982) series,
except unemployment, and postwar U.S. quarterly real GNP.
The natural logarithms of all series were analyzed, except for
the interest rate series, which was analyzed in level form.
181
Empirical evidence for the Nelson–Plosser data can be found
in Zivot and Andrews (1992), Perron (1997), Nunes et al.
(1997), Murray and Zivot (1998), and Murray and Nelson
(2000). Although these studies do not prespecify the location
of break, they retain Perron’s (1989) characterization of the
form of break under the alternative hypothesis. In particular,
Perron (1989) specied the changing growth model for quarterly real GNP, the mixed model for real wages and common
stock prices, and the crash model for all remaining series.
However, Perron’s (1989) specication of the form of break
is based on his selection of the great crash of 1929 for all
Nelson–Plosser series and the oil price shock of 1973 for quarterly real GNP series as appropriate break-dates. We argue that
once the location of break is treated as unknown, the form of
break should also be treated as unknown. Therefore, one must
proceed with the most general characterization of the break
under the alternative hypothesis, namely, a simultaneous break
in the intercept and slope.
Empirical evidence for all Nelson and Plosser (1982) series
and quarterly real GNP when the mixed model is used as the
appropriate alternative is presented. The k4t-sig5 method of
Perron and Vogelsang (1992) is used to determine the appropriate order of the lag-truncation parameter in regression (10).
Following Zivot and Andrews (1992), k-max D 8 is used for
the Nelson–Plosser (1982) series and k-max D 12 for quarmin
4C5,
terly real GNP. Table 6 reports the calculated statistics tDF
OtDF 4C5, and FTmax for each series, and the corresponding estimin
bb 4OtDF 5, and
bb 4tDF
51 T
mated break-dates which are denoted by T
max
b
Tb 4FT 5, respectively. The results for regression (10) corresponding to the different estimated break-dates are presented
in Table 7. [The results for common stock prices and real
wages given in table 6 of Zivot and Andrews (1992) were
found to be incorrect. The estimated regression results for real
wages with Tb D 1940 and k D 8 yield the results reported
by Zivot and Andrews (1992), but the t statistic on the ãytƒ8
was found to be 1.2277, which is less than 1.6. Although
the estimated regression results for common stock prices with
Tb D 1936 and k D 1 yield the results reported by Zivot and
Andrews (1992), it was found that with k D 3 the t statistic
on ãytƒ3 is 1.96, which is greater than 1.6. Therefore, in both
cases the k4t-sig5 procedure for choosing the lag-truncation
parameter was not applied correctly.]
min
min
bb 4tDF
4C5 and T
5. If the
First, consider the results for tDF
min
4C5 given by Zivot
asymptotic critical values are used for tDF
and Andrews (1992), the unit-root null is rejected at the 1%
signicance level for real GNP, nominal GNP, and industrial production; at the 2.5% signicance level for real per
capita GNP, common stock prices, and real wages; at the 5%
signicance level for employment, nominal wages, and quarterly real GNP; and at the 10% signicance level for money
stock. The unit-root null for the GNP deator, consumer
prices, velocity, and interest rate series were not rejected. Of
the series for which the unit-root null is rejected, the estimin
bb 4tDF
5 does not coincide with the great
mated break-date T
crash (1929) for real per capita GNP, money stock, common
stock prices, and real wages. In addition, the estimated breakdate for quarterly real GNP does not coincide with the oil
price shock of 1973. The evidence against the unit-root null
182
Journal of Business & Economic Statistics, January 2003
Table 6. Empirical Results for the Nelson–Plosser (1982) Data and Quarterly Real GNP
Series
Downloaded by [Universitas Maritim Raja Ali Haji] at 00:39 13 January 2016
Real GNP
Nominal GNP
Real per capita GNP
Industrial production
Employment
GNP de‘ator
Consumer prices
Nominal wages
Money stock
Velocity
Interest rate
Common stock prices
Real wages
Quarterly real GNP
min
tDF
(C)
ƒ506577(a1c)
ƒ601741(a1b)
ƒ502983(b1 d)
ƒ508192(a1c)
ƒ501995(c10)
ƒ401723
ƒ305120
ƒ502147(c10)
ƒ409709(d1 0)
ƒ309737
ƒ108785
ƒ505152(b1 d)
ƒ504509(b1 d)
ƒ501135(c10)
b (t min )
T
b DF
tODF (C)
b (tO )
T
b DF
FTmax
Tb (FTmax )
1929
1929
1938
1929
1929
1929
1893
1929
1930
1929
1963
1939
1940
1964.IV
ƒ506577(b1b)
ƒ601741(a1a)
ƒ502983(c1 c)
ƒ508192(a1b)
ƒ501995(c1 d)
ƒ401723
ƒ201086
ƒ306338
ƒ409709(d1d)
ƒ309737
ƒ103131
ƒ505015(b1c)
ƒ504509(b1c)
ƒ501135(d1d)
1929
1929
1938
1929
1929
1920
1869
1920
1930
1929
1964
1936
1940
1964.IV
1009804(b1c)
1300122(a1 b)
1002524(b1d)
1200478(a1 b)
1002154(c1d)
805103
803915
909920(c1d)
807065
505704
908964(c1d)
1002942(b1d)
1103206(a1 c)
807649
1929
1929
1938
1929
1929
1920
1863
1920
1930
1929
1964
1939
1940
1964.IV
P ü
O 1 DU t (Tb ) C Œ
O 0 CŒ
O 2t C Œ
O 3 DTt (Tb ) C y
O t ƒ1 C kD1 cO j ãyt ƒj C eO t . The small letters in parentheses
NOTE: Mixed-model regression: yt D Œ
j
that appear as superscript indicate the signi’cance of these statistics. The ’rst letter in the parentheses indicates signi’cance with
respect to the asymptotic critical values, and the second letter indicates signi’cance with respect to the appropriate ’nite sample
critical values. a, b, c, and d indicate signi’cance at the 1%, 2.5%, 5%, and 10% signi’cance level, and . appears if the statistic is not
signi’cant at the 10% signi’cance level. The asymptotic and ’nite sample critical values of t min (C) are taken from table 4 in Zivot and
DF
Andrews (1992) and table 1 in Perron (1997), respectively. The asymptotic and ’nite sample critical values for OtDF (C ) were obtained
from table 3 in Vogelsang and Perron (1998). The asymptotic and ’nite sample critical values for FTmax were taken from Table 1 in this
article. ‹ 0 D 005 was used.
Table 7. Estimated Regressions for the Nelson–Plosser (1982) Data and Quarterly Real GNP
Series
Real GNP
Tb
kü
1929
8
O0
Œ
306921
(506850)
Nominal GNP
1929
8
504047
(602240)
Real per capita GNP
1938
2
302484
(503017)
Industrial production
1929
8
01339
(403925)
Employment
1929
8
501709
(502386)
GNP De‘ator
1929
5
06601
(402638)
1920
5
02720
(204134)
Consumer prices
1893
5
03813
(303150)
1869
0
03109
(307658)
Nominal wages
1929
7
201348
(502841)
1920
7
09733
(304607)
Money stock
1930
8
04831
(504325)
Velocity
1929
1
03540
(306305)
Interest rate
1963
3
04049
(108000)
1964
0
02403
(102593)
Common stock prices 1939
1
05320
(502391)
1936
3
05743
(502399)
Real wages
1940
3
108163
(504785)
Quarterly real GNP
1964.IV 12 103289
(501524)
O1
Œ
O2
Œ
O3
Œ
ƒ01793
00241
00045
(ƒ402164)
(401459)
(100059)
ƒ02799
00224
00115
(ƒ405282)
(300520)
(107811)
01326
00010
00072
(400687)
(07412)
(301672)
ƒ03239
00331
00012
(ƒ501854)
(506933)
(09001)
ƒ00747
ƒ00015
00098
(ƒ308435)
(405430) (ƒ106187)
ƒ00899
00061
00010
(301706)
(306491)
(100358)
ƒ00997
ƒ00026
00060
(402269)
(400137) (ƒ200290)
ƒ00205
ƒ00009
00031
(ƒ101960) (ƒ08578)
(202153)
ƒ00077
ƒ00193
00204
(ƒ02035) (ƒ208023)
(209934)
ƒ01640
ƒ00004
00177
(ƒ308359)
(404476) (ƒ01847)
ƒ01516
ƒ00133
00211
(ƒ403287)
(404541) (ƒ300775)
ƒ01053
ƒ00023
00212
(ƒ304871)
(406864) (ƒ109341)
ƒ00473
ƒ00041
00060
(ƒ107643) (ƒ303186)
(306149)
ƒ00231
ƒ00016
01594
(ƒ00987) (ƒ07641)
(208927)
ƒ01287
ƒ00004
02157
(ƒ05393) (ƒ01989)
(304392)
ƒ01942
00078
00261
(ƒ205958)
(407365)
(406067)
ƒ02783
00090
00252
(ƒ306409)
(408941)
(409949)
00842
00086
00047
(403846)
(503509)
(303869)
ƒ00001
00171
00015
(401128)
(408642) (ƒ107395)
O
02352
(ƒ506577)
05057
(ƒ601741)
05490
(ƒ502983)
03005
(ƒ508192)
04914
(ƒ501995)
07831
(ƒ401723)
09042
(ƒ206100)
08921
(ƒ305120)
09528
(ƒ201086)
06581
(ƒ502147)
08357
(ƒ306338)
06893
(ƒ409709)
07635
(ƒ309737)
09047
(ƒ108785)
09421
(ƒ103131)
06038
(ƒ505152)
05702
(ƒ505015)
03892
(ƒ504509)
08218
(ƒ501135)
O
S(e)
.0503
.0680
.0522
.0876
.0289
.0435
.0418
.0361
.0520
.0542
.0535
.0419
.0636
.2481
.2475
.1397
.1392
.0307
.0086
P ü
O 1 DU t (Tb ) C Œ
O 0 CŒ
O 2t C Œ
O 3 DTt (Tb ) C y
O t ƒ1 C kD1 cO j ãyt ƒj C eO t . Tb is the chosen break-date,
NOTE: Mixed-model regression: yt D Œ
j
k ü is the value of the lag-truncation parameter chosen according to the k (t -sig) procedure of Perron and Vogelsang (1992) with k O tests
max D 8 for all series except quarterly real GNP, for which k -max D 12. The t statistics appear in parentheses. The t statistic for
O is the estimated standard deviation of the regression error.
the hypothesis that D 1. S(e)
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Sen: Unit-Root Tests
is weakened if the nite sample critical values given by Permin
4C5 with lag-truncation parameter
ron (1997) are used for tDF
selected using the k4t-sig5 method. With the nite sample critical values, the unit-root null for employment, nominal wages,
money stock, and quarterly real GNP cannot be rejected.
Because a break is not allowed under the unit-root null
hypothesis, these results should be compared to the results
of Zivot and Andrews (1992). The unit-root null hypothesis is rejected here for money stock, real wages, and quarterly real GNP in addition to all series for which Zivot and
Andrews (1992) rejected the unit-root null. Further, the present
evidence is stronger for real per capita GNP. Unlike that of
Zivot and Andrews (1992), the break-date estimated here for
real per capita GNP is 1938, for money stock is 1930, for
common stock prices is 1939, and for quarterly real GNP is
1964, fourth quarter. The estimated break-date coincides with
the great crash of 1929 for real GNP, nominal GNP, industrial production, employment, and nominal wages. In cases for
which the unit-root null is rejected, one would like to determine the signicance of the trend function coefcients Œ0 ,
min
bb 4tDF
5. Given that the location
Œ1 , Œ2 , and Œ3 evaluated at T
min
bb 4tDF 5, the asymptotic distribution
of break is estimated by T
of t statistics for H0 2 Œi D 0, denoted by tŒO i , (i D 01 11 21 3)
may not be well approximated by the standard normal distribution. In unreported simulations, the empirical distribution of
tŒO i (i D 01 11 21 3) corresponding to the estimated regression at
bb 4t min 5 was examined. The results indicate that the empiriT
DF
cal distributions of tŒO i (i D 01 21 3) are symmetric, have mean
close to zero, but have a slightly larger variance compared to
the standard normal distribution. On the other hand, the empirical distribution of tŒO 1 is bimodal with a much larger variance
compared to the standard normal distribution. [It was found
that the empirical distributions of tŒO i (for i D 01 11 21 3) evalubb 4OtDF 5 and T
bb 4FTmax 5 exhibit similar behavior.] Results
ated at T
pertaining to the empirical distribution of tŒO i (i D 01 11 21 3) are
available on request. As a rst approximation, the signicance
of the intercept coefcient (Œ0 ), the trend coefcient (Œ2 ), and
the slope break coefcient (Œ3 ) using the critical values from
a standard normal distribution are assessed. [However, the signicance of the intercept break coefcient (Œ1 ) using the standard normal critical values is not determined.] The estimated
coefcients and their respective t statistics are presented in
Table 7. It was found that tŒO 0 and tŒO 2 are signicant for all
series, with the sole exception of Œ2 for real per capita GNP,
and tŒO 3 is signicant for nominal GNP, real per capita GNP,
money stock, common stock prices, real wages, and quarterly
real GNP.
Second, the results for tODF 4C5 are qualitatively similar to
min
4C5. The critical values for OtDF 4C5, both asympthose of tODF
totic and nite sample with lag-truncation parameter chosen
according to the k4t-sig5 method, can be found in Vogelsang
and Perron (1998). Based on the asymptotic critical values, the
unit-root null is rejected at the 1% signicance level for nominal GNP and industrial production; at the 2.5% signicance
level for real GNP, common stock prices, and real wages; at
the 5% signicance level for real per capita GNP and employment; and at the 10% signicance level for money stock and
quarterly real GNP. The unit-root null cannot be rejected for
GNP deator, consumer prices, nominal wages, velocity, and
183
interest rate. The results for all series are in agreement with
min
4C5, with the exception of
the results corresponding to tDF
nominal wages. The estimated break-date Tbb 4tODF 5 is the same
min
bb 4tDF
5 for all series for which the unit-root is rejected,
as T
except common stock prices, for which the estimated breakdate is 1936. The use of nite sample critical values weakens
the evidence against the unit-root null somewhat.
Finally, consider the results with FTmax . The critical values
for FTmax , both asymptotic and nite sample, are obtained from
Table 1. Based on the asymptotic critical values, the joint null
hypothesis can be rejected at the 1% signicance level f