NONLINEAR ADJUSMENT OF THE PURCHASING POWER PARITY IN INDONESIA. Abstract

Transcription

1 NONLINEAR ADJUSMEN OF HE PURCHASING POWER PARIY IN INDONESIA ii Kani Lesari, Jae Kim and Param Silvapulle Deparmen of Economerics and Business Saisics Monash Universiy Vic. 345, Ausralia Absrac his paper models he dynamics of adjusmen process o Indonesian long run purchasing power pariy (PPP) relaive o US, Japan and Singapore by employing a non-linear framework, which is recenly shown o be appropriae in he presence of ransacion coss associaed wih inernaional rade. Using monhly observaions from January 979 o June 3 (pos-breon Woods period), covering he managed- and free-floaing regimes in Indonesia, he real exchange raes were esed for heir mean-revering properies. A large number of sudies found he real exchange series o be mean-avering and persisen, creaing PPP puzzles. Using he linear framework many aemped o resolve hese puzzles unsuccessfully. Moivaed by he success of recen sudies on PPP, applying a non-linear ESAR o model he adjusmen process, we esed for mean-revering properies of all hree real exchange raes for small and large deviaions from he long-run equilibrium. We find ha he small deviaions are non-saionary, persisen and hey can even be explosive, while he large deviaions are saionary wih adjusmen process being very fas, making he overall adjusmen process mean-revering. Keywords: Purchasing Power Pariy, ESAR model, Mean-reversion JEL classificaion: F3, F3, C5, C

2 . Inroducion I is well known ha inernaional linkages for foreign exchange, goods and capial markes play a key role in he process ha deermines he exchange rae. he naure of he equilibrium implied by hese linkages and he speed a which his equilibrium is aained have imporan implicaions for he abiliy of governmens o pursue independen domesic moneary policies. I is widely acceped ha he more inegraed he inernaional markes around he world he greaer is he prospecive difficuly in pursuing independen domesic moneary policies. An imperaive quesion we aemp o answer in his paper is how o es for marke inegraion. Despie he availabiliy of a number of inernaional pariy condiions ha can be used in such esing, his paper deals only wih he purchasing power pariy (PPP) condiion, which measures he exen of inegraion beween goods and foreign exchange markes across counries. his heory posulaes ha he nominal exchange rae beween wo naional currencies adjuss o offse he excess of domesic inflaion over foreign inflaion, keeping he real exchange rae unchanged. his convenional PPP heory is unlikely o be valid if uncerainy is allowed and an explici role for expecaion is inroduced. he main objecive of his paper is o examine he validiy of he PPP hypohesis in he long run in Indonesia by focusing on he cross-currencies: Indonesian Rupiah-US Dollar, Indonesian Rupiah-Japanese Yen and Indonesian Rupiah-Singapore Dollar using a nonlinear framework. he Unied Saes and Japan were chosen since boh are Indonesia s major rading parners, while Singapore is chosen in order o examine wheher regional inegraion helps o achieve he long-run PPP relaionship. Since Indonesia has adoped rade proecion policies and limied he openness of is domesic markes, he ransacion cos of rades is expeced o be large. Consequenly, he Indonesian marke may no be fully inegraed wih he res of he world. According o an emerging lieraure on nonlinear coinegraion, i may be possible o model he adjusmen process o long-run PPP for Indonesia using he mean-revering exponenial smooh ransiion auoregression (ESAR) nonlinear model, he deails of which will be discussed in he nex secion. Furher, he nonlinear framework adoped in his paper is expeced o resolve wo purchasing power pariy puzzles observed by a large number of sudies in he lieraure. he firs is he non-saionariy propery of he real exchange rae (Rogoff, 996 and aylor e. al, ) and he second PPP puzzle is he high degree of persisence in he real exchange rae (Rogoff, 996). esing for he validiy of various forms of PPP is much sudied in he empirical lieraure on inernaional finance. hese include an absolue form - he mos resriced - and wo

3 unresriced forms, wih one being parly resriced and he oher fully unresriced. Many sudies used convenional uni roo procedures for esing he validiy of PPP and failed o find evidence supporing i. Subsequenly, several sudies aemped o es his hypohesis using panel coinegraion and fracional inegraion mehodologies and found evidence in favour of he PPP heory. See Frankel and Rose (996) and aylor and Sarno (998) for he former and Diebold e al. (99) and Cheung and Lai (993) for he laer. Moivaed by he work of Enders and Grangers (998), Lesari, Kim and Silvapulle (3) examined he validiy of he Purchasing Power Pariy (PPP) using non-linear hreshold auoregressive (AR) models. he resuls suppored he fully unresriced and parially unresriced forms of PPP for Indonesia-U.S and Indonesia-Singapore exchange raes, while no evidence was found in supporing he resriced form. he assumpion under he resriced form is ha he nominal exchange rae fully adjuss for excess domesic inflaion over foreign inflaion. According o he heory, his adjused series - defined as he real exchange raes - is expeced o be saionary. Conrary o he expecaion, all hree real exchange raes were found o be non-saionary even in he nonlinear AR framework. However, i has been recenly argued ha he lack of empirical evidence supporing Purchasing Power Pariy is due o facors such as ransacion coss, axaion, subsidies, acual or hreaened rade resricions, he exisence of nonraded goods, imperfec compeiion, foreign exchange marke inervenions, and he differenial composiion of marke baskes and hence he price indices across counries. Subsequenly, an alernaive framework for he empirical analysis of he PPP ha allows for fricions in commodiy rades has emerged. Dumas (99) and Sercu, Uppal, and Van Hulle (995) pu forward heoreical argumens and developed equilibrium models of exchange rae deerminaion in he presence of ransacion coss and showed ha he adjusmen of real exchange raes owards he PPP is a non-linear mean-revering process. A few empirical sudies have suppored he long-run Purchasing Power Pariy (aylor, 988 and Lohian and aylor, 996) when he sample period covered a long ime span including he pre- and he pos-breon Woods era. he resuls were somewha mixed when he recen floaing period was examined. Using he sandard uni roo ess, Corbae and Ouliaris (988) canno rejec he presence of a uni roo in he real exchange rae in he managed-floa regime, providing he evidence agains he PPP hypohesis in he long run. In conras, Hakkio (984) and Papell (997) have found srong suppor for PPP hypohesis using panel daa. However, in a simulaion sudy, aylor, Peel and Sarno () found ha he panel uni roo null hypohesis is over-rejeced, casing doubs on he panel coinegraion resuls, while he convenional uni roo es was found o have low power in small samples. he overall findings of hese sudies

4 promped researchers o come up wih an alernaive nonlinear framework for he empirical analysis of he PPP ha allows for marke fricions in he commodiy rade and also o resolve he puzzles. Some of hese sudies are briefly discussed below. Michael, Nobay and Peel (997) invesigaed nonlineariies in he long-run Purchasing Power Pariy relaionship for he US, UK, France, Germany and Japan. hey employed he exponenial smooh ransiion auoregression (ESAR) o model he adjusmen process o longrun PPP and es for he mean-revering propery of real exchange raes. hen, hey applied impulse response analysis o examine he dynamic adjusmens of he long-run PPP. he resuls showed ha four major real bilaeral dollar exchange raes could be characerised by nonlinear mean-revering behaviour during he inerwar period. Chen and Wu (), on he oher hand, re-examined he long run PPP for US-Japan and US-aiwan using he nonlinear framework. Japan and aiwan were chosen because hey have insiued a coninuing policy of financial marke liberalisaion and experienced rapid growh, which has lead o increasingly srong ies o he US. he empirical analysis is based on monhly daa of spo exchange raes and consumer price indices for he US, Japan, and aiwan. he sample period spans January 974 o December 997 for Japan, and January 98 o December 997 for aiwan. hey employed he ESAR and found ha he parameer esimaes of he ESAR model revealed aypical behaviour of adjusmen process for PPP deviaions, his being random walk behaviour for small deviaions and fas adjusmen (mean-revering) for large deviaions from he PPP, which will be invesigaed in our sudy. Baum, Barkoulas and Caglayan () also sudied he nonlinear adjusmen of he deviaions from he long-run PPP during he pos-breon Woods period. hey sudied 7 counries of US rading parners and found he evidence of a mean revering dynamic process for sizable deviaions from PPP in several counries. Using generalised impulse response funcions hey also found evidence supporing nonlinear dynamic srucure, bu convergence o long-run PPP in he pos-breon Woods era was found o be very slow. here is a parallel sudy by aylor, Peel and Sarno () on esing nonlinear mean revering real exchange raes over he pos-breon Woods period for he UK, Germany, France, and Japan. hey esed he univariae model of he PPP, in he nonlinear framework and argued ha he ESAR model is more appropriae for modelling he real exchange rae movemens since i capures he symmeric behaviour of is deviaions well. heir resuls showed ha hese counries bilaeral real exchange raes were characerised by a nonlinear mean revering process during he floaing rae period since

5 his paper is organised as follows: Secion briefly oulines he Smooh ransiion Auoregressive (SAR) model, paricularly ESAR model. Secion 3 presens various hypoheses and he ess relaed o SAR framework. Secion 4 discusses he esimaion of ESAR model and he difficulies arising from i. Secion 5 oulines he ess for various diagnosic checks. Secion 6 discusses he condiions for he nonlinear mean-revering adjusmen owards he long run PPP. Secion 7 describes he daa series used in his sudy and defines he variables and models o be used in subsequen empirical analysis. Secion 8 repors and analyses he empirical resuls. Some concluding commens are made in secion 9.. Smooh ransiion Auoregressive (SAR) Models Granger and eräsvira (993) argued ha he nonlinear adjusmen process can be characerised my smooh ransiion auoregressive (SAR) models, he reasons for his are given below. he SAR model of order p, for a ime series + ϕx + ( θ + θx ) G( s ; γ,, has he following specificaion y y = ϕ c) + u () where x ( y y,..., y ), ϕ ( ϕ ϕ,..., ϕ ), θ ( θ θ,..., θ ) are unknown =, p =, p =, p parameer vecors, G(.) is a ransiion funcion which is coninuous and bounded by zero and one, s is he ransiion variable and c is he hreshold parameer. I is assumed ha u ~ n. i. d(, σ ). he s may be a single sochasic variable, for example, an elemen of x, or a linear combinaion of sochasic variables or a deerminisic variable such as a linear ime rend. In he SAR model, he ransiion variable s is generally assumed o be he lagged endogenous variable, ha is, s = for a cerain ineger d > (eräsvira, 994). his model can be y d exended by allowing exogenous variables z as addiional regressors, and indeed one of hem can be he ransiion variable. In his case, he model is called he smooh ransiion regression (SR) model (see eräsvira, 998 for deails). he SAR model can be inerpreed as a regime swiching model wih wo regimes, associaed wih he wo exreme values of he ransiion funcion, which are G( s ; γ, c) = and G( s ; γ, c) =, wih he ransiion from one regime o he oher being gradual. he regime ha occurs a ime variable s and he associaed value of G( s ; γ, c). can be deermined by he observable In he SAR model, however, he adjusmen owards he equilibrium akes place a every poin in ime, bu he speed varies wih he size of he deviaions from he long run PPP. In 4

6 conras wih non-linear AR model developed by ong (99) in ha he regime changes occur abruply, as argued before, while in he SAR hey occur gradually. Michael, Nobay and Peel (997) saed ha SAR model is more aracive han he AR model in describing he nonlinear adjusmen process for he following reasons: firs, he adjusmen process is generally expeced o be smooh raher han being discree. Second, even if economic agens make only dichoomous decisions, i is highly likely ha hese decisions are made a differen poins in ime. herefore, in he aggregaed processes, he change in regime expeced o be coninuous and smooh raher han discree and abrup. Finally, he modelling and he saisical inference procedures are more fully developed for SAR models han for AR models. he fac ha he AR model arises due o disconinuiy a each of hreshold-parameer values complicaes esing for lineariy null hypohesis agains nonlinear AR alernaives. here are wo alernaive forms for he ransiion funcion G(.). he firs is he logisic funcion, which can be wrien as { + exp[ γ ( s )]} G( s ; γ, c) = c, γ > () where γ measures he smoohness of ransiion from one regime o anoher and c is he hreshold value for s, which indicaes he halfway poin beween wo regimes. Equaion () combined wih Equaion () yields he Logisic SAR (LSAR) model, in which here are wo regimes, hese being he appreciaing and depreciaing currencies in he foreign exchange marke. hey have differen dynamics wih he speed of adjusmens varying wih he exen of he deviaion from he equilibrium. he ransiion funcion of LSAR is of S-shape around c and monoonically increasing in s, yielding an asymmeric adjusmen process owards he equilibrium, depending on wheher or no hese deviaions are above or below he equilibrium. he second is he exponenial funcion, which can be wrien as [ ( s ) ] G( s ; γ, c) = exp γ c, γ > (3) where, γ, as in LSAR, measures he speed of ransiion from one regime o anoher and c represens he locaion for he hreshold values for y. Equaion () combined wih Equaion (3) yields he Exponenial SAR (ESAR) model. he ransiion funcion of ESAR is symmeric and of U-shape around c. he ESAR model suggess ha he ime series in he upper and lower regimes have raher similar dynamics. he ESAR funcion in (3) defines a ransiion funcion abou c where G (.) is sill bounded beween and. As G(.) approaches eiher or he equaion (3) reduces o a linear model. 5

7 he LSAR and ESAR models describe differen dynamics of exchange rae behaviour. he main difference beween hese wo SAR models is he discrepancies in he reacion of agens o shocks of he same size wih opposie signs. he ESAR models imply a symmeric U- shaped response of he exchange rae abou he hreshold parameer wih respec o posiive and negaive shocks of he same magniude. he asymmeries of S-shaped LSAR responses, on he oher hand, migh be he resul of differences in he reacions of he agens o hese shocks. 3. esing Hypoheses in he SAR Framework Before proceeding o building SAR-ype nonlinear models, an imporan sep o carry ou is o conduc various hypoheses in order o find saisically significance evidence supporing nolineariy hypohesis. his involves esing for lineariy agains SAR, misspecificaion esing and diagnosic checks, some of which are briefly oulined in his secion. esing lineariy agains SAR esing for lineariy agains SAR is he firs sep owards building SAR models. eräsvira (994, 998) derived a lineariy es agains SAR. o explain his, firs define G * = G /, where G is he ransiion funcion defined in (3). Subracing ½ from G is done only for he derivaion of lineariy es. Now, rewrie () as y = ϕ c) + u * + ϕx + ( θ + θx ) G ( s ; γ, (4) wih previous noaions being reained alhough ϕ and ϕ have changed. he assumpion u ~ n. i. d(, σ ) is made in order o derive he disribuion of es saisics. hen, he condiional log-likelihood funcion of he model is given as l ( ϕ, θ, γ, c; y x, s ) = α logσ u (5) = σ = he null hypohesis H : γ of lineariy in (4) is esed agains he alernaive of nonlineariy = hypohesis ha H : γ. However, here is an idenificaion problem ha arises in esing > hese hypoheses since he model is idenified under he alernaive bu no under he null hypohesis. In order o resolve his problem, he ransiion funcion G(.) is replaced by is hirdorder aylor approximaion, so he model becomes y = β x s + e. (6) x + βx s + βx s + β3 3 6

8 Noe ha a higher order approximaion can be used. he lineariy hypohesis H : γ in (4) is = equivalen o he null H : β = β = β 3 = in (6). In small samples, he use of he F -es is recommended, because i has beer size properies han he χ version - LM es, which may heavily oversized in small samples (see Granger and eräsvira, 993; chaper 7). Boh, he χ and F -es versions, can be compued by means of wo auxiliary linear regressions. he F -es based on (6) can be compued as follows:. Esimae he model under he null hypohesis of lineariy by regressing y on x. Compue he residuals ê and he sum of squared residuals, say, SSR =. eˆ =. Esimae he auxiliary regression of y on x and x, i =,, 3. Compue he residuals ê i s eˆ = and he sum of squared residuals, say, SSR =. 3. Compue ( SSR SSR ) / 3p F = (7) SSR /( 4 p ) where p is he number of explanaory variables. Under he null hypohesis F -saisic has approximaely an F disribuion wih3 p and 4 p degrees of freedom. Selecing he ransiion variable he nex sage is o selec he appropriae ransiion variable o be used in he SAR model and he mos suiable form of he ransiion funcion. he appropriae ransiion variable in he SAR model can be deermined wihou specifying he form of he ransiion funcion. I is done by compuing he F saisic for several candidae ransiion variables (say) s s,..., s,, n and selecing he one for which he p -value of he es saisic is he smalles. he raionale behind his procedure is ha he es should have he maximum power, and, in his case, i means ha he alernaive model is correcly acceped. In oher words, he correc ransiion variable is used (Van Dijk, 999). 7

9 Selecing he ransiion funcion If he SAR nonlineariy is acceped and he appropriae ransiion variable has been seleced, he final decision o be made is o choose he mos suiable form of he ransiion funcion G( s ; γ, c). he choice o be made is beween he logisic funcion () and he exponenial funcion (3). eräsvira (994, 998) suggess o use a decision rule based on a sequence of ess nesed wihin he null hypohesis corresponding o F. He proposes o es he following hypoheses: H 4 : 4 = β, H β = / β, (8) 3 : 3 4 = H : β = / β 4 = β3 =, in y = β x s + e (9) 3 x + βx s + β x s + β3x s + β4 4 using he LM -ype ess. If H β is rejeced, hen LSAR model is seleced. Acceping 4 : 4 = H β and rejecing H β = / β imply ha he ESAR model in appropriae, : 4 = 3 : 3 4 = while acceping boh H β and H β = / β, bu rejecing : 4 = 3 : 3 4 = H β = / β = β imply ha he LSAR model is appropriae. However, Granger and : 4 3 = eräsvira (993) and eräsvira (994) argued ha sric applicaion of his sequence of ess can lead o he wrong conclusion. hen, hey recommended ha one should compue he p - values for all hese F -ess and choose he SAR model on he basis of he lowes p -value. herefore, if he rejecion of H β or H β = / β = β is accompanied by he 4 : 4 = : 4 3 = lowes p -value, hen he LSAR model is chosen. On he oher hand, if he rejecion of H β = / β is accompanied by he lowes p -value, hen he ESAR model is chosen. 3 : 3 4 = Furhermore, Van Dijk (999) and Van Dijk, eräsvira and Francis () saed ha an alernaive procedure for selecing he ransiion funcion proposed by Escribano and Jorda (999) is superior o ha developed by eräsvira (994). Escribano and Jorda (999) suggesed esing he following hypoheses: 8

10 H E : β = β 4 = and () H L : β = β3 =, and heir recommendaion is o selec he LSAR(ESAR) model if he minimum p -value is obained for H H ). L ( E 4. Esimaion Afer he ransiion variable s and he ransiion funcion G( s ; γ, c) have been seleced, he nex sage is esimaing he unknown parameers in he SAR model. he esimaion of he parameers in he SAR model is carried ou by he nonlinear leas squares (NLS) mehod. ha is, he parameers θ = ϕ, ϕ, γ, c) can be esimaed as ( ˆ θ = arg min Q where F( ; θ ) is he skeleon of he model x θ ( θ ) = arg min θ = ( y F( x ; θ )) () F θ = c) + u. () ( x ; ) ϕ + ϕx + ( θ + θx ) G( s ; γ, Under he assumpion ha he error u is normally disribued, he NLS is equivalen o maximum likelihood. If u does no follow a normal disribuion, he NLS esimaes are quasimaximum likelihood esimaes. herefore, under cerain regulariy condiions, he NLS esimaors are consisen and asympoically normal, ha is, ( ˆ θ ) (, ) θ N (3) C where θ denoes he se of rue parameer values. he asympoic covariance-marix C of θˆ can be esimaed consisenly as ˆ ˆ ˆ A B A, where  is he Hessian evaluaed a θˆ Aˆ = = q ( ˆ) θ = ˆ (4) ( F( x ; ˆ) θ F( x ; ˆ) θ F( x ; ˆ) θ u ) = wih q ( ˆ) θ ( y F( x ; ˆ)) θ, F( x ; ˆ) θ = F( x ; ˆ) θ / θ, and Bˆ is he ouer-produc of he gradien = ˆ B = ( ˆ) θ ( ˆ) θ = ˆ ( ; ˆ) θ ( ; ˆ) θ q q u F x F x (5) = = 9

11 he model can be esimaed using any convenional nonlinear opimisaion procedure (Van Dijk, 999; Van Dijk e. al., ). Concenraing on he sum of squares funcion Van Dijk (999) and Van Dijk, eräsvira and Francis () argued ha he problems arising from esimaion of he model can be simplified by concenraing on he sum of squares funcion. he SAR model is linear in auoregressive parameers ϕ and ϕ, when he parameers γ and c in he ransiion funcion are known and fixed. herefore, condiional upon γ and c, esimaes of ϕ = ϕ, ϕ ) can be obained by ordinary leas squares (OLS) as ( ˆ ϕ( γ, c) = x ( γ, c) x ( γ, c) x ( γ, c) y = = where x ( γ, c) = ( x ( G( s ; γ, c)), x G( s ; γ, c) ). he noaion ˆ ϕ ( γ, c) is used o indicae ha he esimae of ϕˆ is condiional upon γ and c. herefore, he sum of squares funcion Q (θ ) can be concenraed wih respec o ϕ and ϕ as (6) Q ( γ, c) = ( y ϕ( γ, c) x ( γ, c)) = (7) his will reduce he dimensionaliy of he nonlinear leas squares esimaion, since Q ( γ, c) will be minimized wih respec o only wo parameers γ and c. Saring values Saring values for he nonlinear opimisaion can be obained by wo-dimensional grid search over γ and c. Replacing he ransiion funcion wih γ ( ;, ) = + exp n G s γ c ( s c) (8) n ˆ σ s i= where σˆ is he sample sandard deviaion of s, which makes γ o be approximaely scale-free. s he se of grid values for he locaion parameer c can be chosen from sample perceniles of he ransiion variable s. his guaranees ha he values of he ransiion funcion conain enough sample variaion for each choice of γ and c. If he ransiion funcion remains almos consan in he whole sample, he momen marix of he regression in (6) is ill-condiioned, and he esimaion procedure fails (Van Dijk e. al., ).

12 Esimaing γ As menioned in he previous secion, he smoohness of he ransiion beween wo regimes is characerised by γ. When he value of his parameer is large, i is difficul o obain an accurae esimae of he smoohness of he ransiion beween he wo regimes. I is because, for such large values of γ, he SAR model becomes similar o a hreshold model. o obain an accurae esimae of γ, many observaions in he immediae neighbourhood of c is needed. Because, even he large changes in γ have only a small effec on he shape of ransiion funcion. herefore, he esimae of γ can be raher imprecise and ofen appear o be insignifican when i is judged by he -saisic (Van Dijk, 999 and Van Dijk e. al., ). 5. Diagnosic Checking Afer esimaing he parameers in he SAR model, he nex sep is o conduc specificaion esing o evaluae he fied model. Various diagnosic checks need o be done o ensure ha here is no residual auocorrelaion, no remaining nonlineariy and parameer consancy. esing for residual auocorrelaion Eirheim and eräsvira (996) proposed he following es for serial independence in he residual. Consider he SAR model of order k wih auo-correlaed errors: y = F( ; θ ) + ε (9) x where x = (, ~ x ), ~ x ( y,..., y ) as before and F ( ; θ ) is he skeleon of he model = p x given in (). An LM -es for q -h order serial dependence of ε can be obained as nr, where R is he coefficien of deerminaion of he auxiliary regression of εˆ on F x ; ˆ) θ = F( x ; ˆ) θ / θ wih θ = ϕ, ϕ, γ, c) and q lagged residuals ( ( ˆ ε,..., ˆ ε q. he symbol ^ indicaes ha he relevan quaniies are he esimaes under he null hypohesis of serial independence of ε. he resulan es saisic, denoed as LM SI (q), is asympoically χ disribued wih q degrees of freedom.

13 Van Dijk (999) and Van Dijk, eräsvira and Francis () saed ha his es saisic is a generalisaion of he LM -es for serial correlaion in an AR( p) model of Godfrey-Breusch- Pagan (979), which is based on he following auxiliary regression: ˆ eˆ = α y α p y p + βε β qε q + ν () ˆ where εˆ are he residuals of he AR ( p) model. Noe ha for a linear AR( p) model, p F( x ; θ ) = φ y and F( x ; ˆ) θ / θ = ( y,... y ). i= i i p esing for remaining nonlineariy Anoher diagnosic check is o es wheher he esimaed model successfully capured he nonlinear feaures of he ime series enirely. o do his, we can apply a es for no remaining nonlineariy o an esimaed auxiliary model. he naural approach is o specify he alernaive hypohesis of remaining nonlineariy as he presence of an addiional regime. For insance, esing he hypohesis ha a wo-regime model is adequae agains he alernaive ha a hird regime is necessary. Eirheim and eräsvira (996) develop an LM saisic o es a wo-regime SAR model agains he alernaive of he following addiive SAR model: y = ϕ ) + u () ' x + ( θ ' x ) G( s ; γ, c ) + ( ψ ' x ) H ( s ; γ, c he wo-regime model ha has been esimaed is assumed o have G(.) as ransiion funcion. herefore, he hypohesis o be esed concerns he quesion wheher or no exending he model wih ( ψ ' ) H (.) is appropriae. he null hypohesis of a wo regime model is eiher H γ x : = or H : ψ. Again, his es suffers from a similar idenificaion problem as encounered in = esing he null hypohesis of lineariy agains he alernaive of a wo regime SAR model in secion (3.). Similarly, he soluion o his idenificaion problem is replaced he ransiion funcion H s ; γ, ) by a aylor series approximaion around he poin γ. Using a hird- ( c order approximaion, he resulan approximaion o model () is ' ' ' y = β + β x s + β x s + β x s + e () where he parameers ' x + ( θ ' x ) G( s ; γ, c) β i, =,, 3, are funcions of he parameers i hypohesis of no addiional nonlinear srucure or 3 3 = ψ,γ and. he null H γ in () is equivalen o : = c H β ' : β = β = 3 = in (). he es saisic can be compued as nr from he auxiliary

14 regression of he residuals (obained from esimaed model under he null hypohesis) on he parial derivaives of he regression funcion wih respec o he parameers in he wo-regime model θ,γ and c, evaluaed under he null hypohesis, and he auxiliary regressors x s, i =,,3. he resulan F saisic has an asympoic χ disribuion wih 3 p degrees of freedom. i 6. Nonlinear Adjusmen o he Long Run Purchasing Power Pariy As has been argued before, because of he ransacion coss he adjusmens o posiive and negaive deviaions from he long-run PPP equilibrium are expeced o be same. Michael, Nobay and Peel (997) argued ha he ESAR model is more appropriae for modelling PPP deviaions, since i has symmeric adjusmens o posiive and negaive deviaions of he same magniudes. Incorporaing he equaions () and (3), he ESAR model for he deviaions from he PPP is modelled as follows: [ ( s c) ] p p * * π jν j + ( k + π jν j ) { exp γ } u j= j= ν = k + + (3) where v is a saionary and ergodic process, u ~ n. i. d(, σ ), and γ >. As menioned above, he ransiion funcion G(.) is U-shaped and he parameer γ deermines he speed of he ransiion process beween he wo exreme regimes. he middle regime corresponds o yielding G =, and hen (3) becomes a linear AR( p) model: s = c, p ν = k + π jv j + u (4) j= he ouer regime corresponds o s = ±, yielding G =, and hen (3) again becomes an AR( p) model, bu wih a differen se of parameers: p * * ν = k + k + ( π j + π j ) ν j + u (5) j= For esing purposes, i is convenien o reparameerize he ESAR model in (3) as follows: p p * * * ν = k + λν + φ jν j + ( k + λ ν + φ jν j= A saionary process is ergodic if i is asympoically independen, ha is any wo random variables posiioned far apar in he sequence are almos independenly disribued (see Hayashi, F., : p., for deails). j= j ) 3

15 [ ( c) ] { exp s + u γ } (6) he crucial parameers in (6) are λ and λ which deermine wheher or no he small and large deviaions respecively are mean-revering. he effec of ransacion coss on he real exchange raes suggess ha he larger he deviaion from long-run PPP equilibrium, he sronger he endency o move back o he equilibrium. his implies ha while * λ is possible, he condiions ha λ * < and λ + λ * < should be saisfied for he process o be global saionary. Under hese condiions, for small deviaions, may follow a uni roo or even exhibi explosive behaviour, bu for large deviaions he process is mean-revering (Michael, Nobay and Peel, 997). y he analysis based on he ESAR model above has implicaions for he convenional coinegraion es of PPP, which is based on a linear AR( p ) model, wrien below as an augmened Dickey-Fuller regression: p = + + λ ν φ j ν j j= ν k + u (7) Assuming ha he rue process for ν is given by he nonlinear model (6), hen esimaes of he * parameer λ in (7) will end o lie beween λ and ( λ + λ ). Hence, he null hypohesis H : λ (no linear coinegraion) may no be rejeced agains he saionary alernaive = hypohesis H : λ, even hough he rue nonlinear process is globally sable wih λ + λ * <. < his shows ha he failure o rejec he uni roo hypohesis on he basis of a linear model does no necessarily invalidae he long-run PPP (Michael e. al., 997 and aylor e. al. ). 7. Daa Series he daa series used in his sudy are monhly observaions from January 979 o June 3 aken from he IMF s Inernaional Financial Saisics CD-ROM. he ime period covers he managed-floaing and free-floaing regimes in Indonesia and i appear o be long enough o es for he PPP condiion as a long-run relaionship. he nominal exchange raes used in his sudy are he Indonesian Rupiah agains US-Dollar, Japanese Yen, and Singaporean Dollar. he domesic price is Indonesian CPI and he foreign prices are US, Japan and Singapore CPI series. he relaive price is defined as he raio of domesic price o foreign price. 4

16 As discussed in Lesari, Kim, and Silvapulle (3), he PPP models can be classified ino hree differen forms, namely, he univariae, bivariae and mulivariae models, depending on he naure of resricion(s) imposed. he empirical resuls in his sudy are based on invesigaing he appropriaeness of ESAR models for he deviaions from PPP defined by all hree univariae, bivariae and mulivariae models. When empirical sudies did no find suppor for he univariae model hey sudied less resriced forms, bivariae and mulivariae models. he univariae model (as argued before) is he real exchange rae which is he nominal exchange rae fully adjused o offse excess domesic inflaion over foreign inflaion. In he bivaraie model, he nominal exchange rae is allowed o parially adjus o his excess inflaion, while, in he mulivariae model, he nominal exchange rae is allowed o respond o domesic and foreign inflaion raes separaely. Empirical sudies of bivariae and mulivariae models have emerged as here was no suppor found for he fully resriced form of PPP he univariae model. 8. Empirical Resuls and Analysis Empirical analysis is carried ou in differen sages, which are given below. Lineariy es Resuls Alhough he nonlinear ESAR has been recommended for modelling he deviaions from PPP, in his sudy, esing for he null hypohesis of lineariy agains SAR was done firs o find ou wheher he non-linear framework is more appropriae o model he process han he linear counerpar. Having rejeced he lineariy, esing was hen done agains LSAR and ESAR models separaely. he reason is o find empirical suppor for he ESAR model, among ohers. he lineariy es is carried ou wih differen values of delay parameer (d), wih d ranging from o. able repors he lineariy es resuls of he hypoheses given in (9). In he univariae case, he resuls indicae ha he ESAR process wih he delay parameer d = is an appropriae represenaion of he adjusmen of he deviaions from he long-run PPP equilibrium for all hree real exchange raes and all he unresriced models wih wo excepions. hese being for he mulivariae model of Indonesia-US exchange rae, ESAR process wih d = 4 d =. and for he univariae model of Indonesia-Japan exchange rae, ESAR process wih 5

17 Esimaion of ESAR Models he hree real exchange rae (demeaned) series are ploed in Figure. I can be seen ha here are wo jumps in he 8 s due o he devaluaion in March 983 and Sepember 986. he big jump in he 9 s is due o he Asian financial crisis. he movemen of hese series over he sample period clearly indicae ha hey are mean-revering. However, employing sandard and hreshold uni roo ess, Lesari, Kim and Silvapulle (3) found ha he real exchange rae series are non-saionary, mean-avering. Closely analysing he behaviour of he series, he small deviaions from he long run PPP are found o be persisen, while he large ones o be revering back o he mean very fas. Furher, he movemens are fairly symmeric around he mean. hese observaions are consisen wih he ESAR model discussed in Secion. In wha follows, ESAR is fied o all hree exchange rae series under all hree assumpions briefly discussed in he previous secion. In he univariae model, he ransiion parameer γ esimaes for all series were found o be quie big (see able ). I was 4.68 for he Indonesia-US real exchange rae, 6.5 for he Indonesia-Japan real exchange rae and 9.33 for Indonesia-Singapore real exchange rae. hese esimaed values indicae ha he real exchange raes have a high speed of adjusmen owards he long-run PPP equilibrium. Figures (a), 3(a) and 4(a) show he esimaed ransiion funcions for he real exchange raes. Since all he γ values in he univariae model are found o be significanly differen from zero, i can be said ha he ESAR model can represen he adjusmen process owards he long-run equilibrium of PPP well. Furhermore, he residuals are found o follow a whie noise process, as indicaed by he p-value associaed wih Q-saisics a lag 6 for all real exchange raes. However, he ARCH effecs appear o be presen in all cases. esing for he mean-reversion propery of he series, he Indonesia-Singapore real exchange rae was found o have explosive behaviour in he lower regime as λ >, while he Indonesia-U.S. and Indonesia-Japan real exchange raes were found o have uni roos in he lower regime as λ = was found o be rue in boh cases (see able 3). Furher, all hree real * exchange rae series were found o have saionary behaviour in he upper regime as λ < in all hree cases. However, he sabiliy condiion λ + λ * < ha all exchange raes have saionary mean-revering behaviour overall. is saisfied in all cases. I can be said In he bivariae model of PPP, able repors ha he Indonesia-Japan exchange rae was found o be characerised by a small γ,.3, while he Indonesia-US and Indonesia-Singapore exchange raes were by large γ s, 3.73 and 4.86 respecively. hese resuls sugges ha he 6

18 Indonesia-Japan exchange rae has a low speed of adjusmen owards he long-run equilibrium of PPP, while Indonesia-US and Indonesia-Singapore exchange raes have high speeds of adjusmens owards he long-run equilibriums of PPP. Figure (b), 3(b) and 4(b) show he paerns of ransiion funcions for he bivariae models of PPP. he γ values for Indonesia-US and Indonesia-Singapore exchange raes were found o be saisically differen from zero indicaing ha he ESAR model can be used o model he adjusmen process owards he longrun equilibrium of PPP of he real exchange raes, while he γ value for he Indonesia-Japan exchange rae is found o be saisically equal o zero. However, his resul should no be inerpreed as evidence agains he ESAR model, because he esimae sandard error of γ is raher imprecise in general and ofen appears o be insignifican when judged by is -saisic (see Frances and van Dijk,, pp.9). Furhermore, he residuals are also found o follow a whie noise process, indicaed by he p-value associaed wih Q-saisics a lag 6 for all real exchange raes. As wih univariae models, he ARCH effecs appear o be presen in all cases. Furhermore, able 3 showed ha all cases have explosive behaviour in he lower regime as λ >, and he sabiliy condiion λ + λ * < is saisfied. Clearly, all exchange raes have saionary mean-revering behaviour. Now, urning o he resuls of esimaing he mulivariae (fully unresriced) models of PPP, he Indonesia-Japan exchange rae is found o be characerised by a small γ of.43, while Indonesia-U.S. and Indonesia-Singapore exchange raes by large γ s,.6 and.89 respecively. hese resuls sugges ha he Indonesia-Japan exchange rae has low speeds of adjusmens owards he long-run equilibrium of PPP, while Indonesia-US and Indonesia- Singapore exchange raes have high speeds of adjusmen owards he long-run equilibriums of PPP (see able ). Figure (c), 3(c), and 4(c) show he ransiion funcions for he mulivariae models of PPP. However, no all he γ values in he mulivariae model are significanly differen from zero, such as in he Indonesia-Japan exchange rae which has γ value equal o zero. he ESAR model, sill, can represen he adjusmen process owards he long-run equilibrium of PPP for he mulivariae model. As in oher models, he residuals are found o follow a whie noise process as indicaed by he p-value associaed wih Q-saisics a lag 6 for all cases and he ARCH effecs are presen in all cases. Furher, an explosive behaviour in he lower regime ( λ > ) is found in Indonesia-Japan and Indonesia-Singapore exchange raes, while he Indonesia-US exchange rae has a uni roo in he lower regime ( λ = ). All exchange raes show mean revering behaviour in he overall adjusmen process as indicaed by λ + λ * < (see able 3). 7

19 9. Conclusion his paper models he dynamics of he adjusmen process o Indonesian long run purchasing power pariy relaive o US, Japan and Singapore by employing a non-linear framework, which is recenly shown o be appropriae in he presence of ransacion coss associaed wih inernaional rade. Using monhly observaions from January 979 o June (pos-breon Woods period), covering he managed- and free-floaing regimes in Indonesia, he real exchange raes were esed for mean-revering properies. he daa series used includes he domesic price (which is Indonesian CPI) and he foreign prices (he US, Japan and Singapore CPI series). he relaive price is defined as he raio of domesic price o foreign price. he real exchange rae is defined as he difference beween he nominal exchange raes and he relaive price raio. A large number of sudies found ha he real exchange series are mean-avering and persisen, creaing PPP puzzles. Using he linear framework many aemped o resolve hese puzzles unsuccessfully. Moivaed by he success of recen sudies on PPP, applying he nonlinear ESAR o model he adjusmen process, we esed for mean-revering properies of all hree real exchange raes for small and large deviaions from he long-run equilibria. We find ha he small deviaions are non-saionary, persisen and i can even be explosive, while he large deviaions saionary wih fas adjusmen, making he overall adjusmen process mean-revering. hese resuls are consisen wih he previous findings. Furher, he real exchange rae implied by PPP is a very resriced form of PPP condiion. We also examined less resriced and fully unresriced forms of PPP and found he resuls are sronger han hose for he resriced form. I is noeworhy ha he nonlinear ESAR model helps o resolve he wo PPP puzzles, which many empirical sudies made considerable effors o resolve for many decades unsuccessfully. 8

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22 Figure : (a) Demeaned and derended Indonesia-U.S. Real Exchange Rae EQU (b) Demeaned and derended Indonesia-Japan Real Exchange Rae EQJ (c) Demeaned and derended Indonesia-Singapore Real Exchange Rae EQS

23 able : Lineariy ess Resul PPP Models d p-value F- es Model US as foreign counry Univariae ESAR Bivariae ESAR Mulivariae ESAR Japan as foreign counry Univariae ESAR Bivariae ESAR Mulivariae ESAR Singapore as foreign counry Univariae ESAR Bivariae..965 ESAR Mulivariae ESAR Noes: - he opimal d chosen is which gives he lowes he p-value of he lineariy es over he range d Mor informaion

24 able : Esimaion of he ESAR models Series γ Sd. Err c Q(6) ARCH(6) AR(p) US as foreign counry Univariae Bivariae Mulivariae Japan as foreign counry Univariae Bivariae Mulivariae Singapore as foreign counry Univariae Bivariae Mulivariae Noes: - Q(6) is Ljung-Box saisics for residual auocorrelaion for lag six. - ARCH(6) is Engle s ARCH-LM es for ARCH wih lags six. - ARp(p) ischosen on he basis of serial correlaion ess. 3

25 able 3: Esimaion of he ESAR models (coninued) PPP Models λ λ * λ * λ + λ (s.e) (s.e) (s.e) (F_sa) US as foreign counry Univariae (.4) (.97) (.34) (6.4695) Bivariae (.7) (.95) (.45) (5.78) Mulivariae (.37) (.376) (.46) (6.766) Japan as foreign counry Univariae (.84) (.748) (.) (7.557) Bivariae (.77) (.83) (.773) (4.87) Mulivariae (.4) (.847) (.65) (5.436) Singapore as foreign counry Univariae (.) (.783) (.7) (7.38) Bivariae (.8) (.487) (.86) (5.865) Mulivariae (.34) (.858) (.37) (4.986) Noes: - λ is obained from eq. (5.7) * - λ and λ are obained from eq. (5.6) 4

26 Figure. ransiion Funcion for Indonesia-U.S. Exchange Rae (a) Univariae Model (γ=4.68) (b) Bivariae Model (γ=3.7) (c) Mulivariae Model (γ=.6)

27 Figure 3. ransiion Funcion for Indonesia-Japan Exchange Rae (a) Univariae Model (γ=6.5) (b) Bivariae Model (γ=.3) (c) Mulivariae Model (γ=.43)

28 Figure 4. ransiion Funcion for Indonesia-Singapore Exchange Rae (a) Univariae Model (γ=9.33) (b) Bivariae Model (γ=4.86) (c) Mulivariae Model (γ=.89)