Potential Pitfalls in Determining Multiple Structural ...

Potential Pitfalls in Determining Multiple Structural Changes with an Application to Purchasing Power Parity Author(s): Ruxandra Prodan Source: Journal of Business & Economic Statistics, Vol. 26, No. 1 (Jan., 2008), pp. 50-65 Published by: Taylor & Francis, Ltd. on behalf of American Statistical Association Stable URL: http://www.jstor.org/stable/27638961 Accessed: 20-07-2015 15:44 UTC

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in Determining

Pitfalls

Potential

Multiple

With an Application Changes to Purchasing Power Parity

Structural

Ruxandra

Prodan of Economics,

University of Houston,

Department

Houston, TX 77004

([email protected])

investigate the empirical performance of theBai and Perron multiple structural change tests and show that the use of their critical values may cause severe size distortions in persistent series. To correct these

We

size distortions while maintaining a good power, we implement the Bai structural change test and extend it in two directions, developing a new procedure to choose the number of breaks and a restricted version that specifically models breaks that imply mean or trend reversion. Using our new methodologies, we reconcile the results of unit root tests with those of tests for structural change when testing for long-run purchasing power parity. KEY WORDS:

Persistent series; Restricted structural change; Size distortions; Tests with good size and power.

1.

INTRODUCTION

the sequential method on data that includes certain configura tions of changes (especially breaks of opposite sign). To im

Structural change is of central importance to statisticalmod eling of time series. Disregarding this aspect could lead to in accurate

or

forecasts

about

inferences

economic

variety

contexts.

of modeling

The

change was developed by Chow for many

popular

was

years,

classic

relationships.

test for a structural

(1960). His testingprocedure,

extended

to cover most

economet

ricmodels of interest. Its important limitation,however, is that the break date must be known a priori. In the early 1990s, this problem was solved by several authors, themost significant studies being provided by An

or else

consistent

estimates

of

serial-correlation

the sequential

breaks

procedure

down,

of squared residuals. This study conducts a thorough analysis of theBai and Per ron

structural

multiple

tests. We

change

new

provide

evidence

that in persistent series the use of asymptotic critical values, calculated under the null of iid errors, can be inadequate. Be cause it is known thatmany processes of interest in empiri cal macroeconomics and internationalfinance tend to be highly

drews (1993) and Andrews and Ploberger (1994). They devel oped testswhich use data-based procedures to determine the location of the break and derived theirasymptotic distribution. All of these studies require either full specification of the dy namics

when

theyproposed what we call theBai and Perron procedure: First, look at the sequential test. If 0 against 1 break is rejected, con tinuewith the sequential testuntil the firstfailure to reject. If 0 against 1 is not rejected, then look at the global tests to see if one break is present. If at least one break is found, decide the number of breaks based upon a sequential examination of i + 1 against i breaks with statistics constructed using estimates of the break dates obtained from a global minimization of the sum

Due to the importance of structural stability,much literaturehas been devoted to obtaining powerful testsfor structuralchange in a

the power

prove

our results

persistent,

para

of severe

are of immediate

size distortions

practical interest.Among many examples, Rudebusch (1993), meters, a poor handling of any of these leading to undesirable Diebold and Senhadji (1996), Lothian and Taylor (1996), and tests properties in finite samples. Vogelsang (1997) developed estimated models where theAR coefficients are (2002) a an Taylor unknown date in the trend function for detecting break at an and .82 for quarterly real GNP, U.S. .82, .93, .89, postwar a of dynamic univariate time series, allowing for trending and unit

nual

root regressors.

estimating

structural

multiple

changes,

We

at un

occurring

known dates, in a linearmodel estimated by least squares. They also addressed the problem of testing formultiple structural changes,

proposing

a

sequential

procedure,

which

was

first in

or

errors

across

segments,

a higher

the annual

long-run

dollar-sterling

first show

real exchange that their

rates,

respectively.

sequential

procedure

may

cause

se

some

size

results

for multiple

structural

generating process is an AR(1) a simulation

change

tests,

their data

with a coefficient of .5. Con with

experiment

persistent

processes,

we

argue that their sequential test suffers from large size distor tions (for a persistence of .9we find on average a size of 21% for testswith a nominal size of 5%). We proceed to analyze the

lems associated with these tests. Although Bai and Perron (1998) recommended trimming of 5% of the data, they later argued that in the presence of serial correlation and/or hetero in the data

real GNP,

vere size distortions if the data are highly persistent. The exis tence of these size distortions has not been documented previ Whereas Bai and Perron (2006) reported ously in the literature.

ducting

troduced by Chong (1995, 2001) and Bai (1997). Bai and Perron (2003a,b, 2006) provided new evidence on the adequacy of theirmethods and also identified several prob

geneity

U.S.

dollar

long-run

ceived considerable attention. Bai and Perron (1998) consid

ered

long-run

real exchange rate, and 16 industrialized countries' (average)

The development of tests for one structural change in the data led to the next question: could there be more than one? The problem of testing formultiple structuralchanges has re

?

2008

Journal

American

of Business

trim

Statistical

January

ming is needed. They also acknowledged the lack of power of

Association

& Economic 2008,

Statistics

Vol. 26, No.

1

DOI 10.1198/073500107000000304 50


Prodan:

Perron

and

Bai

51

Multiple Structural Changes

Determining

For

procedure.

this pro

processes,

persistent

cedure, although improving the power on breaks of opposite of

the evidence

increases

severely

sign,

com

distortions

size

of

combination

tests, we

several

of "size."

instead

use

will

a

With

the term

"frequency of

persistence

.9, using

theBai and Perron procedure we find an average frequency of a rejections of the null of no break of 40%, which represents the sequential

from

increase

considerable

procedure.

We have identified an importantproblem related to the em pirical performance ofmultiple structuralchange tests.We next a solution

propose

to this problem.

correct

To

for the size

dis

tortions,we implement theBai (1999) likelihood ratio type test. The Bai likelihood ratio test is a sequential testwhere thebreaks are being chosen globally. The essential feature of this test is its meth easy applicability which allows us to use straightforward to calculate

ods

values.

critical

bootstrap

these methods,

Using

the testhas an approximately correct size for any level of per sistence.Whereas theBai likelihood ratio testworks fairlywell on data

one

that incorporate

structural

or two

change

structural

on changes thatoccur in the same direction, it has low power data that include breaks of opposite sign. To increase the power of the test on such configurations of changes, we extend the Bai likelihood ratio test in two direc tions.

to increase

First,

we

the power,

that

imply

of this

the power or

mean

trend

test when

reversion

are

two or more

changes

we

propose

present,

a restricted version of this test that specifically models these types of breaks (the restrictedprocedure). The simulation ex we periments show that by applying the restricted procedure the power

increase

substantially

of

performance

We

conclude

these

that, using

new methodologies

and boot

strap critical values, themodified Bai testhas reasonable power properties and a quite satisfactoryperformance in determining the number grees and

of breaks

international

for the sample of interest

of persistence

finance

tests considerably

stricted

sizes,

break

in the long-run

literature. increase

In addition, power

over

sizes,

and

de

we

show

unrestricted

that re tests

when the truealternative is restricted structuralchange and also preserve good size when the data include one or two breaks of the same

sign.

We next illustrate thepractical importance of these results by using

long-run

real

exchange

rates. We

are

to resolve

able

start by

documenting

contradictory evidence of purchasing power parity (PPP) from the application of unit root tests and multiple structuralchange

one

only

contradiction.

show

We

the restricted

Using

breaks

in this particular

ing

to reconcile

of opposite sign while, a correct size, we are able

root and

structural

maintain

case,

from unit

the results

tests for five out of six countries.

change

The rest of the article is structured as follows: Section 2 presents theBai and Perron structuralchange model and doc uments the size distortions. In Section 3 we implement the Bai (1999) test formultiple structuralchanges and develop new methodologies thatproduce the best size and power properties of this test.We also present the results of simulations analyzing and

the size

the test. Section

of

the power

an

4 presents

em

pirical application, where we show the inconsistencies between root

unit

and

structural

change

tests when

for purchas

testing

ing power parity, and we also demonstrate the properties of the new

tests. Some

are contained

remarks

concluding

in Section

5.

2. BAI AND PERRON STRUCTURAL CHANGE TESTS 2.1

and Methodology

The Basic Model

Bai and Perron (1998) considered multiple structuralchanges in a linear regressionmodel, which is estimated by minimizing the sum

residuals.

of squared

We consider the following multiple linear regression with m breaks (m+ 1 regimes): t=

yt=j
m +

1,...,

1, To

Tj-i +

= 0, and Tm+\

=

1,..., Tj,

(1)

T.

In thismodel, yt is the observed dependent variable at time t;xt (p x 1) and Zt (q x 1) are vectors of covariates, and ? and

8j

(j

?

cients;

m

1,..., and

-f 1 ) are

of both

pure

vectors

the corresponding

ut is the disturbance

sider models

at time

structural

t. In our

change,

where

change,

only

some

=

of coeffi

study

where

gression coefficients are subject to change (p structural

macroeconomics

tests for six countries.

change

procedure, which further increases power on data that include

tests on

these

breaks of opposite sign, especially in the case of trendingdata, beyond themodified Bai procedure. However, as our simulation results show, although the power is improved, the frequency of on average 10%. rejections for any level of persistence becomes

structural

that these contradictions are due both to the size distortions of theBai and Perron procedure and to the low power of these tests on breaks of opposite sign. Using better sized tests that still maintain reasonable power (themodified Bai procedure), we

a similar method

propose

ology to theBai and Perron procedure for situationswhere it is difficult to reject the hypothesis of 0 versus 1 but it is not diffi cult to reject thenull hypothesis of 0 versus a higher number of breaks, which we call themodified Bai procedure. Next, to fur ther increase

root and

unit

a pared with the sequential test.As this procedure is actually of rejections"

countries. Testing forpossible mean shiftsby applying methods proposed by Bai and Perron, we find contradictions between

we

all

con the re

0), and partial are

of the coefficients

sub

ject to change (parameter vector ? is not subject to shifts and is estimated using the entire sample). The models of Bai and Perron allow heterogeneity in the regression errors, but theydo not provide methods for estimating thisheterogeneity. Using Bai and Perron's method, based on the least squares principle,

we

are

able

to estimate

the regression

coefficients

along with the break points, when T observations are available. Bai and Perron (2003a) provided a detailed discussion. In the case

of

a pure

structural

break

model

(p

=

0), where

all

co

efficients are subject to change, for each possible ra-partition are obtained estimators of The data cover real exchange rates for 16 industrialized coun by (T\,..., Tm) the least squares 8j trieswith theUnited States as the base country, startingfrorr minimizing the sum of square residuals. Then the estimated break points are the ones forwhich 1870 and ending in 1998. Using tests for unit roots, Lopez Murray, and Papell (2005) and Papell and Prodan (2006) founc 1 = ,Tm), (2) ST(T\,... (ri,...,rm) arg 'min a combined evidence of some variant of PPP for 14 out of K Tu...,Tm tests.


Journal of Business

52 where

...,

St(T\,

Tm)

sum

the

denotes

of

residuals.

squared

Since theminimization takes place over all possible partitions, are

estimators

the break-point

minimizers.

global

and Per

Bai

ron (2003 a) used a very efficient algorithm for estimating the break points which is based on dynamic programming tech niques. In the partial structuralbreak model case (p > 0), the dynamic programming method to obtain global minimizers of the sum of squared residuals cannot be applied directly. They estimate the Sfs over the subsamples defined by the break points, but the estimate of ? depends on thefinal optimal parti tion

Tm).

(Ti,...,

A central result derived by Bai and Perron (1998) is that the break fraction X?? ti/T converges to its true value ?? at the fast rate T, making we can

break

the estimated

fraction

the rest of

estimate

Therefore,

superconsistent. which

the parameters,

converge to their truevalues at rate F1/2, taking the break dates as known.

The Bai and Perron procedure allows for the estimation of intervals

the confidence

and

the parameters

under

very

general

conditions regarding the structure of the data and the errors across

Their

segments.

the nature

concerning

assumptions

of

the errors in relation to the regressors [xt,Zt) are of two kinds: in the first,when no lagged dependent variable is allowed in on

{xt, Zt}, the conditions low

and

correlation

substantial

an estimate

the residuals

of the covariance

are

quite

heteroscedasticity.

matrix

and al general In this case,

to heteroscedastic

robust

ityand autocorrelation can be constructed using themethod of Andrews (1991), applied to the vector {zt?t}. The second case allows lagged dependent variables as regressors, but no serial correlation

in the errors

is permitted

{ut}.

In both

the as

cases,

sumptions are general enough to allow different distributions for both

the regressors

and

the errors

in each

segment.

The determination of the existence of structural change and the selection of the number of breaks depend on the values of test

various

statistics

and Perron discussed versus

a fixed

sequential

number

when

the break

dates

are

estimated.

Bai

three types of tests: a test of no break of breaks,

a double

a

test, and

maximum

test.

First, they considered a supF type test of no structural break (m= 0) versus m = k breaks. The test is swpFt(k; q) = ,A.it; q),

Fj(k{,

where

X\,...,

A? minimize

the global

of squared residuals [according to (2)]. Next,

pothesis

of

no

structural

break

against

an

sum

the null hy

unknown

number

of breaks given some upper bound is tested by double maxi

mum

tests. Bai

and Perron

considered

two

statistics:

UDm?x

=

& Economic

Statistics,

January 2008

test the null of a single break versus two breaks, and so forth. This process is repeated until the statistics fail to reject the null hypothesis of no additional breaks. The estimated number of breaks is equal to the number of rejections. Bai and Perron argued that even if the sequential procedure works

in selecting

best

configurations

of breaks,

the number

of changes

when

(e.g.,

are

there

certain

are present

two changes

and the value of the coefficient returns to its original value af ter the second break) when thismethod breaks down. In these cases, it is often difficult to reject the null hypothesis of 0 ver sus 1 break but it is not difficult to reject the null hypothesis of 0 versus a higher number of breaks. Consequently, theypro posed theBai and Perron procedure, which leads to thebest re

sults in empirical applications: First look at the sequential test; if 0 against 1 break is rejected, continue with the sequential testuntil the firstfailure to reject. If 0 against 1 is not rejected, then look at UDmax orWDmax tests to see if at least one break is present. If the null of no breaks is rejected, then the num ber of breaks can be determined by looking at the sequential supFt(l + l\i) statistics constructed using global minimizers for the break dates [ignore the testF(1|0) and select m such that the tests supFt(I + 1 \i) are insignificantfor l>m\. All of the above

test statistics

mentioned

have

nonstandard

asymptotic

distributions, and Bai and Perron (2003b) provided the critical values for a trimming (s) ranging from .05 to .25 and a value for q ranging from 1 to 10. 2.2

Evidence

of Size Distortions

There has been very little previous research that studies the size of tests formultiple structural changes. Bai and Per ron (2006) provided size results for a range of data generating processes

and

sistence

measured

for

tests

the various

for structural

change

sug

gested. However, their results were limited to series with low persistence, specifically an AR(1) model with the level of per In this article,

coefficient of .5. by an autoregressive we make that many the case relevant models

from an economic point of view are highly persistent under the null

of no

structural

change.

Therefore,

we

have

a

reason

to

doubt the accuracy of the critical values for such highly per sistent processes. We illustrate this by extending the previous simulation evidence for themultiple structuralchange tests to processes

with

larger

persistence.

We

consider

only

the size

of

the sequential method and the frequency of rejections of the Bai and Perron procedure rather than focusing on the size of all individual tests.

on the rnaxi^jt^MSupFfCfc; q), where M is an upper bound 2.2.1 Construction of the Size Simulations. We conduct number of possible breaks, and also a version of this statistic, Monte Carlo experiments for an AR(1) data generating process, to that such that WDmax, marginal supFt(k\ q) applies weights m. a with and without trend: are across of values they proposed Finally, p-values equal test for i versus i + 1 breaks, labeled Ft(l + l\l). For this test, yt= m. We will call this themodified Bai procedure. trend function. 3.1.2 The Restricted Version. The rationale formoving The estimatedmodel is similar toBai and Perron, (1). It also beyond the sequential procedure, in both the Bai and Perron allows for pure and partial structuralchanges. For the purpose and the Bai test, is to improve power when the data contain of this studywe estimate the following specification of (1): m

breaks

k

yt= c+ (?t)+ J2 ?tDUtt+ J2 PM-i + Uu

(8)

where DU\t ? 1 if t> Tbt and 0 otherwise, for values of the breakpoints Tbt. The lag length,k, is chosen by the Schwarz InformationCri terion (SIC), which involves minimizing the function of the residual sum of squares combined with a penalty for a large number

of parameters

in the previous

regressions.

We

set

the

upper bound on the number of structuralchanges (m) to 5.We

of opposite

now

sign. We

propose

a restricted

version

of

theBai test thathas the potential to produce furtherpower im provements. If under the alternative the series includes breaks of opposite sign, designing a specific restriction that captures

this configuration raises the possibility of improving power on those specific data. Consequently, we restrict the coefficients on the dummy variables thatdepict thebreaks in (8) toproduce a constant

mean

or trend: m

I>

= 0.

?=i


(9)

Journal of Business

56

This imposes the restriction that themean following the last break is equal to themean prior to the firstbreak.We propose the following methodology to choose the number of restricted breaks: test the hypothesis of no break versus a fixed number of restrictedbreaks. If any supFt (I) (for 1 = 0 versus i = 2,..., k

breaks, where k ismaximum number of breaks considered) is significant, then thenumber of breaks can be decided upon a se ? 2,..., k breaks quential examination of theFt(l + 1 \t)for I and Ft(2\0). Select the number of breaks, m, such that the tests > Ft(i + 1 \i) and Fr(2|0) are insignificant for i m.We will call this the restricted

Finite Sample Performance: Power Analysis

3.2

3.2.1

sider an AR(1)

case,

con

we

data generating process, with and without trend [eq. (3)], varying the level of persistence measured by the au a =

coefficient

toregressive

.5,

0,

.6,

.7,

.8,

.9,

.95,

In all

.99.

of these cases the sample size is T = 125 and 5,000 replica tions are used with st= iidN(0, 1).We consider estimating four specifications: First, we estimate (8), with and without trend. we

Next

estimate

the same

but we

equations,

also

add

the re

striction in (9). For all these cases we use a trimming (s) of 5%. One of the arguments in favor of the Bai likelihood ratio test is thatone does not lose important informationby using a high trimming,as a trimmingof 5% will lead to a correct size, unlike in the Bai

and Perron

test case.

Critical values are calculated using parametric bootstrap methods as described inBerkowitz and Killian (2000). Gener ally we assume that theunderlying process follows a stationary finite-order

of the form

autoregression

= et, (10)

A(L)yt

January 2008

of

the value

of

the nuisance

over

improvement

the Bai

a,

parameter

and

Perron

which

procedure.

is We

could interpret5% rejections as 10% rejections, as we use mul tiple tests,which would produce approximately correct rejec tion frequencies. It is also possible to choose critical values that are smaller than 5%, for each application of the tests, to obtain an actual size of 5% of the overall procedure, unrestricted or In the case for a =

of

the unrestricted

model

rate

.9, the rejection

and

is 11.5%.

For

nontrend trending

data and a persistence of .9, the rejection rate is 10.3%. We find that the restricted testsperform better: fornontrending data and

in the previous

As

Correction.

Size

gardless a large

ing data,

and

Size

Statistics,

application of the test. In both the unrestricted and restricted versions of the tests, the frequency of rejections associated with the use of bootstrap critical values is approximately 10%, re

restricted.

procedure.

& Economic

a =

.9, the rejection

rate

is 9.8%.

The

tests perform

restricted

even better in the case of trendingdata, with a rejection rate of 8.9%.

These results demonstrate that the bootstrap distribution can be a much better approximation to thefinite-sample distribution than the asymptotic distribution when the persistence is high. In contrastwith theBai and Perron procedure, the size does not increase even

as

as

the persistence

the autoregressive

of the data coefficient

process

generating becomes

close

rises,

to unity.

3.2.2 Size Adjusted Power Analysis. We have shown that the size distortions of multiple structural change tests can be corrected by the use of appropriately adjusted critical values. The issue becomes the loss in power implied by such gains in the size

performance

of

the

test. We

propose

to improve the power of the Bai test.First, we procedure that improves the power of the Bai test,which we call themodified Bai procedure. velop a restrictedversion of this new procedure proves

the power

when

the breaks

have

certain

two methods

develop a new likelihood ratio Second, we de thatfurtherim configurations,

~ iid = 0 and < oc.Y = (yu ..., as breaks of opposite sign (the restrictedprocedure). Bai (1999) et N(0, 1)with E(et) E(e]) reported power simulation results for his sup LR test in com yr)' denotes the observed data. A(L) is an invertiblepolynomial as in the lag operator. The AR(p) model may be bootstrapped parison with Bai and Perron's conditional test supFt(l + l\l), follows: First determine the optimal AR(p) model, using the using asymptotic critical values. He pointed out thateven if the Schwarz criterion.Next, estimate the parameters A(L) for the conditional procedure does a satisfactory job, the sup LR tests the show an improved His optimal model. Following, to determine thefinite-sample distri analyzed performance. experiment bution of the statisticsunder thenull hypothesis of no structural sequential to a data generating and itwas restricted procedure, a persistence conduct with of only .5. In contrast, we change, use the optimal AR model with iidN(0, a2) innova process to construct

tions size

of the data,

a pseudo where

a2

to the actual of size equal sample innovation variance is the estimated

of theAR model. Then calculate the bootstrap parameter esti mates A*(L) and compute the statistics of interest.The critical values

are

taken

from

the sorted

vector

of 5,000

replicated

sta

tistics.

We present the results in Table 2. We report both the size of theBai likelihood ratio test and the frequency of rejections of the null of no break of themodified Bai procedure. For the Bai likelihood ratio testwe report the total size in the column "Size," as the probability of rejecting the null against the al ternative of one break, and for themodified Bai procedure we report the frequency of rejections corresponding to each num as well

ber of breaks

If we use the Bai cal

values,

we

obtain

as

the total

in the column

"Frequency."

likelihood ratio test and bootstrap criti an

average

size

of 5%.

The

use

of our

newly developed modified Bai procedure, however, leads to an increased frequency of rejections, which is due to themultiple

power experiments on highly persistent data using our newly developed

procedures.

Our power simulation experiments address the following is sues: (a) comparison between themodified Bai procedure and

the restricted procedure performances, and (b) size adjusted power of themodified Bai procedure as the function of the level of persistence. Because theBai likelihood ratio testhas no power on data that include breaks of opposite sign, we do not report simulation results for the original sequential Bai method. Within Monte Carlo experiments we consider the following data generating processes (DGP):

m

yt= ayt-i+ (?t)+ J2 ?tDU? + ?t'

(11}

The power of themultiple structural change tests is investi gated by constructing experiments with artificial data under a true alternative hypothesis where the data process is stationary


Prodan:

Determining

Multiple

2.

Table rates (%)

Rejection

yt

yt

ayt-\

=

+?ta

+

ayt-\

?t +

Size

of the modified

Bai

procedure

?tb

and

the restricted procedure Bai

The modified

Sequential

DGP =

57

Structural Changes

procedure

Size

1 break

2 breaks

.0

5.0

4.7

2.2

1.3

1.3

.5

5.1

4.9

2.0

1.4

1.3

1.4

11.0

.6

5.2

4.9

2.1

1.6

1.4

1.5

11.5

.7

4.7

4.9

2.0

1.7

1.6

1.3

11.5

.8

5.3

4.9

2.1

1.8

1.7

1.0

11.5

.9

5.1

4.9

2.1

1.9

1.7

.95

4.7

4.9

2.3

2.0

1.5

1.0

11.7

.99

5.2

4.9

2.2

1.7

1.5

1.2

11.5

.0

4.5

4.7

1.8

1.1

1.1

1.1

.5

5.2

4.7

1.9

1.5

1.1

.9

10.1

.6

5.0

4.7

1.8

1.5

1.3

.8

10.1

.7

5.2

4.7

1.5

1.3

.9

10.4

.8

5.4

4.7

1.5

1.3

.9

10.3

.9

5.5

4.7

2.2

1.3

1.3

.9

10.4

.95

5.1

4.9

2.0

1.4

1.3

1.4

11.0

.99

4.9

4.9

2.1

1.6

1.4

1.5

11.5

4 breaks

3 breaks

2.0 1.9

5 breaks

Frequency 10.4

.9

11.5

.9

9.8

2 breaks rates (%)

Rejection

DGP yt

=

yt

=

Size +

ayt-i

eta

+

ayt-i+?t

The

(restricted)

etb

2 breaks

1.8

5.3

3.5

.5

5.1

3.6

2.0

.6

5.6

3.9

2.2

.7

5.4

4.0

2.3

.8

5.6

4.0

2.5

sumptions.

specify

1.6

8.6

1.8

1.5

8.9

1.4

9.5

2.0 1.8 2.1

1.4

9.5

1.4

10.0 9.8

4.3

2.4

1.8

1.3

5.2

4.2

2.4

1.9

1.4

9.9

.99

4.8

4.3

2.5

2.0

1.4

10.2

.0

5.1

3.5

1.5

1.7

1.4

8.1

.5

4.6

3.5

1.9

1.8

1.6

8.8

.6

5.5

3.6

1.9

1.8

1.4

8.7

.7

5.2

3.8

2.0

1.7

1.4

8.9

.8

5.1

3.9

2.3

1.7

1.2

9.1

.9

4.7

4.2

2.2

1.5

1.0

8.9

.95

5.3

4.3

2.0

1.7

1.0

9.0

.99

4.5

4.2

2.1

1.3

1.0

8.6

allowing

data

1.7

5.4

for one,

two, or four changes

in

regardless of the number of breaks included in theDGP, and (3) finding at least as many breaks as are included in theDGP. In Tables 3 and 4 we present the results for (1), (2), and (3), but laterdescribe in detail only (1), thepower of the testdetermined by tabulating how often the procedure selects the exact number of break points present in theDGP. We

Frequency

.9

the intercept [/= 1, 2 and, respectively, 4 in (11)]. The power of a test is normally analyzed by tabulating how often the null is rejected when it is false.When testing formul tiple structural change tests, the definition of the power is am biguous. The power can be defined as (1) finding the exact num ber of breaks included in theDGP, (2) finding at least one break

generated

5 breaks

.95

NOTE: a Estimate without includinga time trend. Estimate includinga time trend.

The

4 breaks

3 breaks

.0

These resultsare based on 5,000Monte Carlo simulations.The sample size isT =

(or trend-stationary),

restricted procedure

are based

on

the nuisance

economically parameter

as plausible a = and .6, .7,

125 observationswith trimming(s) of 5%.

= .8, the trend slope y? being

.3, corresponding

.01, and themagnitude of the breaks as to a standard

deviation

of the residuals

of .223,which covers most of the cases found in the annual real exchange assume a

rate data change

analyzed

in the mean

in the next

section.

of approximately

We

therefore

1.3 standard

de

viations of the residuals. Perron and Vogelsang (1992) reported a change in themean of 1.2 standard deviations, arguing that this value is likely in practical instances. The timing of thebreak is set at 1/3 and 2/3 of the sample. In all of these cases the sample size is T = 125 and 5,000 replica tions are used with st = iidN(0, .223).We calculate bootstrap critical values as shown in Section 3.2.1 and report results for the 5% nominal size. Two versions of the test are estimated: the unrestricted [eq. (8)] and the restrictedmodel [eqs. (8) and (9)].


58 Table

3. Size Bai

The modified

of the modified

power

adjusted

Bai

procedure

and the restricted procedure

yt

?t

.3

= =

0X

+

+0DUt

+ 0xDUXt + = .3 .3, 02

ayt-i

=

otyt-\ + 0\DUit = = -.3 3, 02 9X

yt

yt

=

+

ayt-i

case

1: 0i =

yf= o^-i

case 2: 0i =

+

62DUlt

+

0xDUXt + 02DU2t = = -.3, 03

.3, 03

63DU3t + = -.3

04DU4t

+

=

04

=

.0

.0

99.0

99.0

.1

.1

.0

94.9

94.9

74.0

1.2

.6

.3

.5

76.6

76.6

.7

.5

87.9

74.6

1.3

66.8

63.6

13.3

71.8

3.2

59.1

2.0

.7

45.9

2.5

1.4

1.3

51.8

51.1

.0

80.8

4.7

2.6

2.4

90.5

90.5

.1

72.1

3.6

2.1

2.2

80.1

80

.0

59.7

2.5

1.4

1.7

65.3

65.3

st

-.3

1.6

1.2

.0

.5

.8

47.1

5.3

53.7

52.4

.0

.1

.5

37.4

4.1

42.1

41.5

.0

.2

.5

33.7

3.9

38.3

37.6

.0

.8

55.8

5.1

64.1

60.9

42.2

4.5

50.8

46.7

35.2

3.9

41.4

39.1

.0

2.4 1.2

2.9 1.7

.6

(1)


=

2 breaks ayt-\

+

+6DUt

6= 3

3 breaks

25.4

et

13.4 3.5

=

yt ayt-i + 0xDUlt = = .3 3, e2 ex

+

yt ayt-X + 0xDUXt + = = -3 3, e2 ex

ex

= =

+ 0xDUXt + = -.\5 02

et,

+

02DU2t

ayt-1 3,

+

02DU2t

=

yt

et

+

02DU2t

et

.3, 02

=

-.3,

03

=

.3, 04

=

2: 0! =

.3, 02

=

.3, 03

=

-.3,

04

=

-.3

.2

.5

.1

.1

.1

.1

.0

.0

.0

.0

.0

.0

.0

.0

.0

.0

.0

.0

11.1 9.9

1.8

3.7

65.9

8.4

1.7

3.7

42.2

17.5

1.4

32.6

8.2

.7

.3

2.9

.2

.0

2.4

4.3

.5

.4

1.3

56.8

8.4

.2

1.0

48.5

7.5

49.7

7.7 8.5

.2

case

3.9

74.2

-.3

+ 0xDUXt+ 02DU2t + 03DU3t + 0ADUM+ et yt= otyt-X

5 breaks

2.4 1.5

12.1 1: 0X =

4 breaks

78.2

yt= otyt-x+ 0xDUXt+ 02DU2t + 63DU3t + 6ADUAt+ ?t case

5 breaks

.0

DGP yt

4 breaks

1.1

.3, 04

-.3,

(3)

(2)

3 breaks

1.2

et

+

2 breaks

.0

The

structural change

including

97.9

+ 0iDi/lf + 02DU2t + 03ZW3r+ 04^4* + et =

January 2008

93.5

st

+

e2DU2t

.3, 62

.3, 02

data

Statistics,

(1) 1break

uyt-\

& Economic

on nontrending

procedure

DGP = yt = 0

Journal of Business

.8

10.5

3.5

59.3

4.3

1.7

53.7

8.2

4.4

1.2

50.3

10.0

These results are based on 5,000 Monte Carlo simulations.The sample size is T = 125 observations with trimming(e) of 5%. We report threeways to calculate power: (1) finding theexact numberof breaks included in theDGP, (2) findingat leastone break regardlessof thenumber of breaks included in theDGP, and (3) findingat least as many breaks as are included in theDGP [fortherestrictedprocedurewe reportonly (1)].

NOTE:

To address these issues we consider first a stationaryDGP where we account for different configurations of breaks in the intercept [eq. (11)]. The results are presented in Table 3. There are some general features of themodified Bai proce dure and the restricted procedure: As persistent,

there

is a monotonie

the errors become

increase

in power.

Also,

less the

power of the test decreases as the number of the breaks in cluded in the data generating process increases. Due to the lack

of

space,

we

mainly

report

power

results

for a =

.7 (de

tailed results for differentvalues of a are presented in each ta ble). The modified Bai procedure has very good power when the DGP is stationarywith one break (93.5%), andmoderate power

with two breaks of the same sign (59.1%). Also thepower of the modified Bai procedure is fairly good (72.1%) when data are generated with two breaks of opposite sign, and lower (37.4% and 42.2% for case 1 and 2, respectively) when data are gener ated with four opposite breaks. The erated

restricted data

procedure

include

one

has

permanent

very

low power

change

when

the gen

or two changes

that

occur in the same direction. This is important for empirical models because the testswill not provide evidence of restricted structural

change

when,

in fact,

the actual

structural

change

is

not consistent with the restriction. Ifwe consider a process that includes breaks that are equal and of opposite sign,we find an increase in thepower of the restrictedprocedure over themodi


Prodan:

4.

Table

The modified

Size Bai

Structural

Multiple

Determining

adjusted

59

Changes

of the modified

power

Bai

procedure

and the restricted procedure

a

yt= ayt-i + (?t) + ODUt + et 0= 3 yt= ayt-i + (?t) + OxDUXt+ 02DU2t + et =

.3, 02

=

.3

+ (?t) + OxDUXt+ 02DU2t + et yt= otyt-\ 01

=

.3, 02

=

including

structural change

(1)

procedure

DGP

0X

on trending data

-.3

1break

(2)

2 breaks

3 breaks

58.7

4.8

1.5

.9

53.8

4.5

1.4

.9

45.6

3.9

1.8

3.9

32.2

6.4

4.5

3.1

30.3

5.4

4.0

2.3

26.6

4.8

3.2

65.0

26.2

2.0

55.3

20.5

2.3

32.3

4 breaks

1.3

5 breaks 66.5

66.5

61.5

61.5

53.7

53.7

4.5

51.5

47.6

3.9

46.7

43.6

3.5

40.4

38.1

.9 1.1

.5 1.2

(3)

.3

94.0

29.0

1.3

80.6

25.3

56.9

24.6

15.8

3.5

2.3

3.0

yt= ayt-i + (?t) + OxDUXt+ 02DU2t + 03D?/3i+ 04D?/4?+ st

.6

36.2

3.8

15.8

6.5

62.9

22.3

case

.2

25.1

3.0

14.5

5.6

48.4

20.1

.0

21.1

2.3

11.9

5.8

41.1

17.7

1: 0! =

.3, 02

=

-.3,

=

03

.3, 04

=

-.3

yt= ctyt-\+ (?t) + OiDUu + 02DU2t + 03DU3t + 04Dt/4i+ ?t case

The

2: 0i =

.3, 02

=

.3, 03

=

-.3,

04

=

1.8

-.3

53.0

5.9

13.3

6.5

80.5

19.8

.4

30.9

4.5

12.2

6.4

54.4

18.6

.2

18.3

3.1

10.3

4.9

36.8

15.2

(1)


DGP

2 breaks

yt= ayt-i + (?t) + 0DUt + et 0= 3 yt= ayt-\ + (?t) + 0xDUlt + 02DU2t + et =

0X

3,

02

=

.3

yt= ayt-i + (?t) + 0xDUXt+ 02DU2t + st =

0X

3,

02

=

-3

yt= ayt-i + (?t) + 0xDUXt+ 02DU2t + et 0X

=

3,

02 --.15

case

1: 0i =

.3, 02

=

-.3,

03

=

.3, 04

=

yt= etyt-x+ (?t) + OxDUXt+ 02DU2t + 03DU3t + 04DU4t + et case 2: 0X =

.3, 02

=

.3, 03

=

-.3,

04

=

-.3

5 breaks

26.4

2.3

4.8

16.6

25.3

2.2

3.8

12.8

22.4

1.9

3.7

5.2

4.5

10.8

16.3

4.4

4.1

8.3

14.3

3.9

3.7

6.7

10.6

75.7

12.6

2.5

4.3

70.6

11.2

1.9

3.7

58.7

9.2

1.4

3.5

69.3

16.8

1.2

1.1

66.7

14.2

.5

-.3

4 breaks

19.0

44.15

yt= ayt-i + (?t) + 0xDUXt+ 02DU2t + 03DU3t + 04DU4t + et

3 breaks

.2

6.45 1.5

.1 50.2

.9 .3 8.3

.3

.8

44.3

7.1

.2

.7

39.5

6.5 7.7

8.0

3.8

55.1

2.3

1.9

45.3

7.5

33.4

6.0

.7

.9

These results are based on 5,000 Monte Carlo simulations.The sample size is T = 125 observations with trimming(e) of 5%. We report threeways to calculate power: (1) findingtheexact numberof breaks included in theDGP, (2) findingat leastone break regardlessof thenumberof breaks included in theDGP, and (3) findingat least as many breaks as are included in theDGP [forthe restrictedprocedurewe reportonly (1)].

NOTE:

fledBai procedure ofmore than 10%, especially for cases with highly persistent errors. If the breaks are unequal and of op posite sign, the power of the restricted tests significantly de creases.

We proceed to apply theprevious tests, including a time trend in the estimation,

to trend-stationary

data

that also

include

one,

two, or four changes in the intercept (Table 4). The power of the modified Bai procedure and the restrictedprocedure is smaller than in the case of nontrending generated data. Considering a =

.7, the modified

Bai

procedure

has moderate

power

when

theDGP is stationarywith one break (53.8%), and low power with two breaks of the same sign (30.3%). Its power is even lowerwhen data are generated with two breaks of opposite sign

(20.5%). When the data include four breaks of opposite sign, the power of the test is insignificant,only 14.5% and 12.2% for case

1 and

2, respectively.

In this case,

the procedure

mistak

enly finds two breaks with a higher probability than finding the correct number of breaks (25.1% and 30.9% for case 1 and 2, respectively). The restricted procedure substantially improves power over themodified Bai procedure if the process is consistent with breaks thatare equal and of opposite sign.With two breaks, the power of the restricted procedure increases by approximately 50% in comparison with themodified Bai procedure for all lev els of persistence (for a = .7 the power of restricted tests is 70.6%). With fourbreaks, thepower of the restrictedprocedure


60

Journal of Business

& Economic

January 2008

Statistics,

increases by approximately 30% over themodified Bai proce Norway, Portugal, Spain, Sweden, Switzerland, United King dure. The restrictedprocedure has insignificantpower when the dom, and United States. The real dollar rate is calcu exchange lated as follows: generated data include two changes that occur in the same di rection, but ithas small power when data include one break or = q (12) e+p*-p, unequal breaks of opposite sign.

4.

where q is the logarithm of the real exchange rate, e is the log arithm of the nominal exchange rate (the dollar price of the for

EMPIRICAL APPLICATION: PURCHASING POWER PARITY

eign

A large literaturehas emerged on testing the long-runvalidity of purchasing power parity (PPP), or equivalently the stationar ity of real

rates,

exchange

modern

using

time

economet

series

rics techniques. Inspired by the obvious failure of PPP to hold in the shortrun following the end of theBretton-Woods system, testing for long-run PPP became synonymous with testing the unit

root null

the stationary

against

rates (nominal exchange rates adjusted fornational price differ The

entials).

post-Bretton-Woods

for univariate to panel

methods

univariate

to be

methods

informative. or

for monthly

methods

for annual

is too

however,

period,

Attention

short

has moved

short-horizon

quarterly

and

data.

long-horizon

Although the investigation of time series properties of long horizon real exchange rates has gained an importantplace in PPP studies, the analysis of a century (ormore) of real exchange rates raises both conceptual and methodological issues. Follow ing Cassel (1922), the dominant view of PPP is that of long

run mean

reversion.

From

that perspective,

evidence

can

of PPP

be found by rejecting the unit root null in real exchange rates a

against

level

run

stationary

real

exchange

rate movements.

The

best

theories

of

long

known

of

these

is usually

model,

as implying trending real exchange rates.With evidence

tation,

of

the Balassa-Samuelson

data,

long-horizon

to consider

the Balassa-Samuelson

theories,

With

necessary

it becomes

however,

alternative.

interpreted

this interpre can

view

be

found

by rejecting the unit root null in real exchange rates against a trend stationary alternative. Papell and Prodan (2006) provided a detailed discussion on these two theories on PPP. They called "Trend

Power

Purchasing

Samuelson

the evidence

Parity"

of the Balassa

view.

We focus on the investigation of the time series properties of real

long-horizon

the unit

testing

rates.

exchange

root null

against

is a large

There

a level-

and/or

literature

on

trend-stationary

alternative for real exchange rates.Abuaf and Jorion (1990) and Lothian and Taylor (1996) found evidence of long-runPPP by rejecting

unit

roots

in favor

of

level-stationary

rates using Augmented-Dickey-Fuller

real

exchange

(ADF) tests.Cuddington and Liang (2000) and Lothian and Taylor (2000) investigated the implications of allowing for a linear trend in the Lothian and Taylor (2000) data. We

use

annual

latter are measured

nominal as

exchange

consumer

rates and price indices. The or GDP defla price deflators

tors, depending on their availability. The data were obtained fromTaylor (2002) and were updated by increasing the sample (using International Financial Statistics data from 2001). The data cover a set of 16 industrialized countries with theUnited States as thebase country, startfrom 1870 for the longest series, and end in 1998. The countries areAustralia, Belgium, Canada,

Denmark,

Finland,

France,

Germany,

Italy, Japan, Netherlands,

a.ndpt

are

and/?*

alternative.

trend-stationary if we

the logarithms

When

of the U.S.

for structural

testing

a one-time

find

however,

and

in the mean

change

change, or

trend

of the real exchange rate, long-run (T)PPP does not hold. The situation

in real exchange

alternative

currency),

the foreign price levels, respectively. We test the validity of long-run purchasing power parity us ing structural change tests. The evidence of PPP or TPPP re quires the rejection of the unit root null against a stationaryor

more

becomes

the changes

multiple are offsetting,

real

when

complicated

structural

experience

changes

returns

the series

rates exchange or trend. If

in the mean

to a constant

mean

(trend) and long-run (T)PPP holds. If the changes are not off setting,either because theyact in the same direction or because they act in opposite directions but are of differentmagnitude, the series

does

not

return

to a constant

mean

(trend)

and

long

run (T)PPP does not hold. The results from these two tests,unit root and

structural

to lead

have

change,

to the same

conclusion

to provide a unified evidence of PPP or TPPP. Previous research has provided evidence of PPP and TPPP using conventional unit root tests.Using the Elliott, Rothen

berg, and Stock (1996) generalized least squares version of the test,Taylor (2002) rejected the unit Dickey-Fuller (DF-GLS) root null in favor of either PPP or TPPP for 11 out of 16 in dustrialized countries with theU.S. dollar as numeraire at the 5% level. Lopez et al. (2005) argued that the reason forTay

lor's strong rejections lies in the lag selection procedures used, which tend to produce short lag lengths.As shown by Ng and Perron (1995, 2001), techniques thatproduce short lag lengths have low power forADF tests and are badly sized forDF-GLS tests. Using ADF testswith general-to-specific lag selection,

they found evidence of either PPP or TPPP by rejecting theunit root null for nine countries at the 5% level. They also reported the same number of rejections using DF-GLS testswithM AIC lag selection.

Papell Canada,

and Prodan Denmark,

(2006)

the seven countries,

analyzed

Netherlands,

Japan,

Portugal,

Switzerland,

United Kingdom, and United States, where Lopez et al. (2005) could not reject the unit root null in favor of either level or trend stationarity.They argued thatbecause ADF

corporate

a time

trend have

low power

very

tests thatdo not in to

reject

unit

roots

with trendingdata, previous results from conventional unit root tests can be interpretedas providing evidence of PPP for the eight countries for which the unit root null is rejected in fa vor of thePPP alternative: Belgium, Germany, Finland, France, Italy, Norway,

Spain,

and

and

Sweden,

of TPPP

for one

coun

try,Australia, forwhich the unit root null is rejected in favor of the TPPP, but not the PPP, alternative. To test for PPP (or TPPP) while allowing for structural change, Papell and Pro dan (2006) developed unit root tests that restrict the coeffi cients on the dummy variables that depict the breaks to pro duce

a

long-run

constant

mean

or


trend. Using

restricted

unit

Prodan:

61


Determining

5. Bai

Table

and Perron

Country

Time

Australia

1870-1998 1880-1996 1880-1998 1880-1998 1881-1998 1880-1998 1885-1998 1880-1998 1870-1998 1870-1998 1890-1998 1880-1998 1880-1998 1892-1998 1870-1998

Germany Finland France Japan

Italy Netherlands Norway Portugal Spain Sweden Switzerland

United Kingdom

Break

1919

dates

-.29

1972

.42

1917 1971

-.28 1.12

1972

1947, 1970 1920 1919 1920, 1946

.40

1919

-.43

1972

Time

Australia

1870-1998 1880-1996 1880-1998 1880-1998 1881-1998 1880-1998 1885-1998 1880-1998 1870-1998 1870-1998 1890-1998 1880-1998 1880-1998 1892-1998 1870-1998

Belgium Denmark Germany Finland France Japan

Italy Netherlands Norway Portugal Spain Sweden Switzerland

United Kingdom

five more

Break

span

-.07

.75

.74

-.07

-.10

1970

1918, 1979 1914, 1944 1920, 1940 1918 1970 1923

.78

.13

1919, 1947 1918, 1945

-.31,

re

to the previous

.440, -.230

.012

.41

.009

root and

structural

.012 .004

.04, -.06

.91

.001

-.20

.65

.003

.70

-.002

-.13

.83

.004

.11

-.14

.82

.002

.38, -.27

.54

.007

.21,-.17

.75

-.09

change

.62 .74

-.13,-.14

.82

1945

added

.004

-.60, -.57

dates

1918 1918

.008

-.63

.78

.11

1916

are

rejections

Break

1970

.47

Trend

dates

1916

.001

.51

-.21,

1947, 1972

.65

No-trend

Country

tests,

rates Trend

dates

Break

span

Parametric

root

and real exchange

No-trend

Nonparametric

Belgium Denmark

procedure

tests. We

report

.69

.006

.77

.008

.74

.001

results

for the non

sults: evidence of PPP restricted structuralchange forPortugal parametric procedure with a 5% level of significance and a trim and United Kingdom and evidence of TPPP restricted structural ming (e) of 15%. Due to the severe size distortions of theBai we next use the modified and Perron Bai procedure. change forDenmark, Japan, and Switzerland. They concluded conventional

that using

and

restricted

unit

root

tests,

evidence

of some variant of PPP can be found for 14 of the 16 coun tries. The

Netherlands

ity, reversion and Canada

power par quasi purchasing experiences mean one structural to a changing with change, no evidence of any vari is the only country where

ant of PPP was found. In this article we investigate the (T)PPP hypothesis by us ing tests change two

of

with real changes long-run tests allow for a maximum

structural for multiple rate data. Since the unit

breaks,

we

impose

ex

root

the

same

constraints

on

the

struc

tural change tests.Whereas theBai and Perron procedure still exhibits large size distortions and themodified Bai procedure remains breaks

somewhat has

correct

oversized, size; we

the restricted get a correct

size

procedure of 5%

with

two

on average,

(column 1, Table 2) and also we do not find spurious breaks (column 1,Tables 3 and 4). We firstuse theBai and Perron pro cedure and find six contradictions between the results of unit

procedure,

Finally, to improve thepower of themodified Bai procedure on breaks of opposite sign,we apply the restrictedprocedure. For themodified Bai procedure and the restrictedprocedure, we re

port results for a 5% level of significance and a trimming (s) of 15%. We test the 15 cases where previous literaturehas found ev idence of regime-wise (trend) stationarity (Canada is the only countrywhich we do not consider furthermore).The results of

these tests are presented inTables 5, 6, and 7. We divide the evidence found for the real exchange rates into

five categories: PPP, TPPP, restrictedPPP, restrictedTPPP, and quasi PPP. Within each category,we investigate the consistency of unit root and structuralchange test resultswhen testingfor a variant of PPP. We startwith the eight countries where there is previous ev idence of PPP with unit root tests.Using


theBai

and Perron

Journal of Business

62 6. The modified

Table

Bai

yt= c + T!L\ ?iDUtt + E/=i Break Critical

Hypothesis

values

procedure

and real exchange

yt= c + (?t) + Y?Li Critical

Ptyt-i+ *t

dates

Country

testing

Australia

Otol 1 to2

13.50

14.26

7.89

15.27

1.78

16.91

5.53

15.02

Otol lto2 Otol lto2

Denmark

France

Germany

Japan

.27

.73

.000

1918

.12

.80

.002

.35

.56

.006

.70

.004

.72

.001

12.13

17.42*

1916

15.13

Oto 1 lto2

6.39

12.67

5.81

14.63

5.90

16.27

6.22

14.39

1.94

11.38

21.76*

14.58

Otol 1 to2

9.20 5.94

Otol 1 to2

14.26*

13.08

5.89

Otol 1 to2

Otol Otol 1 to2

Switzerland

Otol 1 to2

UK

find

that only

the restricted

procedure,

1939 1945

the two

5.91*

13.81

17.28*

13.91

17.13

14.53

15.73

18.53

9.25

15.54

13.41

15.11

16.68

4.23

15.11

9.41

11.85

14.47*

14.13

4.45

13.70

12.91

14.31

18.50*

14.78

7.87

13.33

12.23

16.28

.28

1970

6.14

12.09

13.32

14.81

10.42

15.71

18.87*

14.86

6.81

11.56

4.46

13.18

4.20

14.82

8.03

13.23

16.76*

15.01

5.07

17.65

5.82

12.69

17.38*

15.70

one

country,

.13

1970 1946 1984

France,

.08

we

solve

countries

where

there

restricted

structural

find

is previous

change,

.68

.10

experiences

this contradiction,

.80

there

fore solving the contradiction.We find no evidence of struc tural change evidence forUnited Kingdom with the Bai and Perron procedure, which is not consistent with the results of the restrictedunit root tests.Using themodified Bai procedure,

14.65*

13.95

8.99

14.07

11.22

15.07

10.83

14.76

.36

.27

.55

.006

.52

.002

-.002

.29

1918 1947

14.76

23.59*

evidence of one break, inconsistentwith the evidence of PPP restricted structuralchange. Using themodified Bai procedure, we find no evidence of breaks. Applying the restricted proce of

.78

.09

evidence of PPP restricted structuralchange: Portugal and the United Kingdom. Using theBai and Perron procedure, we find contradictions in both cases: In the case of Portugal we find

evidence

.72

.29

ing evidence of restricted structuralchange consistent with PPP (Fig. 1A). For the countrywhere TPPP holds, Australia, we do not find any evidence of structuralchange. consider

.67

.09

(one) structuralchange. This result implies thatwe find both ev idence of PPP and structuralchange inconsistentwith PPP. Us ing themodified Bai procedure, we also find one break. Finally,

find

1939 1945

12.80*

11.65

Oto 1

we

.61

15.18

8.24

Sweden

dure,

.14

15.54

7.16

16.36*

Otol 1 to2

next

1972

13.56

8.80

16.70

Otol 1 to2

Spain

We

13.54

9.21

13.91

3.99

1912 1918

Portugal

using

17.39*

12.24

13.00

Norway

we

-.38

12.76

10.19

Netherlands

procedure,

1912 1918

12.73

23.95*

4.66

Oto 1 lto2

Italy

11.89

25.60*

14.24

.55

dates

-.61

11.19

.30

Break

1912 1918

8.19 12.32

,28

January 2008

Piyt-i + ut

0iDU?t + ?ii

values

16.45*

Otol lto2

Finland

Statistic

Statistics,

rates

Statistics

Belgium

& Economic

,25

1918 1945

,21 .19

.14

1972

we find evidence of two breaks of opposite sign.Whereas this result is potentially consistent with restricted structuralchange, thebreaks could have differentmagnitudes. We finally solve the contradiction, finding evidence of restricted structural change with the restrictedprocedure (Fig. 1A). We then consider the three countries where there is previous evidence of TPPP restricted structuralchange: Japan,Denmark, and Switzerland. cedure do

leads

not find

For

all

these

to inconsistent evidence

of

countries,

results: structural

For

the Bai Japan

change.

and Perron and Denmark,

In these

cases,

pro we cor

recting for size with themodified Bai procedure does not affect the result. Increasing the power with the restrictedprocedure, we solve the contradiction for Japan, finding evidence of re stricted structuralchange, but we still find no evidence of struc tural change in the case of Denmark. Second, for Switzerland, we find evidence of two breaks of opposite sign. If the breaks

of opposite sign are equal, the resultwould be consistent with the restricted unit root test result. Using the better sized test, themodified Bai procedure, we find evidence of one structural change, which leads to the conclusion that the previous finding


Prodan:


Determining

63

Table yt


HEf=1?^i+E/=i^H'+?i,

Statistics

Country

7. The

= o

E?=i^

Critical

Break

values

dates

and real exchange yf

rates

:+(?t)+IT=i

Statistic

OiDUtt+zLi

Critical

Break

values

dates

Piyt-i+ut, E?=i

= Oe o

Australia

1918 1932

10.10

21.33*

Belgium

,24

.71

32.10*

1912 1918

19.05

.24

.43

.60

.003

.52

.001

.56

.001

.73

.000

.69

.005

.43

Denmark Finland

21.12*

France

1912 1918 1916 1984

17.19

22.06*

19.55

Germany

,09

26.33*

1912 1918 1916 1972

18.20

.69

23.73*

20.10

.09

1939 1945

18.94

24.04*

Italy

.57

.29 ,29

.72

.29

23.85* 23.03*

,09 .09

1939 1945 1929 1970

20.48

,29

Japan

.32 ,32

22.95

.29 ,29 ,14 .14

Netherlands Norway

1919 1984

20.32

20.60*

Portugal

.15

.70

24.36*

21.71

.15

1918 1984

.15

1945 1972 1946 1984

.13

,000

.15

Spain Sweden Switzerland

23.84*

UK

23.78

1946 1976

19.73

.11

.63

Perron

size

procedure's

distortion.

22.94*

21.68

.11

of two breaks of opposite sign could be the result of the Bai and

20.68

Furthermore,

the

using

.64

.003

.67

.000

.13 .10 .10

of the countries where we found evidence of PPP or TPPP re stricted structural change experienced a depreciation of their exchange rate against the dollar associated with the twoWorld

restricted procedure, we find evidence of restricted structural Wars (Portugal,United Kingdom, France, and Switzerland). On change, solving the contradiction (Fig. IB). ev For the case of the Netherlands, where there is previous the other hand, the collapse of the Bretton-Woods system, in idence of quasi purchasing power parity (with one structural 1971, triggereda positive shock inUnited Kingdom, Japan, and change)

unit

using

root tests, we

also

find evidence

of one

struc

turalchange using both theBai and Perron and themodified Bai The

procedure.

restricted

not provide

does

procedure

any

evi

dence of restricted structuralchange, which is again consistent with the quasi PPP hypothesis of one structuralchange. Comparing the results from unit root and structural change tests, we

find

six contradictions

among

the

15 countries

Because

we

do

not

consider

the conventional

and

these

results

the restricted

as

a contradiction.

unit

root

tests have

a good size on data that include structural changes which are inconsistentwith the PPP hypothesis, we conclude that these contradictions are due either to the size distortions of the Bai and Perron procedure or to the low power of themodified Bai

procedure on data with breaks of opposite sign. Using the re stricted

procedure,

which

increases

the power

on data

that

in

clude breaks of opposite signwhile maintaining a correct size, we

are able There

to solve

are numerous

the potential

to cause

five out of six contradictions. political shocks

and

economic

to real exchange

causing

factors

that have

rates. We

explore

possible explanations for some of the structuralchanges found with both the unrestricted and the restricted procedure. Most

a

strong

appreciation

of

their

exchange

rates against the dollar. Following the establishment of flexible nominal

rates,

exchange

these

countries

a return

experienced

to

the original level (or trend) of theirreal exchange rates.

5. CONCLUSION

con

sidered. In cases such as Finland and Italy,we find one spike (breaks of opposite sign that are very close to each other).We also find in both cases evidence of restricted structuralchange; therefore

Switzerland,

It is common tural changes

in empirical when

analyzing

work

to test for one

macroeconomic

or more

and

struc

international

finance data. Bai and Perron (1998) considered estimating and testing formultiple structural changes, proposing a sequential

procedure thatfinds breaks by splittingof the sample. In later studies, they proposed a procedure that improves the power of the Bai likelihood ratio test on special configurations of changes, mostly breaks of equal and opposite sign (we call it theBai and Perron procedure). The purpose of this article is (1) to identifya potential seri ous problem in the use of these tests,and (2) to propose meth ods to solve this problem. We show that the Bai and Perron sequential tests sufferfrom severe size distortions, rejecting the no-structural-change null hypothesis too oftenwhen it is true, if the data are highly persistent. The problem is amplified by the use of theBai and Perron procedure. For both theBai and Per ron sequential method and theirprocedure, the size distortions


64

Journal of Business

& Economic

January 2008

Statistics,

(a) PPP restricted structuralchange rate

real exchange

UK-US 0.8

real exchange

Portugal-US

rate

-4.6

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 1872

1890

1908

1926

1962

1944

1891

1980 France-US

-1.3

1907

1923

1939

1955

1971

1987

real exchange rate

-1.4 -1.5 -1.6 -1.7 H -1.8 -1.9 -2.0 -2.1 -2.2

iiiiiiiiiiiiiiiiiiniiiiiiiiiiiiiiiiiiiiiiiiiiniiiiiiiiiiiiniiiiHiiiiiiiiiiiiiiiiiiiiiiiiiiiniiiiiiiiiiiiiiiiiiiii! 1899 1916 1967 1882 1933 1950 1984

(b) TPPP restricted structuralchange Japan-US

rate

real exchange

Switzerland-US 0.00

real exchange rate

-0.25

-0.50

-0.75

-1.00

-1.25

-7.0

-1

"1-50 IIIIIIIIIIIIIIIIIIIIIHIIIIIIIIIIIIIINIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii 1967 1983 1894 1910 1926 1942 1958 1974 1990 1887 1903 1919 1935 1951 Figure

(frequency

of rejections)

1. Evidence

become

more

of PPP

severe

or TPPP

restricted

as the persistence

increases. If there is a possibility that the data contain breaks of equal and opposite sign, one is faced with a trade-offbetween testswith very low power to detect such breaks (the sequential test) and tests thatare very badly oversized (theBai and Perron procedure).

To correct for size distortions,we implement theBai (1999) likelihood ratio test. Its essential feature lies in its easy applica bility, allowing for straightforwardcalculation of bootstrapped critical values. We show how finite sample bootstrappingmeth ods

can

achieve

correct

size

regardless

of the persistence

in the

structural change

(-RER;-restricted).

data. However, similarly to the Bai and Perron tests, the Bai likelihood ratio testhas low power to detect breaks of equal and opposite sign.We extend the Bai test in two directions: First, we

propose

a

methodology,

similar

to the Bai

and

Perron

pro

cedure, thatwe call themodified Bai procedure. Second, to fur ther improve

the power

on

these

specific

changes,

we

propose

a

restriction that specifically models data including two or more changes which implymean or trend reversion,which we call the restrictedprocedure. The frequency of rejections of the null of both themodified Bai procedure and the restrictedprocedure does not increase with the persistence of the data. The power


Prodan:

Multiple

Determining

simulations

Structural Changes

a reasonable

show

power

65 -

for the mod

performance

ifiedBai procedure and a substantially increased power for the restrictedprocedure, especially in the case of trendingdata. We illustrate the practical importance of these results by an alyzing

real

long-run

rates,

exchange

using

the longest

of

span

available historical data for 16 industrialized countries with the United States as the base country.We test the validity of long run (trend) purchasing power parity using structural change tests. We

our

then compare

results

with

results

previous

from

unit root tests (conventional and restricted structural change). Evidence of PPP or TPPP requires the rejection of the unit root null

against

a

or trend

stationary

stationary

In addi

alternative.

in the mean of real ex change testing for structural or several rates, if we find a one-time change, changes

tion, when change

occurring either in the same direction or in opposite direction but of differentmagnitudes, long-run (T)PPP does not hold. If the changes

are

the series

offsetting,

returns

to a constant

mean

(trend) and long-run (T)PPP holds. Comparing results from the unit root tests and the Bai and Perron procedure, we find six contradictions among the 15 countries:

France,

Portugal,

UK,

Japan,

Switzerland,

and Den

mark. These contradictions are due to both size distortions and the low power of the testswith breaks of equal and opposite sign.Using better sized tests (modified Bai procedure), we are able

to resolve

only

one

out of six contradictions.

By

using

tests

that increase the power on data that include breaks of opposite signwhile maintaining a correct size (restrictedprocedure), we are able

to resolve

five out of six contradictions.

ACKNOWLEDGMENTS I am grateful to Bruce Hansen, Tim Hopper, Lutz Kilian, Ulrich Mueller, Chris Murray, Bent Sorensen, and especially David Papell, as well as to participants in seminars at theUni versity of Alabama, Bates College, University of Connecti cut,University ofHouston, Kansas State University, University ofMississippi, University of Texas at Arlington, Texas Camp Econometrics IX, and the 2003 Southern Economics Meetings for helpful comments and discussions. I also thankPierre Per ron forproviding me with his Gauss program. [Received April 2004. Revised September 2006.]

-

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