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7232019 Economic Modelling Further Evidence on the PPP Analysis of the Australian Dollar- Non-linearities Fractional Integration and Structu
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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