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In this paper, we propose a new augmented Dickey–Fuller-type test for unit roots which accounts for two structural breaks. We consider two different specifications: (a) two breaks in the level of a trending data series and (b) two breaks in the level and slope of a trending data series. The breaks whose time of occurrence is assumed to be unknown are modeled as innovational outliers and thus take effect gradually. Using Monte Carlo simulations, we showthat our proposed test has correct size, stable power, and identifies the structural breaks accurately.<br /
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Faculty of Business and Law
School of Accounting, Economics and Finance
ECONOMICS SERIES
SWP 2009/11
A New Unit Root Test with Two Structural
Breaks in Level and Slope at Unknown Time
Paresh Kumar Narayan and Stephan Popp
The working papers are a series of manuscripts in their draft form. Please do not
quote without obtaining the author’s consent as these works are in their draft form.
The views expressed in this paper are those of the author and not necessarily
endorsed by the School or IBISWorld Pty Ltd.
A New Unit Root Test with Two Structural
Breaks in Level and Slope at Unknown Time
Paresh Kumar Narayan, School of Accounting,
Economics and Finance, Deakin UNiversity.
Stephan Popp, Department of Economics,
University of Duisburg-Essen, Germany.
Abstract
In this paper we propose a new ADF-type test for unit roots which
accounts for two structural breaks. We consider two di¤erent speci…-
cations: (a) two breaks in the level of a trending series; and (b) two
breaks in the level and slope of trending data. The breaks whose time
of occurance is assumed to be unknown are modelled as innovational
outliers and thus take e¤ect gradually. Using Monte Carlo simula-
tions, we show that our proposed test has correct size, stable power,
and identi…es the structural breaks accurately.
1
1 Introduction
The unit root hypothesis has both theoretical and empirical implications for
economic theory and modelling. This is one reason for the popularity of
unit root tests and a key motivation for methodological innovations. Perron
(1989) showed that ignoring a structural break, as is the case with the Dickey
and Fuller (DF), can lead to the false acceptance of the unit root null hypoth-
esis. The e¤ect of structural breaks on the performance of the DF unit root
test is discussed intensively in the literature. This branch of the literature
emphasizes the power reductions of the DF-test if a break occurs under the
alternative hypothesis (see, for instance, Perron, 1989; and Rappoport and
Reichlin, 1989). In order to handle this problem, Perron (1989) augments
the ADF test regression with dummy variables accounting for the break.
In this paper our goal is to extend the literature on unit root tests with
structural breaks. Extensions of Perron (1989) have been made by Zivot and
Andrews (ZA, 1992) and Perron (1997), inter alia, through accounting for
an endogenous structural break, and by Lumsdaine and Papell (LP, 1997)
through accounting for two structural breaks. However, Lee and Strazicich
(LS, 2001, 2003) show that these ADF-type unit root tests which either do
not allow for a break under the null as ZA and LP or model the break
as an innovational outlier (IO) as Perron (1997) su¤er from severe spurious
2
rejections in …nite samples when a break is present under the null hypothesis.
Because the spurious rejections are not present in the case of a known break
point, LS (2001) identify the inaccurate estimation of the break date as
source of the spurious rejections. Judging it di¢ cult to …nd a convenient
remedy to the problem of spurious rejections for ADF-type unit root tests,
LS (2003, 2004) follow a di¤erent route by proposing a minimum Lagrange
Multiplier (LM) unit root test which do not su¤er from spurious rejections of
this kind. Though, Popp (2008) has pointed out that these spurious rejections
are not a general feature of ADF-type unit root tests. Rather, the root of the
problem of spurious rejections is that the parameters of the test regression
have di¤erent interpretations under the null and alternative hypothesis, cf.
Schmidt and Phillips (SP, 1992), which is crucial since the parameters have
implications for the selection of the structural break date. Following SP
(1992), this can be avoided by formulating the data generating process (DGP)
as an unobserved components model which allows us to generate a new ADF-
type unit root test for the case of IOs. An interesting feature of the new test
is that the critical values of the test assuming unknown break dates converges
with increasing sample size to the critical values when the break points are
known.
We organise the balance of the paper as follows. In section 2, we discuss
our proposed new test. In section 3, we assess the size and power properties
3
of our test. Because the spurious rejections are a feature especially in …nite
samples, we show the favorable properies of the new test by Monte Carlo
simulations. In section 4, we demonstrate the applicability of our new test
using the Nelson and Plosser dataset and an up-dated post-war dataset that
includes 32 macroeconomic data series for the USA. In section 5, we provide
some concluding remarks.
2 Models and test statistics
Following SP (1992), we consider an unobserved components model to rep-
resent the DGP. The DGP of a time series ythas two components, a deter-
ministic component (dt)and a stochastic component (ut), as follows:
yt=dt+ut;(1)
ut=ut1+"t;(2)
"t= (L)et=A(L)1B(L)et;(3)
with etsiid(0; 2
e). It is assumed that the roots of the lag polynomials
A(L)and B(L)which are of order pand q, respectively, lie outside the unit
circle.
We consider two di¤erent speci…cations both for trending data: one allows
4
for two breaks in level (denoted model 1 or M1) and the other allows for
two breaks in level as well as slope (denoted model 2 or M2). Both model
speci…cations di¤er in how the deterministic component dtis dened:
dM1
t=+t + (L)1DU 0
1;t +2DU 0
2;t;(4)
dM2
t=+t + (L)1DU 0
1;t +2DU 0
2;t +1DT 0
1;t +2DT 0
2;t;(5)
with
DU 0
i;t = 1(t > T 0
B;i); DT 0
i;t = 1(t > T 0
B;i)(tT0
B;i); i = 1;2:(6)
Here, T0
B;i,i= 1;2, denote the true break dates. The parameters iand
iindicate the magnitude of the level and slope breaks, respectively. The
inclusion of (L)in (4) and (5) enables the breaks to occur slowly over
time. Speci…cally, it is assumed that the series responds to shocks to the
trend function the way it reacts to shocks to the innovation process et, see
Vogelsang and Perron (VP, 1998). This approach is called the IO model.
The IO-type test regressions for M1 and M2 to test the unit root null
hypothesis can be derived by merging the structural model (1)-(5). The test
5
equation for M1 has the following form:
yM1
t=yt1+1+t+1D(T0
B)1;t +2D(T0
B)2;t +
+1DU 0
1;t1+2DU 0
2;t1+
k
X
j=1
jytj+et;(7)
with 1= (1)1[(1 )+]+0(1)1(1 ),0 (1)1being the
mean lag, = (1)1(1 ),=1,i=iand D(T0
B)i;t = 1(t=
T0
B;i + 1),i= 1;2.
The IO-type test regression for M2 is as follows:
yM2
t=yt1++t+1D(T0
B)1;t +2D(T0
B)2;t +
1DU 0
1;t1
+
2DU 0
2;t1+
1DT 0
1;t1+
2DT 0
2;t1+
k
X
j=1
jytj+et;(8)
where i= (i+i)
i= (ii)and
i=i,i= 1;2.
In order to test the unit root null hypothesis of = 1 against the alter-
native hypothesis of  < 1, we use the t-statistics of ^, denoted t^, in (7) and
(8).
It is worth noting that in contrast to the well-known Perron-type test
regressions for the one break case, see e.g. equations (5.1) and (5.2) in VP
(1998), the dummy variables DU 0
i;t and DT 0
i;t are lagged in (7) and (8). How-
ever, for given break dates, both the Perron-type test regressions (augmented
6
to two breaks) and the test regressions formulated in (7) and (8) produce
identical t-values t^.1Despite this fact, we favor the use of (7) and (8) be-
cause the coe¢ cients of the impulse dummy variable D(T0
B)i;t,ifor M1 and
ifor M2, solely comprise the break parameters iand i. This is essential
in the situation of an unknown break date in which we want to identify the
timing of the break on the basis of estimates of the break parameters.
Because we assume that the true break dates are unknown, T0
B;i in equa-
tions (7) and (8) has to be substituted by their estimates ^
TB;i,i= 1;2, in
order to conduct the unit root test. The break dates can be selected si-
multaneously following a grid search procedure. Therefor, we conduct the
test regressions for every potential break point combination (TB;1; TB;2) and
choose that points in time as break dates for which the joint signi…cance of
the impulse dummy variable coe¢ cients is maximised, i.e.
^
TB;1;^
TB;2=
8
>
>
<
>
>
:
arg max F^
1;^
2, for model 1
arg max F^1;^2, for model 2
:(9)
Alternatively, we use a sequential procedure comparable to Kapetanios
(2005). In a …rst step, we search for a single break which we select according
to the maximum absolute t-value of the break dummy coe¢ cient 1for M1
1So, the asymptotic results for the Perron-test in the case of a known break date apply
also to the new test.
7
and 1for M2 under the restriction of 2=2= 0 for M1 and 2=
2=
2= 0 for M2:
^
TB;1=
8
>
>
>
<
>
>
>
:
arg max
TB;1
jt^
1(TB;1)j, for model 1
arg max
TB;1
jt^1(TB;1)j, for model 2
:(10)
So, in the …rst step, the test procedure reduces to the case described in
Popp (2008). Under the restriction of the …rst break ^
TB;1, we estimate the
second break date ^
TB;2analogously to the …rst break. The results of the
simultaneous and the sequential procedure do not di¤er much. So, we prefer
the sequential procedure because it is far less computationally intensive. In
the grid search case, we compute the test statistic approximately T2times
compared to approximately 2Tfor the sequential procedure.
As discussed intensively by VP (1998) for the one break case, the Perron-
type test statistics are invariant under the null hypothesis to a break in
level and slope asymptotically as well as in …nite samples when the break
point is known. Because, as mentioned above, the procedure proposed by
VP (1998) generalized to the two break case and the new procedure are
identical for known break dates, the invariance results apply to the new unit
root test. However, when the break dates are unknown, the invariance to
level shifts for the Perron-type test no longer holds in …nite samples leading
8
to considerable spurious rejections of the unit root null hypothesis, see VP
(1998) and LS (2001). Moreover, the Perron-type test statistic capable of
trend breaks is no longer invariant to breaks in slope neither in …nite samples
nor asymptotically, see VP (1998). In contrast, the invariance to level and
slope breaks holds for the minimum LM unit root test proposed by LS (2003).
Because the spurious rejection property of existent ADF-type tests is
primarily a problem in …nite samples and for this reason a major drawback
of their applicability, one main goal of the present paper is to show that the
new ADF-type test are (approximately) invariant to level and slope breaks
in …nite samples by means of Monte Carlo simulations whose results are
summarized in the following section.
3 Monte Carlo simulation results
All simulations were carried out in GAUSS 8.0. The series ytis generated
according to (1)-(3) togerther with (4) for M1 and (5) for M2 assuming the
innovation process etto be standard normally distributed, etn:i:d:(0;1).
For et, samples of size T+ 50 are generated, of which the …rst 50 observa-
tions are then discarded. Because our main focus is on the e¤ect of varying
break magnitudes on the test performance, we adopt the assumption made
in comparable studies by VP (1998), Harvey et al. (2001) and LS (2001) and
9
set (L) = 1. The tests are conducted using (7) and (8) always assuming
the appropriate lag order of k= 0 to be known.
3.1 Critical values
The critical values (CVs) are based on 50000 replications. For the M1- and
M2-type tests, we calculate the CVs at the 1 per cent, 5 per cent, and 10
per cent levels for both the case of known and unknown break dates which
we denote CVexo and CVendo, respectively. We generate CVs for sample
sizes of T= 50,100,300, and 500. All CVs are calculated assuming no
break, i.e. i= 0 in (4) for M1 and i=i= 0 in (5) for M2, i= 1;2. For
the case of known break dates, we generate the dummy variables in (7) and
(8) according to T0
B;i = [0
iT],i= 1;2,[:]: greatest integer function, with
the break fraction 0= (0
1; 0
2) = (0:2;0:4),(0:2;0:6),(0:2;0:8),(0:4;0:6),
(0:4;0:8), and (0:6;0:8). For the case of unknown break points, we determine
the break dates assuming that there exist two periods for M1 and three
periods for M2 between the …rst and second break. The CVs for the case of
known break dates are reported in Tables 1 for M1 and 2 for M2 and in the
case of unknown break dates in Table 3.
It can be observed that CVexo vary only slightly with the break fraction
0and that the CVexos for di¤erent break fractions converge as Tincreases
10
from T= 50 to T= 500. Furthermore, it can be seen that CVexo converges
sharply to CVendo for the respective model with the sample size. This feature
can be motivated in the following way. If the unit root test for unknown break
dates is invariant to the break magnitude and the probability of detecting
the true break dates goes to 1 with increasing break magnitude, i.e. for
su¢ ciently large breaks we always identify the break dates correctly which
corresponds to the situation of knowing the break dates, the distribution of
the test statistic for known break dates has to coincide with the distribution
of the test statistic for unknown break dates and consequently CVendo is
equal to CVexo.
Both the break dates estimation accuracy and the invariance to level and
slope breaks will be shown in the next subsection for the new unit root test.
3.2 Finite sample size
Because of the great computational burden, the simulations of the empirical
size and power are based only on 5000 replications. For the size and power
simulations, is set to 1 and 0.9, respectively. The results for the size e¤ects
are reported in Tables 4 and 5 for models M1 and M2, respectively. We
calculate the empirical size and power for the case of 0= (0:4;0:6) and
sample sizes of T= 50,100,300, and 500. We also generate results for
11
various combinations of the break fractions 0
1and 0
2using CVendo in Table
3 which turn out to be qualitatively equal.2This is evidence that the unit
root test for unknown break points do not depend considerably on the break
fraction parameters in …nite samples.
We calculate the empirical size and power of the new unit root test for the
case where the true break date is exogenously given (denoted ’exo’in Table
4 and 5) and for unknown break dates where we detect the break dates
endogenously (denoted ’endo’). Furthermore, because of the relationship
between CVexo and CVendo, we use CVexo for test decision in the unknown
break dates case (denoted ’endoCVexo’). Thereby, we are able to show the
correspondence of CVexo and CVendo.
The performance of the new test for M1 and M2 are similar. In the case of
the exogenous break test the empirical size is independent of the magnitude
of the breaks close to the nominal 5 per cent level proving the invariance
to level and slope break for known break dates. The empirical size of the
endogenous break test is also close to the nominal 5 per cent level in the
case of a small break, but as the break magnitude increases the empirical
size decreases slightly. The endogenous break test using CVexo, however, is
a little bit oversized for small breaks and small sample sizes, but the size
2Due to space considerations, we only report results for the case 0= (0:4;0:6); the
rest of the results are available from the authors upon request.
12
converges to the 5 per cent nominal level with increasing break and sample
size. The ability of the test to identify both breaks simultaneously is high
even for medium sized breaks. Because we assume the realistic case of a …xed
break size (independent of the sample size T), the probability decreases with
the sample size as can be expected.
3.3 Empirical power
The empirical power of M1 and M2 are reported in the second half of Table
4 and in Table 6, respectively. The power of the exogenous break test and
the endogenous break test do not di¤er substantially. This means that the
additional information about the timing of the break do not augment the
power of the test considerably. This is in contrast to the statement of Perron
(1997) that a procedure imposing no a priori information with respect to the
choice of the break date has relatively low power.
Moreover, the power of the test converges to 100 per cent with increasing
sample size showing the consistency of the test. The results also reveal that
the probability of detecting the true break date goes rapidly to 100 per cent
with increasing break magnitude.
13
4 Application
In this section, we demonstrate the applicability of our proposed new models
M1 and M2. We use two datasets on the US macroeconomic variables. The
rst dataset is the famous and commonly used Nelson and Plosser dataset.
The second dataset is one that we compile from the International Financial
Statistics, published by the International Monetary Fund.
There are two main di¤erences between the Nelson and Plosser dataset
and our new dataset. First, the Nelson and Plosser dataset considers data
that includes the World War period, while our new dataset considers data
in the post-war period. Our dataset is also the most up-to-date: the Nelson-
Plosser dataset ends in 1970 while our dataset ends in either 2006 or in most
cases 2007. It follows that the new dataset captures the most recent (over
the last three to four decades) developments in the US economy, which may
have implications for unit root testing. In any case, our aim here is not to
draw on the economic theory that motivates a test for a unit root, rather it
is to merely demonstrate the applicability of our test. The second di¤erence
is that Nelson and Plosser consider only 14 macroeconomic series, while the
new dataset allows us to test for unit roots in 32 macroeconomic variables.
We begin with a discussion of results obtained from the Nelson and Plosser
dataset. The results are reported in Table 7. Results from M1 reveal that
14
we are able to reject the unit root null hypothesis for GNP at the 1 per
cent level, and for industrial production and the unemployment rate at the
10 per cent level. Finally, results from M2 reveal that we are able to reject
the unit root null hypothesis for real GNP, industrial production, and real
wage rates, all at the 5 per cent level. Taken together, results from our two
models are able to reject the unit root null hypothesis for six out of 14 series,
representing about 43 per cent of the variables considered.
In Table 8, we report results from our new dataset. All data series are
converted into logarithmic form before the empirical analysis. The presenta-
tion of results is as follows. Column 1 lists the data series, column 2 contains
results for M1, while column 3 contains results from the M2 model. For each
of these respective models, test statistics for the null of a unit root, structural
breaks, and optimal lag lengths are presented. The optimal lag length kis
obtained by using the procedure suggested by Hall (1994).
Beginning with the M1 model, we …nd that we are able to reject the unit
root null hypothesis for the unemployment rate, exports, the mortgage rate,
and the export price index at the 10 per cent level, and for the T-bill rate at
the 1 per cent level.
In the case of M2, we …nd that we are able to reject the unit root null
hypothesis for M2, industrial production, the PPI (for capital equipment)
and consumer goods at the 10 per cent level, for the unemployment rate,
15
mortgage rate and the NASDAQ index at the 5 per cent level, and for the
CPI and the T-Bill rate at the 1 per cent level.
In sum, we …nd that based on models M1 and M2, we are able to reject
the unit root null hypothesis for 13 of the 32 series. This represents about
41 per cent of the US macroeconomic series considered here. It is worth
highlighting here that it is up to the applied researcher to choose the best
model, which, in our view, should be dictated by economic theory.
5 Concluding remarks
In this paper, we proposed a new test for unit roots that is ‡exible enough
to allow for at most two structural breaks in the level and trend of a data
series. More speci…cally, we considered two di¤erent models for trending
data: model 1 allows for two breaks in the level of the series and model 2
accounts for two breaks in the level and slope.
The key features of our test are that it is a ADF-type innovational out-
lier unit root test for which we specify the data generating process as an
unobserved components model, and breaks are allowed under both the null
and alternative hypotheses. Using Monte Carlo simulations, we showed that
our proposed test has correct size, stable power, and identi…es the structural
breaks accurately.
16
We demonstrated the applicability of our unit root test through under-
taking two exercises: one based on the Nelson and Plosser dataset and the
other based on an updated post-war dataset. Using the new dataset, we
found that tests based on models 1 and 2 taken together were able to reject
the unit root null hypothesis for 13 of the 32 US macroeconomic series.
17
References
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Harvey, D., S. Leybourne, and P. Newbold (2001): “Innovational
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Kapetanios, G. (2005): Unit-root testing against the alternative hypoth-
esis of up to m structural breaks,”Journal of Time Series Analysis, 26(1),
123–133.
Lee, J., and M. Strazicich (2001): “Break Point Estimation and Spu-
rious Rejections with Endogenous Unit Root Tests,” Oxford Bulletin of
Economics and Statistics, 63(5), 535558.
(2003): Minimum Lagrange Multiplier Unit Root Test With Two
Structural Breaks,”The Review of Economics and Statistics, 85(4), 1082–
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(2004): “Minimum LM Unit Root Test With One Structural Break,”
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versity.
18
Lumsdaine, R., and D. Papell (1997): “Multiple Trend Break and the
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19
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20
Table 1: 1%, 5% and 10% critical values for exogenous two break test, Model
1, 50000 replications
2= 0:42= 0:62= 0:8
T 11% 5% 10% 1% 5% 10% 1% 5% 10%
50 0:2-4.953 -4.194 -3.826 -4.842 -4.127 -3.777 -4.895 -4.178 -3.827
0:4- - - -4.850 -4.148 -3.780 -4.872 -4.145 -3.788
0:6- - - - - - -4.922 -4.191 -3.823
100 0:2-4.760 -4.113 -3.787 -4.738 -4.077 -3.733 -4.761 -4.112 -3.785
0:4- - - -4.745 -4.078 -3.741 -4.715 -4.087 -3.743
0:6- - - - - - -4.736 -4.112 -3.785
300 0:2-4.664 -4.073 -3.770 -4.615 -4.037 -3.727 -4.642 -4.051 -3.754
0:4- - - -4.620 -4.036 -3.721 -4.621 -4.039 -3.724
0:6- - - - - - -4.650 -4.067 -3.754
500 0:2-4.640 -4.069 -3.759 -4.612 -4.024 -3.728 -4.624 -4.064 -3.755
0:4- - - -4.600 -4.024 -3.713 -4.603 -4.023 -3.717
0:6- - - - - - -4.611 -4.058 -3.755
Table 2: 1%, 5% and 10% critical values for exogenous two break test, Model
2, 50000 replications
2= 0:42= 0:62= 0:8
T 11% 5% 10% 1% 5% 10% 1% 5% 10%
50 0:2-5.401 -4.609 -4.221 -5.635 -4.866 -4.501 -5.390 -4.616 -4.231
0:4- - - -5.591 -4.876 -4.499 -5.645 -4.882 -4.507
0:6- - - - - - -5.380 -4.631 -4.251
100 0:2-5.232 -4.577 -4.237 -5.404 -4.768 -4.450 -5.252 -4.602 -4.252
0:4- - - -5.430 -4.782 -4.457 -5.387 -4.784 -4.462
0:6- - - - - - -5.246 -4.574 -4.231
300 0:2-5.135 -4.537 -4.224 -5.276 -4.720 -4.421 -5.163 -4.557 -4.236
0:4- - - -5.279 -4.724 -4.420 -5.297 -4.712 -4.418
0:6- - - - - - -5.140 -4.549 -4.238
500 0:2-5.125 -4.541 -4.233 -5.251 -4.699 -4.410 -5.126 -4.544 -4.239
0:4- - - -5.273 -4.712 -4.415 -5.271 -4.712 -4.409
0:6- - - - - - -5.136 -4.534 -4.219
21
Table 3: 1%, 5% and 10% critical values for endogenous two break test
(computed under the assumption of no breaks), 50000 replications
M1 M2
T1% 5% 10% 1% 5% 10%
50 -5.259 -4.514 -4.143 -5.949 -5.181 -4.789
100 -4.958 -4.316 -3.980 -5.576 -4.937 -4.596
300 -4.731 -4.136 -3.825 -5.318 -4.741 -4.430
500 -4.672 -4.081 -3.772 -5.287 -4.692 -4.396
22
Table 4: 5 percent rejection frequency with nominal 5 percent signi…cance level and probability of detecting the true
break date, M1, 0= (0:4;0:6), 5000 replications
empirical size (= 1) empirical power (= 0:9)
T  exo endo endoCVexo P(^
TB=T0
B)exo endo endoCVexo P(^
TB=T0
B)
50 0 0.050 0.050 0.089 0.003 0.067 0.068 0.128 0.003
50 3 0.045 0.042 0.073 0.498 0.066 0.061 0.102 0.474
50 5 0.047 0.027 0.051 0.970 0.068 0.043 0.077 0.955
50 10 0.045 0.024 0.045 1.000 0.059 0.028 0.059 1.000
50 20 0.046 0.023 0.046 1.000 0.066 0.031 0.066 1.000
100 0 0.050 0.050 0.083 0.000 0.147 0.136 0.199 0.000
100 3 0.047 0.038 0.063 0.411 0.133 0.102 0.155 0.397
100 5 0.055 0.034 0.057 0.969 0.139 0.087 0.140 0.960
100 10 0.050 0.030 0.050 1.000 0.139 0.083 0.139 1.000
100 20 0.055 0.031 0.055 1.000 0.133 0.083 0.133 1.000
300 0 0.050 0.050 0.061 0.000 0.789 0.762 0.801 0.000
300 3 0.048 0.044 0.055 0.278 0.789 0.690 0.737 0.284
300 5 0.048 0.039 0.049 0.943 0.791 0.733 0.781 0.938
300 10 0.046 0.036 0.046 1.000 0.775 0.731 0.775 1.000
300 20 0.045 0.037 0.045 1.000 0.780 0.733 0.780 1.000
500 0 0.050 0.050 0.052 0.000 0.998 0.997 0.998 0.000
500 3 0.052 0.051 0.056 0.236 0.999 0.992 0.993 0.218
500 5 0.046 0.044 0.047 0.921 0.997 0.994 0.994 0.919
500 10 0.048 0.045 0.048 1.000 0.998 0.998 0.998 1.000
500 20 0.050 0.047 0.050 1.000 0.999 0.998 0.999 1.000
23
Table 5: 5 percent rejection frequency with nominal 5 percent signi…cance
level and probability of detecting the true break date, M2, 0= (0:4;0:6),
5000 replications
empirical size (= 1)
T   exo endo endoCVexo P(^
TB=T0
B)
50 0 0 0.050 0.050 0.092 0.003
50 0 5 0.048 0.033 0.053 0.455
50 0 10 0.053 0.026 0.049 0.893
50 5 0 0.052 0.026 0.057 0.948
50 5 5 0.050 0.026 0.050 1.000
50 5 10 0.053 0.026 0.053 1.000
50 10 0 0.051 0.023 0.051 1.000
50 10 5 0.054 0.028 0.054 1.000
50 10 10 0.053 0.025 0.053 1.000
100 0 0 0.050 0.050 0.069 0.001
100 0 5 0.051 0.068 0.079 0.292
100 0 10 0.049 0.029 0.040 0.763
100 5 0 0.051 0.036 0.051 0.955
100 5 5 0.047 0.033 0.047 1.000
100 5 10 0.049 0.037 0.048 1.000
100 10 0 0.049 0.037 0.049 1.000
100 10 5 0.053 0.038 0.053 1.000
100 10 10 0.050 0.038 0.050 1.000
300 0 0 0.050 0.050 0.056 0.000
300 0 5 0.059 0.091 0.093 0.097
300 0 10 0.055 0.031 0.035 0.405
300 5 0 0.057 0.052 0.058 0.940
300 5 5 0.055 0.050 0.055 1.000
300 5 10 0.056 0.050 0.056 1.000
300 10 0 0.052 0.044 0.052 1.000
300 10 5 0.051 0.045 0.051 1.000
300 10 10 0.058 0.050 0.058 1.000
500 0 0 0.050 0.050 0.057 0.000
500 0 5 0.049 0.069 0.071 0.057
500 0 10 0.051 0.017 0.021 0.250
500 5 0 0.055 0.045 0.055 0.920
500 5 5 0.048 0.040 0.048 0.999
500 5 10 0.056 0.049 0.056 1.000
500 10 0 0.054 0.046 0.054 1.000
500 10 5 0.053 0.047 0.053 1.000
500 10 10 0.053 0.045 0.053 1.000
24
Table 6: Empirical power of the M2 model
empirical power (= 0:9)
T   exo endo endoCVexo P(^
TB=T0
B)
50 0 0 0.060 0.055 0.104 0.003
50 0 5 0.063 0.042 0.065 0.414
50 0 10 0.067 0.029 0.059 0.871
50 5 0 0.064 0.041 0.074 0.931
50 5 5 0.063 0.032 0.063 1.000
50 5 10 0.065 0.030 0.065 1.000
50 10 0 0.070 0.031 0.070 1.000
50 10 5 0.057 0.026 0.057 1.000
50 10 10 0.066 0.035 0.066 1.000
100 0 0 0.089 0.105 0.136 0.002
100 0 5 0.087 0.087 0.100 0.246
100 0 10 0.090 0.050 0.069 0.729
100 5 0 0.095 0.072 0.098 0.942
100 5 5 0.085 0.062 0.084 1.000
100 5 10 0.100 0.072 0.100 1.000
100 10 0 0.093 0.068 0.093 1.000
100 10 5 0.090 0.063 0.090 1.000
100 10 10 0.095 0.070 0.095 1.000
300 0 0 0.587 0.570 0.597 0.000
300 0 5 0.594 0.229 0.240 0.075
300 0 10 0.592 0.287 0.304 0.331
300 5 0 0.582 0.545 0.574 0.932
300 5 5 0.595 0.565 0.594 0.998
300 5 10 0.590 0.560 0.590 1.000
300 10 0 0.591 0.561 0.591 1.000
300 10 5 0.586 0.559 0.586 1.000
300 10 10 0.598 0.567 0.598 1.000
500 0 0 0.974 0.958 0.970 0.000
500 0 5 0.974 0.339 0.352 0.036
500 0 10 0.976 0.459 0.471 0.197
500 5 0 0.968 0.952 0.959 0.911
500 5 5 0.974 0.966 0.974 0.997
500 5 10 0.974 0.964 0.974 0.999
500 10 0 0.974 0.969 0.974 1.000
500 10 5 0.972 0.963 0.972 1.000
500 10 10 0.971 0.964 0.971 1.000
25
Table 7: Results of two-break unit root test, Nelson-Plosser data
M1 M2
Nr. Series Sample Ttest statistic TB1 TB2 k test statistic TB1 TB2 k
1 Real GDP 1909-1970 62 -3.680 1929 1931 1 -5.597 1921 1938 2
2 Nominal GNP 1909-1970 62 -6.396 1929 1941 1 -3.705 1921 1940 1
3 Real per Capita GNP 1909-1970 62 -3.491 1929 1931 1 -5.529 1921 1938 2
4 Industrial Production 1860-1970 111 -4.3101920 1931 0 -4.632 1920 1931 3
5 Employment 1890-1970 81 -2.002 1931 1945 1 -2.145 1931 1945 0
6 Unemployment 1890-1970 81 -4.1301917 1922 3 -3.703 1917 1923 3
7 GNP De‡ator 1889-1970 82 -2.777 1916 1920 5 -2.749 1916 1920 5
8 Comsumer Prices 1860-1970 111 -1.582 1916 1920 3 -2.733 1916 1920 5
9 Wages 1900-1970 71 -1.636 1920 1931 1 -3.160 1920 1940 1
10 Real Wages 1900-1970 71 -1.622 1931 1945 0 -5.565 1931 1940 3
11 Money Stock 1889-1970 82 -2.029 1920 1931 1 -3.191 1920 1931 1
12 Velocity 1869-1970 102 -2.886 1941 1945 0 -4.228 1917 1941 1
13 Bond Yield 1900-1970 71 0.026 1921 1932 0 -0.247 1917 1931 0
14 Common Stock Prices 1871-1970 100 -1.928 1931 1937 0 -4.215 1931 1942 3
26
Table 8: Results of two-break unit root test using logarithmized data
M 1 M 2
N r. Se r ie s Sa m p l e Tte s t s t at i st ic T B 1 T B 2 k t e st st a ti st ic T B 1 T B 2 k
1 R e er b a s ed o n r e l. C P 1 9 8 0- 2 0 0 6 2 7 -1. 7 1 3 1 9 6 8 1 9 7 4 0 -2 .2 6 2 1 9 6 8 19 7 3 0
2 R e er ba s ed o n R N U L C 1 9 7 5 -2 0 07 3 3 - 2 .7 8 7 1 9 85 1 99 9 4 -3 . 22 3 19 8 5 1 9 9 9 0
3 M1 1959-2007 49 -1.167 1985 1994 3 -3.113 1985 1994 0
4 M2 1959-2007 49 -2.643 1970 1974 1 -4.8481970 1986 5
5 M3 1959-2005 47 2.357 1968 1994 0 1.778 1969 1994 0
6 G r o ss sa vi n g 19 4 8 -2 0 0 6 5 9 - 2 .5 9 3 1 9 7 2 19 8 3 0 - 4. 5 4 7 1 9 7 7 19 8 3 5
7 G r o ss na t io n a l i n c om e 19 4 8 - 20 0 6 5 9 0 . 30 8 19 8 1 1 9 9 0 4 -2 .9 5 8 19 72 1 9 8 1 4
8 G r o ss do m e s ti c pr o d uc t 1 94 8 - 20 0 7 6 0 0 . 06 7 1 96 1 1 9 8 1 5 -3 . 42 9 19 7 2 19 8 1 4
9 GDP De‡ator 1948-2007 60 -2.504 1973 1976 4 0.899 1975 1981 2
10 Wages 1948-2007 60 4.678 1967 1981 0 -3.130 1967 1981 0
11 In d u s tr ia l pr o d uc t io n 19 5 0 -2 0 0 7 5 8 - 1 .3 6 0 1 9 7 4 19 7 9 5 -5. 0 36 1974 1983 4
12 C ru d e pe t ro l e um p r o d uc t io n 19 4 8 -2 0 0 6 5 9 - 2. 2 2 1 1 9 6 5 19 7 7 0 - 4. 2 4 0 1 9 73 1 9 8 8 4
13 N on - ag ri cu l tu ra l em p lo ym e nt 1 94 8 -2 00 7 60 -2 .5 8 7 1 96 5 19 8 1 0 - 3 .4 64 19 74 1 98 1 1
14 U ne m p lo ym e nt ra te 19 4 8- 2 00 7 60 -4 .3 37 1974 1981 4 -5.374 1974 1983 5
15 E xp or t s 1 9 4 8- 2 0 07 6 0 -4. 3 3 11972 1978 1 -4.168 1972 1981 1
16 E xp or t pr ic e in de x 1 94 8 - 20 0 7 60 -4 . 15 7 1972 1974 2 -4.485 1972 1975 2
17 Imp orts 1948-2007 60 -1.944 1973 1975 0 -0.775 1973 1981 2
18 Im p or t pr ic e in de x 1 9 48 - 2 0 07 6 0 -1. 8 1 7 1 9 7 3 19 7 8 3 0 . 1 89 1 1 9 73 1 9 8 0 1
19 3- m o n th s T- B il l rat e 19 7 5- 2 0 07 3 3 - 6. 8 03  1991 1999 5 -9.412 1996 1999 1
20 B an k pr i m e loa n ra t e 1 94 8 - 2 00 7 6 0 - 1 .7 4 5 1 9 7 4 1 9 84 4 -7. 4 36  1980 1984 1
21 M ortgage rate 1972-2007 36 -4.1581979 1985 4 -5.686 1982 1998 4
22 B on d yi e ld , 3 y ea r 19 4 8 -2 0 0 7 6 0 - 1 .2 7 4 1 9 8 0 19 8 5 3 - 4. 6 9 1 1 9 8 5 19 9 3 2
23 Sh a r e pr ic e s i n d ex 1 9 4 8- 2 0 0 7 60 -2 .6 7 0 1 9 8 2 1 99 5 0 - 2 .9 8 9 1 9 8 2 19 8 5 0
24 N A SD A Q co m p o s it e ind e x 1 97 2 -2 0 0 7 3 6 - 5 .6 8 1  1987 1999 0 -5.459 1989 1998 4
25 A M E X av er a g e ind e x 1 9 7 1 -2 0 0 7 3 7 1 .0 3 9 1 9 8 1 1 98 3 5 - 0 .9 6 7 1 9 8 1 1 98 4 5
26 S& P in du s t ri a ls in d ex 1 9 4 8 -2 0 0 7 6 0 - 2. 1 7 5 1 9 69 1 9 7 3 0 - 3. 5 6 5 1 9 73 1 9 8 2 4
27 P P I / W P I 19 4 8 -2 0 0 7 60 -3 .8 9 7 1 9 7 2 1 9 7 8 3 - 4. 8 4 91972 1976 1
28 P P I: C a p i ta l e q ui p m e nt 19 4 8 - 20 0 7 60 -3 .6 4 2 1 9 7 3 19 7 6 2 -4 .0 9 6 19 73 1 9 76 2
29 W P I : …n i s he d go o d s 1 94 8 - 20 0 7 6 0 - 3 .1 1 8 1 9 7 3 1 97 8 3 - 1 .4 8 3 1 9 7 3 1 98 1 1
30 C P I: a l l it e m s 19 4 8 -2 0 0 7 60 -2 .3 3 5 1 9 7 2 19 7 4 5 -6 . 7 45  1968 1974 1
31 C P I: …ni sh e d go o d s 1 94 8 - 20 0 7 60 -4 . 39 4 1972 1978 1 -5.1641972 1976 1
32 In d u s tr ia l go o d s ind e x 19 4 8 -2 0 0 7 6 0 - 3 .6 5 2 1 9 7 3 19 8 6 5 - 2. 2 1 4 1 9 73 1 9 8 1 1
27
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Recently, Perron has carried out tests of the unit-root hypothesis against the alternative hypothesis of trend stationarity with a break in the trend occurring at the Great Crash of 1929 or at the 1973 oil-price shock. His analysis covers the Nelson-Plosser macroeconomic data series as well as a postwar quarterly real gross national product (GNP) series. His tests reject the unit-root null hypothesis for most of the series. This article takes issue with the assumption used by Perron that the Great Crash and the oil-price shock can be treated as exogenous events. A variation of Perron's test is considered in which the breakpoint is estimated rather than fixed. We argue that this test is more appropriate than Perron's because it circumvents the problem of data-mining. The asymptotic distribution of the estimated breakpoint test statistic is determined. The data series considered by Perron are reanalyzed using this test statistic. The empirical results make use of the asymptotics developed for the test statistic as well as extensive finite-sample corrections obtained by simulation. The effect on the empirical results of fat-tailed and temporally dependent innovations is investigated. In brief, by treating the breakpoint as endogenous, we find that there is less evidence against the unit-root hypothesis than Perron finds for many of the data series but stronger evidence against it for several of the series, including the Nelson-Plosser industrial-production, nominal-GNP, and real-GNP series.
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The Perron test which is based on a Dickey–Fuller test regression is a commonly employed approach to test for a unit root in the presence of a structural break of unknown timing. In the case of an innovational outlier (IO), the Perron test tends to exhibit spurious rejections in finite samples when the break occurs under the null hypothesis. In the present paper, a new Perron-type IO unit root test is developed. It is shown in Monte Carlo experiments that the new test does not over-reject the null hypothesis. Even for the case of a level and slope break for trending data, the empirical size is near its nominal level. The test distribution equals the case of a known break date. Furthermore, the test is able to identify the true break date very accurately even for small breaks. As an application serves the Nelson–Plosser data set.
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This paper investigates whether macroeconomic time series are better characterized as stationary fluctuations around a deterministic trend or as non-stationary processes that have no tendency to return to a deterministic path. Using long historical time series for the U.S. we are unable to reject the hypothesis that these series are non-stationary stochastic processes with no tendency to return to a trend line. Based on these findings and an unobserved components model for output that decomposes fluctuations into a secular or growth component and a cyclical component we infer that shocks to the former, which we associate with real disturbances, contribute substantially to the variation in observed output. We conclude that macroeconomic models that focus on monetary disturbances as a source of purely transitory fluctuations may never be successful in explaining a large fraction of output variation and that stochastic variation due to real factors is an essential element of any model of macroeconomic fluctuations.
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This study first reexamines the findings of Perron (1989) regarding the claim that most macroeconomic time series are best construed as stationary fluctuations around a deterministic trend function if allowance is made for the possibility of a shift in the intercept of the trend function in 1929 (a crash) and a shift in slope in 1973 (a slowdown in growth). Unlike that previous study, the date of possible change is not fixed a priori but is considered as unknown. We consider various methods to select the break points and the asymptotic and finite sample distributions of the corresponding statistics. A detailed discussion about the choice of the truncation lag parameter in the autoregression and of its effect on the critical values is also included. Most of the rejections reported in Perron (1989) are confirmed using this approach. Secondly, this paper investigates an international data set of post-war quarterly real GNP (or GDP) series for the G-7 countries. Our results are compared and contrasted to those of Banerjee et al. (1992) and Zivot and Andrews (1992). In contrast to the theoretical results contained in these papers, we derive the limiting distribution of the sequential test without trimming.
Article
Ever since Nelson and Plosser (1982) found evidence in favor of the unit - root hypothesis for 13 long - term annual macro series, observed unit - root behavior has been equated with persistence in the economy. Perron (1989) questioned this interpretation, arguing instead that the "observed" behavior may indicate failure to account for structural change. Zivot and Andrews (1992) restored confidence in the unit - root hypothesis by incorporating an endogenous break point into the specification. By allowing for the possibility of two endogenous break points, we find more evidence against the unit - root hypothesis than Zivot and Andrews, but less than Perron. © 2000 by the President and Fellows of Harvard College and the Massachusetts Institute of Technolog
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[eng] Transportation costs and monopoly location in presence of regional disparities. . This article aims at analysing the impact of the level of transportation costs on the location choice of a monopolist. We consider two asymmetric regions. The heterogeneity of space lies in both regional incomes and population sizes: the first region is endowed with wide income spreads allocated among few consumers whereas the second one is highly populated however not as wealthy. Among the results, we show that a low transportation costs induces the firm to exploit size effects through locating in the most populated region. Moreover, a small transport cost decrease may induce a net welfare loss, thus allowing for regional development policies which do not rely on inter-regional transportation infrastructures. cost decrease may induce a net welfare loss, thus allowing for regional development policies which do not rely on inter-regional transportation infrastructures. [fre] Cet article d�veloppe une statique comparative de l'impact de diff�rents sc�narios d'investissement (projet d'infrastructure conduisant � une baisse mod�r�e ou � une forte baisse du co�t de transport inter-r�gional) sur le choix de localisation d'une entreprise en situation de monopole, au sein d'un espace int�gr� compos� de deux r�gions aux populations et revenus h�t�rog�nes. La premi�re r�gion, faiblement peupl�e, pr�sente de fortes disparit�s de revenus, tandis que la seconde, plus homog�ne en termes de revenu, repr�sente un march� potentiel plus �tendu. On montre que l'h�t�rog�n�it� des revenus constitue la force dominante du mod�le lorsque le sc�nario d'investissement privil�gi� par les politiques publiques conduit � des gains substantiels du point de vue du co�t de transport entre les deux r�gions. L'effet de richesse, lorsqu'il est associ� � une forte disparit� des revenus, n'incite pas l'entreprise � exploiter son pouvoir de march� au d�triment de la r�gion l
Article
The authors consider unit root tests that allow a shift in trend at an unknown time. They focus on the additive outlier approach but also give results for the innovational outlier approach. Various methods of choosing the break date are considered. New limiting distributions are derived, including the case where a shift in trend occurs under the unit root null hypothesis. Limiting distributions are invariant to mean shifts but not to slope shifts. Simulations are used to assess finite sample size and power. The authors focus on the effects of a break under the null and the choice of break date. Copyright 1998 by Economics Department of the University of Pennsylvania and the Osaka University Institute of Social and Economic Research Association.