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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 de…ned:

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= (ii)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

Hall, A. (1994): “Testing for a Unit Root in Time Series with Pretest Data-

Based Model Selection,”Journal of Business and Economic Statistics, 12,

461–470.

Harvey, D., S. Leybourne, and P. Newbold (2001): “Innovational

Outlier Unit Root Tests with an Endogenously Determined Break in

Level,”Oxford Bulletin of Economics and Statistics, 63(5), 559–575.

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), 535–558.

(2003): “Minimum Lagrange Multiplier Unit Root Test With Two

Structural Breaks,”The Review of Economics and Statistics, 85(4), 1082–

1089.

(2004): “Minimum LM Unit Root Test With One Structural Break,”

Working Paper 04-17, Department of Economics, Appalachian State Uni-

versity.

18

Lumsdaine, R., and D. Papell (1997): “Multiple Trend Break and the

Unit-Root Hypothesis,”The Review of Economics and Statistics, 79, 212–

218.

Perron, P. (1989): “The Great Crash, the Oil Price Shock, and the Unit

Root Hypothesis,”Econometrica, 57, 1361 –1401.

(1997): “Further Evidence on Breaking Trend Functions in Macro-

economic Variables,”Journal of Econometrics, 80, 355–385.

Popp, S. (2008): “New Innovational Outlier Unit Root Test With a Break at

an Unknown Time,” Journal of Statistical Computation and Simulation,

forthcoming.

Rappoport, P., and L. Reichlin (1989): “Segmented Trends and Non-

stationary Time Series,”Economic Journal, 99(395), 168–177.

Schmidt, P., and P. Phillips (1992): “LM Tests for a Unit Root in

the Presence of Deterministic Trends,”Oxford Bulletin of Economics and

Statistics, 54(3), 257–287.

Vogelsang, T., and P. Perron (1998): “Additional Tests for a Unit

Root Allowing for a Break in the Trend Function at an Unknown Time,”

International Economic Review, 39(4), 1073–1100.

19

Zivot, E., and D. Andrews (1992): “Further Evidence on the Great

Crash, the Oil-Price Shock, and the Unit-Root Hypothesis,” Journal of

Business and Economic Statistics, 10(3), 251–270.

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