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7232019 Economic Modelling Further Evidence on the PPP Analysis of the Australian Dollar- Non-linearities Fractional Integration and Structu
httpslidepdfcomreaderfulleconomic-modelling-further-evidence-on-the-ppp-analysis-of-the-australian-dollar- 19
Further evidence on the PPP analysis of the Australian dollar Non-linearitiesfractional integration and structural changes
Juan C Cuestas a Luiacutes A Gil-Alana b
a Nottingham Trent University UK b University of Navarra Spain
a b s t r a c ta r t i c l e i n f o
Article history
Accepted 14 May 2009
JEL classi 1047297cation
C32
F15
Keywords
PPP
Real exchange rate
Unit roots
Non-linearities
The aim of this paper is to analyse the empirical ful1047297lment of the Purchasing Power Parity (PPP) theory for
the Australian dollar In order to do so we have applied recently developed unit root tests that account for
asymmetric adjustment towards the equilibrium [Kapetanios G Shin Y Snell A 2003 ldquoTesting for a unit
root in the nonlinear STAR frameworkrdquo Journal of Econometrics 112 359ndash379] and fractional integration in
the context of structural changes [Robinson PM 1994 ldquoEf 1047297cient tests of nonstationary hypothesesrdquo Journal
of the American Statistical Association 89 1420ndash1437 Gil-Alana LA 2008 ldquoFractional integration and
structural breaks at unknown periods of timerdquo Journal of Time Series Analysis 29 163ndash185] Although our
results point to the rejection of the PPP hypothesis we 1047297nd that the degree of persistence of shocks to the
Australian dollar decreases after the 1985 currency crisis
copy 2009 Elsevier BV All rights reserved
1 Introduction
There is no doubt that Purchasing Power Parity (PPP) analysis has
become one of the most controversial topics within international
economics Although the existing literature is vast empirical
contributions have yielded contradictory results given that they are
dependent on the countries period analysed and econometric
techniques employed Although PPP is a theory of exchange rate
determination its empiricalful1047297lment has several policy implications1047297rst many macroeconomic models assume a constant real exchange
rate in equilibrium which has consequences when the policy makers
base their decisions on this type of modelling second as Wei and
Parsley (1995) claim the PPP hypothesis can be considered as a
measure of the degree of economic integration between countries
since this theory assumes perfect mobility of goods and an absence of
trade barriers 1047297nally Faria and Leoacuten-Ledesma (2003 2005) claimthat a misaligned real exchange rate (RER) might affect growth and
unemployment thus a proper assessment of the degree of over-
valuation or undervaluation of the currencies is important to help
promote long term economic growthThe PPP theory establishes that the nominal exchange rate
between two currencies should be equal to the price index ratio
between the countries so that any change in the price relationship
will affect the nominal exchange rate to keep the purchasing power
between both countries unaffected that is
E t = P 4t
P t
eth1THORN
where P t is the foreign price index P t is the national price index and
E t is the nominal exchange rate between both currencies
This implies that the RER
Q t = P
t
E t P 4t
= 1 eth2THORN
Relations such as Eqs (1) and (2) are known as the absolute
version of the PPP hypothesis This implies that the same quantity of
goods can be purchased in both countries with the same amount of
money However this is not always true since differences in
productivity and transport costs may affect the relationship between
exchange rates and prices Then it is possible to de1047297ne a less
restrictive version of the PPP hypothesis known as the relative PPP In
this case the RER should be equal to a constant different from 1
meaning that exchange rates react proportionally to changes in the
price ratio instead of identically
Economic Modelling 26 (2009) 1184ndash1192
Juan Carlos Cuestas gratefully acknowledges 1047297nancial support from the CICYT
project ECO2008-05908-C02-01ECON Juan Carlos Cuestas is a member of the INTECO
research group Luis A Gil-Alana acknowledges1047297nancial support fromthe Ministerio de
Ciencia y Tecnologia (ECO2008-03035 ECON Y FINANZAS Spain) and the Secretariacutea de
Estado de Cooperacioacuten Internacional (Ayudas de movilidad a artistas investigadores y
cientiacute1047297cos 2008) The usual disclaimer applies
Corresponding authorNottingham Business School Division of EconomicsNottingham
Trent University Burton Street NG1 4BU Nottingham UK Tel +44 1158484328
E-mail address juancuestasntuacuk (JC Cuestas)
0264-9993$ ndash see front matter copy 2009 Elsevier BV All rights reserved
doi101016jeconmod200905006
Contents lists available at ScienceDirect
Economic Modelling
j o u r n a l h o m e p a g e w w w e l s ev i e r c o m l o c a t e e c m o d
7232019 Economic Modelling Further Evidence on the PPP Analysis of the Australian Dollar- Non-linearities Fractional Integration and Structu
httpslidepdfcomreaderfulleconomic-modelling-further-evidence-on-the-ppp-analysis-of-the-australian-dollar- 29
Within the empirical literature on the PPP theory it is well known
that short run deviation may prevent the PPP hypothesis from holding
true However one may expect the RER to return to its equilibrium
value after a shock in the long run Statistically this implies that the
RER should be a mean reverting process for the PPP hypothesis to be
ful1047297lled which makes unit root testing to be an appealing econo-
metric approach for this purpose
Sarno and Taylor (2002) Taylor et al (2001) and Taylor (2006)
among others provide summaries of the main contributions to the
literature from which it is possible to highlight several facts 1047297rst
authors have applied different techniques since the 70s in order to
capture the true data generating process (DGP) of the real exchange
rates second it appears that the results have been quite
controversial and 1047297nally that the most recent contributions focus
on the consideration of non-linearities and fractional integration
alternatives In this vein Taylor et al (2001) identi1047297ed two main
paradoxes relating to RER behaviour First the relationship bet-
ween nominal exchange rates and prices implied by the PPP theory
is not in many cases a cointegrating one second how is it possible
to observe a high volatility of exchange rates in the short termwith the low speed of mean reversion generally observed in this
variable
The failure of the literature to provide good answers to the
aforementioned questions have been related to the low power of the
(unit-root)tests applied to analyse the empirical ful1047297lment of PPP1 It
appears that increasing the data sample creates additional problems
such as structural changes that may affect the power of the tests if
these are not taken into account (see Perron and Phillips1987 West
1987 among others) Therefore these kind of non-linearities in the
deterministic components may increase the probability of Type II
error with traditional unit root tests (eg DickeyndashFuller type)
Furthermore non-linearities in the RERlong runpath maybe present
in the form of an asymmetric speed of adjustment towards the
equilibrium ie the further the RER deviates from the long runequilibrium value the faster the speed of mean reversion is expected
to be From an econometric viewpoint that may imply an
autoregressive parameter dependent on the values of the variable
The non-linear behaviour of exchange rates has been acknowledged
by several authors such as Dumas (1994) Michael et al (1997)
Sarno et al (2004) Juvenal and Taylor (2008) and Cuestas (2009)
among many others
In addition as pointed out by several authors such as Diebold et al
(1991) Cheung and Lai (1993) and Gil-Alana (2000) among others
the real exchange rate may be characterised as a slow mean reversion
process and traditional unit root may not be able to distinguish this
type of process from a unit root Thus according to these authors
fractional integration may be an alternative viable route to the
analysis of the RER dynamics As mentioned above the question of
interest is to determine if deviations from PPP are transitory or
permanent which translated to the fractionally integrated literature
means to determine if the order of integration is smallerthan 1 (mean
reversion) or equal to or higher than 1 (no mean reversion) Applying
RS techniques to daily rates for the British pound French franc and
Deutsche mark Booth et al (1982) found evidence of fractional
integration during the 1047298exible exchange rate period (1973ndash1979)
Cheung (1993) also found similar evidence in foreign exchange
markets during the managed 1047298oating regime On the other hand
Baum et al (1999) estimated ARFIMA models for real exchange rates
in the post-Bretton Woods era and found almost no evidence to
support long run PPP Additional papers on exchange rate dynamics
using fractional integration are Fang et al (1994) Crato and Ray(2000) and Wang (2004)
In another contribution Henry and Olekalns (2002) analyse the
Australian RERlong run behaviour in the context of structural changes
and fractional integration These authors apply Robinsons (1994) and
Geweke and Porter-Hudaks (1983) fractional integration tests and the
Vogelsangs (1997) unit root test with structural changes but they are
not able to 1047297nd evidence of mean reversion in the RER Australian
dollar Similar results are obtained by Darneacute and Hoarau (2008) who
applied the Perron and Rodriacuteguezs (2003) unit root test with
structural changes to the Australian dollar Contrary to these results
Cuestas and Regis (2008) found that the Australian RER is stationary
around a non-linear deterministic trend by means of applying the
Bierens (1997) unit root tests which proposes a Chebyshev
polynomial approximation for the non-linear trend However theresults are dependent on the selection of the order of the polynomials
since there is no unique way of doing it Additionally in the context of
PPP it is dif 1047297cult to give an economic interpretation of the non-linear
trend
The aim of the present paper is to provide further evidence of the
PPP ful1047297lment in Australia and complement the previous literatureon
the PPP analysis by applying different techniques that address the
aforementioned facts ie fractional integration and non-linearities
that may affect the power of the traditional (linear) unit root tests
First we apply the upgraded versions of linear unit root tests
propossed by Ng and Perron (2001) second we account for
asymmetric speed of mean reversion and apply the Kapetanios et al
(KSS 2003) unit root test which generalises the alternative hypoth-
esis to a globally stationary exponential smooth transition (ESTAR)
1 Unit root tests most commonly employed in the literature (Dickey and Fuller 1979
Phillips and Perron 1988 Kwiatkowski et al 1992 etc) have very low power against
trend-stationarity (DeJong et al 1992) structural breaks (Perron 1989 Campbell and
Perron 1991) regime-switching ( Nelson et al 2001) or fractional integration
(Diebold and Rudebusch 1991 Hassler and Wolters 1995 Lee and Schmidt 1996)
Fig 1 Australian RER
1185 JC Cuestas LA Gil-Alana Economic Modelling 26 (2009) 1184ndash1192
7232019 Economic Modelling Further Evidence on the PPP Analysis of the Australian Dollar- Non-linearities Fractional Integration and Structu
httpslidepdfcomreaderfulleconomic-modelling-further-evidence-on-the-ppp-analysis-of-the-australian-dollar- 39
process Finally in order to take into account the existence of
structural changes and fractional integration at the same time we
apply Robinsons (1994) and Gil-Alanas (2008) fractional integration
tests in the context of structural changes
The remainder of this paper is organised as follows The next
section describes the data Section 3 summarises the econometric
techniques applied to empirically assess the ful1047297lment of PPP in
Australia and presents the results Section 4 concludes the paper
2 The data
The data for this empirical analysis consists of quarterly observa-
tions of the Australian real effective exchange rate computed as a
trade weighted index from 19702 to 20083 obtained from the
Central Bank of Australia web page (httpwwwrbagovau) A plot
of the data is displayed in Fig 1 From this graph it is possible to
highlight two mainstylised facts the existence of a structural change
in theseriesmdash presumablyat 1985 coincidingwith the currency crisis
(Darneacute and Hoarau 2008) mdash that needs to be accounted for and it
appears that the real value of the currency tends to revert to its
equilibrium very slowly after a shock which is suggestive of the fact
that the DGP may be fractionally integrated (Henry and Olekalns
2002)
3 Econometric methodology and results
31 Methodology
In this section we brie1047298
y describe some of the methods that will beemployed in this article in order to test for the empirical ful1047297lment of
the PPP theory
The 1047297rst tests presented are those attributable to Ng and Perron
(2001) who propose several modi1047297cations to existing unit root tests
in order to improve their size and power in particular with relatively
small samples The authors present the following tests MZ α and MZ t which arethe modi1047297ed versions of the Phillips (1987) and Phillips and
Perron (1988) Z α and Z t tests the MSB which is related to the
Bhargava (1986) R1 test and 1047297nally the MP t test which is a modi1047297ed
version of the Elliot et al (1996) Point Optimal Test These new tests
incorporate a Modi1047297ed Information Criterion (MIC) to select the lag
length in the auxiliary regression given that according to Ng and
Perron (2001) the Akaike and Schwartz Information Criteria tend to
select a low lag order Additionally these authors propose a General-
ised Least Squares method of detrending the data in order to improve
the power of the tests
Secondly as KSS point out traditional (linear) unit root tests
may fail to reject the null hypothesis when the DGP is non-linear If
the speed of adjustment is asymmetric ie it actually depends on
the degree of misalignment from the equilibrium DickeyndashFuller
type tests may incorrectly conclude that the series is a unit root
when in fact is a non-linear globally stationary process In this case
we may de1047297ne a DGP with two regimes that is an inner regime
where the variable is assumed to be I (1) and an outer regime
where the variable may or may not be a unit root The transition
between regimes is smooth rather than sudden Thus KSS propose a
unit root test to analyse the order of integration of the variable in
Fig 2 Estimates of d in model (6) and (7) across T b
Table 1
NgndashPerron and KSS unit root test results
Test Statistic CV (5) CV (10)
MZ α minus222737 minus810000 minus570000
MZ t minus105215 minus198000 minus162000
MSB 047237 023300 027500
MP t 109753 317000 445000
t NLD minus195642 minus291689 minus263526
Note Theorderof lagto computethe tests hasbeen chosen using themodi1047297edAIC (MAIC)
suggested by Ng andPerron (2001) The NgndashPerron tests includean interceptwhereas theKSS test has been applied to the demeaned data t NLD say The critical values for the Ngndash
Perron tests have been taken from Ng and Perron (2001) whereas those for the KSS have
been obtained by Monte Carlo simulations with 50000 replications
1186 JC Cuestas LA Gil-Alana Economic Modelling 26 (2009) 1184ndash1192
7232019 Economic Modelling Further Evidence on the PPP Analysis of the Australian Dollar- Non-linearities Fractional Integration and Structu
httpslidepdfcomreaderfulleconomic-modelling-further-evidence-on-the-ppp-analysis-of-the-australian-dollar- 49
the outer regime bearing in mind that the process is globally
stationary In other words
yt = β yt minus1 + yt minus1F θ yt minus1eth THORN + et eth3THORN
where ε t is iid(0 σ 2) and F (θ yt minus1) is the transition function which
is assumed to be exponential (ESTAR)
F θ yt minus1eth THORN = 1minus exp minusθ y2t minus 1
n o eth4THORN
with θ N0
In practice it is common to reparameterise Eq (3) as
Δ yt = α yt minus1 + γ yt minus1 1minus exp minusθ y2t minus 1
n o + et eth5THORN
in order to apply the test The null hypothesis H 0 θ =0 is tested
against the alternative H 1 θ N0 ie we test whether the variable is
an I (1) process in the outer regime From an economic viewpoint
and in the context of exchange rates this implies that the further
the RER deviates from the equilibrium the faster will be the speed
of mean reversion towards the fundamental equilibrium In addition
the existence of trade barriers may create a central threshold where
transactions are not pro1047297table and arbitrage does not clear the
market mdash unit root process in the inner regime whereas for large
deviations the pro1047297ts from arbitrage are greater than the cost and
the arbitrage mechanism brings the exchange to the inner regimeMoreover according to Taylor and Peel (2000) among others an
ESTAR function is appropriate to model exchange rates given that
this type of equation assumes that the effects of the shock on the
variable are symmetric in the sense that these effects do not depend
on the sign of the shock
On the other hand many test statistics have been developed in
recent years for fractional integration They can be parametric or
semiparametric and they can be speci1047297ed in the time domain or in the
frequency domain In this article we employ two parametric
approaches that allow us to incorporate structural breaks The 1047297rst
one is the well-known Robinson tests (1994) that we specify in a way
that permit us to include deterministicbroken trends In particular we
consider a model of form
yt = α 1I t b T beth THORN + α 2I t zT beth THORN + xt eth6THORN
1minusLeth THORNd xt = ut ut asympI 0eth THORN eth7THORN
where yt is the observed time series (RER) I ( x) is the indicator
function L is the lag operator (Lxt = xt minus1) d is a real value and ut isI (0) Based on this set-up for a given T b-value we test the null
hypothesis
H 0 d = do eth8THORN
in Eqs (6) and (7) for any real value do The functional form of the
test statistic is described in Appendix A and it was shown by
Robinson (1994) that under very mild regularity conditions a test
of Eq (8) against the one-sided alternatives dbdo (dNdo) follows
the standard N (0 1) distribution
The method presented just above imposes the same degree of
integration before and after the break-date Then we also perform a
recent procedure developed by Gil-Alana (2008) which allows us to
estimate the break-date along with the fractional differencing
parameters which might be different for each subsample This
method is based on minimising the residuals sum squares and is
brie1047298y described in Appendix B In the 1047297nal part of the article a
semiparametric method (Robinson 1995) is conducted on the two
subsamples
32 Empirical results
The results of applying the Ng and Perron (2001) and KSS tests are
reported in Table 1 The lag length has been selected using the
Modi1047297ed Akaike Information Criterion proposed by the former
authors for both tests In both cases the unit root hypothesis cannot
be rejected at conventional signi1047297cance levels Thus according to
these (unit-root) procedures there is no evidence of the PPP
hypothesis for the Australian case
Next we examine the possibility of fractional integration in the
context of non-linear deterministic terms For this purpose we
employ the model given by Eqs (6) and (7) testing H 0 (Eq (8)) for
do-values ranging from 0 to 2 with 001 increments using the
Robinsons (1994) parametric approach for 1047297
xed values of T b Wethen estimate d by choosing the value that produces the lowest
statistic in absolute value moving the break date T b 1-period
forward recursively from T b=40 (1980Q1) to T b=119 (1999Q4)
Fig 2 displays the estimated ds along with the 95 con1047297dence bands
for the two cases of white noise u t (in Fig 2(i)) and autocorrelated
disturbances (in Fig 2(ii)) In the latter case we assume that ut
follows the exponential spectral model of Bloom1047297eld (1973) Thisisa
non-parametric speci1047297cation for the error term that produces
autocorrelations decaying exponentially as in the autoregressive
(AR) case2
We observe in Fig 2 that the estimated values of d have remained
relatively stable across T b with values 1047298uctuating around 1 In fact
the unit root null hypothesis cannot be rejected for any T b We
observe that the only turbulences in the estimators take place when
T b is around 1985 Table 2 displays the estimates of d and the
intercepts for the case of a break at the four quarters in 1985 for the
two cases of white noise and autocorrelated disturbances We see in
this table that practically all the estimates are slightly above 1 The
only exception is the case of the break at 1985Q4 with autocorrelated
disturbances where the estimate of the fractional differencing
parameter is equal to 099 Nevertheless the unit root cannot be
rejected in any single case and thus we do not 1047297nd support for the
PPP hypothesis when using this model
2 See Gil-Alana (2004) for the analysis of fractional integration in the context of
Bloom1047297eld disturbances
Table 2
Estimates of intercepts and fractional differencing parameters with a break in 1985
Break dates White noise disturbances Autocorrelated disturbances
d α 1 α 2 d α 1 α 2
19 85 Q1 105 ( 095 117 ) 150 0 ( 149 7 1503 ) 140 7 ( 140 0 1413 ) 103 ( 085 129) 150 0 (1497 150 3) 140 4 (1398 1410)
19 85 Q2 104 ( 095 117 ) 150 0 ( 149 7 1503 ) 135 4 ( 134 8 1360 ) 103 ( 084 127) 150 0 (1497 150 3) 135 3 (1347 1359 )
19 85 Q3 110 ( 099 125 ) 1501 ( 149 8 150 4) 1556 ( 154 9 1562 ) 106 ( 088 132) 150 0 (1497 150 4) 1546 (1539 1552 )
19 85 Q4 106 ( 097 119) 150 0 ( 149 7 150 4) 143 5 ( 142 9 144 2) 099 ( 082 122) 149 9 (149 6 150 3) 1428 (1422 1435 )
Note In parenthesis the 95 con1047297dence intervals
1187 JC Cuestas LA Gil-Alana Economic Modelling 26 (2009) 1184ndash1192
7232019 Economic Modelling Further Evidence on the PPP Analysis of the Australian Dollar- Non-linearities Fractional Integration and Structu
httpslidepdfcomreaderfulleconomic-modelling-further-evidence-on-the-ppp-analysis-of-the-australian-dollar- 59
Still on this set-up we might be interested in knowing if the
intercept has changed from one subsample to another Thus we can
consider a joint test of the null hypothesis
H 0 α 1 = α 2 and d = do eth9THORNin Eqs (6) and (7) against the alternative
H a α 1 ne α 2 or d ne do eth10THORN
This possibility is not addressed in Robinson (1994) though Gil-
Alana and Robinson (1997) derived a similar Lagrange Multiplier(LM)
test as follows they consider a regression model of form as
yt = β V z t + xt t = 1 2 N eth11THORNand xt given by Eq (7) with the vector partitions z t = ( z At
T z Bt T )T
β = ( β AT β B
T )T In general we want to test H 0 d = do and β B =
β B0 Then a LM statistic may be shown to be r 2 (see Appendix
A) plus
XT
t =1
ˆ ut wT Bt
XT
t =1
wBt wT Bt minus
XT
t =1
wBt wT At
XT
t =1
w At wT At
minus1XT
t =1
w At wT Bt
minus1XT
t =1
ˆ ut wBt
eth12THORN
with wt = (w At T wBt
T )T = (1minus L)do z t
ˆ ut = 1minusLeth THORNdo yt minus ˜ β
T A β
T B0 wt ˜ β A = X
T
t =1
w At wT
At minus1
XT
t =1
w At 1minusLeth THORNdo yt
ˆ σ 2 = T
minus1 PT
t =1
ˆ u2t and r
2 is calculated as in Appendix A but
using the u t just de1047297ned If the dimension of z Bt is qB then we
Fig 4 Sum of squared residuals using the procedure of Gil-Alana
Fig 3 Estimates of d recursively obtained with samples of size T =100
1188 JC Cuestas LA Gil-Alana Economic Modelling 26 (2009) 1184ndash1192
7232019 Economic Modelling Further Evidence on the PPP Analysis of the Australian Dollar- Non-linearities Fractional Integration and Structu
httpslidepdfcomreaderfulleconomic-modelling-further-evidence-on-the-ppp-analysis-of-the-australian-dollar- 69
7232019 Economic Modelling Further Evidence on the PPP Analysis of the Australian Dollar- Non-linearities Fractional Integration and Structu
httpslidepdfcomreaderfulleconomic-modelling-further-evidence-on-the-ppp-analysis-of-the-australian-dollar- 79
asymmetric speed of adjustment towards the equilibrium as well as
fractional integration and structural changes Our results are in line
with previous empirical works in the sense that there is no evidence
of mean reversionin theAustralian dollarRER forthe periodanalysed
However we 1047297nd that the order of integration decreases after 1985
coinciding with the currency crisis This conclusion highlights the fact
that the monetary authorities have managed to decrease the
dependence of the Australian dollar on real shocks that may affect
the value of the currency on a permanent basis
Appendix A
The LM test of Robinson (1994) for testing H 0 (Eq (8)) in Eqs (6)
and (7) is
ˆ r = T 1 =2
ˆ σ 2ˆ Aminus1=2
ˆ a
where T is the sample size and
ˆ a = minus2π
T
XT minus1
j =1
ψ λ j
g λ j ˆ τ
minus1I λ j
ˆ σ 2
= σ 2
ˆ τ = 2π
T XT minus1
j =1
g λ j
ˆ τ minus1I λ
j
ˆ A = 2
T ethXT minus1
j =1
ψ λ j
2minus
XT minus1
j =1
ψ λ j
ˆ e λ j
V
timesXT minus1
j = 1
ˆ e λ j
ˆ e λ j
V
0
1Aminus1
timesXT minus1
j =1
ˆ e λ j
ψ λ j
THORNψ λ j
= log j2sin
λ j
2 j ˆ e λ j
=
A
Aτ log g λ j ˆ τ
λ j =
2π j
T ˆ τ = argmin σ
2τ eth THORN
a and A in the above expressions are obtained through the 1047297rst and
second derivatives of the log-likelihood function with respect to d
(see Robinson 1994 page 1422 for further details) I (l j) is the
periodogram of ut evaluated under the null ie
ˆ ut = 1minusLeth THORNdo yt minus α Vwt
ˆ α = XT
t = 1
wt wt V minus1
XT
t =1
wt 1minusL
eth THORNdo yt wt = 1minusL
eth THORNdo z t
z t = 1I t b T beth THORN 1I t zT beth THORNeth THORN V
and g is a known function related to the spectral density function of ut
f λ σ 2
τ
= σ
2
2π g λ τ eth THORN minus π b λ V π
Appendix B
Gil-Alana (2008) considers the following model
yt = α 1 + xt 1minus
Leth THORNd1
xt = ut t = 1 N
T b
and
yt = α 2 + xt 1minusLeth THORNd2 xt = ut t = T b + 1 N T
where the α 1 and α 2 are the intercepts corresponding to the 1047297rst and
second subsamples respectively d1 and d2 may be real values u t is
I (0) and T b is the time of a break that is supposed to be unknown
This model can be re-parameterized as follows
1minusLeth THORNd1 yt = α 1˜ 1t d1eth THORN + β 1˜ t t d1eth THORN + ut t = 1 N T b
1minus
Leth THORNd2
yt = α 2˜
1t d2eth THORN + β 2˜
t t d2eth THORN + ut t = T b + 1 N
T
Fig 5 Estimates of d based on a semiparametric model for each subsample
1190 JC Cuestas LA Gil-Alana Economic Modelling 26 (2009) 1184ndash1192
7232019 Economic Modelling Further Evidence on the PPP Analysis of the Australian Dollar- Non-linearities Fractional Integration and Structu
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with 1 t (di)=(1minusL)di1 The idea is then to minimise the residual sum
of squares (RSS)
wr t α 1 α 2 β 1 β 2f gXT b
t =1
1minusLeth THORNd 1eth THORN1o yt minus α 1˜ 1t d
1eth THORN1o
+ β 1˜ t
t d
1eth THORN1o
2
+XT
t = T b
frac12 1minusLeth THORNd 1eth THORN2o yt minusα 2˜ 1t d
1eth THORN2o
+ β 2 ˜ t t d
1eth THORN2o
2
In so doing it is necessary to choose a grid for the values of the
differencing parameters d1 and d2 d1o and d2o say Once the estimated
parameters α ˆ 1 α ˆ 2 β ˆ 1 and β ˆ
2 are obtained for partition T b and initial
values d1o(1) and d2o
(1) we plug these values into the objective function
and obtain RSS(T b)=arg mini jRSS(T bd1o(i) d1o
( j)) In order to estimate
the time break T k we obtain the moment that minimises the RSS
where the minimisation is taken over all partitions T 1 T 2 hellip T m such
that T iminusT iminus1ge |ε T | Then it is possible to obtain the regression
parameter estimates as α ˆ i=α ˆ i(T k) and the differencing parameters
d i= d
i(T k) for i =12
Appendix C
The ldquolocalrdquo Whittle estimate of Robinson (1995) is implicitly
de1047297ned by
ˆ d = arg mind log C deth THORN minus 2d 1
m
Xm
j = 1
log λ j
0
1A
for da minus1 = 2 1 = 2eth THORN C deth THORN = 1
m
Xm
j =1
I λ j
λ
2d j λ j =
2π j
T
m
T Y0
where m is a bandwidth parameter number and d isin(minus05 05)
Under 1047297niteness of the fourth moment and other mild conditions
Robinson (1995) proved that
ffiffiffiffiffimp ˆ d minus do
YdN 0 1 = 4eth THORN as T Yinfin
where do is the true value of d This estimator is robust to a certain
degree of conditional heteroskedasticity (Robinson and Henry 1999)
and is more ef 1047297cient than other semi-parametric competitors
References
Baum CF Barkoulas J Caglayan M 1999 Persistence in the international in1047298ationrates Southern Economic Journal 65 900ndash913
Bhargava A 1986 On the theory of testing for unit roots in observed time seriesReview of Economic Studies 53 369ndash384
Bierens HJ 1997 Testing the unit root with drift hypothesis against nonlinear trendstationarity with an application to the US price level and interest rate Journal of Econometrics 81 29ndash64
Bloom1047297eld P 1973 An exponential model in the spectrum of a scalar time series
Biometrika 60 217ndash226Booth GGKaen FR Koveos PE1982RS analysis of foreign exchangemarketsunder
two international monetary regimes Journal of Monetary Economics 10 407ndash415CampbellJY Perron P1991 Pitfalls and opportunities whatmacroeconomists should
know about unit roots NBER Macroeconomics Annual 1141ndash1201Casin P Ceacutespedes LF Sahay R 2004 Commodity currencies and the real exchange
rate Journal of Development Economics 75 239ndash268Cheung YW 1993 Long memory in foreign exchange rates Journal of Business and
Economic Statistics 11 93ndash101Cheung Y-W Lai KS 1993 A fractional cointegration analysis of purchasing power
parity Journal of Business and Economic Statistics 11 103ndash112Crato N Ray BK 2000 Memory in returns and volatilities of futures contracts
Journal of Futures Markets 20 525ndash543Cuestas JC 2009 Purchasing power parity in Central and Eastern European countries
an analysis of unit roots and nonlinearities Applied Economics Letters 16 87ndash94Cuestas JC Regis PJ 2008 Testing for PPP in Australia evidence from unit root tests
against nonlinear trend stationarity alternatives Economics Bulletin 27 1ndash8Darneacute O Hoarau J-F 2008 The purchasing power parity in Australia evidence from
unit root test with structural break Applied Economics Letters 15 203ndash206
DeJong D Nankervis J Savin NE Whiteman CH 1992 Integration versus trendstationarity in time series Econometrica 60 423ndash433
Dickey DA Fuller WA 1979 Distribution of the estimators for autoregressive timeseries with a unit root Journal of the American Statistical Association 74 427ndash431
Diebold FX Rudebusch GD 1991 Is consumption too smooth Long memory and theDeaton paradox The Review of Economics and Statistics 73 1ndash9
DieboldFXInoueA 2001 Longmemory andregimeswitchingJournal of Econometrics105 131ndash159
Diebold FX Husted S Rush M 1991 Real exchange rates under the gold standard Journal of Political Economy 99 1252ndash1271
Dumas B1994 Partial equilibrium versus general equilibrium models of the international
capital market In van der Ploeg F (Ed) The Handbook of International Macro-economics Blackwell Oxford pp 301ndash347 chap 10Elliot G Rothenberg TJ Stock JH 1996 Ef 1047297cient tests for an autoregressive unit root
Econometrica 64 813ndash836Fang HLai KS Lai M1994 Fractal structure in currency futurespricedynamicsJournal
of Futures Markets 14 169ndash181Faria JR Leoacuten-Ledesma MA 2003 Testing the BalassandashSamuelson effect implica-
tions for growth and the PPP Journal of Macroeconomics 25 241ndash253Faria JR Leoacuten-Ledesma MA 2005 Real exchange rate and employment performance
in an open economy Research in Economics 59 67ndash80Geweke J Porter-Hudack S 1983 The estimation and application of long memory
time series models and fractional differencing Journal of Time Series Analysis 4221ndash238
Gil-Alana LA 2000 Mean reversion in the real exchange rates Economics Letters 69285ndash288
Gil-Alana LA 2004 The use of the model of Bloom1047297eld (1973) as an approximation toARMA processes in the context of fractional integration Mathematical andComputer Modelling 39 429ndash436
Gil-AlanaLA 2008 Fractional integrationand structuralbreaks at unknown periods of
time Journal of Time Series Analysis 29 163ndash185Gil-Alana LA Robinson PM 1997 Testing of unit roots and other nonstationary
hypotheses in macroeconomic time series Journal of Econometrics 80 241ndash268GrangerCWJHyung N 2004 Occasional structural breaks and long memory with an
application to the SampP 500 absolute stock return Journal of Empirical Finance 11399ndash421
Hassler U Wolters J 1995 Long memory in in1047298ation rates International evidence Journal of Business and Economic Statistics 13 37ndash45
Henry OT Olekalns N 2002 Does the Australian dollar real exchange rate displaymean reversion Journal of International Money and Finance 21 651ndash666
Juvenal L Taylor MP 20 08 Threshold adjustment of deviations from the law of oneprice Studies in Nonlinear Dynamics and Econometrics 12 Article 8
Kapetanios G Shin Y Snell A 2003 Testing for a unit root in the nonlinear STAR framework Journal of Econometrics 112 359ndash379
Kwiatkowski D Phillips PCB Schmidt P Shin Y 1992 Testing the null hypothesis of stationarity againstthe alternative of a unitrootJournal of Econometrics54159ndash178
Lee D Schmidt P 1996 On the power of the KPSS test of stationarity againstfractionally integrated alternatives Journal of Econometrics 73 285ndash302
Michael P Nobay A Peel D 1997 Transaction costs and nonlinear adjustment in realexchange ratesan empirical investigation Journalof Political Economy 105 862ndash879
Nelson CR Piger J Zivot E 2001 Markov regime-switching and unit root tests Journal of Business Economics and Statistics 19 404ndash415
Ng S Perron P 2001 Lag selection and the construction of unit root tests with goodsize and power Econometrica 69 1519ndash1554
Perron P 1989 The great crash the oil price shock and the unit root hypothesisEconometrica 571361ndash1401
Perron P Phillips PCB 1987 Does GNP have a unit root A reevaluation EconomicsLetters 23 139ndash145
Perron P Rodriacuteguez G 2003 GLS detrending ef 1047297cient unit root tests and structuralchange Journal of Econometrics 115 1ndash27
Phillips PCB 1987 Time series regression with a unit root Econometrica 55 311ndash340Phillips PCB Perron P 1988 lsquoTesting for a unit root in time series regression
Biometrica 75 335ndash346Phillips PCB Shimotsu K 2004 Local Whittle estimation in nonstationary and unit
root cases Annals of Statistics 32 656ndash692Phillips PCB Shimotsu K 2005 Exact local Whittle estimation of fractional
integration Annals of Statistics 33 1890ndash1933
Robinson PM 1994 ldquoEf 1047297cient tests of nonstationary hypotheses Journal of theAmerican Statistical Association 89 1420ndash1437
Robinson PM 1995 Gaussian semi-parametric estimation of long range dependenceAnnals of Statistics 23 1630ndash1661
Robinson PM Henry M 1996 Bandwidth choice in Gaussian semiparametricestimation of long-range dependence In Robinson PM Rosenblatt M (Eds)Athens Conference on Applied Probability in Time Series Analysis Vol II New Yorkpp 220ndash232
Robinson PM Henry M 1999 Long and short memory conditional heteroskedasticityin estimating the memory in levels Econometric Theory 15 299ndash336
Sarno L Taylor MP 2002 Purchasing power parity and the real exchange rate TheEconomics of the Exchange Rate Cambridge University Press Cambridge pp 51ndash96
Sarno L Taylor MP Chowdhury I 2004 Nonlinear dynamics in deviations from thelaw of oneprice a broad-based empiricalstudy Journal of International Money andFinance 23 1ndash25
Taylor MP 2006 Real exchange rates and purchasing power paritymean-reversion ineconomic thought Applied Financial Economics 16 1ndash17
Taylor MP Peel DA 2000 ldquoNonlinear adjustment long-run equilibrium andexchangerate fundamentals Journal of International Moneyand Finance 1933ndash53
1191 JC Cuestas LA Gil-Alana Economic Modelling 26 (2009) 1184ndash1192
7232019 Economic Modelling Further Evidence on the PPP Analysis of the Australian Dollar- Non-linearities Fractional Integration and Structu
httpslidepdfcomreaderfulleconomic-modelling-further-evidence-on-the-ppp-analysis-of-the-australian-dollar- 99
Taylor MP Peel DA Sarno L 2001 Nonlinear mean-reversion in real eschange ratesTowards a solution to the purchasing power parity puzzles International EconomicReview 42 1015ndash1042
Velasco C 1999 Gaussian semiparametric estimation of nonstationary time series Journal of Time Series Analysis 20 87ndash127
Velasco C Robinson PM 2000 Whittle pseudo maximum likelihood estimation fornonstationary time series Journal of the American Statistical Association 951229ndash1243
Vogelsang TJ 1997 Wald-type test for detecting breaks in the trend function of adynamic time series Econometric Theory 13 818ndash849
Wang C 2004 Futures trading activity and predictable foreign exchange movements Journal of Banking and Finance 28 1023ndash1041
Wei S-J Parsley DC 1995 Purchasing power disparity during the 1047298oat rate periodexchange rate volatility trade barriers and other culprits Working Paper 5032NBER
WestKD1987 A noteon thepowerof least squares testsfor a unitroot Economics Letters24 1397ndash1418
1192 JC Cuestas LA Gil-Alana Economic Modelling 26 (2009) 1184ndash1192
7232019 Economic Modelling Further Evidence on the PPP Analysis of the Australian Dollar- Non-linearities Fractional Integration and Structu
httpslidepdfcomreaderfulleconomic-modelling-further-evidence-on-the-ppp-analysis-of-the-australian-dollar- 29
Within the empirical literature on the PPP theory it is well known
that short run deviation may prevent the PPP hypothesis from holding
true However one may expect the RER to return to its equilibrium
value after a shock in the long run Statistically this implies that the
RER should be a mean reverting process for the PPP hypothesis to be
ful1047297lled which makes unit root testing to be an appealing econo-
metric approach for this purpose
Sarno and Taylor (2002) Taylor et al (2001) and Taylor (2006)
among others provide summaries of the main contributions to the
literature from which it is possible to highlight several facts 1047297rst
authors have applied different techniques since the 70s in order to
capture the true data generating process (DGP) of the real exchange
rates second it appears that the results have been quite
controversial and 1047297nally that the most recent contributions focus
on the consideration of non-linearities and fractional integration
alternatives In this vein Taylor et al (2001) identi1047297ed two main
paradoxes relating to RER behaviour First the relationship bet-
ween nominal exchange rates and prices implied by the PPP theory
is not in many cases a cointegrating one second how is it possible
to observe a high volatility of exchange rates in the short termwith the low speed of mean reversion generally observed in this
variable
The failure of the literature to provide good answers to the
aforementioned questions have been related to the low power of the
(unit-root)tests applied to analyse the empirical ful1047297lment of PPP1 It
appears that increasing the data sample creates additional problems
such as structural changes that may affect the power of the tests if
these are not taken into account (see Perron and Phillips1987 West
1987 among others) Therefore these kind of non-linearities in the
deterministic components may increase the probability of Type II
error with traditional unit root tests (eg DickeyndashFuller type)
Furthermore non-linearities in the RERlong runpath maybe present
in the form of an asymmetric speed of adjustment towards the
equilibrium ie the further the RER deviates from the long runequilibrium value the faster the speed of mean reversion is expected
to be From an econometric viewpoint that may imply an
autoregressive parameter dependent on the values of the variable
The non-linear behaviour of exchange rates has been acknowledged
by several authors such as Dumas (1994) Michael et al (1997)
Sarno et al (2004) Juvenal and Taylor (2008) and Cuestas (2009)
among many others
In addition as pointed out by several authors such as Diebold et al
(1991) Cheung and Lai (1993) and Gil-Alana (2000) among others
the real exchange rate may be characterised as a slow mean reversion
process and traditional unit root may not be able to distinguish this
type of process from a unit root Thus according to these authors
fractional integration may be an alternative viable route to the
analysis of the RER dynamics As mentioned above the question of
interest is to determine if deviations from PPP are transitory or
permanent which translated to the fractionally integrated literature
means to determine if the order of integration is smallerthan 1 (mean
reversion) or equal to or higher than 1 (no mean reversion) Applying
RS techniques to daily rates for the British pound French franc and
Deutsche mark Booth et al (1982) found evidence of fractional
integration during the 1047298exible exchange rate period (1973ndash1979)
Cheung (1993) also found similar evidence in foreign exchange
markets during the managed 1047298oating regime On the other hand
Baum et al (1999) estimated ARFIMA models for real exchange rates
in the post-Bretton Woods era and found almost no evidence to
support long run PPP Additional papers on exchange rate dynamics
using fractional integration are Fang et al (1994) Crato and Ray(2000) and Wang (2004)
In another contribution Henry and Olekalns (2002) analyse the
Australian RERlong run behaviour in the context of structural changes
and fractional integration These authors apply Robinsons (1994) and
Geweke and Porter-Hudaks (1983) fractional integration tests and the
Vogelsangs (1997) unit root test with structural changes but they are
not able to 1047297nd evidence of mean reversion in the RER Australian
dollar Similar results are obtained by Darneacute and Hoarau (2008) who
applied the Perron and Rodriacuteguezs (2003) unit root test with
structural changes to the Australian dollar Contrary to these results
Cuestas and Regis (2008) found that the Australian RER is stationary
around a non-linear deterministic trend by means of applying the
Bierens (1997) unit root tests which proposes a Chebyshev
polynomial approximation for the non-linear trend However theresults are dependent on the selection of the order of the polynomials
since there is no unique way of doing it Additionally in the context of
PPP it is dif 1047297cult to give an economic interpretation of the non-linear
trend
The aim of the present paper is to provide further evidence of the
PPP ful1047297lment in Australia and complement the previous literatureon
the PPP analysis by applying different techniques that address the
aforementioned facts ie fractional integration and non-linearities
that may affect the power of the traditional (linear) unit root tests
First we apply the upgraded versions of linear unit root tests
propossed by Ng and Perron (2001) second we account for
asymmetric speed of mean reversion and apply the Kapetanios et al
(KSS 2003) unit root test which generalises the alternative hypoth-
esis to a globally stationary exponential smooth transition (ESTAR)
1 Unit root tests most commonly employed in the literature (Dickey and Fuller 1979
Phillips and Perron 1988 Kwiatkowski et al 1992 etc) have very low power against
trend-stationarity (DeJong et al 1992) structural breaks (Perron 1989 Campbell and
Perron 1991) regime-switching ( Nelson et al 2001) or fractional integration
(Diebold and Rudebusch 1991 Hassler and Wolters 1995 Lee and Schmidt 1996)
Fig 1 Australian RER
1185 JC Cuestas LA Gil-Alana Economic Modelling 26 (2009) 1184ndash1192
7232019 Economic Modelling Further Evidence on the PPP Analysis of the Australian Dollar- Non-linearities Fractional Integration and Structu
httpslidepdfcomreaderfulleconomic-modelling-further-evidence-on-the-ppp-analysis-of-the-australian-dollar- 39
process Finally in order to take into account the existence of
structural changes and fractional integration at the same time we
apply Robinsons (1994) and Gil-Alanas (2008) fractional integration
tests in the context of structural changes
The remainder of this paper is organised as follows The next
section describes the data Section 3 summarises the econometric
techniques applied to empirically assess the ful1047297lment of PPP in
Australia and presents the results Section 4 concludes the paper
2 The data
The data for this empirical analysis consists of quarterly observa-
tions of the Australian real effective exchange rate computed as a
trade weighted index from 19702 to 20083 obtained from the
Central Bank of Australia web page (httpwwwrbagovau) A plot
of the data is displayed in Fig 1 From this graph it is possible to
highlight two mainstylised facts the existence of a structural change
in theseriesmdash presumablyat 1985 coincidingwith the currency crisis
(Darneacute and Hoarau 2008) mdash that needs to be accounted for and it
appears that the real value of the currency tends to revert to its
equilibrium very slowly after a shock which is suggestive of the fact
that the DGP may be fractionally integrated (Henry and Olekalns
2002)
3 Econometric methodology and results
31 Methodology
In this section we brie1047298
y describe some of the methods that will beemployed in this article in order to test for the empirical ful1047297lment of
the PPP theory
The 1047297rst tests presented are those attributable to Ng and Perron
(2001) who propose several modi1047297cations to existing unit root tests
in order to improve their size and power in particular with relatively
small samples The authors present the following tests MZ α and MZ t which arethe modi1047297ed versions of the Phillips (1987) and Phillips and
Perron (1988) Z α and Z t tests the MSB which is related to the
Bhargava (1986) R1 test and 1047297nally the MP t test which is a modi1047297ed
version of the Elliot et al (1996) Point Optimal Test These new tests
incorporate a Modi1047297ed Information Criterion (MIC) to select the lag
length in the auxiliary regression given that according to Ng and
Perron (2001) the Akaike and Schwartz Information Criteria tend to
select a low lag order Additionally these authors propose a General-
ised Least Squares method of detrending the data in order to improve
the power of the tests
Secondly as KSS point out traditional (linear) unit root tests
may fail to reject the null hypothesis when the DGP is non-linear If
the speed of adjustment is asymmetric ie it actually depends on
the degree of misalignment from the equilibrium DickeyndashFuller
type tests may incorrectly conclude that the series is a unit root
when in fact is a non-linear globally stationary process In this case
we may de1047297ne a DGP with two regimes that is an inner regime
where the variable is assumed to be I (1) and an outer regime
where the variable may or may not be a unit root The transition
between regimes is smooth rather than sudden Thus KSS propose a
unit root test to analyse the order of integration of the variable in
Fig 2 Estimates of d in model (6) and (7) across T b
Table 1
NgndashPerron and KSS unit root test results
Test Statistic CV (5) CV (10)
MZ α minus222737 minus810000 minus570000
MZ t minus105215 minus198000 minus162000
MSB 047237 023300 027500
MP t 109753 317000 445000
t NLD minus195642 minus291689 minus263526
Note Theorderof lagto computethe tests hasbeen chosen using themodi1047297edAIC (MAIC)
suggested by Ng andPerron (2001) The NgndashPerron tests includean interceptwhereas theKSS test has been applied to the demeaned data t NLD say The critical values for the Ngndash
Perron tests have been taken from Ng and Perron (2001) whereas those for the KSS have
been obtained by Monte Carlo simulations with 50000 replications
1186 JC Cuestas LA Gil-Alana Economic Modelling 26 (2009) 1184ndash1192
7232019 Economic Modelling Further Evidence on the PPP Analysis of the Australian Dollar- Non-linearities Fractional Integration and Structu
httpslidepdfcomreaderfulleconomic-modelling-further-evidence-on-the-ppp-analysis-of-the-australian-dollar- 49
the outer regime bearing in mind that the process is globally
stationary In other words
yt = β yt minus1 + yt minus1F θ yt minus1eth THORN + et eth3THORN
where ε t is iid(0 σ 2) and F (θ yt minus1) is the transition function which
is assumed to be exponential (ESTAR)
F θ yt minus1eth THORN = 1minus exp minusθ y2t minus 1
n o eth4THORN
with θ N0
In practice it is common to reparameterise Eq (3) as
Δ yt = α yt minus1 + γ yt minus1 1minus exp minusθ y2t minus 1
n o + et eth5THORN
in order to apply the test The null hypothesis H 0 θ =0 is tested
against the alternative H 1 θ N0 ie we test whether the variable is
an I (1) process in the outer regime From an economic viewpoint
and in the context of exchange rates this implies that the further
the RER deviates from the equilibrium the faster will be the speed
of mean reversion towards the fundamental equilibrium In addition
the existence of trade barriers may create a central threshold where
transactions are not pro1047297table and arbitrage does not clear the
market mdash unit root process in the inner regime whereas for large
deviations the pro1047297ts from arbitrage are greater than the cost and
the arbitrage mechanism brings the exchange to the inner regimeMoreover according to Taylor and Peel (2000) among others an
ESTAR function is appropriate to model exchange rates given that
this type of equation assumes that the effects of the shock on the
variable are symmetric in the sense that these effects do not depend
on the sign of the shock
On the other hand many test statistics have been developed in
recent years for fractional integration They can be parametric or
semiparametric and they can be speci1047297ed in the time domain or in the
frequency domain In this article we employ two parametric
approaches that allow us to incorporate structural breaks The 1047297rst
one is the well-known Robinson tests (1994) that we specify in a way
that permit us to include deterministicbroken trends In particular we
consider a model of form
yt = α 1I t b T beth THORN + α 2I t zT beth THORN + xt eth6THORN
1minusLeth THORNd xt = ut ut asympI 0eth THORN eth7THORN
where yt is the observed time series (RER) I ( x) is the indicator
function L is the lag operator (Lxt = xt minus1) d is a real value and ut isI (0) Based on this set-up for a given T b-value we test the null
hypothesis
H 0 d = do eth8THORN
in Eqs (6) and (7) for any real value do The functional form of the
test statistic is described in Appendix A and it was shown by
Robinson (1994) that under very mild regularity conditions a test
of Eq (8) against the one-sided alternatives dbdo (dNdo) follows
the standard N (0 1) distribution
The method presented just above imposes the same degree of
integration before and after the break-date Then we also perform a
recent procedure developed by Gil-Alana (2008) which allows us to
estimate the break-date along with the fractional differencing
parameters which might be different for each subsample This
method is based on minimising the residuals sum squares and is
brie1047298y described in Appendix B In the 1047297nal part of the article a
semiparametric method (Robinson 1995) is conducted on the two
subsamples
32 Empirical results
The results of applying the Ng and Perron (2001) and KSS tests are
reported in Table 1 The lag length has been selected using the
Modi1047297ed Akaike Information Criterion proposed by the former
authors for both tests In both cases the unit root hypothesis cannot
be rejected at conventional signi1047297cance levels Thus according to
these (unit-root) procedures there is no evidence of the PPP
hypothesis for the Australian case
Next we examine the possibility of fractional integration in the
context of non-linear deterministic terms For this purpose we
employ the model given by Eqs (6) and (7) testing H 0 (Eq (8)) for
do-values ranging from 0 to 2 with 001 increments using the
Robinsons (1994) parametric approach for 1047297
xed values of T b Wethen estimate d by choosing the value that produces the lowest
statistic in absolute value moving the break date T b 1-period
forward recursively from T b=40 (1980Q1) to T b=119 (1999Q4)
Fig 2 displays the estimated ds along with the 95 con1047297dence bands
for the two cases of white noise u t (in Fig 2(i)) and autocorrelated
disturbances (in Fig 2(ii)) In the latter case we assume that ut
follows the exponential spectral model of Bloom1047297eld (1973) Thisisa
non-parametric speci1047297cation for the error term that produces
autocorrelations decaying exponentially as in the autoregressive
(AR) case2
We observe in Fig 2 that the estimated values of d have remained
relatively stable across T b with values 1047298uctuating around 1 In fact
the unit root null hypothesis cannot be rejected for any T b We
observe that the only turbulences in the estimators take place when
T b is around 1985 Table 2 displays the estimates of d and the
intercepts for the case of a break at the four quarters in 1985 for the
two cases of white noise and autocorrelated disturbances We see in
this table that practically all the estimates are slightly above 1 The
only exception is the case of the break at 1985Q4 with autocorrelated
disturbances where the estimate of the fractional differencing
parameter is equal to 099 Nevertheless the unit root cannot be
rejected in any single case and thus we do not 1047297nd support for the
PPP hypothesis when using this model
2 See Gil-Alana (2004) for the analysis of fractional integration in the context of
Bloom1047297eld disturbances
Table 2
Estimates of intercepts and fractional differencing parameters with a break in 1985
Break dates White noise disturbances Autocorrelated disturbances
d α 1 α 2 d α 1 α 2
19 85 Q1 105 ( 095 117 ) 150 0 ( 149 7 1503 ) 140 7 ( 140 0 1413 ) 103 ( 085 129) 150 0 (1497 150 3) 140 4 (1398 1410)
19 85 Q2 104 ( 095 117 ) 150 0 ( 149 7 1503 ) 135 4 ( 134 8 1360 ) 103 ( 084 127) 150 0 (1497 150 3) 135 3 (1347 1359 )
19 85 Q3 110 ( 099 125 ) 1501 ( 149 8 150 4) 1556 ( 154 9 1562 ) 106 ( 088 132) 150 0 (1497 150 4) 1546 (1539 1552 )
19 85 Q4 106 ( 097 119) 150 0 ( 149 7 150 4) 143 5 ( 142 9 144 2) 099 ( 082 122) 149 9 (149 6 150 3) 1428 (1422 1435 )
Note In parenthesis the 95 con1047297dence intervals
1187 JC Cuestas LA Gil-Alana Economic Modelling 26 (2009) 1184ndash1192
7232019 Economic Modelling Further Evidence on the PPP Analysis of the Australian Dollar- Non-linearities Fractional Integration and Structu
httpslidepdfcomreaderfulleconomic-modelling-further-evidence-on-the-ppp-analysis-of-the-australian-dollar- 59
Still on this set-up we might be interested in knowing if the
intercept has changed from one subsample to another Thus we can
consider a joint test of the null hypothesis
H 0 α 1 = α 2 and d = do eth9THORNin Eqs (6) and (7) against the alternative
H a α 1 ne α 2 or d ne do eth10THORN
This possibility is not addressed in Robinson (1994) though Gil-
Alana and Robinson (1997) derived a similar Lagrange Multiplier(LM)
test as follows they consider a regression model of form as
yt = β V z t + xt t = 1 2 N eth11THORNand xt given by Eq (7) with the vector partitions z t = ( z At
T z Bt T )T
β = ( β AT β B
T )T In general we want to test H 0 d = do and β B =
β B0 Then a LM statistic may be shown to be r 2 (see Appendix
A) plus
XT
t =1
ˆ ut wT Bt
XT
t =1
wBt wT Bt minus
XT
t =1
wBt wT At
XT
t =1
w At wT At
minus1XT
t =1
w At wT Bt
minus1XT
t =1
ˆ ut wBt
eth12THORN
with wt = (w At T wBt
T )T = (1minus L)do z t
ˆ ut = 1minusLeth THORNdo yt minus ˜ β
T A β
T B0 wt ˜ β A = X
T
t =1
w At wT
At minus1
XT
t =1
w At 1minusLeth THORNdo yt
ˆ σ 2 = T
minus1 PT
t =1
ˆ u2t and r
2 is calculated as in Appendix A but
using the u t just de1047297ned If the dimension of z Bt is qB then we
Fig 4 Sum of squared residuals using the procedure of Gil-Alana
Fig 3 Estimates of d recursively obtained with samples of size T =100
1188 JC Cuestas LA Gil-Alana Economic Modelling 26 (2009) 1184ndash1192
7232019 Economic Modelling Further Evidence on the PPP Analysis of the Australian Dollar- Non-linearities Fractional Integration and Structu
httpslidepdfcomreaderfulleconomic-modelling-further-evidence-on-the-ppp-analysis-of-the-australian-dollar- 69
7232019 Economic Modelling Further Evidence on the PPP Analysis of the Australian Dollar- Non-linearities Fractional Integration and Structu
httpslidepdfcomreaderfulleconomic-modelling-further-evidence-on-the-ppp-analysis-of-the-australian-dollar- 79
asymmetric speed of adjustment towards the equilibrium as well as
fractional integration and structural changes Our results are in line
with previous empirical works in the sense that there is no evidence
of mean reversionin theAustralian dollarRER forthe periodanalysed
However we 1047297nd that the order of integration decreases after 1985
coinciding with the currency crisis This conclusion highlights the fact
that the monetary authorities have managed to decrease the
dependence of the Australian dollar on real shocks that may affect
the value of the currency on a permanent basis
Appendix A
The LM test of Robinson (1994) for testing H 0 (Eq (8)) in Eqs (6)
and (7) is
ˆ r = T 1 =2
ˆ σ 2ˆ Aminus1=2
ˆ a
where T is the sample size and
ˆ a = minus2π
T
XT minus1
j =1
ψ λ j
g λ j ˆ τ
minus1I λ j
ˆ σ 2
= σ 2
ˆ τ = 2π
T XT minus1
j =1
g λ j
ˆ τ minus1I λ
j
ˆ A = 2
T ethXT minus1
j =1
ψ λ j
2minus
XT minus1
j =1
ψ λ j
ˆ e λ j
V
timesXT minus1
j = 1
ˆ e λ j
ˆ e λ j
V
0
1Aminus1
timesXT minus1
j =1
ˆ e λ j
ψ λ j
THORNψ λ j
= log j2sin
λ j
2 j ˆ e λ j
=
A
Aτ log g λ j ˆ τ
λ j =
2π j
T ˆ τ = argmin σ
2τ eth THORN
a and A in the above expressions are obtained through the 1047297rst and
second derivatives of the log-likelihood function with respect to d
(see Robinson 1994 page 1422 for further details) I (l j) is the
periodogram of ut evaluated under the null ie
ˆ ut = 1minusLeth THORNdo yt minus α Vwt
ˆ α = XT
t = 1
wt wt V minus1
XT
t =1
wt 1minusL
eth THORNdo yt wt = 1minusL
eth THORNdo z t
z t = 1I t b T beth THORN 1I t zT beth THORNeth THORN V
and g is a known function related to the spectral density function of ut
f λ σ 2
τ
= σ
2
2π g λ τ eth THORN minus π b λ V π
Appendix B
Gil-Alana (2008) considers the following model
yt = α 1 + xt 1minus
Leth THORNd1
xt = ut t = 1 N
T b
and
yt = α 2 + xt 1minusLeth THORNd2 xt = ut t = T b + 1 N T
where the α 1 and α 2 are the intercepts corresponding to the 1047297rst and
second subsamples respectively d1 and d2 may be real values u t is
I (0) and T b is the time of a break that is supposed to be unknown
This model can be re-parameterized as follows
1minusLeth THORNd1 yt = α 1˜ 1t d1eth THORN + β 1˜ t t d1eth THORN + ut t = 1 N T b
1minus
Leth THORNd2
yt = α 2˜
1t d2eth THORN + β 2˜
t t d2eth THORN + ut t = T b + 1 N
T
Fig 5 Estimates of d based on a semiparametric model for each subsample
1190 JC Cuestas LA Gil-Alana Economic Modelling 26 (2009) 1184ndash1192
7232019 Economic Modelling Further Evidence on the PPP Analysis of the Australian Dollar- Non-linearities Fractional Integration and Structu
httpslidepdfcomreaderfulleconomic-modelling-further-evidence-on-the-ppp-analysis-of-the-australian-dollar- 89
with 1 t (di)=(1minusL)di1 The idea is then to minimise the residual sum
of squares (RSS)
wr t α 1 α 2 β 1 β 2f gXT b
t =1
1minusLeth THORNd 1eth THORN1o yt minus α 1˜ 1t d
1eth THORN1o
+ β 1˜ t
t d
1eth THORN1o
2
+XT
t = T b
frac12 1minusLeth THORNd 1eth THORN2o yt minusα 2˜ 1t d
1eth THORN2o
+ β 2 ˜ t t d
1eth THORN2o
2
In so doing it is necessary to choose a grid for the values of the
differencing parameters d1 and d2 d1o and d2o say Once the estimated
parameters α ˆ 1 α ˆ 2 β ˆ 1 and β ˆ
2 are obtained for partition T b and initial
values d1o(1) and d2o
(1) we plug these values into the objective function
and obtain RSS(T b)=arg mini jRSS(T bd1o(i) d1o
( j)) In order to estimate
the time break T k we obtain the moment that minimises the RSS
where the minimisation is taken over all partitions T 1 T 2 hellip T m such
that T iminusT iminus1ge |ε T | Then it is possible to obtain the regression
parameter estimates as α ˆ i=α ˆ i(T k) and the differencing parameters
d i= d
i(T k) for i =12
Appendix C
The ldquolocalrdquo Whittle estimate of Robinson (1995) is implicitly
de1047297ned by
ˆ d = arg mind log C deth THORN minus 2d 1
m
Xm
j = 1
log λ j
0
1A
for da minus1 = 2 1 = 2eth THORN C deth THORN = 1
m
Xm
j =1
I λ j
λ
2d j λ j =
2π j
T
m
T Y0
where m is a bandwidth parameter number and d isin(minus05 05)
Under 1047297niteness of the fourth moment and other mild conditions
Robinson (1995) proved that
ffiffiffiffiffimp ˆ d minus do
YdN 0 1 = 4eth THORN as T Yinfin
where do is the true value of d This estimator is robust to a certain
degree of conditional heteroskedasticity (Robinson and Henry 1999)
and is more ef 1047297cient than other semi-parametric competitors
References
Baum CF Barkoulas J Caglayan M 1999 Persistence in the international in1047298ationrates Southern Economic Journal 65 900ndash913
Bhargava A 1986 On the theory of testing for unit roots in observed time seriesReview of Economic Studies 53 369ndash384
Bierens HJ 1997 Testing the unit root with drift hypothesis against nonlinear trendstationarity with an application to the US price level and interest rate Journal of Econometrics 81 29ndash64
Bloom1047297eld P 1973 An exponential model in the spectrum of a scalar time series
Biometrika 60 217ndash226Booth GGKaen FR Koveos PE1982RS analysis of foreign exchangemarketsunder
two international monetary regimes Journal of Monetary Economics 10 407ndash415CampbellJY Perron P1991 Pitfalls and opportunities whatmacroeconomists should
know about unit roots NBER Macroeconomics Annual 1141ndash1201Casin P Ceacutespedes LF Sahay R 2004 Commodity currencies and the real exchange
rate Journal of Development Economics 75 239ndash268Cheung YW 1993 Long memory in foreign exchange rates Journal of Business and
Economic Statistics 11 93ndash101Cheung Y-W Lai KS 1993 A fractional cointegration analysis of purchasing power
parity Journal of Business and Economic Statistics 11 103ndash112Crato N Ray BK 2000 Memory in returns and volatilities of futures contracts
Journal of Futures Markets 20 525ndash543Cuestas JC 2009 Purchasing power parity in Central and Eastern European countries
an analysis of unit roots and nonlinearities Applied Economics Letters 16 87ndash94Cuestas JC Regis PJ 2008 Testing for PPP in Australia evidence from unit root tests
against nonlinear trend stationarity alternatives Economics Bulletin 27 1ndash8Darneacute O Hoarau J-F 2008 The purchasing power parity in Australia evidence from
unit root test with structural break Applied Economics Letters 15 203ndash206
DeJong D Nankervis J Savin NE Whiteman CH 1992 Integration versus trendstationarity in time series Econometrica 60 423ndash433
Dickey DA Fuller WA 1979 Distribution of the estimators for autoregressive timeseries with a unit root Journal of the American Statistical Association 74 427ndash431
Diebold FX Rudebusch GD 1991 Is consumption too smooth Long memory and theDeaton paradox The Review of Economics and Statistics 73 1ndash9
DieboldFXInoueA 2001 Longmemory andregimeswitchingJournal of Econometrics105 131ndash159
Diebold FX Husted S Rush M 1991 Real exchange rates under the gold standard Journal of Political Economy 99 1252ndash1271
Dumas B1994 Partial equilibrium versus general equilibrium models of the international
capital market In van der Ploeg F (Ed) The Handbook of International Macro-economics Blackwell Oxford pp 301ndash347 chap 10Elliot G Rothenberg TJ Stock JH 1996 Ef 1047297cient tests for an autoregressive unit root
Econometrica 64 813ndash836Fang HLai KS Lai M1994 Fractal structure in currency futurespricedynamicsJournal
of Futures Markets 14 169ndash181Faria JR Leoacuten-Ledesma MA 2003 Testing the BalassandashSamuelson effect implica-
tions for growth and the PPP Journal of Macroeconomics 25 241ndash253Faria JR Leoacuten-Ledesma MA 2005 Real exchange rate and employment performance
in an open economy Research in Economics 59 67ndash80Geweke J Porter-Hudack S 1983 The estimation and application of long memory
time series models and fractional differencing Journal of Time Series Analysis 4221ndash238
Gil-Alana LA 2000 Mean reversion in the real exchange rates Economics Letters 69285ndash288
Gil-Alana LA 2004 The use of the model of Bloom1047297eld (1973) as an approximation toARMA processes in the context of fractional integration Mathematical andComputer Modelling 39 429ndash436
Gil-AlanaLA 2008 Fractional integrationand structuralbreaks at unknown periods of
time Journal of Time Series Analysis 29 163ndash185Gil-Alana LA Robinson PM 1997 Testing of unit roots and other nonstationary
hypotheses in macroeconomic time series Journal of Econometrics 80 241ndash268GrangerCWJHyung N 2004 Occasional structural breaks and long memory with an
application to the SampP 500 absolute stock return Journal of Empirical Finance 11399ndash421
Hassler U Wolters J 1995 Long memory in in1047298ation rates International evidence Journal of Business and Economic Statistics 13 37ndash45
Henry OT Olekalns N 2002 Does the Australian dollar real exchange rate displaymean reversion Journal of International Money and Finance 21 651ndash666
Juvenal L Taylor MP 20 08 Threshold adjustment of deviations from the law of oneprice Studies in Nonlinear Dynamics and Econometrics 12 Article 8
Kapetanios G Shin Y Snell A 2003 Testing for a unit root in the nonlinear STAR framework Journal of Econometrics 112 359ndash379
Kwiatkowski D Phillips PCB Schmidt P Shin Y 1992 Testing the null hypothesis of stationarity againstthe alternative of a unitrootJournal of Econometrics54159ndash178
Lee D Schmidt P 1996 On the power of the KPSS test of stationarity againstfractionally integrated alternatives Journal of Econometrics 73 285ndash302
Michael P Nobay A Peel D 1997 Transaction costs and nonlinear adjustment in realexchange ratesan empirical investigation Journalof Political Economy 105 862ndash879
Nelson CR Piger J Zivot E 2001 Markov regime-switching and unit root tests Journal of Business Economics and Statistics 19 404ndash415
Ng S Perron P 2001 Lag selection and the construction of unit root tests with goodsize and power Econometrica 69 1519ndash1554
Perron P 1989 The great crash the oil price shock and the unit root hypothesisEconometrica 571361ndash1401
Perron P Phillips PCB 1987 Does GNP have a unit root A reevaluation EconomicsLetters 23 139ndash145
Perron P Rodriacuteguez G 2003 GLS detrending ef 1047297cient unit root tests and structuralchange Journal of Econometrics 115 1ndash27
Phillips PCB 1987 Time series regression with a unit root Econometrica 55 311ndash340Phillips PCB Perron P 1988 lsquoTesting for a unit root in time series regression
Biometrica 75 335ndash346Phillips PCB Shimotsu K 2004 Local Whittle estimation in nonstationary and unit
root cases Annals of Statistics 32 656ndash692Phillips PCB Shimotsu K 2005 Exact local Whittle estimation of fractional
integration Annals of Statistics 33 1890ndash1933
Robinson PM 1994 ldquoEf 1047297cient tests of nonstationary hypotheses Journal of theAmerican Statistical Association 89 1420ndash1437
Robinson PM 1995 Gaussian semi-parametric estimation of long range dependenceAnnals of Statistics 23 1630ndash1661
Robinson PM Henry M 1996 Bandwidth choice in Gaussian semiparametricestimation of long-range dependence In Robinson PM Rosenblatt M (Eds)Athens Conference on Applied Probability in Time Series Analysis Vol II New Yorkpp 220ndash232
Robinson PM Henry M 1999 Long and short memory conditional heteroskedasticityin estimating the memory in levels Econometric Theory 15 299ndash336
Sarno L Taylor MP 2002 Purchasing power parity and the real exchange rate TheEconomics of the Exchange Rate Cambridge University Press Cambridge pp 51ndash96
Sarno L Taylor MP Chowdhury I 2004 Nonlinear dynamics in deviations from thelaw of oneprice a broad-based empiricalstudy Journal of International Money andFinance 23 1ndash25
Taylor MP 2006 Real exchange rates and purchasing power paritymean-reversion ineconomic thought Applied Financial Economics 16 1ndash17
Taylor MP Peel DA 2000 ldquoNonlinear adjustment long-run equilibrium andexchangerate fundamentals Journal of International Moneyand Finance 1933ndash53
1191 JC Cuestas LA Gil-Alana Economic Modelling 26 (2009) 1184ndash1192
7232019 Economic Modelling Further Evidence on the PPP Analysis of the Australian Dollar- Non-linearities Fractional Integration and Structu
httpslidepdfcomreaderfulleconomic-modelling-further-evidence-on-the-ppp-analysis-of-the-australian-dollar- 99
Taylor MP Peel DA Sarno L 2001 Nonlinear mean-reversion in real eschange ratesTowards a solution to the purchasing power parity puzzles International EconomicReview 42 1015ndash1042
Velasco C 1999 Gaussian semiparametric estimation of nonstationary time series Journal of Time Series Analysis 20 87ndash127
Velasco C Robinson PM 2000 Whittle pseudo maximum likelihood estimation fornonstationary time series Journal of the American Statistical Association 951229ndash1243
Vogelsang TJ 1997 Wald-type test for detecting breaks in the trend function of adynamic time series Econometric Theory 13 818ndash849
Wang C 2004 Futures trading activity and predictable foreign exchange movements Journal of Banking and Finance 28 1023ndash1041
Wei S-J Parsley DC 1995 Purchasing power disparity during the 1047298oat rate periodexchange rate volatility trade barriers and other culprits Working Paper 5032NBER
WestKD1987 A noteon thepowerof least squares testsfor a unitroot Economics Letters24 1397ndash1418
1192 JC Cuestas LA Gil-Alana Economic Modelling 26 (2009) 1184ndash1192
7232019 Economic Modelling Further Evidence on the PPP Analysis of the Australian Dollar- Non-linearities Fractional Integration and Structu
httpslidepdfcomreaderfulleconomic-modelling-further-evidence-on-the-ppp-analysis-of-the-australian-dollar- 39
process Finally in order to take into account the existence of
structural changes and fractional integration at the same time we
apply Robinsons (1994) and Gil-Alanas (2008) fractional integration
tests in the context of structural changes
The remainder of this paper is organised as follows The next
section describes the data Section 3 summarises the econometric
techniques applied to empirically assess the ful1047297lment of PPP in
Australia and presents the results Section 4 concludes the paper
2 The data
The data for this empirical analysis consists of quarterly observa-
tions of the Australian real effective exchange rate computed as a
trade weighted index from 19702 to 20083 obtained from the
Central Bank of Australia web page (httpwwwrbagovau) A plot
of the data is displayed in Fig 1 From this graph it is possible to
highlight two mainstylised facts the existence of a structural change
in theseriesmdash presumablyat 1985 coincidingwith the currency crisis
(Darneacute and Hoarau 2008) mdash that needs to be accounted for and it
appears that the real value of the currency tends to revert to its
equilibrium very slowly after a shock which is suggestive of the fact
that the DGP may be fractionally integrated (Henry and Olekalns
2002)
3 Econometric methodology and results
31 Methodology
In this section we brie1047298
y describe some of the methods that will beemployed in this article in order to test for the empirical ful1047297lment of
the PPP theory
The 1047297rst tests presented are those attributable to Ng and Perron
(2001) who propose several modi1047297cations to existing unit root tests
in order to improve their size and power in particular with relatively
small samples The authors present the following tests MZ α and MZ t which arethe modi1047297ed versions of the Phillips (1987) and Phillips and
Perron (1988) Z α and Z t tests the MSB which is related to the
Bhargava (1986) R1 test and 1047297nally the MP t test which is a modi1047297ed
version of the Elliot et al (1996) Point Optimal Test These new tests
incorporate a Modi1047297ed Information Criterion (MIC) to select the lag
length in the auxiliary regression given that according to Ng and
Perron (2001) the Akaike and Schwartz Information Criteria tend to
select a low lag order Additionally these authors propose a General-
ised Least Squares method of detrending the data in order to improve
the power of the tests
Secondly as KSS point out traditional (linear) unit root tests
may fail to reject the null hypothesis when the DGP is non-linear If
the speed of adjustment is asymmetric ie it actually depends on
the degree of misalignment from the equilibrium DickeyndashFuller
type tests may incorrectly conclude that the series is a unit root
when in fact is a non-linear globally stationary process In this case
we may de1047297ne a DGP with two regimes that is an inner regime
where the variable is assumed to be I (1) and an outer regime
where the variable may or may not be a unit root The transition
between regimes is smooth rather than sudden Thus KSS propose a
unit root test to analyse the order of integration of the variable in
Fig 2 Estimates of d in model (6) and (7) across T b
Table 1
NgndashPerron and KSS unit root test results
Test Statistic CV (5) CV (10)
MZ α minus222737 minus810000 minus570000
MZ t minus105215 minus198000 minus162000
MSB 047237 023300 027500
MP t 109753 317000 445000
t NLD minus195642 minus291689 minus263526
Note Theorderof lagto computethe tests hasbeen chosen using themodi1047297edAIC (MAIC)
suggested by Ng andPerron (2001) The NgndashPerron tests includean interceptwhereas theKSS test has been applied to the demeaned data t NLD say The critical values for the Ngndash
Perron tests have been taken from Ng and Perron (2001) whereas those for the KSS have
been obtained by Monte Carlo simulations with 50000 replications
1186 JC Cuestas LA Gil-Alana Economic Modelling 26 (2009) 1184ndash1192
7232019 Economic Modelling Further Evidence on the PPP Analysis of the Australian Dollar- Non-linearities Fractional Integration and Structu
httpslidepdfcomreaderfulleconomic-modelling-further-evidence-on-the-ppp-analysis-of-the-australian-dollar- 49
the outer regime bearing in mind that the process is globally
stationary In other words
yt = β yt minus1 + yt minus1F θ yt minus1eth THORN + et eth3THORN
where ε t is iid(0 σ 2) and F (θ yt minus1) is the transition function which
is assumed to be exponential (ESTAR)
F θ yt minus1eth THORN = 1minus exp minusθ y2t minus 1
n o eth4THORN
with θ N0
In practice it is common to reparameterise Eq (3) as
Δ yt = α yt minus1 + γ yt minus1 1minus exp minusθ y2t minus 1
n o + et eth5THORN
in order to apply the test The null hypothesis H 0 θ =0 is tested
against the alternative H 1 θ N0 ie we test whether the variable is
an I (1) process in the outer regime From an economic viewpoint
and in the context of exchange rates this implies that the further
the RER deviates from the equilibrium the faster will be the speed
of mean reversion towards the fundamental equilibrium In addition
the existence of trade barriers may create a central threshold where
transactions are not pro1047297table and arbitrage does not clear the
market mdash unit root process in the inner regime whereas for large
deviations the pro1047297ts from arbitrage are greater than the cost and
the arbitrage mechanism brings the exchange to the inner regimeMoreover according to Taylor and Peel (2000) among others an
ESTAR function is appropriate to model exchange rates given that
this type of equation assumes that the effects of the shock on the
variable are symmetric in the sense that these effects do not depend
on the sign of the shock
On the other hand many test statistics have been developed in
recent years for fractional integration They can be parametric or
semiparametric and they can be speci1047297ed in the time domain or in the
frequency domain In this article we employ two parametric
approaches that allow us to incorporate structural breaks The 1047297rst
one is the well-known Robinson tests (1994) that we specify in a way
that permit us to include deterministicbroken trends In particular we
consider a model of form
yt = α 1I t b T beth THORN + α 2I t zT beth THORN + xt eth6THORN
1minusLeth THORNd xt = ut ut asympI 0eth THORN eth7THORN
where yt is the observed time series (RER) I ( x) is the indicator
function L is the lag operator (Lxt = xt minus1) d is a real value and ut isI (0) Based on this set-up for a given T b-value we test the null
hypothesis
H 0 d = do eth8THORN
in Eqs (6) and (7) for any real value do The functional form of the
test statistic is described in Appendix A and it was shown by
Robinson (1994) that under very mild regularity conditions a test
of Eq (8) against the one-sided alternatives dbdo (dNdo) follows
the standard N (0 1) distribution
The method presented just above imposes the same degree of
integration before and after the break-date Then we also perform a
recent procedure developed by Gil-Alana (2008) which allows us to
estimate the break-date along with the fractional differencing
parameters which might be different for each subsample This
method is based on minimising the residuals sum squares and is
brie1047298y described in Appendix B In the 1047297nal part of the article a
semiparametric method (Robinson 1995) is conducted on the two
subsamples
32 Empirical results
The results of applying the Ng and Perron (2001) and KSS tests are
reported in Table 1 The lag length has been selected using the
Modi1047297ed Akaike Information Criterion proposed by the former
authors for both tests In both cases the unit root hypothesis cannot
be rejected at conventional signi1047297cance levels Thus according to
these (unit-root) procedures there is no evidence of the PPP
hypothesis for the Australian case
Next we examine the possibility of fractional integration in the
context of non-linear deterministic terms For this purpose we
employ the model given by Eqs (6) and (7) testing H 0 (Eq (8)) for
do-values ranging from 0 to 2 with 001 increments using the
Robinsons (1994) parametric approach for 1047297
xed values of T b Wethen estimate d by choosing the value that produces the lowest
statistic in absolute value moving the break date T b 1-period
forward recursively from T b=40 (1980Q1) to T b=119 (1999Q4)
Fig 2 displays the estimated ds along with the 95 con1047297dence bands
for the two cases of white noise u t (in Fig 2(i)) and autocorrelated
disturbances (in Fig 2(ii)) In the latter case we assume that ut
follows the exponential spectral model of Bloom1047297eld (1973) Thisisa
non-parametric speci1047297cation for the error term that produces
autocorrelations decaying exponentially as in the autoregressive
(AR) case2
We observe in Fig 2 that the estimated values of d have remained
relatively stable across T b with values 1047298uctuating around 1 In fact
the unit root null hypothesis cannot be rejected for any T b We
observe that the only turbulences in the estimators take place when
T b is around 1985 Table 2 displays the estimates of d and the
intercepts for the case of a break at the four quarters in 1985 for the
two cases of white noise and autocorrelated disturbances We see in
this table that practically all the estimates are slightly above 1 The
only exception is the case of the break at 1985Q4 with autocorrelated
disturbances where the estimate of the fractional differencing
parameter is equal to 099 Nevertheless the unit root cannot be
rejected in any single case and thus we do not 1047297nd support for the
PPP hypothesis when using this model
2 See Gil-Alana (2004) for the analysis of fractional integration in the context of
Bloom1047297eld disturbances
Table 2
Estimates of intercepts and fractional differencing parameters with a break in 1985
Break dates White noise disturbances Autocorrelated disturbances
d α 1 α 2 d α 1 α 2
19 85 Q1 105 ( 095 117 ) 150 0 ( 149 7 1503 ) 140 7 ( 140 0 1413 ) 103 ( 085 129) 150 0 (1497 150 3) 140 4 (1398 1410)
19 85 Q2 104 ( 095 117 ) 150 0 ( 149 7 1503 ) 135 4 ( 134 8 1360 ) 103 ( 084 127) 150 0 (1497 150 3) 135 3 (1347 1359 )
19 85 Q3 110 ( 099 125 ) 1501 ( 149 8 150 4) 1556 ( 154 9 1562 ) 106 ( 088 132) 150 0 (1497 150 4) 1546 (1539 1552 )
19 85 Q4 106 ( 097 119) 150 0 ( 149 7 150 4) 143 5 ( 142 9 144 2) 099 ( 082 122) 149 9 (149 6 150 3) 1428 (1422 1435 )
Note In parenthesis the 95 con1047297dence intervals
1187 JC Cuestas LA Gil-Alana Economic Modelling 26 (2009) 1184ndash1192
7232019 Economic Modelling Further Evidence on the PPP Analysis of the Australian Dollar- Non-linearities Fractional Integration and Structu
httpslidepdfcomreaderfulleconomic-modelling-further-evidence-on-the-ppp-analysis-of-the-australian-dollar- 59
Still on this set-up we might be interested in knowing if the
intercept has changed from one subsample to another Thus we can
consider a joint test of the null hypothesis
H 0 α 1 = α 2 and d = do eth9THORNin Eqs (6) and (7) against the alternative
H a α 1 ne α 2 or d ne do eth10THORN
This possibility is not addressed in Robinson (1994) though Gil-
Alana and Robinson (1997) derived a similar Lagrange Multiplier(LM)
test as follows they consider a regression model of form as
yt = β V z t + xt t = 1 2 N eth11THORNand xt given by Eq (7) with the vector partitions z t = ( z At
T z Bt T )T
β = ( β AT β B
T )T In general we want to test H 0 d = do and β B =
β B0 Then a LM statistic may be shown to be r 2 (see Appendix
A) plus
XT
t =1
ˆ ut wT Bt
XT
t =1
wBt wT Bt minus
XT
t =1
wBt wT At
XT
t =1
w At wT At
minus1XT
t =1
w At wT Bt
minus1XT
t =1
ˆ ut wBt
eth12THORN
with wt = (w At T wBt
T )T = (1minus L)do z t
ˆ ut = 1minusLeth THORNdo yt minus ˜ β
T A β
T B0 wt ˜ β A = X
T
t =1
w At wT
At minus1
XT
t =1
w At 1minusLeth THORNdo yt
ˆ σ 2 = T
minus1 PT
t =1
ˆ u2t and r
2 is calculated as in Appendix A but
using the u t just de1047297ned If the dimension of z Bt is qB then we
Fig 4 Sum of squared residuals using the procedure of Gil-Alana
Fig 3 Estimates of d recursively obtained with samples of size T =100
1188 JC Cuestas LA Gil-Alana Economic Modelling 26 (2009) 1184ndash1192
7232019 Economic Modelling Further Evidence on the PPP Analysis of the Australian Dollar- Non-linearities Fractional Integration and Structu
httpslidepdfcomreaderfulleconomic-modelling-further-evidence-on-the-ppp-analysis-of-the-australian-dollar- 69
7232019 Economic Modelling Further Evidence on the PPP Analysis of the Australian Dollar- Non-linearities Fractional Integration and Structu
httpslidepdfcomreaderfulleconomic-modelling-further-evidence-on-the-ppp-analysis-of-the-australian-dollar- 79
asymmetric speed of adjustment towards the equilibrium as well as
fractional integration and structural changes Our results are in line
with previous empirical works in the sense that there is no evidence
of mean reversionin theAustralian dollarRER forthe periodanalysed
However we 1047297nd that the order of integration decreases after 1985
coinciding with the currency crisis This conclusion highlights the fact
that the monetary authorities have managed to decrease the
dependence of the Australian dollar on real shocks that may affect
the value of the currency on a permanent basis
Appendix A
The LM test of Robinson (1994) for testing H 0 (Eq (8)) in Eqs (6)
and (7) is
ˆ r = T 1 =2
ˆ σ 2ˆ Aminus1=2
ˆ a
where T is the sample size and
ˆ a = minus2π
T
XT minus1
j =1
ψ λ j
g λ j ˆ τ
minus1I λ j
ˆ σ 2
= σ 2
ˆ τ = 2π
T XT minus1
j =1
g λ j
ˆ τ minus1I λ
j
ˆ A = 2
T ethXT minus1
j =1
ψ λ j
2minus
XT minus1
j =1
ψ λ j
ˆ e λ j
V
timesXT minus1
j = 1
ˆ e λ j
ˆ e λ j
V
0
1Aminus1
timesXT minus1
j =1
ˆ e λ j
ψ λ j
THORNψ λ j
= log j2sin
λ j
2 j ˆ e λ j
=
A
Aτ log g λ j ˆ τ
λ j =
2π j
T ˆ τ = argmin σ
2τ eth THORN
a and A in the above expressions are obtained through the 1047297rst and
second derivatives of the log-likelihood function with respect to d
(see Robinson 1994 page 1422 for further details) I (l j) is the
periodogram of ut evaluated under the null ie
ˆ ut = 1minusLeth THORNdo yt minus α Vwt
ˆ α = XT
t = 1
wt wt V minus1
XT
t =1
wt 1minusL
eth THORNdo yt wt = 1minusL
eth THORNdo z t
z t = 1I t b T beth THORN 1I t zT beth THORNeth THORN V
and g is a known function related to the spectral density function of ut
f λ σ 2
τ
= σ
2
2π g λ τ eth THORN minus π b λ V π
Appendix B
Gil-Alana (2008) considers the following model
yt = α 1 + xt 1minus
Leth THORNd1
xt = ut t = 1 N
T b
and
yt = α 2 + xt 1minusLeth THORNd2 xt = ut t = T b + 1 N T
where the α 1 and α 2 are the intercepts corresponding to the 1047297rst and
second subsamples respectively d1 and d2 may be real values u t is
I (0) and T b is the time of a break that is supposed to be unknown
This model can be re-parameterized as follows
1minusLeth THORNd1 yt = α 1˜ 1t d1eth THORN + β 1˜ t t d1eth THORN + ut t = 1 N T b
1minus
Leth THORNd2
yt = α 2˜
1t d2eth THORN + β 2˜
t t d2eth THORN + ut t = T b + 1 N
T
Fig 5 Estimates of d based on a semiparametric model for each subsample
1190 JC Cuestas LA Gil-Alana Economic Modelling 26 (2009) 1184ndash1192
7232019 Economic Modelling Further Evidence on the PPP Analysis of the Australian Dollar- Non-linearities Fractional Integration and Structu
httpslidepdfcomreaderfulleconomic-modelling-further-evidence-on-the-ppp-analysis-of-the-australian-dollar- 89
with 1 t (di)=(1minusL)di1 The idea is then to minimise the residual sum
of squares (RSS)
wr t α 1 α 2 β 1 β 2f gXT b
t =1
1minusLeth THORNd 1eth THORN1o yt minus α 1˜ 1t d
1eth THORN1o
+ β 1˜ t
t d
1eth THORN1o
2
+XT
t = T b
frac12 1minusLeth THORNd 1eth THORN2o yt minusα 2˜ 1t d
1eth THORN2o
+ β 2 ˜ t t d
1eth THORN2o
2
In so doing it is necessary to choose a grid for the values of the
differencing parameters d1 and d2 d1o and d2o say Once the estimated
parameters α ˆ 1 α ˆ 2 β ˆ 1 and β ˆ
2 are obtained for partition T b and initial
values d1o(1) and d2o
(1) we plug these values into the objective function
and obtain RSS(T b)=arg mini jRSS(T bd1o(i) d1o
( j)) In order to estimate
the time break T k we obtain the moment that minimises the RSS
where the minimisation is taken over all partitions T 1 T 2 hellip T m such
that T iminusT iminus1ge |ε T | Then it is possible to obtain the regression
parameter estimates as α ˆ i=α ˆ i(T k) and the differencing parameters
d i= d
i(T k) for i =12
Appendix C
The ldquolocalrdquo Whittle estimate of Robinson (1995) is implicitly
de1047297ned by
ˆ d = arg mind log C deth THORN minus 2d 1
m
Xm
j = 1
log λ j
0
1A
for da minus1 = 2 1 = 2eth THORN C deth THORN = 1
m
Xm
j =1
I λ j
λ
2d j λ j =
2π j
T
m
T Y0
where m is a bandwidth parameter number and d isin(minus05 05)
Under 1047297niteness of the fourth moment and other mild conditions
Robinson (1995) proved that
ffiffiffiffiffimp ˆ d minus do
YdN 0 1 = 4eth THORN as T Yinfin
where do is the true value of d This estimator is robust to a certain
degree of conditional heteroskedasticity (Robinson and Henry 1999)
and is more ef 1047297cient than other semi-parametric competitors
References
Baum CF Barkoulas J Caglayan M 1999 Persistence in the international in1047298ationrates Southern Economic Journal 65 900ndash913
Bhargava A 1986 On the theory of testing for unit roots in observed time seriesReview of Economic Studies 53 369ndash384
Bierens HJ 1997 Testing the unit root with drift hypothesis against nonlinear trendstationarity with an application to the US price level and interest rate Journal of Econometrics 81 29ndash64
Bloom1047297eld P 1973 An exponential model in the spectrum of a scalar time series
Biometrika 60 217ndash226Booth GGKaen FR Koveos PE1982RS analysis of foreign exchangemarketsunder
two international monetary regimes Journal of Monetary Economics 10 407ndash415CampbellJY Perron P1991 Pitfalls and opportunities whatmacroeconomists should
know about unit roots NBER Macroeconomics Annual 1141ndash1201Casin P Ceacutespedes LF Sahay R 2004 Commodity currencies and the real exchange
rate Journal of Development Economics 75 239ndash268Cheung YW 1993 Long memory in foreign exchange rates Journal of Business and
Economic Statistics 11 93ndash101Cheung Y-W Lai KS 1993 A fractional cointegration analysis of purchasing power
parity Journal of Business and Economic Statistics 11 103ndash112Crato N Ray BK 2000 Memory in returns and volatilities of futures contracts
Journal of Futures Markets 20 525ndash543Cuestas JC 2009 Purchasing power parity in Central and Eastern European countries
an analysis of unit roots and nonlinearities Applied Economics Letters 16 87ndash94Cuestas JC Regis PJ 2008 Testing for PPP in Australia evidence from unit root tests
against nonlinear trend stationarity alternatives Economics Bulletin 27 1ndash8Darneacute O Hoarau J-F 2008 The purchasing power parity in Australia evidence from
unit root test with structural break Applied Economics Letters 15 203ndash206
DeJong D Nankervis J Savin NE Whiteman CH 1992 Integration versus trendstationarity in time series Econometrica 60 423ndash433
Dickey DA Fuller WA 1979 Distribution of the estimators for autoregressive timeseries with a unit root Journal of the American Statistical Association 74 427ndash431
Diebold FX Rudebusch GD 1991 Is consumption too smooth Long memory and theDeaton paradox The Review of Economics and Statistics 73 1ndash9
DieboldFXInoueA 2001 Longmemory andregimeswitchingJournal of Econometrics105 131ndash159
Diebold FX Husted S Rush M 1991 Real exchange rates under the gold standard Journal of Political Economy 99 1252ndash1271
Dumas B1994 Partial equilibrium versus general equilibrium models of the international
capital market In van der Ploeg F (Ed) The Handbook of International Macro-economics Blackwell Oxford pp 301ndash347 chap 10Elliot G Rothenberg TJ Stock JH 1996 Ef 1047297cient tests for an autoregressive unit root
Econometrica 64 813ndash836Fang HLai KS Lai M1994 Fractal structure in currency futurespricedynamicsJournal
of Futures Markets 14 169ndash181Faria JR Leoacuten-Ledesma MA 2003 Testing the BalassandashSamuelson effect implica-
tions for growth and the PPP Journal of Macroeconomics 25 241ndash253Faria JR Leoacuten-Ledesma MA 2005 Real exchange rate and employment performance
in an open economy Research in Economics 59 67ndash80Geweke J Porter-Hudack S 1983 The estimation and application of long memory
time series models and fractional differencing Journal of Time Series Analysis 4221ndash238
Gil-Alana LA 2000 Mean reversion in the real exchange rates Economics Letters 69285ndash288
Gil-Alana LA 2004 The use of the model of Bloom1047297eld (1973) as an approximation toARMA processes in the context of fractional integration Mathematical andComputer Modelling 39 429ndash436
Gil-AlanaLA 2008 Fractional integrationand structuralbreaks at unknown periods of
time Journal of Time Series Analysis 29 163ndash185Gil-Alana LA Robinson PM 1997 Testing of unit roots and other nonstationary
hypotheses in macroeconomic time series Journal of Econometrics 80 241ndash268GrangerCWJHyung N 2004 Occasional structural breaks and long memory with an
application to the SampP 500 absolute stock return Journal of Empirical Finance 11399ndash421
Hassler U Wolters J 1995 Long memory in in1047298ation rates International evidence Journal of Business and Economic Statistics 13 37ndash45
Henry OT Olekalns N 2002 Does the Australian dollar real exchange rate displaymean reversion Journal of International Money and Finance 21 651ndash666
Juvenal L Taylor MP 20 08 Threshold adjustment of deviations from the law of oneprice Studies in Nonlinear Dynamics and Econometrics 12 Article 8
Kapetanios G Shin Y Snell A 2003 Testing for a unit root in the nonlinear STAR framework Journal of Econometrics 112 359ndash379
Kwiatkowski D Phillips PCB Schmidt P Shin Y 1992 Testing the null hypothesis of stationarity againstthe alternative of a unitrootJournal of Econometrics54159ndash178
Lee D Schmidt P 1996 On the power of the KPSS test of stationarity againstfractionally integrated alternatives Journal of Econometrics 73 285ndash302
Michael P Nobay A Peel D 1997 Transaction costs and nonlinear adjustment in realexchange ratesan empirical investigation Journalof Political Economy 105 862ndash879
Nelson CR Piger J Zivot E 2001 Markov regime-switching and unit root tests Journal of Business Economics and Statistics 19 404ndash415
Ng S Perron P 2001 Lag selection and the construction of unit root tests with goodsize and power Econometrica 69 1519ndash1554
Perron P 1989 The great crash the oil price shock and the unit root hypothesisEconometrica 571361ndash1401
Perron P Phillips PCB 1987 Does GNP have a unit root A reevaluation EconomicsLetters 23 139ndash145
Perron P Rodriacuteguez G 2003 GLS detrending ef 1047297cient unit root tests and structuralchange Journal of Econometrics 115 1ndash27
Phillips PCB 1987 Time series regression with a unit root Econometrica 55 311ndash340Phillips PCB Perron P 1988 lsquoTesting for a unit root in time series regression
Biometrica 75 335ndash346Phillips PCB Shimotsu K 2004 Local Whittle estimation in nonstationary and unit
root cases Annals of Statistics 32 656ndash692Phillips PCB Shimotsu K 2005 Exact local Whittle estimation of fractional
integration Annals of Statistics 33 1890ndash1933
Robinson PM 1994 ldquoEf 1047297cient tests of nonstationary hypotheses Journal of theAmerican Statistical Association 89 1420ndash1437
Robinson PM 1995 Gaussian semi-parametric estimation of long range dependenceAnnals of Statistics 23 1630ndash1661
Robinson PM Henry M 1996 Bandwidth choice in Gaussian semiparametricestimation of long-range dependence In Robinson PM Rosenblatt M (Eds)Athens Conference on Applied Probability in Time Series Analysis Vol II New Yorkpp 220ndash232
Robinson PM Henry M 1999 Long and short memory conditional heteroskedasticityin estimating the memory in levels Econometric Theory 15 299ndash336
Sarno L Taylor MP 2002 Purchasing power parity and the real exchange rate TheEconomics of the Exchange Rate Cambridge University Press Cambridge pp 51ndash96
Sarno L Taylor MP Chowdhury I 2004 Nonlinear dynamics in deviations from thelaw of oneprice a broad-based empiricalstudy Journal of International Money andFinance 23 1ndash25
Taylor MP 2006 Real exchange rates and purchasing power paritymean-reversion ineconomic thought Applied Financial Economics 16 1ndash17
Taylor MP Peel DA 2000 ldquoNonlinear adjustment long-run equilibrium andexchangerate fundamentals Journal of International Moneyand Finance 1933ndash53
1191 JC Cuestas LA Gil-Alana Economic Modelling 26 (2009) 1184ndash1192
7232019 Economic Modelling Further Evidence on the PPP Analysis of the Australian Dollar- Non-linearities Fractional Integration and Structu
httpslidepdfcomreaderfulleconomic-modelling-further-evidence-on-the-ppp-analysis-of-the-australian-dollar- 99
Taylor MP Peel DA Sarno L 2001 Nonlinear mean-reversion in real eschange ratesTowards a solution to the purchasing power parity puzzles International EconomicReview 42 1015ndash1042
Velasco C 1999 Gaussian semiparametric estimation of nonstationary time series Journal of Time Series Analysis 20 87ndash127
Velasco C Robinson PM 2000 Whittle pseudo maximum likelihood estimation fornonstationary time series Journal of the American Statistical Association 951229ndash1243
Vogelsang TJ 1997 Wald-type test for detecting breaks in the trend function of adynamic time series Econometric Theory 13 818ndash849
Wang C 2004 Futures trading activity and predictable foreign exchange movements Journal of Banking and Finance 28 1023ndash1041
Wei S-J Parsley DC 1995 Purchasing power disparity during the 1047298oat rate periodexchange rate volatility trade barriers and other culprits Working Paper 5032NBER
WestKD1987 A noteon thepowerof least squares testsfor a unitroot Economics Letters24 1397ndash1418
1192 JC Cuestas LA Gil-Alana Economic Modelling 26 (2009) 1184ndash1192
7232019 Economic Modelling Further Evidence on the PPP Analysis of the Australian Dollar- Non-linearities Fractional Integration and Structu
httpslidepdfcomreaderfulleconomic-modelling-further-evidence-on-the-ppp-analysis-of-the-australian-dollar- 49
the outer regime bearing in mind that the process is globally
stationary In other words
yt = β yt minus1 + yt minus1F θ yt minus1eth THORN + et eth3THORN
where ε t is iid(0 σ 2) and F (θ yt minus1) is the transition function which
is assumed to be exponential (ESTAR)
F θ yt minus1eth THORN = 1minus exp minusθ y2t minus 1
n o eth4THORN
with θ N0
In practice it is common to reparameterise Eq (3) as
Δ yt = α yt minus1 + γ yt minus1 1minus exp minusθ y2t minus 1
n o + et eth5THORN
in order to apply the test The null hypothesis H 0 θ =0 is tested
against the alternative H 1 θ N0 ie we test whether the variable is
an I (1) process in the outer regime From an economic viewpoint
and in the context of exchange rates this implies that the further
the RER deviates from the equilibrium the faster will be the speed
of mean reversion towards the fundamental equilibrium In addition
the existence of trade barriers may create a central threshold where
transactions are not pro1047297table and arbitrage does not clear the
market mdash unit root process in the inner regime whereas for large
deviations the pro1047297ts from arbitrage are greater than the cost and
the arbitrage mechanism brings the exchange to the inner regimeMoreover according to Taylor and Peel (2000) among others an
ESTAR function is appropriate to model exchange rates given that
this type of equation assumes that the effects of the shock on the
variable are symmetric in the sense that these effects do not depend
on the sign of the shock
On the other hand many test statistics have been developed in
recent years for fractional integration They can be parametric or
semiparametric and they can be speci1047297ed in the time domain or in the
frequency domain In this article we employ two parametric
approaches that allow us to incorporate structural breaks The 1047297rst
one is the well-known Robinson tests (1994) that we specify in a way
that permit us to include deterministicbroken trends In particular we
consider a model of form
yt = α 1I t b T beth THORN + α 2I t zT beth THORN + xt eth6THORN
1minusLeth THORNd xt = ut ut asympI 0eth THORN eth7THORN
where yt is the observed time series (RER) I ( x) is the indicator
function L is the lag operator (Lxt = xt minus1) d is a real value and ut isI (0) Based on this set-up for a given T b-value we test the null
hypothesis
H 0 d = do eth8THORN
in Eqs (6) and (7) for any real value do The functional form of the
test statistic is described in Appendix A and it was shown by
Robinson (1994) that under very mild regularity conditions a test
of Eq (8) against the one-sided alternatives dbdo (dNdo) follows
the standard N (0 1) distribution
The method presented just above imposes the same degree of
integration before and after the break-date Then we also perform a
recent procedure developed by Gil-Alana (2008) which allows us to
estimate the break-date along with the fractional differencing
parameters which might be different for each subsample This
method is based on minimising the residuals sum squares and is
brie1047298y described in Appendix B In the 1047297nal part of the article a
semiparametric method (Robinson 1995) is conducted on the two
subsamples
32 Empirical results
The results of applying the Ng and Perron (2001) and KSS tests are
reported in Table 1 The lag length has been selected using the
Modi1047297ed Akaike Information Criterion proposed by the former
authors for both tests In both cases the unit root hypothesis cannot
be rejected at conventional signi1047297cance levels Thus according to
these (unit-root) procedures there is no evidence of the PPP
hypothesis for the Australian case
Next we examine the possibility of fractional integration in the
context of non-linear deterministic terms For this purpose we
employ the model given by Eqs (6) and (7) testing H 0 (Eq (8)) for
do-values ranging from 0 to 2 with 001 increments using the
Robinsons (1994) parametric approach for 1047297
xed values of T b Wethen estimate d by choosing the value that produces the lowest
statistic in absolute value moving the break date T b 1-period
forward recursively from T b=40 (1980Q1) to T b=119 (1999Q4)
Fig 2 displays the estimated ds along with the 95 con1047297dence bands
for the two cases of white noise u t (in Fig 2(i)) and autocorrelated
disturbances (in Fig 2(ii)) In the latter case we assume that ut
follows the exponential spectral model of Bloom1047297eld (1973) Thisisa
non-parametric speci1047297cation for the error term that produces
autocorrelations decaying exponentially as in the autoregressive
(AR) case2
We observe in Fig 2 that the estimated values of d have remained
relatively stable across T b with values 1047298uctuating around 1 In fact
the unit root null hypothesis cannot be rejected for any T b We
observe that the only turbulences in the estimators take place when
T b is around 1985 Table 2 displays the estimates of d and the
intercepts for the case of a break at the four quarters in 1985 for the
two cases of white noise and autocorrelated disturbances We see in
this table that practically all the estimates are slightly above 1 The
only exception is the case of the break at 1985Q4 with autocorrelated
disturbances where the estimate of the fractional differencing
parameter is equal to 099 Nevertheless the unit root cannot be
rejected in any single case and thus we do not 1047297nd support for the
PPP hypothesis when using this model
2 See Gil-Alana (2004) for the analysis of fractional integration in the context of
Bloom1047297eld disturbances
Table 2
Estimates of intercepts and fractional differencing parameters with a break in 1985
Break dates White noise disturbances Autocorrelated disturbances
d α 1 α 2 d α 1 α 2
19 85 Q1 105 ( 095 117 ) 150 0 ( 149 7 1503 ) 140 7 ( 140 0 1413 ) 103 ( 085 129) 150 0 (1497 150 3) 140 4 (1398 1410)
19 85 Q2 104 ( 095 117 ) 150 0 ( 149 7 1503 ) 135 4 ( 134 8 1360 ) 103 ( 084 127) 150 0 (1497 150 3) 135 3 (1347 1359 )
19 85 Q3 110 ( 099 125 ) 1501 ( 149 8 150 4) 1556 ( 154 9 1562 ) 106 ( 088 132) 150 0 (1497 150 4) 1546 (1539 1552 )
19 85 Q4 106 ( 097 119) 150 0 ( 149 7 150 4) 143 5 ( 142 9 144 2) 099 ( 082 122) 149 9 (149 6 150 3) 1428 (1422 1435 )
Note In parenthesis the 95 con1047297dence intervals
1187 JC Cuestas LA Gil-Alana Economic Modelling 26 (2009) 1184ndash1192
7232019 Economic Modelling Further Evidence on the PPP Analysis of the Australian Dollar- Non-linearities Fractional Integration and Structu
httpslidepdfcomreaderfulleconomic-modelling-further-evidence-on-the-ppp-analysis-of-the-australian-dollar- 59
Still on this set-up we might be interested in knowing if the
intercept has changed from one subsample to another Thus we can
consider a joint test of the null hypothesis
H 0 α 1 = α 2 and d = do eth9THORNin Eqs (6) and (7) against the alternative
H a α 1 ne α 2 or d ne do eth10THORN
This possibility is not addressed in Robinson (1994) though Gil-
Alana and Robinson (1997) derived a similar Lagrange Multiplier(LM)
test as follows they consider a regression model of form as
yt = β V z t + xt t = 1 2 N eth11THORNand xt given by Eq (7) with the vector partitions z t = ( z At
T z Bt T )T
β = ( β AT β B
T )T In general we want to test H 0 d = do and β B =
β B0 Then a LM statistic may be shown to be r 2 (see Appendix
A) plus
XT
t =1
ˆ ut wT Bt
XT
t =1
wBt wT Bt minus
XT
t =1
wBt wT At
XT
t =1
w At wT At
minus1XT
t =1
w At wT Bt
minus1XT
t =1
ˆ ut wBt
eth12THORN
with wt = (w At T wBt
T )T = (1minus L)do z t
ˆ ut = 1minusLeth THORNdo yt minus ˜ β
T A β
T B0 wt ˜ β A = X
T
t =1
w At wT
At minus1
XT
t =1
w At 1minusLeth THORNdo yt
ˆ σ 2 = T
minus1 PT
t =1
ˆ u2t and r
2 is calculated as in Appendix A but
using the u t just de1047297ned If the dimension of z Bt is qB then we
Fig 4 Sum of squared residuals using the procedure of Gil-Alana
Fig 3 Estimates of d recursively obtained with samples of size T =100
1188 JC Cuestas LA Gil-Alana Economic Modelling 26 (2009) 1184ndash1192
7232019 Economic Modelling Further Evidence on the PPP Analysis of the Australian Dollar- Non-linearities Fractional Integration and Structu
httpslidepdfcomreaderfulleconomic-modelling-further-evidence-on-the-ppp-analysis-of-the-australian-dollar- 69
7232019 Economic Modelling Further Evidence on the PPP Analysis of the Australian Dollar- Non-linearities Fractional Integration and Structu
httpslidepdfcomreaderfulleconomic-modelling-further-evidence-on-the-ppp-analysis-of-the-australian-dollar- 79
asymmetric speed of adjustment towards the equilibrium as well as
fractional integration and structural changes Our results are in line
with previous empirical works in the sense that there is no evidence
of mean reversionin theAustralian dollarRER forthe periodanalysed
However we 1047297nd that the order of integration decreases after 1985
coinciding with the currency crisis This conclusion highlights the fact
that the monetary authorities have managed to decrease the
dependence of the Australian dollar on real shocks that may affect
the value of the currency on a permanent basis
Appendix A
The LM test of Robinson (1994) for testing H 0 (Eq (8)) in Eqs (6)
and (7) is
ˆ r = T 1 =2
ˆ σ 2ˆ Aminus1=2
ˆ a
where T is the sample size and
ˆ a = minus2π
T
XT minus1
j =1
ψ λ j
g λ j ˆ τ
minus1I λ j
ˆ σ 2
= σ 2
ˆ τ = 2π
T XT minus1
j =1
g λ j
ˆ τ minus1I λ
j
ˆ A = 2
T ethXT minus1
j =1
ψ λ j
2minus
XT minus1
j =1
ψ λ j
ˆ e λ j
V
timesXT minus1
j = 1
ˆ e λ j
ˆ e λ j
V
0
1Aminus1
timesXT minus1
j =1
ˆ e λ j
ψ λ j
THORNψ λ j
= log j2sin
λ j
2 j ˆ e λ j
=
A
Aτ log g λ j ˆ τ
λ j =
2π j
T ˆ τ = argmin σ
2τ eth THORN
a and A in the above expressions are obtained through the 1047297rst and
second derivatives of the log-likelihood function with respect to d
(see Robinson 1994 page 1422 for further details) I (l j) is the
periodogram of ut evaluated under the null ie
ˆ ut = 1minusLeth THORNdo yt minus α Vwt
ˆ α = XT
t = 1
wt wt V minus1
XT
t =1
wt 1minusL
eth THORNdo yt wt = 1minusL
eth THORNdo z t
z t = 1I t b T beth THORN 1I t zT beth THORNeth THORN V
and g is a known function related to the spectral density function of ut
f λ σ 2
τ
= σ
2
2π g λ τ eth THORN minus π b λ V π
Appendix B
Gil-Alana (2008) considers the following model
yt = α 1 + xt 1minus
Leth THORNd1
xt = ut t = 1 N
T b
and
yt = α 2 + xt 1minusLeth THORNd2 xt = ut t = T b + 1 N T
where the α 1 and α 2 are the intercepts corresponding to the 1047297rst and
second subsamples respectively d1 and d2 may be real values u t is
I (0) and T b is the time of a break that is supposed to be unknown
This model can be re-parameterized as follows
1minusLeth THORNd1 yt = α 1˜ 1t d1eth THORN + β 1˜ t t d1eth THORN + ut t = 1 N T b
1minus
Leth THORNd2
yt = α 2˜
1t d2eth THORN + β 2˜
t t d2eth THORN + ut t = T b + 1 N
T
Fig 5 Estimates of d based on a semiparametric model for each subsample
1190 JC Cuestas LA Gil-Alana Economic Modelling 26 (2009) 1184ndash1192
7232019 Economic Modelling Further Evidence on the PPP Analysis of the Australian Dollar- Non-linearities Fractional Integration and Structu
httpslidepdfcomreaderfulleconomic-modelling-further-evidence-on-the-ppp-analysis-of-the-australian-dollar- 89
with 1 t (di)=(1minusL)di1 The idea is then to minimise the residual sum
of squares (RSS)
wr t α 1 α 2 β 1 β 2f gXT b
t =1
1minusLeth THORNd 1eth THORN1o yt minus α 1˜ 1t d
1eth THORN1o
+ β 1˜ t
t d
1eth THORN1o
2
+XT
t = T b
frac12 1minusLeth THORNd 1eth THORN2o yt minusα 2˜ 1t d
1eth THORN2o
+ β 2 ˜ t t d
1eth THORN2o
2
In so doing it is necessary to choose a grid for the values of the
differencing parameters d1 and d2 d1o and d2o say Once the estimated
parameters α ˆ 1 α ˆ 2 β ˆ 1 and β ˆ
2 are obtained for partition T b and initial
values d1o(1) and d2o
(1) we plug these values into the objective function
and obtain RSS(T b)=arg mini jRSS(T bd1o(i) d1o
( j)) In order to estimate
the time break T k we obtain the moment that minimises the RSS
where the minimisation is taken over all partitions T 1 T 2 hellip T m such
that T iminusT iminus1ge |ε T | Then it is possible to obtain the regression
parameter estimates as α ˆ i=α ˆ i(T k) and the differencing parameters
d i= d
i(T k) for i =12
Appendix C
The ldquolocalrdquo Whittle estimate of Robinson (1995) is implicitly
de1047297ned by
ˆ d = arg mind log C deth THORN minus 2d 1
m
Xm
j = 1
log λ j
0
1A
for da minus1 = 2 1 = 2eth THORN C deth THORN = 1
m
Xm
j =1
I λ j
λ
2d j λ j =
2π j
T
m
T Y0
where m is a bandwidth parameter number and d isin(minus05 05)
Under 1047297niteness of the fourth moment and other mild conditions
Robinson (1995) proved that
ffiffiffiffiffimp ˆ d minus do
YdN 0 1 = 4eth THORN as T Yinfin
where do is the true value of d This estimator is robust to a certain
degree of conditional heteroskedasticity (Robinson and Henry 1999)
and is more ef 1047297cient than other semi-parametric competitors
References
Baum CF Barkoulas J Caglayan M 1999 Persistence in the international in1047298ationrates Southern Economic Journal 65 900ndash913
Bhargava A 1986 On the theory of testing for unit roots in observed time seriesReview of Economic Studies 53 369ndash384
Bierens HJ 1997 Testing the unit root with drift hypothesis against nonlinear trendstationarity with an application to the US price level and interest rate Journal of Econometrics 81 29ndash64
Bloom1047297eld P 1973 An exponential model in the spectrum of a scalar time series
Biometrika 60 217ndash226Booth GGKaen FR Koveos PE1982RS analysis of foreign exchangemarketsunder
two international monetary regimes Journal of Monetary Economics 10 407ndash415CampbellJY Perron P1991 Pitfalls and opportunities whatmacroeconomists should
know about unit roots NBER Macroeconomics Annual 1141ndash1201Casin P Ceacutespedes LF Sahay R 2004 Commodity currencies and the real exchange
rate Journal of Development Economics 75 239ndash268Cheung YW 1993 Long memory in foreign exchange rates Journal of Business and
Economic Statistics 11 93ndash101Cheung Y-W Lai KS 1993 A fractional cointegration analysis of purchasing power
parity Journal of Business and Economic Statistics 11 103ndash112Crato N Ray BK 2000 Memory in returns and volatilities of futures contracts
Journal of Futures Markets 20 525ndash543Cuestas JC 2009 Purchasing power parity in Central and Eastern European countries
an analysis of unit roots and nonlinearities Applied Economics Letters 16 87ndash94Cuestas JC Regis PJ 2008 Testing for PPP in Australia evidence from unit root tests
against nonlinear trend stationarity alternatives Economics Bulletin 27 1ndash8Darneacute O Hoarau J-F 2008 The purchasing power parity in Australia evidence from
unit root test with structural break Applied Economics Letters 15 203ndash206
DeJong D Nankervis J Savin NE Whiteman CH 1992 Integration versus trendstationarity in time series Econometrica 60 423ndash433
Dickey DA Fuller WA 1979 Distribution of the estimators for autoregressive timeseries with a unit root Journal of the American Statistical Association 74 427ndash431
Diebold FX Rudebusch GD 1991 Is consumption too smooth Long memory and theDeaton paradox The Review of Economics and Statistics 73 1ndash9
DieboldFXInoueA 2001 Longmemory andregimeswitchingJournal of Econometrics105 131ndash159
Diebold FX Husted S Rush M 1991 Real exchange rates under the gold standard Journal of Political Economy 99 1252ndash1271
Dumas B1994 Partial equilibrium versus general equilibrium models of the international
capital market In van der Ploeg F (Ed) The Handbook of International Macro-economics Blackwell Oxford pp 301ndash347 chap 10Elliot G Rothenberg TJ Stock JH 1996 Ef 1047297cient tests for an autoregressive unit root
Econometrica 64 813ndash836Fang HLai KS Lai M1994 Fractal structure in currency futurespricedynamicsJournal
of Futures Markets 14 169ndash181Faria JR Leoacuten-Ledesma MA 2003 Testing the BalassandashSamuelson effect implica-
tions for growth and the PPP Journal of Macroeconomics 25 241ndash253Faria JR Leoacuten-Ledesma MA 2005 Real exchange rate and employment performance
in an open economy Research in Economics 59 67ndash80Geweke J Porter-Hudack S 1983 The estimation and application of long memory
time series models and fractional differencing Journal of Time Series Analysis 4221ndash238
Gil-Alana LA 2000 Mean reversion in the real exchange rates Economics Letters 69285ndash288
Gil-Alana LA 2004 The use of the model of Bloom1047297eld (1973) as an approximation toARMA processes in the context of fractional integration Mathematical andComputer Modelling 39 429ndash436
Gil-AlanaLA 2008 Fractional integrationand structuralbreaks at unknown periods of
time Journal of Time Series Analysis 29 163ndash185Gil-Alana LA Robinson PM 1997 Testing of unit roots and other nonstationary
hypotheses in macroeconomic time series Journal of Econometrics 80 241ndash268GrangerCWJHyung N 2004 Occasional structural breaks and long memory with an
application to the SampP 500 absolute stock return Journal of Empirical Finance 11399ndash421
Hassler U Wolters J 1995 Long memory in in1047298ation rates International evidence Journal of Business and Economic Statistics 13 37ndash45
Henry OT Olekalns N 2002 Does the Australian dollar real exchange rate displaymean reversion Journal of International Money and Finance 21 651ndash666
Juvenal L Taylor MP 20 08 Threshold adjustment of deviations from the law of oneprice Studies in Nonlinear Dynamics and Econometrics 12 Article 8
Kapetanios G Shin Y Snell A 2003 Testing for a unit root in the nonlinear STAR framework Journal of Econometrics 112 359ndash379
Kwiatkowski D Phillips PCB Schmidt P Shin Y 1992 Testing the null hypothesis of stationarity againstthe alternative of a unitrootJournal of Econometrics54159ndash178
Lee D Schmidt P 1996 On the power of the KPSS test of stationarity againstfractionally integrated alternatives Journal of Econometrics 73 285ndash302
Michael P Nobay A Peel D 1997 Transaction costs and nonlinear adjustment in realexchange ratesan empirical investigation Journalof Political Economy 105 862ndash879
Nelson CR Piger J Zivot E 2001 Markov regime-switching and unit root tests Journal of Business Economics and Statistics 19 404ndash415
Ng S Perron P 2001 Lag selection and the construction of unit root tests with goodsize and power Econometrica 69 1519ndash1554
Perron P 1989 The great crash the oil price shock and the unit root hypothesisEconometrica 571361ndash1401
Perron P Phillips PCB 1987 Does GNP have a unit root A reevaluation EconomicsLetters 23 139ndash145
Perron P Rodriacuteguez G 2003 GLS detrending ef 1047297cient unit root tests and structuralchange Journal of Econometrics 115 1ndash27
Phillips PCB 1987 Time series regression with a unit root Econometrica 55 311ndash340Phillips PCB Perron P 1988 lsquoTesting for a unit root in time series regression
Biometrica 75 335ndash346Phillips PCB Shimotsu K 2004 Local Whittle estimation in nonstationary and unit
root cases Annals of Statistics 32 656ndash692Phillips PCB Shimotsu K 2005 Exact local Whittle estimation of fractional
integration Annals of Statistics 33 1890ndash1933
Robinson PM 1994 ldquoEf 1047297cient tests of nonstationary hypotheses Journal of theAmerican Statistical Association 89 1420ndash1437
Robinson PM 1995 Gaussian semi-parametric estimation of long range dependenceAnnals of Statistics 23 1630ndash1661
Robinson PM Henry M 1996 Bandwidth choice in Gaussian semiparametricestimation of long-range dependence In Robinson PM Rosenblatt M (Eds)Athens Conference on Applied Probability in Time Series Analysis Vol II New Yorkpp 220ndash232
Robinson PM Henry M 1999 Long and short memory conditional heteroskedasticityin estimating the memory in levels Econometric Theory 15 299ndash336
Sarno L Taylor MP 2002 Purchasing power parity and the real exchange rate TheEconomics of the Exchange Rate Cambridge University Press Cambridge pp 51ndash96
Sarno L Taylor MP Chowdhury I 2004 Nonlinear dynamics in deviations from thelaw of oneprice a broad-based empiricalstudy Journal of International Money andFinance 23 1ndash25
Taylor MP 2006 Real exchange rates and purchasing power paritymean-reversion ineconomic thought Applied Financial Economics 16 1ndash17
Taylor MP Peel DA 2000 ldquoNonlinear adjustment long-run equilibrium andexchangerate fundamentals Journal of International Moneyand Finance 1933ndash53
1191 JC Cuestas LA Gil-Alana Economic Modelling 26 (2009) 1184ndash1192
7232019 Economic Modelling Further Evidence on the PPP Analysis of the Australian Dollar- Non-linearities Fractional Integration and Structu
httpslidepdfcomreaderfulleconomic-modelling-further-evidence-on-the-ppp-analysis-of-the-australian-dollar- 99
Taylor MP Peel DA Sarno L 2001 Nonlinear mean-reversion in real eschange ratesTowards a solution to the purchasing power parity puzzles International EconomicReview 42 1015ndash1042
Velasco C 1999 Gaussian semiparametric estimation of nonstationary time series Journal of Time Series Analysis 20 87ndash127
Velasco C Robinson PM 2000 Whittle pseudo maximum likelihood estimation fornonstationary time series Journal of the American Statistical Association 951229ndash1243
Vogelsang TJ 1997 Wald-type test for detecting breaks in the trend function of adynamic time series Econometric Theory 13 818ndash849
Wang C 2004 Futures trading activity and predictable foreign exchange movements Journal of Banking and Finance 28 1023ndash1041
Wei S-J Parsley DC 1995 Purchasing power disparity during the 1047298oat rate periodexchange rate volatility trade barriers and other culprits Working Paper 5032NBER
WestKD1987 A noteon thepowerof least squares testsfor a unitroot Economics Letters24 1397ndash1418
1192 JC Cuestas LA Gil-Alana Economic Modelling 26 (2009) 1184ndash1192
7232019 Economic Modelling Further Evidence on the PPP Analysis of the Australian Dollar- Non-linearities Fractional Integration and Structu
httpslidepdfcomreaderfulleconomic-modelling-further-evidence-on-the-ppp-analysis-of-the-australian-dollar- 59
Still on this set-up we might be interested in knowing if the
intercept has changed from one subsample to another Thus we can
consider a joint test of the null hypothesis
H 0 α 1 = α 2 and d = do eth9THORNin Eqs (6) and (7) against the alternative
H a α 1 ne α 2 or d ne do eth10THORN
This possibility is not addressed in Robinson (1994) though Gil-
Alana and Robinson (1997) derived a similar Lagrange Multiplier(LM)
test as follows they consider a regression model of form as
yt = β V z t + xt t = 1 2 N eth11THORNand xt given by Eq (7) with the vector partitions z t = ( z At
T z Bt T )T
β = ( β AT β B
T )T In general we want to test H 0 d = do and β B =
β B0 Then a LM statistic may be shown to be r 2 (see Appendix
A) plus
XT
t =1
ˆ ut wT Bt
XT
t =1
wBt wT Bt minus
XT
t =1
wBt wT At
XT
t =1
w At wT At
minus1XT
t =1
w At wT Bt
minus1XT
t =1
ˆ ut wBt
eth12THORN
with wt = (w At T wBt
T )T = (1minus L)do z t
ˆ ut = 1minusLeth THORNdo yt minus ˜ β
T A β
T B0 wt ˜ β A = X
T
t =1
w At wT
At minus1
XT
t =1
w At 1minusLeth THORNdo yt
ˆ σ 2 = T
minus1 PT
t =1
ˆ u2t and r
2 is calculated as in Appendix A but
using the u t just de1047297ned If the dimension of z Bt is qB then we
Fig 4 Sum of squared residuals using the procedure of Gil-Alana
Fig 3 Estimates of d recursively obtained with samples of size T =100
1188 JC Cuestas LA Gil-Alana Economic Modelling 26 (2009) 1184ndash1192
7232019 Economic Modelling Further Evidence on the PPP Analysis of the Australian Dollar- Non-linearities Fractional Integration and Structu
httpslidepdfcomreaderfulleconomic-modelling-further-evidence-on-the-ppp-analysis-of-the-australian-dollar- 69
7232019 Economic Modelling Further Evidence on the PPP Analysis of the Australian Dollar- Non-linearities Fractional Integration and Structu
httpslidepdfcomreaderfulleconomic-modelling-further-evidence-on-the-ppp-analysis-of-the-australian-dollar- 79
asymmetric speed of adjustment towards the equilibrium as well as
fractional integration and structural changes Our results are in line
with previous empirical works in the sense that there is no evidence
of mean reversionin theAustralian dollarRER forthe periodanalysed
However we 1047297nd that the order of integration decreases after 1985
coinciding with the currency crisis This conclusion highlights the fact
that the monetary authorities have managed to decrease the
dependence of the Australian dollar on real shocks that may affect
the value of the currency on a permanent basis
Appendix A
The LM test of Robinson (1994) for testing H 0 (Eq (8)) in Eqs (6)
and (7) is
ˆ r = T 1 =2
ˆ σ 2ˆ Aminus1=2
ˆ a
where T is the sample size and
ˆ a = minus2π
T
XT minus1
j =1
ψ λ j
g λ j ˆ τ
minus1I λ j
ˆ σ 2
= σ 2
ˆ τ = 2π
T XT minus1
j =1
g λ j
ˆ τ minus1I λ
j
ˆ A = 2
T ethXT minus1
j =1
ψ λ j
2minus
XT minus1
j =1
ψ λ j
ˆ e λ j
V
timesXT minus1
j = 1
ˆ e λ j
ˆ e λ j
V
0
1Aminus1
timesXT minus1
j =1
ˆ e λ j
ψ λ j
THORNψ λ j
= log j2sin
λ j
2 j ˆ e λ j
=
A
Aτ log g λ j ˆ τ
λ j =
2π j
T ˆ τ = argmin σ
2τ eth THORN
a and A in the above expressions are obtained through the 1047297rst and
second derivatives of the log-likelihood function with respect to d
(see Robinson 1994 page 1422 for further details) I (l j) is the
periodogram of ut evaluated under the null ie
ˆ ut = 1minusLeth THORNdo yt minus α Vwt
ˆ α = XT
t = 1
wt wt V minus1
XT
t =1
wt 1minusL
eth THORNdo yt wt = 1minusL
eth THORNdo z t
z t = 1I t b T beth THORN 1I t zT beth THORNeth THORN V
and g is a known function related to the spectral density function of ut
f λ σ 2
τ
= σ
2
2π g λ τ eth THORN minus π b λ V π
Appendix B
Gil-Alana (2008) considers the following model
yt = α 1 + xt 1minus
Leth THORNd1
xt = ut t = 1 N
T b
and
yt = α 2 + xt 1minusLeth THORNd2 xt = ut t = T b + 1 N T
where the α 1 and α 2 are the intercepts corresponding to the 1047297rst and
second subsamples respectively d1 and d2 may be real values u t is
I (0) and T b is the time of a break that is supposed to be unknown
This model can be re-parameterized as follows
1minusLeth THORNd1 yt = α 1˜ 1t d1eth THORN + β 1˜ t t d1eth THORN + ut t = 1 N T b
1minus
Leth THORNd2
yt = α 2˜
1t d2eth THORN + β 2˜
t t d2eth THORN + ut t = T b + 1 N
T
Fig 5 Estimates of d based on a semiparametric model for each subsample
1190 JC Cuestas LA Gil-Alana Economic Modelling 26 (2009) 1184ndash1192
7232019 Economic Modelling Further Evidence on the PPP Analysis of the Australian Dollar- Non-linearities Fractional Integration and Structu
httpslidepdfcomreaderfulleconomic-modelling-further-evidence-on-the-ppp-analysis-of-the-australian-dollar- 89
with 1 t (di)=(1minusL)di1 The idea is then to minimise the residual sum
of squares (RSS)
wr t α 1 α 2 β 1 β 2f gXT b
t =1
1minusLeth THORNd 1eth THORN1o yt minus α 1˜ 1t d
1eth THORN1o
+ β 1˜ t
t d
1eth THORN1o
2
+XT
t = T b
frac12 1minusLeth THORNd 1eth THORN2o yt minusα 2˜ 1t d
1eth THORN2o
+ β 2 ˜ t t d
1eth THORN2o
2
In so doing it is necessary to choose a grid for the values of the
differencing parameters d1 and d2 d1o and d2o say Once the estimated
parameters α ˆ 1 α ˆ 2 β ˆ 1 and β ˆ
2 are obtained for partition T b and initial
values d1o(1) and d2o
(1) we plug these values into the objective function
and obtain RSS(T b)=arg mini jRSS(T bd1o(i) d1o
( j)) In order to estimate
the time break T k we obtain the moment that minimises the RSS
where the minimisation is taken over all partitions T 1 T 2 hellip T m such
that T iminusT iminus1ge |ε T | Then it is possible to obtain the regression
parameter estimates as α ˆ i=α ˆ i(T k) and the differencing parameters
d i= d
i(T k) for i =12
Appendix C
The ldquolocalrdquo Whittle estimate of Robinson (1995) is implicitly
de1047297ned by
ˆ d = arg mind log C deth THORN minus 2d 1
m
Xm
j = 1
log λ j
0
1A
for da minus1 = 2 1 = 2eth THORN C deth THORN = 1
m
Xm
j =1
I λ j
λ
2d j λ j =
2π j
T
m
T Y0
where m is a bandwidth parameter number and d isin(minus05 05)
Under 1047297niteness of the fourth moment and other mild conditions
Robinson (1995) proved that
ffiffiffiffiffimp ˆ d minus do
YdN 0 1 = 4eth THORN as T Yinfin
where do is the true value of d This estimator is robust to a certain
degree of conditional heteroskedasticity (Robinson and Henry 1999)
and is more ef 1047297cient than other semi-parametric competitors
References
Baum CF Barkoulas J Caglayan M 1999 Persistence in the international in1047298ationrates Southern Economic Journal 65 900ndash913
Bhargava A 1986 On the theory of testing for unit roots in observed time seriesReview of Economic Studies 53 369ndash384
Bierens HJ 1997 Testing the unit root with drift hypothesis against nonlinear trendstationarity with an application to the US price level and interest rate Journal of Econometrics 81 29ndash64
Bloom1047297eld P 1973 An exponential model in the spectrum of a scalar time series
Biometrika 60 217ndash226Booth GGKaen FR Koveos PE1982RS analysis of foreign exchangemarketsunder
two international monetary regimes Journal of Monetary Economics 10 407ndash415CampbellJY Perron P1991 Pitfalls and opportunities whatmacroeconomists should
know about unit roots NBER Macroeconomics Annual 1141ndash1201Casin P Ceacutespedes LF Sahay R 2004 Commodity currencies and the real exchange
rate Journal of Development Economics 75 239ndash268Cheung YW 1993 Long memory in foreign exchange rates Journal of Business and
Economic Statistics 11 93ndash101Cheung Y-W Lai KS 1993 A fractional cointegration analysis of purchasing power
parity Journal of Business and Economic Statistics 11 103ndash112Crato N Ray BK 2000 Memory in returns and volatilities of futures contracts
Journal of Futures Markets 20 525ndash543Cuestas JC 2009 Purchasing power parity in Central and Eastern European countries
an analysis of unit roots and nonlinearities Applied Economics Letters 16 87ndash94Cuestas JC Regis PJ 2008 Testing for PPP in Australia evidence from unit root tests
against nonlinear trend stationarity alternatives Economics Bulletin 27 1ndash8Darneacute O Hoarau J-F 2008 The purchasing power parity in Australia evidence from
unit root test with structural break Applied Economics Letters 15 203ndash206
DeJong D Nankervis J Savin NE Whiteman CH 1992 Integration versus trendstationarity in time series Econometrica 60 423ndash433
Dickey DA Fuller WA 1979 Distribution of the estimators for autoregressive timeseries with a unit root Journal of the American Statistical Association 74 427ndash431
Diebold FX Rudebusch GD 1991 Is consumption too smooth Long memory and theDeaton paradox The Review of Economics and Statistics 73 1ndash9
DieboldFXInoueA 2001 Longmemory andregimeswitchingJournal of Econometrics105 131ndash159
Diebold FX Husted S Rush M 1991 Real exchange rates under the gold standard Journal of Political Economy 99 1252ndash1271
Dumas B1994 Partial equilibrium versus general equilibrium models of the international
capital market In van der Ploeg F (Ed) The Handbook of International Macro-economics Blackwell Oxford pp 301ndash347 chap 10Elliot G Rothenberg TJ Stock JH 1996 Ef 1047297cient tests for an autoregressive unit root
Econometrica 64 813ndash836Fang HLai KS Lai M1994 Fractal structure in currency futurespricedynamicsJournal
of Futures Markets 14 169ndash181Faria JR Leoacuten-Ledesma MA 2003 Testing the BalassandashSamuelson effect implica-
tions for growth and the PPP Journal of Macroeconomics 25 241ndash253Faria JR Leoacuten-Ledesma MA 2005 Real exchange rate and employment performance
in an open economy Research in Economics 59 67ndash80Geweke J Porter-Hudack S 1983 The estimation and application of long memory
time series models and fractional differencing Journal of Time Series Analysis 4221ndash238
Gil-Alana LA 2000 Mean reversion in the real exchange rates Economics Letters 69285ndash288
Gil-Alana LA 2004 The use of the model of Bloom1047297eld (1973) as an approximation toARMA processes in the context of fractional integration Mathematical andComputer Modelling 39 429ndash436
Gil-AlanaLA 2008 Fractional integrationand structuralbreaks at unknown periods of
time Journal of Time Series Analysis 29 163ndash185Gil-Alana LA Robinson PM 1997 Testing of unit roots and other nonstationary
hypotheses in macroeconomic time series Journal of Econometrics 80 241ndash268GrangerCWJHyung N 2004 Occasional structural breaks and long memory with an
application to the SampP 500 absolute stock return Journal of Empirical Finance 11399ndash421
Hassler U Wolters J 1995 Long memory in in1047298ation rates International evidence Journal of Business and Economic Statistics 13 37ndash45
Henry OT Olekalns N 2002 Does the Australian dollar real exchange rate displaymean reversion Journal of International Money and Finance 21 651ndash666
Juvenal L Taylor MP 20 08 Threshold adjustment of deviations from the law of oneprice Studies in Nonlinear Dynamics and Econometrics 12 Article 8
Kapetanios G Shin Y Snell A 2003 Testing for a unit root in the nonlinear STAR framework Journal of Econometrics 112 359ndash379
Kwiatkowski D Phillips PCB Schmidt P Shin Y 1992 Testing the null hypothesis of stationarity againstthe alternative of a unitrootJournal of Econometrics54159ndash178
Lee D Schmidt P 1996 On the power of the KPSS test of stationarity againstfractionally integrated alternatives Journal of Econometrics 73 285ndash302
Michael P Nobay A Peel D 1997 Transaction costs and nonlinear adjustment in realexchange ratesan empirical investigation Journalof Political Economy 105 862ndash879
Nelson CR Piger J Zivot E 2001 Markov regime-switching and unit root tests Journal of Business Economics and Statistics 19 404ndash415
Ng S Perron P 2001 Lag selection and the construction of unit root tests with goodsize and power Econometrica 69 1519ndash1554
Perron P 1989 The great crash the oil price shock and the unit root hypothesisEconometrica 571361ndash1401
Perron P Phillips PCB 1987 Does GNP have a unit root A reevaluation EconomicsLetters 23 139ndash145
Perron P Rodriacuteguez G 2003 GLS detrending ef 1047297cient unit root tests and structuralchange Journal of Econometrics 115 1ndash27
Phillips PCB 1987 Time series regression with a unit root Econometrica 55 311ndash340Phillips PCB Perron P 1988 lsquoTesting for a unit root in time series regression
Biometrica 75 335ndash346Phillips PCB Shimotsu K 2004 Local Whittle estimation in nonstationary and unit
root cases Annals of Statistics 32 656ndash692Phillips PCB Shimotsu K 2005 Exact local Whittle estimation of fractional
integration Annals of Statistics 33 1890ndash1933
Robinson PM 1994 ldquoEf 1047297cient tests of nonstationary hypotheses Journal of theAmerican Statistical Association 89 1420ndash1437
Robinson PM 1995 Gaussian semi-parametric estimation of long range dependenceAnnals of Statistics 23 1630ndash1661
Robinson PM Henry M 1996 Bandwidth choice in Gaussian semiparametricestimation of long-range dependence In Robinson PM Rosenblatt M (Eds)Athens Conference on Applied Probability in Time Series Analysis Vol II New Yorkpp 220ndash232
Robinson PM Henry M 1999 Long and short memory conditional heteroskedasticityin estimating the memory in levels Econometric Theory 15 299ndash336
Sarno L Taylor MP 2002 Purchasing power parity and the real exchange rate TheEconomics of the Exchange Rate Cambridge University Press Cambridge pp 51ndash96
Sarno L Taylor MP Chowdhury I 2004 Nonlinear dynamics in deviations from thelaw of oneprice a broad-based empiricalstudy Journal of International Money andFinance 23 1ndash25
Taylor MP 2006 Real exchange rates and purchasing power paritymean-reversion ineconomic thought Applied Financial Economics 16 1ndash17
Taylor MP Peel DA 2000 ldquoNonlinear adjustment long-run equilibrium andexchangerate fundamentals Journal of International Moneyand Finance 1933ndash53
1191 JC Cuestas LA Gil-Alana Economic Modelling 26 (2009) 1184ndash1192
7232019 Economic Modelling Further Evidence on the PPP Analysis of the Australian Dollar- Non-linearities Fractional Integration and Structu
httpslidepdfcomreaderfulleconomic-modelling-further-evidence-on-the-ppp-analysis-of-the-australian-dollar- 99
Taylor MP Peel DA Sarno L 2001 Nonlinear mean-reversion in real eschange ratesTowards a solution to the purchasing power parity puzzles International EconomicReview 42 1015ndash1042
Velasco C 1999 Gaussian semiparametric estimation of nonstationary time series Journal of Time Series Analysis 20 87ndash127
Velasco C Robinson PM 2000 Whittle pseudo maximum likelihood estimation fornonstationary time series Journal of the American Statistical Association 951229ndash1243
Vogelsang TJ 1997 Wald-type test for detecting breaks in the trend function of adynamic time series Econometric Theory 13 818ndash849
Wang C 2004 Futures trading activity and predictable foreign exchange movements Journal of Banking and Finance 28 1023ndash1041
Wei S-J Parsley DC 1995 Purchasing power disparity during the 1047298oat rate periodexchange rate volatility trade barriers and other culprits Working Paper 5032NBER
WestKD1987 A noteon thepowerof least squares testsfor a unitroot Economics Letters24 1397ndash1418
1192 JC Cuestas LA Gil-Alana Economic Modelling 26 (2009) 1184ndash1192
7232019 Economic Modelling Further Evidence on the PPP Analysis of the Australian Dollar- Non-linearities Fractional Integration and Structu
httpslidepdfcomreaderfulleconomic-modelling-further-evidence-on-the-ppp-analysis-of-the-australian-dollar- 69
7232019 Economic Modelling Further Evidence on the PPP Analysis of the Australian Dollar- Non-linearities Fractional Integration and Structu
httpslidepdfcomreaderfulleconomic-modelling-further-evidence-on-the-ppp-analysis-of-the-australian-dollar- 79
asymmetric speed of adjustment towards the equilibrium as well as
fractional integration and structural changes Our results are in line
with previous empirical works in the sense that there is no evidence
of mean reversionin theAustralian dollarRER forthe periodanalysed
However we 1047297nd that the order of integration decreases after 1985
coinciding with the currency crisis This conclusion highlights the fact
that the monetary authorities have managed to decrease the
dependence of the Australian dollar on real shocks that may affect
the value of the currency on a permanent basis
Appendix A
The LM test of Robinson (1994) for testing H 0 (Eq (8)) in Eqs (6)
and (7) is
ˆ r = T 1 =2
ˆ σ 2ˆ Aminus1=2
ˆ a
where T is the sample size and
ˆ a = minus2π
T
XT minus1
j =1
ψ λ j
g λ j ˆ τ
minus1I λ j
ˆ σ 2
= σ 2
ˆ τ = 2π
T XT minus1
j =1
g λ j
ˆ τ minus1I λ
j
ˆ A = 2
T ethXT minus1
j =1
ψ λ j
2minus
XT minus1
j =1
ψ λ j
ˆ e λ j
V
timesXT minus1
j = 1
ˆ e λ j
ˆ e λ j
V
0
1Aminus1
timesXT minus1
j =1
ˆ e λ j
ψ λ j
THORNψ λ j
= log j2sin
λ j
2 j ˆ e λ j
=
A
Aτ log g λ j ˆ τ
λ j =
2π j
T ˆ τ = argmin σ
2τ eth THORN
a and A in the above expressions are obtained through the 1047297rst and
second derivatives of the log-likelihood function with respect to d
(see Robinson 1994 page 1422 for further details) I (l j) is the
periodogram of ut evaluated under the null ie
ˆ ut = 1minusLeth THORNdo yt minus α Vwt
ˆ α = XT
t = 1
wt wt V minus1
XT
t =1
wt 1minusL
eth THORNdo yt wt = 1minusL
eth THORNdo z t
z t = 1I t b T beth THORN 1I t zT beth THORNeth THORN V
and g is a known function related to the spectral density function of ut
f λ σ 2
τ
= σ
2
2π g λ τ eth THORN minus π b λ V π
Appendix B
Gil-Alana (2008) considers the following model
yt = α 1 + xt 1minus
Leth THORNd1
xt = ut t = 1 N
T b
and
yt = α 2 + xt 1minusLeth THORNd2 xt = ut t = T b + 1 N T
where the α 1 and α 2 are the intercepts corresponding to the 1047297rst and
second subsamples respectively d1 and d2 may be real values u t is
I (0) and T b is the time of a break that is supposed to be unknown
This model can be re-parameterized as follows
1minusLeth THORNd1 yt = α 1˜ 1t d1eth THORN + β 1˜ t t d1eth THORN + ut t = 1 N T b
1minus
Leth THORNd2
yt = α 2˜
1t d2eth THORN + β 2˜
t t d2eth THORN + ut t = T b + 1 N
T
Fig 5 Estimates of d based on a semiparametric model for each subsample
1190 JC Cuestas LA Gil-Alana Economic Modelling 26 (2009) 1184ndash1192
7232019 Economic Modelling Further Evidence on the PPP Analysis of the Australian Dollar- Non-linearities Fractional Integration and Structu
httpslidepdfcomreaderfulleconomic-modelling-further-evidence-on-the-ppp-analysis-of-the-australian-dollar- 89
with 1 t (di)=(1minusL)di1 The idea is then to minimise the residual sum
of squares (RSS)
wr t α 1 α 2 β 1 β 2f gXT b
t =1
1minusLeth THORNd 1eth THORN1o yt minus α 1˜ 1t d
1eth THORN1o
+ β 1˜ t
t d
1eth THORN1o
2
+XT
t = T b
frac12 1minusLeth THORNd 1eth THORN2o yt minusα 2˜ 1t d
1eth THORN2o
+ β 2 ˜ t t d
1eth THORN2o
2
In so doing it is necessary to choose a grid for the values of the
differencing parameters d1 and d2 d1o and d2o say Once the estimated
parameters α ˆ 1 α ˆ 2 β ˆ 1 and β ˆ
2 are obtained for partition T b and initial
values d1o(1) and d2o
(1) we plug these values into the objective function
and obtain RSS(T b)=arg mini jRSS(T bd1o(i) d1o
( j)) In order to estimate
the time break T k we obtain the moment that minimises the RSS
where the minimisation is taken over all partitions T 1 T 2 hellip T m such
that T iminusT iminus1ge |ε T | Then it is possible to obtain the regression
parameter estimates as α ˆ i=α ˆ i(T k) and the differencing parameters
d i= d
i(T k) for i =12
Appendix C
The ldquolocalrdquo Whittle estimate of Robinson (1995) is implicitly
de1047297ned by
ˆ d = arg mind log C deth THORN minus 2d 1
m
Xm
j = 1
log λ j
0
1A
for da minus1 = 2 1 = 2eth THORN C deth THORN = 1
m
Xm
j =1
I λ j
λ
2d j λ j =
2π j
T
m
T Y0
where m is a bandwidth parameter number and d isin(minus05 05)
Under 1047297niteness of the fourth moment and other mild conditions
Robinson (1995) proved that
ffiffiffiffiffimp ˆ d minus do
YdN 0 1 = 4eth THORN as T Yinfin
where do is the true value of d This estimator is robust to a certain
degree of conditional heteroskedasticity (Robinson and Henry 1999)
and is more ef 1047297cient than other semi-parametric competitors
References
Baum CF Barkoulas J Caglayan M 1999 Persistence in the international in1047298ationrates Southern Economic Journal 65 900ndash913
Bhargava A 1986 On the theory of testing for unit roots in observed time seriesReview of Economic Studies 53 369ndash384
Bierens HJ 1997 Testing the unit root with drift hypothesis against nonlinear trendstationarity with an application to the US price level and interest rate Journal of Econometrics 81 29ndash64
Bloom1047297eld P 1973 An exponential model in the spectrum of a scalar time series
Biometrika 60 217ndash226Booth GGKaen FR Koveos PE1982RS analysis of foreign exchangemarketsunder
two international monetary regimes Journal of Monetary Economics 10 407ndash415CampbellJY Perron P1991 Pitfalls and opportunities whatmacroeconomists should
know about unit roots NBER Macroeconomics Annual 1141ndash1201Casin P Ceacutespedes LF Sahay R 2004 Commodity currencies and the real exchange
rate Journal of Development Economics 75 239ndash268Cheung YW 1993 Long memory in foreign exchange rates Journal of Business and
Economic Statistics 11 93ndash101Cheung Y-W Lai KS 1993 A fractional cointegration analysis of purchasing power
parity Journal of Business and Economic Statistics 11 103ndash112Crato N Ray BK 2000 Memory in returns and volatilities of futures contracts
Journal of Futures Markets 20 525ndash543Cuestas JC 2009 Purchasing power parity in Central and Eastern European countries
an analysis of unit roots and nonlinearities Applied Economics Letters 16 87ndash94Cuestas JC Regis PJ 2008 Testing for PPP in Australia evidence from unit root tests
against nonlinear trend stationarity alternatives Economics Bulletin 27 1ndash8Darneacute O Hoarau J-F 2008 The purchasing power parity in Australia evidence from
unit root test with structural break Applied Economics Letters 15 203ndash206
DeJong D Nankervis J Savin NE Whiteman CH 1992 Integration versus trendstationarity in time series Econometrica 60 423ndash433
Dickey DA Fuller WA 1979 Distribution of the estimators for autoregressive timeseries with a unit root Journal of the American Statistical Association 74 427ndash431
Diebold FX Rudebusch GD 1991 Is consumption too smooth Long memory and theDeaton paradox The Review of Economics and Statistics 73 1ndash9
DieboldFXInoueA 2001 Longmemory andregimeswitchingJournal of Econometrics105 131ndash159
Diebold FX Husted S Rush M 1991 Real exchange rates under the gold standard Journal of Political Economy 99 1252ndash1271
Dumas B1994 Partial equilibrium versus general equilibrium models of the international
capital market In van der Ploeg F (Ed) The Handbook of International Macro-economics Blackwell Oxford pp 301ndash347 chap 10Elliot G Rothenberg TJ Stock JH 1996 Ef 1047297cient tests for an autoregressive unit root
Econometrica 64 813ndash836Fang HLai KS Lai M1994 Fractal structure in currency futurespricedynamicsJournal
of Futures Markets 14 169ndash181Faria JR Leoacuten-Ledesma MA 2003 Testing the BalassandashSamuelson effect implica-
tions for growth and the PPP Journal of Macroeconomics 25 241ndash253Faria JR Leoacuten-Ledesma MA 2005 Real exchange rate and employment performance
in an open economy Research in Economics 59 67ndash80Geweke J Porter-Hudack S 1983 The estimation and application of long memory
time series models and fractional differencing Journal of Time Series Analysis 4221ndash238
Gil-Alana LA 2000 Mean reversion in the real exchange rates Economics Letters 69285ndash288
Gil-Alana LA 2004 The use of the model of Bloom1047297eld (1973) as an approximation toARMA processes in the context of fractional integration Mathematical andComputer Modelling 39 429ndash436
Gil-AlanaLA 2008 Fractional integrationand structuralbreaks at unknown periods of
time Journal of Time Series Analysis 29 163ndash185Gil-Alana LA Robinson PM 1997 Testing of unit roots and other nonstationary
hypotheses in macroeconomic time series Journal of Econometrics 80 241ndash268GrangerCWJHyung N 2004 Occasional structural breaks and long memory with an
application to the SampP 500 absolute stock return Journal of Empirical Finance 11399ndash421
Hassler U Wolters J 1995 Long memory in in1047298ation rates International evidence Journal of Business and Economic Statistics 13 37ndash45
Henry OT Olekalns N 2002 Does the Australian dollar real exchange rate displaymean reversion Journal of International Money and Finance 21 651ndash666
Juvenal L Taylor MP 20 08 Threshold adjustment of deviations from the law of oneprice Studies in Nonlinear Dynamics and Econometrics 12 Article 8
Kapetanios G Shin Y Snell A 2003 Testing for a unit root in the nonlinear STAR framework Journal of Econometrics 112 359ndash379
Kwiatkowski D Phillips PCB Schmidt P Shin Y 1992 Testing the null hypothesis of stationarity againstthe alternative of a unitrootJournal of Econometrics54159ndash178
Lee D Schmidt P 1996 On the power of the KPSS test of stationarity againstfractionally integrated alternatives Journal of Econometrics 73 285ndash302
Michael P Nobay A Peel D 1997 Transaction costs and nonlinear adjustment in realexchange ratesan empirical investigation Journalof Political Economy 105 862ndash879
Nelson CR Piger J Zivot E 2001 Markov regime-switching and unit root tests Journal of Business Economics and Statistics 19 404ndash415
Ng S Perron P 2001 Lag selection and the construction of unit root tests with goodsize and power Econometrica 69 1519ndash1554
Perron P 1989 The great crash the oil price shock and the unit root hypothesisEconometrica 571361ndash1401
Perron P Phillips PCB 1987 Does GNP have a unit root A reevaluation EconomicsLetters 23 139ndash145
Perron P Rodriacuteguez G 2003 GLS detrending ef 1047297cient unit root tests and structuralchange Journal of Econometrics 115 1ndash27
Phillips PCB 1987 Time series regression with a unit root Econometrica 55 311ndash340Phillips PCB Perron P 1988 lsquoTesting for a unit root in time series regression
Biometrica 75 335ndash346Phillips PCB Shimotsu K 2004 Local Whittle estimation in nonstationary and unit
root cases Annals of Statistics 32 656ndash692Phillips PCB Shimotsu K 2005 Exact local Whittle estimation of fractional
integration Annals of Statistics 33 1890ndash1933
Robinson PM 1994 ldquoEf 1047297cient tests of nonstationary hypotheses Journal of theAmerican Statistical Association 89 1420ndash1437
Robinson PM 1995 Gaussian semi-parametric estimation of long range dependenceAnnals of Statistics 23 1630ndash1661
Robinson PM Henry M 1996 Bandwidth choice in Gaussian semiparametricestimation of long-range dependence In Robinson PM Rosenblatt M (Eds)Athens Conference on Applied Probability in Time Series Analysis Vol II New Yorkpp 220ndash232
Robinson PM Henry M 1999 Long and short memory conditional heteroskedasticityin estimating the memory in levels Econometric Theory 15 299ndash336
Sarno L Taylor MP 2002 Purchasing power parity and the real exchange rate TheEconomics of the Exchange Rate Cambridge University Press Cambridge pp 51ndash96
Sarno L Taylor MP Chowdhury I 2004 Nonlinear dynamics in deviations from thelaw of oneprice a broad-based empiricalstudy Journal of International Money andFinance 23 1ndash25
Taylor MP 2006 Real exchange rates and purchasing power paritymean-reversion ineconomic thought Applied Financial Economics 16 1ndash17
Taylor MP Peel DA 2000 ldquoNonlinear adjustment long-run equilibrium andexchangerate fundamentals Journal of International Moneyand Finance 1933ndash53
1191 JC Cuestas LA Gil-Alana Economic Modelling 26 (2009) 1184ndash1192
7232019 Economic Modelling Further Evidence on the PPP Analysis of the Australian Dollar- Non-linearities Fractional Integration and Structu
httpslidepdfcomreaderfulleconomic-modelling-further-evidence-on-the-ppp-analysis-of-the-australian-dollar- 99
Taylor MP Peel DA Sarno L 2001 Nonlinear mean-reversion in real eschange ratesTowards a solution to the purchasing power parity puzzles International EconomicReview 42 1015ndash1042
Velasco C 1999 Gaussian semiparametric estimation of nonstationary time series Journal of Time Series Analysis 20 87ndash127
Velasco C Robinson PM 2000 Whittle pseudo maximum likelihood estimation fornonstationary time series Journal of the American Statistical Association 951229ndash1243
Vogelsang TJ 1997 Wald-type test for detecting breaks in the trend function of adynamic time series Econometric Theory 13 818ndash849
Wang C 2004 Futures trading activity and predictable foreign exchange movements Journal of Banking and Finance 28 1023ndash1041
Wei S-J Parsley DC 1995 Purchasing power disparity during the 1047298oat rate periodexchange rate volatility trade barriers and other culprits Working Paper 5032NBER
WestKD1987 A noteon thepowerof least squares testsfor a unitroot Economics Letters24 1397ndash1418
1192 JC Cuestas LA Gil-Alana Economic Modelling 26 (2009) 1184ndash1192
7232019 Economic Modelling Further Evidence on the PPP Analysis of the Australian Dollar- Non-linearities Fractional Integration and Structu
httpslidepdfcomreaderfulleconomic-modelling-further-evidence-on-the-ppp-analysis-of-the-australian-dollar- 79
asymmetric speed of adjustment towards the equilibrium as well as
fractional integration and structural changes Our results are in line
with previous empirical works in the sense that there is no evidence
of mean reversionin theAustralian dollarRER forthe periodanalysed
However we 1047297nd that the order of integration decreases after 1985
coinciding with the currency crisis This conclusion highlights the fact
that the monetary authorities have managed to decrease the
dependence of the Australian dollar on real shocks that may affect
the value of the currency on a permanent basis
Appendix A
The LM test of Robinson (1994) for testing H 0 (Eq (8)) in Eqs (6)
and (7) is
ˆ r = T 1 =2
ˆ σ 2ˆ Aminus1=2
ˆ a
where T is the sample size and
ˆ a = minus2π
T
XT minus1
j =1
ψ λ j
g λ j ˆ τ
minus1I λ j
ˆ σ 2
= σ 2
ˆ τ = 2π
T XT minus1
j =1
g λ j
ˆ τ minus1I λ
j
ˆ A = 2
T ethXT minus1
j =1
ψ λ j
2minus
XT minus1
j =1
ψ λ j
ˆ e λ j
V
timesXT minus1
j = 1
ˆ e λ j
ˆ e λ j
V
0
1Aminus1
timesXT minus1
j =1
ˆ e λ j
ψ λ j
THORNψ λ j
= log j2sin
λ j
2 j ˆ e λ j
=
A
Aτ log g λ j ˆ τ
λ j =
2π j
T ˆ τ = argmin σ
2τ eth THORN
a and A in the above expressions are obtained through the 1047297rst and
second derivatives of the log-likelihood function with respect to d
(see Robinson 1994 page 1422 for further details) I (l j) is the
periodogram of ut evaluated under the null ie
ˆ ut = 1minusLeth THORNdo yt minus α Vwt
ˆ α = XT
t = 1
wt wt V minus1
XT
t =1
wt 1minusL
eth THORNdo yt wt = 1minusL
eth THORNdo z t
z t = 1I t b T beth THORN 1I t zT beth THORNeth THORN V
and g is a known function related to the spectral density function of ut
f λ σ 2
τ
= σ
2
2π g λ τ eth THORN minus π b λ V π
Appendix B
Gil-Alana (2008) considers the following model
yt = α 1 + xt 1minus
Leth THORNd1
xt = ut t = 1 N
T b
and
yt = α 2 + xt 1minusLeth THORNd2 xt = ut t = T b + 1 N T
where the α 1 and α 2 are the intercepts corresponding to the 1047297rst and
second subsamples respectively d1 and d2 may be real values u t is
I (0) and T b is the time of a break that is supposed to be unknown
This model can be re-parameterized as follows
1minusLeth THORNd1 yt = α 1˜ 1t d1eth THORN + β 1˜ t t d1eth THORN + ut t = 1 N T b
1minus
Leth THORNd2
yt = α 2˜
1t d2eth THORN + β 2˜
t t d2eth THORN + ut t = T b + 1 N
T
Fig 5 Estimates of d based on a semiparametric model for each subsample
1190 JC Cuestas LA Gil-Alana Economic Modelling 26 (2009) 1184ndash1192
7232019 Economic Modelling Further Evidence on the PPP Analysis of the Australian Dollar- Non-linearities Fractional Integration and Structu
httpslidepdfcomreaderfulleconomic-modelling-further-evidence-on-the-ppp-analysis-of-the-australian-dollar- 89
with 1 t (di)=(1minusL)di1 The idea is then to minimise the residual sum
of squares (RSS)
wr t α 1 α 2 β 1 β 2f gXT b
t =1
1minusLeth THORNd 1eth THORN1o yt minus α 1˜ 1t d
1eth THORN1o
+ β 1˜ t
t d
1eth THORN1o
2
+XT
t = T b
frac12 1minusLeth THORNd 1eth THORN2o yt minusα 2˜ 1t d
1eth THORN2o
+ β 2 ˜ t t d
1eth THORN2o
2
In so doing it is necessary to choose a grid for the values of the
differencing parameters d1 and d2 d1o and d2o say Once the estimated
parameters α ˆ 1 α ˆ 2 β ˆ 1 and β ˆ
2 are obtained for partition T b and initial
values d1o(1) and d2o
(1) we plug these values into the objective function
and obtain RSS(T b)=arg mini jRSS(T bd1o(i) d1o
( j)) In order to estimate
the time break T k we obtain the moment that minimises the RSS
where the minimisation is taken over all partitions T 1 T 2 hellip T m such
that T iminusT iminus1ge |ε T | Then it is possible to obtain the regression
parameter estimates as α ˆ i=α ˆ i(T k) and the differencing parameters
d i= d
i(T k) for i =12
Appendix C
The ldquolocalrdquo Whittle estimate of Robinson (1995) is implicitly
de1047297ned by
ˆ d = arg mind log C deth THORN minus 2d 1
m
Xm
j = 1
log λ j
0
1A
for da minus1 = 2 1 = 2eth THORN C deth THORN = 1
m
Xm
j =1
I λ j
λ
2d j λ j =
2π j
T
m
T Y0
where m is a bandwidth parameter number and d isin(minus05 05)
Under 1047297niteness of the fourth moment and other mild conditions
Robinson (1995) proved that
ffiffiffiffiffimp ˆ d minus do
YdN 0 1 = 4eth THORN as T Yinfin
where do is the true value of d This estimator is robust to a certain
degree of conditional heteroskedasticity (Robinson and Henry 1999)
and is more ef 1047297cient than other semi-parametric competitors
References
Baum CF Barkoulas J Caglayan M 1999 Persistence in the international in1047298ationrates Southern Economic Journal 65 900ndash913
Bhargava A 1986 On the theory of testing for unit roots in observed time seriesReview of Economic Studies 53 369ndash384
Bierens HJ 1997 Testing the unit root with drift hypothesis against nonlinear trendstationarity with an application to the US price level and interest rate Journal of Econometrics 81 29ndash64
Bloom1047297eld P 1973 An exponential model in the spectrum of a scalar time series
Biometrika 60 217ndash226Booth GGKaen FR Koveos PE1982RS analysis of foreign exchangemarketsunder
two international monetary regimes Journal of Monetary Economics 10 407ndash415CampbellJY Perron P1991 Pitfalls and opportunities whatmacroeconomists should
know about unit roots NBER Macroeconomics Annual 1141ndash1201Casin P Ceacutespedes LF Sahay R 2004 Commodity currencies and the real exchange
rate Journal of Development Economics 75 239ndash268Cheung YW 1993 Long memory in foreign exchange rates Journal of Business and
Economic Statistics 11 93ndash101Cheung Y-W Lai KS 1993 A fractional cointegration analysis of purchasing power
parity Journal of Business and Economic Statistics 11 103ndash112Crato N Ray BK 2000 Memory in returns and volatilities of futures contracts
Journal of Futures Markets 20 525ndash543Cuestas JC 2009 Purchasing power parity in Central and Eastern European countries
an analysis of unit roots and nonlinearities Applied Economics Letters 16 87ndash94Cuestas JC Regis PJ 2008 Testing for PPP in Australia evidence from unit root tests
against nonlinear trend stationarity alternatives Economics Bulletin 27 1ndash8Darneacute O Hoarau J-F 2008 The purchasing power parity in Australia evidence from
unit root test with structural break Applied Economics Letters 15 203ndash206
DeJong D Nankervis J Savin NE Whiteman CH 1992 Integration versus trendstationarity in time series Econometrica 60 423ndash433
Dickey DA Fuller WA 1979 Distribution of the estimators for autoregressive timeseries with a unit root Journal of the American Statistical Association 74 427ndash431
Diebold FX Rudebusch GD 1991 Is consumption too smooth Long memory and theDeaton paradox The Review of Economics and Statistics 73 1ndash9
DieboldFXInoueA 2001 Longmemory andregimeswitchingJournal of Econometrics105 131ndash159
Diebold FX Husted S Rush M 1991 Real exchange rates under the gold standard Journal of Political Economy 99 1252ndash1271
Dumas B1994 Partial equilibrium versus general equilibrium models of the international
capital market In van der Ploeg F (Ed) The Handbook of International Macro-economics Blackwell Oxford pp 301ndash347 chap 10Elliot G Rothenberg TJ Stock JH 1996 Ef 1047297cient tests for an autoregressive unit root
Econometrica 64 813ndash836Fang HLai KS Lai M1994 Fractal structure in currency futurespricedynamicsJournal
of Futures Markets 14 169ndash181Faria JR Leoacuten-Ledesma MA 2003 Testing the BalassandashSamuelson effect implica-
tions for growth and the PPP Journal of Macroeconomics 25 241ndash253Faria JR Leoacuten-Ledesma MA 2005 Real exchange rate and employment performance
in an open economy Research in Economics 59 67ndash80Geweke J Porter-Hudack S 1983 The estimation and application of long memory
time series models and fractional differencing Journal of Time Series Analysis 4221ndash238
Gil-Alana LA 2000 Mean reversion in the real exchange rates Economics Letters 69285ndash288
Gil-Alana LA 2004 The use of the model of Bloom1047297eld (1973) as an approximation toARMA processes in the context of fractional integration Mathematical andComputer Modelling 39 429ndash436
Gil-AlanaLA 2008 Fractional integrationand structuralbreaks at unknown periods of
time Journal of Time Series Analysis 29 163ndash185Gil-Alana LA Robinson PM 1997 Testing of unit roots and other nonstationary
hypotheses in macroeconomic time series Journal of Econometrics 80 241ndash268GrangerCWJHyung N 2004 Occasional structural breaks and long memory with an
application to the SampP 500 absolute stock return Journal of Empirical Finance 11399ndash421
Hassler U Wolters J 1995 Long memory in in1047298ation rates International evidence Journal of Business and Economic Statistics 13 37ndash45
Henry OT Olekalns N 2002 Does the Australian dollar real exchange rate displaymean reversion Journal of International Money and Finance 21 651ndash666
Juvenal L Taylor MP 20 08 Threshold adjustment of deviations from the law of oneprice Studies in Nonlinear Dynamics and Econometrics 12 Article 8
Kapetanios G Shin Y Snell A 2003 Testing for a unit root in the nonlinear STAR framework Journal of Econometrics 112 359ndash379
Kwiatkowski D Phillips PCB Schmidt P Shin Y 1992 Testing the null hypothesis of stationarity againstthe alternative of a unitrootJournal of Econometrics54159ndash178
Lee D Schmidt P 1996 On the power of the KPSS test of stationarity againstfractionally integrated alternatives Journal of Econometrics 73 285ndash302
Michael P Nobay A Peel D 1997 Transaction costs and nonlinear adjustment in realexchange ratesan empirical investigation Journalof Political Economy 105 862ndash879
Nelson CR Piger J Zivot E 2001 Markov regime-switching and unit root tests Journal of Business Economics and Statistics 19 404ndash415
Ng S Perron P 2001 Lag selection and the construction of unit root tests with goodsize and power Econometrica 69 1519ndash1554
Perron P 1989 The great crash the oil price shock and the unit root hypothesisEconometrica 571361ndash1401
Perron P Phillips PCB 1987 Does GNP have a unit root A reevaluation EconomicsLetters 23 139ndash145
Perron P Rodriacuteguez G 2003 GLS detrending ef 1047297cient unit root tests and structuralchange Journal of Econometrics 115 1ndash27
Phillips PCB 1987 Time series regression with a unit root Econometrica 55 311ndash340Phillips PCB Perron P 1988 lsquoTesting for a unit root in time series regression
Biometrica 75 335ndash346Phillips PCB Shimotsu K 2004 Local Whittle estimation in nonstationary and unit
root cases Annals of Statistics 32 656ndash692Phillips PCB Shimotsu K 2005 Exact local Whittle estimation of fractional
integration Annals of Statistics 33 1890ndash1933
Robinson PM 1994 ldquoEf 1047297cient tests of nonstationary hypotheses Journal of theAmerican Statistical Association 89 1420ndash1437
Robinson PM 1995 Gaussian semi-parametric estimation of long range dependenceAnnals of Statistics 23 1630ndash1661
Robinson PM Henry M 1996 Bandwidth choice in Gaussian semiparametricestimation of long-range dependence In Robinson PM Rosenblatt M (Eds)Athens Conference on Applied Probability in Time Series Analysis Vol II New Yorkpp 220ndash232
Robinson PM Henry M 1999 Long and short memory conditional heteroskedasticityin estimating the memory in levels Econometric Theory 15 299ndash336
Sarno L Taylor MP 2002 Purchasing power parity and the real exchange rate TheEconomics of the Exchange Rate Cambridge University Press Cambridge pp 51ndash96
Sarno L Taylor MP Chowdhury I 2004 Nonlinear dynamics in deviations from thelaw of oneprice a broad-based empiricalstudy Journal of International Money andFinance 23 1ndash25
Taylor MP 2006 Real exchange rates and purchasing power paritymean-reversion ineconomic thought Applied Financial Economics 16 1ndash17
Taylor MP Peel DA 2000 ldquoNonlinear adjustment long-run equilibrium andexchangerate fundamentals Journal of International Moneyand Finance 1933ndash53
1191 JC Cuestas LA Gil-Alana Economic Modelling 26 (2009) 1184ndash1192
7232019 Economic Modelling Further Evidence on the PPP Analysis of the Australian Dollar- Non-linearities Fractional Integration and Structu
httpslidepdfcomreaderfulleconomic-modelling-further-evidence-on-the-ppp-analysis-of-the-australian-dollar- 99
Taylor MP Peel DA Sarno L 2001 Nonlinear mean-reversion in real eschange ratesTowards a solution to the purchasing power parity puzzles International EconomicReview 42 1015ndash1042
Velasco C 1999 Gaussian semiparametric estimation of nonstationary time series Journal of Time Series Analysis 20 87ndash127
Velasco C Robinson PM 2000 Whittle pseudo maximum likelihood estimation fornonstationary time series Journal of the American Statistical Association 951229ndash1243
Vogelsang TJ 1997 Wald-type test for detecting breaks in the trend function of adynamic time series Econometric Theory 13 818ndash849
Wang C 2004 Futures trading activity and predictable foreign exchange movements Journal of Banking and Finance 28 1023ndash1041
Wei S-J Parsley DC 1995 Purchasing power disparity during the 1047298oat rate periodexchange rate volatility trade barriers and other culprits Working Paper 5032NBER
WestKD1987 A noteon thepowerof least squares testsfor a unitroot Economics Letters24 1397ndash1418
1192 JC Cuestas LA Gil-Alana Economic Modelling 26 (2009) 1184ndash1192
7232019 Economic Modelling Further Evidence on the PPP Analysis of the Australian Dollar- Non-linearities Fractional Integration and Structu
httpslidepdfcomreaderfulleconomic-modelling-further-evidence-on-the-ppp-analysis-of-the-australian-dollar- 89
with 1 t (di)=(1minusL)di1 The idea is then to minimise the residual sum
of squares (RSS)
wr t α 1 α 2 β 1 β 2f gXT b
t =1
1minusLeth THORNd 1eth THORN1o yt minus α 1˜ 1t d
1eth THORN1o
+ β 1˜ t
t d
1eth THORN1o
2
+XT
t = T b
frac12 1minusLeth THORNd 1eth THORN2o yt minusα 2˜ 1t d
1eth THORN2o
+ β 2 ˜ t t d
1eth THORN2o
2
In so doing it is necessary to choose a grid for the values of the
differencing parameters d1 and d2 d1o and d2o say Once the estimated
parameters α ˆ 1 α ˆ 2 β ˆ 1 and β ˆ
2 are obtained for partition T b and initial
values d1o(1) and d2o
(1) we plug these values into the objective function
and obtain RSS(T b)=arg mini jRSS(T bd1o(i) d1o
( j)) In order to estimate
the time break T k we obtain the moment that minimises the RSS
where the minimisation is taken over all partitions T 1 T 2 hellip T m such
that T iminusT iminus1ge |ε T | Then it is possible to obtain the regression
parameter estimates as α ˆ i=α ˆ i(T k) and the differencing parameters
d i= d
i(T k) for i =12
Appendix C
The ldquolocalrdquo Whittle estimate of Robinson (1995) is implicitly
de1047297ned by
ˆ d = arg mind log C deth THORN minus 2d 1
m
Xm
j = 1
log λ j
0
1A
for da minus1 = 2 1 = 2eth THORN C deth THORN = 1
m
Xm
j =1
I λ j
λ
2d j λ j =
2π j
T
m
T Y0
where m is a bandwidth parameter number and d isin(minus05 05)
Under 1047297niteness of the fourth moment and other mild conditions
Robinson (1995) proved that
ffiffiffiffiffimp ˆ d minus do
YdN 0 1 = 4eth THORN as T Yinfin
where do is the true value of d This estimator is robust to a certain
degree of conditional heteroskedasticity (Robinson and Henry 1999)
and is more ef 1047297cient than other semi-parametric competitors
References
Baum CF Barkoulas J Caglayan M 1999 Persistence in the international in1047298ationrates Southern Economic Journal 65 900ndash913
Bhargava A 1986 On the theory of testing for unit roots in observed time seriesReview of Economic Studies 53 369ndash384
Bierens HJ 1997 Testing the unit root with drift hypothesis against nonlinear trendstationarity with an application to the US price level and interest rate Journal of Econometrics 81 29ndash64
Bloom1047297eld P 1973 An exponential model in the spectrum of a scalar time series
Biometrika 60 217ndash226Booth GGKaen FR Koveos PE1982RS analysis of foreign exchangemarketsunder
two international monetary regimes Journal of Monetary Economics 10 407ndash415CampbellJY Perron P1991 Pitfalls and opportunities whatmacroeconomists should
know about unit roots NBER Macroeconomics Annual 1141ndash1201Casin P Ceacutespedes LF Sahay R 2004 Commodity currencies and the real exchange
rate Journal of Development Economics 75 239ndash268Cheung YW 1993 Long memory in foreign exchange rates Journal of Business and
Economic Statistics 11 93ndash101Cheung Y-W Lai KS 1993 A fractional cointegration analysis of purchasing power
parity Journal of Business and Economic Statistics 11 103ndash112Crato N Ray BK 2000 Memory in returns and volatilities of futures contracts
Journal of Futures Markets 20 525ndash543Cuestas JC 2009 Purchasing power parity in Central and Eastern European countries
an analysis of unit roots and nonlinearities Applied Economics Letters 16 87ndash94Cuestas JC Regis PJ 2008 Testing for PPP in Australia evidence from unit root tests
against nonlinear trend stationarity alternatives Economics Bulletin 27 1ndash8Darneacute O Hoarau J-F 2008 The purchasing power parity in Australia evidence from
unit root test with structural break Applied Economics Letters 15 203ndash206
DeJong D Nankervis J Savin NE Whiteman CH 1992 Integration versus trendstationarity in time series Econometrica 60 423ndash433
Dickey DA Fuller WA 1979 Distribution of the estimators for autoregressive timeseries with a unit root Journal of the American Statistical Association 74 427ndash431
Diebold FX Rudebusch GD 1991 Is consumption too smooth Long memory and theDeaton paradox The Review of Economics and Statistics 73 1ndash9
DieboldFXInoueA 2001 Longmemory andregimeswitchingJournal of Econometrics105 131ndash159
Diebold FX Husted S Rush M 1991 Real exchange rates under the gold standard Journal of Political Economy 99 1252ndash1271
Dumas B1994 Partial equilibrium versus general equilibrium models of the international
capital market In van der Ploeg F (Ed) The Handbook of International Macro-economics Blackwell Oxford pp 301ndash347 chap 10Elliot G Rothenberg TJ Stock JH 1996 Ef 1047297cient tests for an autoregressive unit root
Econometrica 64 813ndash836Fang HLai KS Lai M1994 Fractal structure in currency futurespricedynamicsJournal
of Futures Markets 14 169ndash181Faria JR Leoacuten-Ledesma MA 2003 Testing the BalassandashSamuelson effect implica-
tions for growth and the PPP Journal of Macroeconomics 25 241ndash253Faria JR Leoacuten-Ledesma MA 2005 Real exchange rate and employment performance
in an open economy Research in Economics 59 67ndash80Geweke J Porter-Hudack S 1983 The estimation and application of long memory
time series models and fractional differencing Journal of Time Series Analysis 4221ndash238
Gil-Alana LA 2000 Mean reversion in the real exchange rates Economics Letters 69285ndash288
Gil-Alana LA 2004 The use of the model of Bloom1047297eld (1973) as an approximation toARMA processes in the context of fractional integration Mathematical andComputer Modelling 39 429ndash436
Gil-AlanaLA 2008 Fractional integrationand structuralbreaks at unknown periods of
time Journal of Time Series Analysis 29 163ndash185Gil-Alana LA Robinson PM 1997 Testing of unit roots and other nonstationary
hypotheses in macroeconomic time series Journal of Econometrics 80 241ndash268GrangerCWJHyung N 2004 Occasional structural breaks and long memory with an
application to the SampP 500 absolute stock return Journal of Empirical Finance 11399ndash421
Hassler U Wolters J 1995 Long memory in in1047298ation rates International evidence Journal of Business and Economic Statistics 13 37ndash45
Henry OT Olekalns N 2002 Does the Australian dollar real exchange rate displaymean reversion Journal of International Money and Finance 21 651ndash666
Juvenal L Taylor MP 20 08 Threshold adjustment of deviations from the law of oneprice Studies in Nonlinear Dynamics and Econometrics 12 Article 8
Kapetanios G Shin Y Snell A 2003 Testing for a unit root in the nonlinear STAR framework Journal of Econometrics 112 359ndash379
Kwiatkowski D Phillips PCB Schmidt P Shin Y 1992 Testing the null hypothesis of stationarity againstthe alternative of a unitrootJournal of Econometrics54159ndash178
Lee D Schmidt P 1996 On the power of the KPSS test of stationarity againstfractionally integrated alternatives Journal of Econometrics 73 285ndash302
Michael P Nobay A Peel D 1997 Transaction costs and nonlinear adjustment in realexchange ratesan empirical investigation Journalof Political Economy 105 862ndash879
Nelson CR Piger J Zivot E 2001 Markov regime-switching and unit root tests Journal of Business Economics and Statistics 19 404ndash415
Ng S Perron P 2001 Lag selection and the construction of unit root tests with goodsize and power Econometrica 69 1519ndash1554
Perron P 1989 The great crash the oil price shock and the unit root hypothesisEconometrica 571361ndash1401
Perron P Phillips PCB 1987 Does GNP have a unit root A reevaluation EconomicsLetters 23 139ndash145
Perron P Rodriacuteguez G 2003 GLS detrending ef 1047297cient unit root tests and structuralchange Journal of Econometrics 115 1ndash27
Phillips PCB 1987 Time series regression with a unit root Econometrica 55 311ndash340Phillips PCB Perron P 1988 lsquoTesting for a unit root in time series regression
Biometrica 75 335ndash346Phillips PCB Shimotsu K 2004 Local Whittle estimation in nonstationary and unit
root cases Annals of Statistics 32 656ndash692Phillips PCB Shimotsu K 2005 Exact local Whittle estimation of fractional
integration Annals of Statistics 33 1890ndash1933
Robinson PM 1994 ldquoEf 1047297cient tests of nonstationary hypotheses Journal of theAmerican Statistical Association 89 1420ndash1437
Robinson PM 1995 Gaussian semi-parametric estimation of long range dependenceAnnals of Statistics 23 1630ndash1661
Robinson PM Henry M 1996 Bandwidth choice in Gaussian semiparametricestimation of long-range dependence In Robinson PM Rosenblatt M (Eds)Athens Conference on Applied Probability in Time Series Analysis Vol II New Yorkpp 220ndash232
Robinson PM Henry M 1999 Long and short memory conditional heteroskedasticityin estimating the memory in levels Econometric Theory 15 299ndash336
Sarno L Taylor MP 2002 Purchasing power parity and the real exchange rate TheEconomics of the Exchange Rate Cambridge University Press Cambridge pp 51ndash96
Sarno L Taylor MP Chowdhury I 2004 Nonlinear dynamics in deviations from thelaw of oneprice a broad-based empiricalstudy Journal of International Money andFinance 23 1ndash25
Taylor MP 2006 Real exchange rates and purchasing power paritymean-reversion ineconomic thought Applied Financial Economics 16 1ndash17
Taylor MP Peel DA 2000 ldquoNonlinear adjustment long-run equilibrium andexchangerate fundamentals Journal of International Moneyand Finance 1933ndash53
1191 JC Cuestas LA Gil-Alana Economic Modelling 26 (2009) 1184ndash1192
7232019 Economic Modelling Further Evidence on the PPP Analysis of the Australian Dollar- Non-linearities Fractional Integration and Structu
httpslidepdfcomreaderfulleconomic-modelling-further-evidence-on-the-ppp-analysis-of-the-australian-dollar- 99
Taylor MP Peel DA Sarno L 2001 Nonlinear mean-reversion in real eschange ratesTowards a solution to the purchasing power parity puzzles International EconomicReview 42 1015ndash1042
Velasco C 1999 Gaussian semiparametric estimation of nonstationary time series Journal of Time Series Analysis 20 87ndash127
Velasco C Robinson PM 2000 Whittle pseudo maximum likelihood estimation fornonstationary time series Journal of the American Statistical Association 951229ndash1243
Vogelsang TJ 1997 Wald-type test for detecting breaks in the trend function of adynamic time series Econometric Theory 13 818ndash849
Wang C 2004 Futures trading activity and predictable foreign exchange movements Journal of Banking and Finance 28 1023ndash1041
Wei S-J Parsley DC 1995 Purchasing power disparity during the 1047298oat rate periodexchange rate volatility trade barriers and other culprits Working Paper 5032NBER
WestKD1987 A noteon thepowerof least squares testsfor a unitroot Economics Letters24 1397ndash1418
1192 JC Cuestas LA Gil-Alana Economic Modelling 26 (2009) 1184ndash1192
7232019 Economic Modelling Further Evidence on the PPP Analysis of the Australian Dollar- Non-linearities Fractional Integration and Structu
httpslidepdfcomreaderfulleconomic-modelling-further-evidence-on-the-ppp-analysis-of-the-australian-dollar- 99
Taylor MP Peel DA Sarno L 2001 Nonlinear mean-reversion in real eschange ratesTowards a solution to the purchasing power parity puzzles International EconomicReview 42 1015ndash1042
Velasco C 1999 Gaussian semiparametric estimation of nonstationary time series Journal of Time Series Analysis 20 87ndash127
Velasco C Robinson PM 2000 Whittle pseudo maximum likelihood estimation fornonstationary time series Journal of the American Statistical Association 951229ndash1243
Vogelsang TJ 1997 Wald-type test for detecting breaks in the trend function of adynamic time series Econometric Theory 13 818ndash849
Wang C 2004 Futures trading activity and predictable foreign exchange movements Journal of Banking and Finance 28 1023ndash1041
Wei S-J Parsley DC 1995 Purchasing power disparity during the 1047298oat rate periodexchange rate volatility trade barriers and other culprits Working Paper 5032NBER
WestKD1987 A noteon thepowerof least squares testsfor a unitroot Economics Letters24 1397ndash1418
1192 JC Cuestas LA Gil-Alana Economic Modelling 26 (2009) 1184ndash1192