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

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

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Further evidence on the PPP analysis of the Australian dollar Non-linearitiesfractional integration and structural changes

Juan C Cuestas a Luiacutes A Gil-Alana b

a Nottingham Trent University UK b University of Navarra Spain

a b s t r a c ta r t i c l e i n f o

Article history

Accepted 14 May 2009

JEL classi 1047297cation

C32

F15

Keywords

PPP

Real exchange rate

Unit roots

Non-linearities

The aim of this paper is to analyse the empirical ful1047297lment of the Purchasing Power Parity (PPP) theory for

the Australian dollar In order to do so we have applied recently developed unit root tests that account for

asymmetric adjustment towards the equilibrium [Kapetanios G Shin Y Snell A 2003 ldquoTesting for a unit

root in the nonlinear STAR frameworkrdquo Journal of Econometrics 112 359ndash379] and fractional integration in

the context of structural changes [Robinson PM 1994 ldquoEf 1047297cient tests of nonstationary hypothesesrdquo Journal

of the American Statistical Association 89 1420ndash1437 Gil-Alana LA 2008 ldquoFractional integration and

structural breaks at unknown periods of timerdquo Journal of Time Series Analysis 29 163ndash185] Although our

results point to the rejection of the PPP hypothesis we 1047297nd that the degree of persistence of shocks to the

Australian dollar decreases after the 1985 currency crisis

copy 2009 Elsevier BV All rights reserved

1 Introduction

There is no doubt that Purchasing Power Parity (PPP) analysis has

become one of the most controversial topics within international

economics Although the existing literature is vast empirical

contributions have yielded contradictory results given that they are

dependent on the countries period analysed and econometric

techniques employed Although PPP is a theory of exchange rate

determination its empiricalful1047297lment has several policy implications1047297rst many macroeconomic models assume a constant real exchange

rate in equilibrium which has consequences when the policy makers

base their decisions on this type of modelling second as Wei and

Parsley (1995) claim the PPP hypothesis can be considered as a

measure of the degree of economic integration between countries

since this theory assumes perfect mobility of goods and an absence of

trade barriers 1047297nally Faria and Leoacuten-Ledesma (2003 2005) claimthat a misaligned real exchange rate (RER) might affect growth and

unemployment thus a proper assessment of the degree of over-

valuation or undervaluation of the currencies is important to help

promote long term economic growthThe PPP theory establishes that the nominal exchange rate

between two currencies should be equal to the price index ratio

between the countries so that any change in the price relationship

will affect the nominal exchange rate to keep the purchasing power

between both countries unaffected that is

E t = P 4t

P t

eth1THORN

where P t is the foreign price index P t is the national price index and

E t is the nominal exchange rate between both currencies

This implies that the RER

Q t = P

t

E t P 4t

= 1 eth2THORN

Relations such as Eqs (1) and (2) are known as the absolute

version of the PPP hypothesis This implies that the same quantity of

goods can be purchased in both countries with the same amount of

money However this is not always true since differences in

productivity and transport costs may affect the relationship between

exchange rates and prices Then it is possible to de1047297ne a less

restrictive version of the PPP hypothesis known as the relative PPP In

this case the RER should be equal to a constant different from 1

meaning that exchange rates react proportionally to changes in the

price ratio instead of identically

Economic Modelling 26 (2009) 1184ndash1192

Juan Carlos Cuestas gratefully acknowledges 1047297nancial support from the CICYT

project ECO2008-05908-C02-01ECON Juan Carlos Cuestas is a member of the INTECO

research group Luis A Gil-Alana acknowledges1047297nancial support fromthe Ministerio de

Ciencia y Tecnologia (ECO2008-03035 ECON Y FINANZAS Spain) and the Secretariacutea de

Estado de Cooperacioacuten Internacional (Ayudas de movilidad a artistas investigadores y

cientiacute1047297cos 2008) The usual disclaimer applies

Corresponding authorNottingham Business School Division of EconomicsNottingham

Trent University Burton Street NG1 4BU Nottingham UK Tel +44 1158484328

E-mail address juancuestasntuacuk (JC Cuestas)

0264-9993$ ndash see front matter copy 2009 Elsevier BV All rights reserved

doi101016jeconmod200905006

Contents lists available at ScienceDirect

Economic Modelling

j o u r n a l h o m e p a g e w w w e l s ev i e r c o m l o c a t e e c m o d



Within the empirical literature on the PPP theory it is well known

that short run deviation may prevent the PPP hypothesis from holding

true However one may expect the RER to return to its equilibrium

value after a shock in the long run Statistically this implies that the

RER should be a mean reverting process for the PPP hypothesis to be

ful1047297lled which makes unit root testing to be an appealing econo-

metric approach for this purpose

Sarno and Taylor (2002) Taylor et al (2001) and Taylor (2006)

among others provide summaries of the main contributions to the

literature from which it is possible to highlight several facts 1047297rst

authors have applied different techniques since the 70s in order to

capture the true data generating process (DGP) of the real exchange

rates second it appears that the results have been quite

controversial and 1047297nally that the most recent contributions focus

on the consideration of non-linearities and fractional integration

alternatives In this vein Taylor et al (2001) identi1047297ed two main

paradoxes relating to RER behaviour First the relationship bet-

ween nominal exchange rates and prices implied by the PPP theory

is not in many cases a cointegrating one second how is it possible

to observe a high volatility of exchange rates in the short termwith the low speed of mean reversion generally observed in this

variable

The failure of the literature to provide good answers to the

aforementioned questions have been related to the low power of the

(unit-root)tests applied to analyse the empirical ful1047297lment of PPP1 It

appears that increasing the data sample creates additional problems

such as structural changes that may affect the power of the tests if

these are not taken into account (see Perron and Phillips1987 West

1987 among others) Therefore these kind of non-linearities in the

deterministic components may increase the probability of Type II

error with traditional unit root tests (eg DickeyndashFuller type)

Furthermore non-linearities in the RERlong runpath maybe present

in the form of an asymmetric speed of adjustment towards the

equilibrium ie the further the RER deviates from the long runequilibrium value the faster the speed of mean reversion is expected

to be From an econometric viewpoint that may imply an

autoregressive parameter dependent on the values of the variable

The non-linear behaviour of exchange rates has been acknowledged

by several authors such as Dumas (1994) Michael et al (1997)

Sarno et al (2004) Juvenal and Taylor (2008) and Cuestas (2009)

among many others

In addition as pointed out by several authors such as Diebold et al

(1991) Cheung and Lai (1993) and Gil-Alana (2000) among others

the real exchange rate may be characterised as a slow mean reversion

process and traditional unit root may not be able to distinguish this

type of process from a unit root Thus according to these authors

fractional integration may be an alternative viable route to the

analysis of the RER dynamics As mentioned above the question of

interest is to determine if deviations from PPP are transitory or

permanent which translated to the fractionally integrated literature

means to determine if the order of integration is smallerthan 1 (mean

reversion) or equal to or higher than 1 (no mean reversion) Applying

RS techniques to daily rates for the British pound French franc and

Deutsche mark Booth et al (1982) found evidence of fractional

integration during the 1047298exible exchange rate period (1973ndash1979)

Cheung (1993) also found similar evidence in foreign exchange

markets during the managed 1047298oating regime On the other hand

Baum et al (1999) estimated ARFIMA models for real exchange rates

in the post-Bretton Woods era and found almost no evidence to

support long run PPP Additional papers on exchange rate dynamics

using fractional integration are Fang et al (1994) Crato and Ray(2000) and Wang (2004)

In another contribution Henry and Olekalns (2002) analyse the

Australian RERlong run behaviour in the context of structural changes

and fractional integration These authors apply Robinsons (1994) and

Geweke and Porter-Hudaks (1983) fractional integration tests and the

Vogelsangs (1997) unit root test with structural changes but they are

not able to 1047297nd evidence of mean reversion in the RER Australian

dollar Similar results are obtained by Darneacute and Hoarau (2008) who

applied the Perron and Rodriacuteguezs (2003) unit root test with

structural changes to the Australian dollar Contrary to these results

Cuestas and Regis (2008) found that the Australian RER is stationary

around a non-linear deterministic trend by means of applying the

Bierens (1997) unit root tests which proposes a Chebyshev

polynomial approximation for the non-linear trend However theresults are dependent on the selection of the order of the polynomials

since there is no unique way of doing it Additionally in the context of

PPP it is dif 1047297cult to give an economic interpretation of the non-linear

trend

The aim of the present paper is to provide further evidence of the

PPP ful1047297lment in Australia and complement the previous literatureon

the PPP analysis by applying different techniques that address the

aforementioned facts ie fractional integration and non-linearities

that may affect the power of the traditional (linear) unit root tests

First we apply the upgraded versions of linear unit root tests

propossed by Ng and Perron (2001) second we account for

asymmetric speed of mean reversion and apply the Kapetanios et al

(KSS 2003) unit root test which generalises the alternative hypoth-

esis to a globally stationary exponential smooth transition (ESTAR)

1 Unit root tests most commonly employed in the literature (Dickey and Fuller 1979

Phillips and Perron 1988 Kwiatkowski et al 1992 etc) have very low power against

trend-stationarity (DeJong et al 1992) structural breaks (Perron 1989 Campbell and

Perron 1991) regime-switching ( Nelson et al 2001) or fractional integration

(Diebold and Rudebusch 1991 Hassler and Wolters 1995 Lee and Schmidt 1996)

Fig 1 Australian RER

1185 JC Cuestas LA Gil-Alana Economic Modelling 26 (2009) 1184ndash1192



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




the outer regime bearing in mind that the process is globally

stationary In other words

yt = β yt minus1 + yt minus1F θ yt minus1eth THORN + et eth3THORN

where ε t is iid(0 σ 2) and F (θ yt minus1) is the transition function which

is assumed to be exponential (ESTAR)

F θ yt minus1eth THORN = 1minus exp minusθ y2t minus 1

n o eth4THORN

with θ N0

In practice it is common to reparameterise Eq (3) as

Δ yt = α yt minus1 + γ yt minus1 1minus exp minusθ y2t minus 1

n o + et eth5THORN

in order to apply the test The null hypothesis H 0 θ =0 is tested

against the alternative H 1 θ N0 ie we test whether the variable is

an I (1) process in the outer regime From an economic viewpoint

and in the context of exchange rates this implies that the further

the RER deviates from the equilibrium the faster will be the speed

of mean reversion towards the fundamental equilibrium In addition

the existence of trade barriers may create a central threshold where

transactions are not pro1047297table and arbitrage does not clear the

market mdash unit root process in the inner regime whereas for large

deviations the pro1047297ts from arbitrage are greater than the cost and

the arbitrage mechanism brings the exchange to the inner regimeMoreover according to Taylor and Peel (2000) among others an

ESTAR function is appropriate to model exchange rates given that

this type of equation assumes that the effects of the shock on the

variable are symmetric in the sense that these effects do not depend

on the sign of the shock

On the other hand many test statistics have been developed in

recent years for fractional integration They can be parametric or

semiparametric and they can be speci1047297ed in the time domain or in the

frequency domain In this article we employ two parametric

approaches that allow us to incorporate structural breaks The 1047297rst

one is the well-known Robinson tests (1994) that we specify in a way

that permit us to include deterministicbroken trends In particular we

consider a model of form

yt = α 1I t b T beth THORN + α 2I t zT beth THORN + xt eth6THORN

1minusLeth THORNd xt = ut ut asympI 0eth THORN eth7THORN

where yt is the observed time series (RER) I ( x) is the indicator

function L is the lag operator (Lxt = xt minus1) d is a real value and ut isI (0) Based on this set-up for a given T b-value we test the null

hypothesis

H 0 d = do eth8THORN

in Eqs (6) and (7) for any real value do The functional form of the

test statistic is described in Appendix A and it was shown by

Robinson (1994) that under very mild regularity conditions a test

of Eq (8) against the one-sided alternatives dbdo (dNdo) follows

the standard N (0 1) distribution

The method presented just above imposes the same degree of

integration before and after the break-date Then we also perform a

recent procedure developed by Gil-Alana (2008) which allows us to

estimate the break-date along with the fractional differencing

parameters which might be different for each subsample This

method is based on minimising the residuals sum squares and is

brie1047298y described in Appendix B In the 1047297nal part of the article a

semiparametric method (Robinson 1995) is conducted on the two

subsamples

32 Empirical results

The results of applying the Ng and Perron (2001) and KSS tests are

reported in Table 1 The lag length has been selected using the

Modi1047297ed Akaike Information Criterion proposed by the former

authors for both tests In both cases the unit root hypothesis cannot

be rejected at conventional signi1047297cance levels Thus according to

these (unit-root) procedures there is no evidence of the PPP

hypothesis for the Australian case

Next we examine the possibility of fractional integration in the

context of non-linear deterministic terms For this purpose we

employ the model given by Eqs (6) and (7) testing H 0 (Eq (8)) for

do-values ranging from 0 to 2 with 001 increments using the

Robinsons (1994) parametric approach for 1047297

xed values of T b Wethen estimate d by choosing the value that produces the lowest

statistic in absolute value moving the break date T b 1-period

forward recursively from T b=40 (1980Q1) to T b=119 (1999Q4)

Fig 2 displays the estimated ds along with the 95 con1047297dence bands

for the two cases of white noise u t (in Fig 2(i)) and autocorrelated

disturbances (in Fig 2(ii)) In the latter case we assume that ut

follows the exponential spectral model of Bloom1047297eld (1973) Thisisa

non-parametric speci1047297cation for the error term that produces

autocorrelations decaying exponentially as in the autoregressive

(AR) case2

We observe in Fig 2 that the estimated values of d have remained

relatively stable across T b with values 1047298uctuating around 1 In fact

the unit root null hypothesis cannot be rejected for any T b We

observe that the only turbulences in the estimators take place when

T b is around 1985 Table 2 displays the estimates of d and the

intercepts for the case of a break at the four quarters in 1985 for the

two cases of white noise and autocorrelated disturbances We see in

this table that practically all the estimates are slightly above 1 The

only exception is the case of the break at 1985Q4 with autocorrelated

disturbances where the estimate of the fractional differencing

parameter is equal to 099 Nevertheless the unit root cannot be

rejected in any single case and thus we do not 1047297nd support for the

PPP hypothesis when using this model

2 See Gil-Alana (2004) for the analysis of fractional integration in the context of

Bloom1047297eld disturbances

Table 2

Estimates of intercepts and fractional differencing parameters with a break in 1985

Break dates White noise disturbances Autocorrelated disturbances

d α 1 α 2 d α 1 α 2

19 85 Q1 105 ( 095 117 ) 150 0 ( 149 7 1503 ) 140 7 ( 140 0 1413 ) 103 ( 085 129) 150 0 (1497 150 3) 140 4 (1398 1410)

19 85 Q2 104 ( 095 117 ) 150 0 ( 149 7 1503 ) 135 4 ( 134 8 1360 ) 103 ( 084 127) 150 0 (1497 150 3) 135 3 (1347 1359 )

19 85 Q3 110 ( 099 125 ) 1501 ( 149 8 150 4) 1556 ( 154 9 1562 ) 106 ( 088 132) 150 0 (1497 150 4) 1546 (1539 1552 )

19 85 Q4 106 ( 097 119) 150 0 ( 149 7 150 4) 143 5 ( 142 9 144 2) 099 ( 082 122) 149 9 (149 6 150 3) 1428 (1422 1435 )

Note In parenthesis the 95 con1047297dence intervals




Still on this set-up we might be interested in knowing if the

intercept has changed from one subsample to another Thus we can

consider a joint test of the null hypothesis

H 0 α 1 = α 2 and d = do eth9THORNin Eqs (6) and (7) against the alternative

H a α 1 ne α 2 or d ne do eth10THORN

This possibility is not addressed in Robinson (1994) though Gil-

Alana and Robinson (1997) derived a similar Lagrange Multiplier(LM)

test as follows they consider a regression model of form as

yt = β V z t + xt t = 1 2 N eth11THORNand xt given by Eq (7) with the vector partitions z t = ( z At

T z Bt T )T

β = ( β AT β B

T )T In general we want to test H 0 d = do and β B =

β B0 Then a LM statistic may be shown to be r 2 (see Appendix

A) plus

XT

t =1

ˆ ut wT Bt

XT

t =1

wBt wT Bt minus

XT

t =1

wBt wT At

XT

t =1

w At wT At

minus1XT

t =1

w At wT Bt

minus1XT

t =1

ˆ ut wBt

eth12THORN

with wt = (w At T wBt

T )T = (1minus L)do z t

ˆ ut = 1minusLeth THORNdo yt minus ˜ β

T A β

T B0 wt ˜ β A = X

T

t =1

w At wT

At minus1

XT

t =1

w At 1minusLeth THORNdo yt

ˆ σ 2 = T

minus1 PT

t =1

ˆ u2t and r

2 is calculated as in Appendix A but

using the u t just de1047297ned If the dimension of z Bt is qB then we

Fig 4 Sum of squared residuals using the procedure of Gil-Alana

Fig 3 Estimates of d recursively obtained with samples of size T =100






asymmetric speed of adjustment towards the equilibrium as well as

fractional integration and structural changes Our results are in line

with previous empirical works in the sense that there is no evidence

of mean reversionin theAustralian dollarRER forthe periodanalysed

However we 1047297nd that the order of integration decreases after 1985

coinciding with the currency crisis This conclusion highlights the fact

that the monetary authorities have managed to decrease the

dependence of the Australian dollar on real shocks that may affect

the value of the currency on a permanent basis

Appendix A

The LM test of Robinson (1994) for testing H 0 (Eq (8)) in Eqs (6)

and (7) is

ˆ r = T 1 =2

ˆ σ 2ˆ Aminus1=2

ˆ a

where T is the sample size and

ˆ a = minus2π

T

XT minus1

j =1

ψ λ j

g λ j ˆ τ

minus1I λ j

ˆ σ 2

= σ 2

ˆ τ = 2π

T XT minus1

j =1

g λ j

ˆ τ minus1I λ

j

ˆ A = 2

T ethXT minus1

j =1

ψ λ j

2minus

XT minus1

j =1

ψ λ j

ˆ e λ j

V

timesXT minus1

j = 1

ˆ e λ j

ˆ e λ j

V

0

1Aminus1

timesXT minus1

j =1

ˆ e λ j

ψ λ j

THORNψ λ j

= log j2sin

λ j

2 j ˆ e λ j

=

A

Aτ log g λ j ˆ τ

λ j =

2π j

T ˆ τ = argmin σ

2τ eth THORN

a and A in the above expressions are obtained through the 1047297rst and

second derivatives of the log-likelihood function with respect to d

(see Robinson 1994 page 1422 for further details) I (l j) is the

periodogram of ut evaluated under the null ie

ˆ ut = 1minusLeth THORNdo yt minus α Vwt

ˆ α = XT

t = 1

wt wt V minus1

XT

t =1

wt 1minusL

eth THORNdo yt wt = 1minusL

eth THORNdo z t

z t = 1I t b T beth THORN 1I t zT beth THORNeth THORN V

and g is a known function related to the spectral density function of ut

f λ σ 2

τ

= σ

2

2π g λ τ eth THORN minus π b λ V π

Appendix B

Gil-Alana (2008) considers the following model

yt = α 1 + xt 1minus

Leth THORNd1

xt = ut t = 1 N

T b

and

yt = α 2 + xt 1minusLeth THORNd2 xt = ut t = T b + 1 N T

where the α 1 and α 2 are the intercepts corresponding to the 1047297rst and

second subsamples respectively d1 and d2 may be real values u t is

I (0) and T b is the time of a break that is supposed to be unknown

This model can be re-parameterized as follows

1minusLeth THORNd1 yt = α 1˜ 1t d1eth THORN + β 1˜ t t d1eth THORN + ut t = 1 N T b

1minus

Leth THORNd2

yt = α 2˜

1t d2eth THORN + β 2˜

t t d2eth THORN + ut t = T b + 1 N

T

Fig 5 Estimates of d based on a semiparametric model for each subsample




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

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Bierens HJ 1997 Testing the unit root with drift hypothesis against nonlinear trendstationarity with an application to the US price level and interest rate Journal of Econometrics 81 29ndash64

Bloom1047297eld P 1973 An exponential model in the spectrum of a scalar time series

Biometrika 60 217ndash226Booth GGKaen FR Koveos PE1982RS analysis of foreign exchangemarketsunder

two international monetary regimes Journal of Monetary Economics 10 407ndash415CampbellJY Perron P1991 Pitfalls and opportunities whatmacroeconomists should

know about unit roots NBER Macroeconomics Annual 1141ndash1201Casin P Ceacutespedes LF Sahay R 2004 Commodity currencies and the real exchange

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Economic Statistics 11 93ndash101Cheung Y-W Lai KS 1993 A fractional cointegration analysis of purchasing power

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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




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




Within the empirical literature on the PPP theory it is well known

that short run deviation may prevent the PPP hypothesis from holding

true However one may expect the RER to return to its equilibrium

value after a shock in the long run Statistically this implies that the

RER should be a mean reverting process for the PPP hypothesis to be

ful1047297lled which makes unit root testing to be an appealing econo-

metric approach for this purpose

Sarno and Taylor (2002) Taylor et al (2001) and Taylor (2006)

among others provide summaries of the main contributions to the

literature from which it is possible to highlight several facts 1047297rst

authors have applied different techniques since the 70s in order to

capture the true data generating process (DGP) of the real exchange

rates second it appears that the results have been quite

controversial and 1047297nally that the most recent contributions focus

on the consideration of non-linearities and fractional integration

alternatives In this vein Taylor et al (2001) identi1047297ed two main

paradoxes relating to RER behaviour First the relationship bet-

ween nominal exchange rates and prices implied by the PPP theory

is not in many cases a cointegrating one second how is it possible

to observe a high volatility of exchange rates in the short termwith the low speed of mean reversion generally observed in this

variable

The failure of the literature to provide good answers to the

aforementioned questions have been related to the low power of the

(unit-root)tests applied to analyse the empirical ful1047297lment of PPP1 It

appears that increasing the data sample creates additional problems

such as structural changes that may affect the power of the tests if

these are not taken into account (see Perron and Phillips1987 West

1987 among others) Therefore these kind of non-linearities in the

deterministic components may increase the probability of Type II

error with traditional unit root tests (eg DickeyndashFuller type)

Furthermore non-linearities in the RERlong runpath maybe present

in the form of an asymmetric speed of adjustment towards the

equilibrium ie the further the RER deviates from the long runequilibrium value the faster the speed of mean reversion is expected

to be From an econometric viewpoint that may imply an

autoregressive parameter dependent on the values of the variable

The non-linear behaviour of exchange rates has been acknowledged

by several authors such as Dumas (1994) Michael et al (1997)

Sarno et al (2004) Juvenal and Taylor (2008) and Cuestas (2009)

among many others

In addition as pointed out by several authors such as Diebold et al

(1991) Cheung and Lai (1993) and Gil-Alana (2000) among others

the real exchange rate may be characterised as a slow mean reversion

process and traditional unit root may not be able to distinguish this

type of process from a unit root Thus according to these authors

fractional integration may be an alternative viable route to the

analysis of the RER dynamics As mentioned above the question of

interest is to determine if deviations from PPP are transitory or

permanent which translated to the fractionally integrated literature

means to determine if the order of integration is smallerthan 1 (mean

reversion) or equal to or higher than 1 (no mean reversion) Applying

RS techniques to daily rates for the British pound French franc and

Deutsche mark Booth et al (1982) found evidence of fractional

integration during the 1047298exible exchange rate period (1973ndash1979)

Cheung (1993) also found similar evidence in foreign exchange

markets during the managed 1047298oating regime On the other hand

Baum et al (1999) estimated ARFIMA models for real exchange rates

in the post-Bretton Woods era and found almost no evidence to

support long run PPP Additional papers on exchange rate dynamics

using fractional integration are Fang et al (1994) Crato and Ray(2000) and Wang (2004)

In another contribution Henry and Olekalns (2002) analyse the

Australian RERlong run behaviour in the context of structural changes

and fractional integration These authors apply Robinsons (1994) and

Geweke and Porter-Hudaks (1983) fractional integration tests and the

Vogelsangs (1997) unit root test with structural changes but they are

not able to 1047297nd evidence of mean reversion in the RER Australian

dollar Similar results are obtained by Darneacute and Hoarau (2008) who

applied the Perron and Rodriacuteguezs (2003) unit root test with

structural changes to the Australian dollar Contrary to these results

Cuestas and Regis (2008) found that the Australian RER is stationary

around a non-linear deterministic trend by means of applying the

Bierens (1997) unit root tests which proposes a Chebyshev

polynomial approximation for the non-linear trend However theresults are dependent on the selection of the order of the polynomials

since there is no unique way of doing it Additionally in the context of

PPP it is dif 1047297cult to give an economic interpretation of the non-linear

trend

The aim of the present paper is to provide further evidence of the

PPP ful1047297lment in Australia and complement the previous literatureon

the PPP analysis by applying different techniques that address the

aforementioned facts ie fractional integration and non-linearities

that may affect the power of the traditional (linear) unit root tests

First we apply the upgraded versions of linear unit root tests

propossed by Ng and Perron (2001) second we account for

asymmetric speed of mean reversion and apply the Kapetanios et al

(KSS 2003) unit root test which generalises the alternative hypoth-

esis to a globally stationary exponential smooth transition (ESTAR)

1 Unit root tests most commonly employed in the literature (Dickey and Fuller 1979

Phillips and Perron 1988 Kwiatkowski et al 1992 etc) have very low power against

trend-stationarity (DeJong et al 1992) structural breaks (Perron 1989 Campbell and

Perron 1991) regime-switching ( Nelson et al 2001) or fractional integration

(Diebold and Rudebusch 1991 Hassler and Wolters 1995 Lee and Schmidt 1996)

Fig 1 Australian RER












2 The data












2002)


31 Methodology



the PPP theory


























Table 1





MSB 047237 023300 027500

MP t 109753 317000 445000















n o eth4THORN

with θ N0



n o + et eth5THORN




























hypothesis















subsamples























(AR) case2
















Table 2



d α 1 α 2 d α 1 α 2

19 85 Q1 105 ( 095 117 ) 150 0 ( 149 7 1503 ) 140 7 ( 140 0 1413 ) 103 ( 085 129) 150 0 (1497 150 3) 140 4 (1398 1410)

19 85 Q2 104 ( 095 117 ) 150 0 ( 149 7 1503 ) 135 4 ( 134 8 1360 ) 103 ( 084 127) 150 0 (1497 150 3) 135 3 (1347 1359 )

19 85 Q3 110 ( 099 125 ) 1501 ( 149 8 150 4) 1556 ( 154 9 1562 ) 106 ( 088 132) 150 0 (1497 150 4) 1546 (1539 1552 )

19 85 Q4 106 ( 097 119) 150 0 ( 149 7 150 4) 143 5 ( 142 9 144 2) 099 ( 082 122) 149 9 (149 6 150 3) 1428 (1422 1435 )














T z Bt T )T

β = ( β AT β B



A) plus

XT

t =1

ˆ ut wT Bt

XT

t =1

wBt wT Bt minus

XT

t =1

wBt wT At

XT

t =1

w At wT At

minus1XT

t =1

w At wT Bt

minus1XT

t =1

ˆ ut wBt

eth12THORN




T A β

T B0 wt ˜ β A = X

T

t =1

w At wT

At minus1

XT

t =1


ˆ σ 2 = T

minus1 PT

t =1

ˆ u2t and r



















Appendix A


and (7) is

ˆ r = T 1 =2

ˆ σ 2ˆ Aminus1=2

ˆ a


ˆ a = minus2π

T

XT minus1

j =1

ψ λ j

g λ j ˆ τ

minus1I λ j

ˆ σ 2

= σ 2

ˆ τ = 2π

T XT minus1

j =1

g λ j

ˆ τ minus1I λ

j

ˆ A = 2

T ethXT minus1

j =1

ψ λ j

2minus

XT minus1

j =1

ψ λ j

ˆ e λ j

V

timesXT minus1

j = 1

ˆ e λ j

ˆ e λ j

V

0

1Aminus1

timesXT minus1

j =1

ˆ e λ j

ψ λ j

THORNψ λ j

= log j2sin

λ j

2 j ˆ e λ j

=

A


λ j =

2π j

T ˆ τ = argmin σ

2τ eth THORN






ˆ α = XT

t = 1

wt wt V minus1

XT

t =1

wt 1minusL


eth THORNdo z t



f λ σ 2

τ

= σ

2


Appendix B



Leth THORNd1

xt = ut t = 1 N

T b

and







1minus

Leth THORNd2

yt = α 2˜



T






of squares (RSS)


t =1


1eth THORN1o

+ β 1˜ t

t d

1eth THORN1o

2

+XT

t = T b


1eth THORN2o

+ β 2 ˜ t t d

1eth THORN2o

2













d i= d

i(T k) for i =12

Appendix C


de1047297ned by


m

Xm

j = 1

log λ j

0

1A


m

Xm

j =1

I λ j

λ

2d j λ j =

2π j

T

m

T Y0









References














































































2 The data












2002)


31 Methodology



the PPP theory


























Table 1





MSB 047237 023300 027500

MP t 109753 317000 445000















n o eth4THORN

with θ N0



n o + et eth5THORN




























hypothesis















subsamples























(AR) case2
















Table 2



d α 1 α 2 d α 1 α 2

19 85 Q1 105 ( 095 117 ) 150 0 ( 149 7 1503 ) 140 7 ( 140 0 1413 ) 103 ( 085 129) 150 0 (1497 150 3) 140 4 (1398 1410)

19 85 Q2 104 ( 095 117 ) 150 0 ( 149 7 1503 ) 135 4 ( 134 8 1360 ) 103 ( 084 127) 150 0 (1497 150 3) 135 3 (1347 1359 )

19 85 Q3 110 ( 099 125 ) 1501 ( 149 8 150 4) 1556 ( 154 9 1562 ) 106 ( 088 132) 150 0 (1497 150 4) 1546 (1539 1552 )

19 85 Q4 106 ( 097 119) 150 0 ( 149 7 150 4) 143 5 ( 142 9 144 2) 099 ( 082 122) 149 9 (149 6 150 3) 1428 (1422 1435 )














T z Bt T )T

β = ( β AT β B



A) plus

XT

t =1

ˆ ut wT Bt

XT

t =1

wBt wT Bt minus

XT

t =1

wBt wT At

XT

t =1

w At wT At

minus1XT

t =1

w At wT Bt

minus1XT

t =1

ˆ ut wBt

eth12THORN




T A β

T B0 wt ˜ β A = X

T

t =1

w At wT

At minus1

XT

t =1


ˆ σ 2 = T

minus1 PT

t =1

ˆ u2t and r



















Appendix A


and (7) is

ˆ r = T 1 =2

ˆ σ 2ˆ Aminus1=2

ˆ a


ˆ a = minus2π

T

XT minus1

j =1

ψ λ j

g λ j ˆ τ

minus1I λ j

ˆ σ 2

= σ 2

ˆ τ = 2π

T XT minus1

j =1

g λ j

ˆ τ minus1I λ

j

ˆ A = 2

T ethXT minus1

j =1

ψ λ j

2minus

XT minus1

j =1

ψ λ j

ˆ e λ j

V

timesXT minus1

j = 1

ˆ e λ j

ˆ e λ j

V

0

1Aminus1

timesXT minus1

j =1

ˆ e λ j

ψ λ j

THORNψ λ j

= log j2sin

λ j

2 j ˆ e λ j

=

A


λ j =

2π j

T ˆ τ = argmin σ

2τ eth THORN






ˆ α = XT

t = 1

wt wt V minus1

XT

t =1

wt 1minusL


eth THORNdo z t



f λ σ 2

τ

= σ

2


Appendix B



Leth THORNd1

xt = ut t = 1 N

T b

and







1minus

Leth THORNd2

yt = α 2˜



T






of squares (RSS)


t =1


1eth THORN1o

+ β 1˜ t

t d

1eth THORN1o

2

+XT

t = T b


1eth THORN2o

+ β 2 ˜ t t d

1eth THORN2o

2













d i= d

i(T k) for i =12

Appendix C


de1047297ned by


m

Xm

j = 1

log λ j

0

1A


m

Xm

j =1

I λ j

λ

2d j λ j =

2π j

T

m

T Y0









References












































































n o eth4THORN

with θ N0



n o + et eth5THORN




























hypothesis















subsamples























(AR) case2
















Table 2



d α 1 α 2 d α 1 α 2

19 85 Q1 105 ( 095 117 ) 150 0 ( 149 7 1503 ) 140 7 ( 140 0 1413 ) 103 ( 085 129) 150 0 (1497 150 3) 140 4 (1398 1410)

19 85 Q2 104 ( 095 117 ) 150 0 ( 149 7 1503 ) 135 4 ( 134 8 1360 ) 103 ( 084 127) 150 0 (1497 150 3) 135 3 (1347 1359 )

19 85 Q3 110 ( 099 125 ) 1501 ( 149 8 150 4) 1556 ( 154 9 1562 ) 106 ( 088 132) 150 0 (1497 150 4) 1546 (1539 1552 )

19 85 Q4 106 ( 097 119) 150 0 ( 149 7 150 4) 143 5 ( 142 9 144 2) 099 ( 082 122) 149 9 (149 6 150 3) 1428 (1422 1435 )














T z Bt T )T

β = ( β AT β B



A) plus

XT

t =1

ˆ ut wT Bt

XT

t =1

wBt wT Bt minus

XT

t =1

wBt wT At

XT

t =1

w At wT At

minus1XT

t =1

w At wT Bt

minus1XT

t =1

ˆ ut wBt

eth12THORN




T A β

T B0 wt ˜ β A = X

T

t =1

w At wT

At minus1

XT

t =1


ˆ σ 2 = T

minus1 PT

t =1

ˆ u2t and r



















Appendix A


and (7) is

ˆ r = T 1 =2

ˆ σ 2ˆ Aminus1=2

ˆ a


ˆ a = minus2π

T

XT minus1

j =1

ψ λ j

g λ j ˆ τ

minus1I λ j

ˆ σ 2

= σ 2

ˆ τ = 2π

T XT minus1

j =1

g λ j

ˆ τ minus1I λ

j

ˆ A = 2

T ethXT minus1

j =1

ψ λ j

2minus

XT minus1

j =1

ψ λ j

ˆ e λ j

V

timesXT minus1

j = 1

ˆ e λ j

ˆ e λ j

V

0

1Aminus1

timesXT minus1

j =1

ˆ e λ j

ψ λ j

THORNψ λ j

= log j2sin

λ j

2 j ˆ e λ j

=

A


λ j =

2π j

T ˆ τ = argmin σ

2τ eth THORN






ˆ α = XT

t = 1

wt wt V minus1

XT

t =1

wt 1minusL


eth THORNdo z t



f λ σ 2

τ

= σ

2


Appendix B



Leth THORNd1

xt = ut t = 1 N

T b

and







1minus

Leth THORNd2

yt = α 2˜



T






of squares (RSS)


t =1


1eth THORN1o

+ β 1˜ t

t d

1eth THORN1o

2

+XT

t = T b


1eth THORN2o

+ β 2 ˜ t t d

1eth THORN2o

2













d i= d

i(T k) for i =12

Appendix C


de1047297ned by


m

Xm

j = 1

log λ j

0

1A


m

Xm

j =1

I λ j

λ

2d j λ j =

2π j

T

m

T Y0









References















































































T z Bt T )T

β = ( β AT β B



A) plus

XT

t =1

ˆ ut wT Bt

XT

t =1

wBt wT Bt minus

XT

t =1

wBt wT At

XT

t =1

w At wT At

minus1XT

t =1

w At wT Bt

minus1XT

t =1

ˆ ut wBt

eth12THORN




T A β

T B0 wt ˜ β A = X

T

t =1

w At wT

At minus1

XT

t =1


ˆ σ 2 = T

minus1 PT

t =1

ˆ u2t and r



















Appendix A


and (7) is

ˆ r = T 1 =2

ˆ σ 2ˆ Aminus1=2

ˆ a


ˆ a = minus2π

T

XT minus1

j =1

ψ λ j

g λ j ˆ τ

minus1I λ j

ˆ σ 2

= σ 2

ˆ τ = 2π

T XT minus1

j =1

g λ j

ˆ τ minus1I λ

j

ˆ A = 2

T ethXT minus1

j =1

ψ λ j

2minus

XT minus1

j =1

ψ λ j

ˆ e λ j

V

timesXT minus1

j = 1

ˆ e λ j

ˆ e λ j

V

0

1Aminus1

timesXT minus1

j =1

ˆ e λ j

ψ λ j

THORNψ λ j

= log j2sin

λ j

2 j ˆ e λ j

=

A


λ j =

2π j

T ˆ τ = argmin σ

2τ eth THORN






ˆ α = XT

t = 1

wt wt V minus1

XT

t =1

wt 1minusL


eth THORNdo z t



f λ σ 2

τ

= σ

2


Appendix B



Leth THORNd1

xt = ut t = 1 N

T b

and







1minus

Leth THORNd2

yt = α 2˜



T






of squares (RSS)


t =1


1eth THORN1o

+ β 1˜ t

t d

1eth THORN1o

2

+XT

t = T b


1eth THORN2o

+ β 2 ˜ t t d

1eth THORN2o

2













d i= d

i(T k) for i =12

Appendix C


de1047297ned by


m

Xm

j = 1

log λ j

0

1A


m

Xm

j =1

I λ j

λ

2d j λ j =

2π j

T

m

T Y0









References

















































































Appendix A


and (7) is

ˆ r = T 1 =2

ˆ σ 2ˆ Aminus1=2

ˆ a


ˆ a = minus2π

T

XT minus1

j =1

ψ λ j

g λ j ˆ τ

minus1I λ j

ˆ σ 2

= σ 2

ˆ τ = 2π

T XT minus1

j =1

g λ j

ˆ τ minus1I λ

j

ˆ A = 2

T ethXT minus1

j =1

ψ λ j

2minus

XT minus1

j =1

ψ λ j

ˆ e λ j

V

timesXT minus1

j = 1

ˆ e λ j

ˆ e λ j

V

0

1Aminus1

timesXT minus1

j =1

ˆ e λ j

ψ λ j

THORNψ λ j

= log j2sin

λ j

2 j ˆ e λ j

=

A


λ j =

2π j

T ˆ τ = argmin σ

2τ eth THORN






ˆ α = XT

t = 1

wt wt V minus1

XT

t =1

wt 1minusL


eth THORNdo z t



f λ σ 2

τ

= σ

2


Appendix B



Leth THORNd1

xt = ut t = 1 N

T b

and







1minus

Leth THORNd2

yt = α 2˜



T






of squares (RSS)


t =1


1eth THORN1o

+ β 1˜ t

t d

1eth THORN1o

2

+XT

t = T b


1eth THORN2o

+ β 2 ˜ t t d

1eth THORN2o

2













d i= d

i(T k) for i =12

Appendix C


de1047297ned by


m

Xm

j = 1

log λ j

0

1A


m

Xm

j =1

I λ j

λ

2d j λ j =

2π j

T

m

T Y0









References















































































Appendix A


and (7) is

ˆ r = T 1 =2

ˆ σ 2ˆ Aminus1=2

ˆ a


ˆ a = minus2π

T

XT minus1

j =1

ψ λ j

g λ j ˆ τ

minus1I λ j

ˆ σ 2

= σ 2

ˆ τ = 2π

T XT minus1

j =1

g λ j

ˆ τ minus1I λ

j

ˆ A = 2

T ethXT minus1

j =1

ψ λ j

2minus

XT minus1

j =1

ψ λ j

ˆ e λ j

V

timesXT minus1

j = 1

ˆ e λ j

ˆ e λ j

V

0

1Aminus1

timesXT minus1

j =1

ˆ e λ j

ψ λ j

THORNψ λ j

= log j2sin

λ j

2 j ˆ e λ j

=

A


λ j =

2π j

T ˆ τ = argmin σ

2τ eth THORN






ˆ α = XT

t = 1

wt wt V minus1

XT

t =1

wt 1minusL


eth THORNdo z t



f λ σ 2

τ

= σ

2


Appendix B



Leth THORNd1

xt = ut t = 1 N

T b

and







1minus

Leth THORNd2

yt = α 2˜



T






of squares (RSS)


t =1


1eth THORN1o

+ β 1˜ t

t d

1eth THORN1o

2

+XT

t = T b


1eth THORN2o

+ β 2 ˜ t t d

1eth THORN2o

2













d i= d

i(T k) for i =12

Appendix C


de1047297ned by


m

Xm

j = 1

log λ j

0

1A


m

Xm

j =1

I λ j

λ

2d j λ j =

2π j

T

m

T Y0









References







































































of squares (RSS)


t =1


1eth THORN1o

+ β 1˜ t

t d

1eth THORN1o

2

+XT

t = T b


1eth THORN2o

+ β 2 ˜ t t d

1eth THORN2o

2













d i= d

i(T k) for i =12

Appendix C


de1047297ned by


m

Xm

j = 1

log λ j

0

1A


m

Xm

j =1

I λ j

λ

2d j λ j =

2π j

T

m

T Y0









References














































































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