global non-linear stationary alternatives

Transcription

1 Tesing for non-saionary hypoheses agains local and global non-linear saionary alernaives Param Silvapulle and Roland Shami Deparmen of Economerics and Business Saisics Monash Universiy Caulfield Campus, Ausralia 3415 ABSTRACT This paper develops a sequenial procedure for esing non-saionary hypoheses agains local and global non-linear saionary alernaives. This involves: (i) developing a es agains a local non-linear ESTAR saionary alernaive, and (ii) developing a es agains a global ESTAR nonlinear saionary alernaive process. The null hypohesis of non-saionary is esed agains he local ESTAR non-linear saionary alernaive; if he null is rejeced, hen esing agains he global ESTAR non-linear saionary alernaive process follows. These ess were applied o Indonesian real exchange raes, agains is major rading parners - U.S, Japan and Singapore. The Indonesian governmen occasionally inervened in he foreign exchange markes, and consequenly, here were noable srucural breaks in real exchanges raes o be sudied in his paper. This paper inegraes hree ideas - local saionariy, global saionariy and srucural breaks - in developing ess o esablish he saionary properies of real exchange raes. Applicaion of hese ess provides concree evidence in esablishing he global non-linear saionariy of all hree real exchange raes series. These ess will have much wider applicaion han hose provided in his paper. Keywords: ESTAR model, local saionariy, nuisance parameers, global saionariy, srucural breaks, real exchange raes JEL classificaion: F32, F31, C22, C5

2 1. Inroducion The sandard linear auoregressive moving average (ARMA) framework has long been used o esablish wheher or no ime series are saionary. However, here is a growing dissaisfacion among applied researchers regarding he use of his framework o invesigae he saionary behaviour of economic and financial ime series. Much of his disconen is due o heoreical predicions of saionariy of some variables - in several areas of economics such as inernaional moneary economics and macro-finance - no being suppored by ADF-ype ess. For example, he finding ha real exchange raes have uni roos caused apprehension among economiss, waning o use he long run purchasing power pariy (PPP) equilibrium relaionship in exchange rae deerminaion. Anoher noable example is he finding of uni roos in real ineres raes, and hus several economiss argued ha heories relaed o real business cycles and modern economic growh are hard o reain. Recenly, in order o resolve his puzzle, numerous researchers developed ess ha are more powerful han he ADF es in rejecing he uni roo hypohesis. One srand of lieraure focused on he use of panel daa and rejeced he hypohesis of join uni roos in a group of real exchange raes agains he saionary alernaive hypohesis. See, for example, Im, Pesaran and Shin (2002). Anoher srand of lieraure focused on fracional inegraion (Cheung and Lai, 1993) o esablish he saionariy of ime series. Recenly, in esing for he saionariy of economic and financial variables, several empirical sudies exploied he fac ha many ime series are largely non-linear and ha he ADF-ype uni roos do no have power agains hese models. See, for example, Killian and Taylor (2001) and Taylor e al. (2002). Furher, economies generally go hrough various phases of business cycles, and hence many economic and financial variables exhibi deepness and seepness asymmeries. Using hreshold auoregressive (TAR) models o capure hese asymmeries, Enders and Granger (1998) developed procedures o es agains alernaives of asymmeric saionary roos. In hese models, he regime swiching occurs abruply, and he sudies ha used hese models o es he saionariy of real exchange raes have had limied success in supporing he saionary hypohesis. See, for example, Enders and Dibooglu (2001), and Lesrari, Silvapulle and 1

3 Kim (2003). Moreover, several heoreical and empirical economiss have argued ha here are cases in which he ransformaion or swiching occurs smoohly raher han abruply as occurs in TAR models, and hus he applicaion of smooh ransiion auoregressive (STAR) framework ha can capure such smooh swiching from one regime o oher has become very popular. I has been argued in he recen lieraure ha he lack of empirical evidence supporing PPP appears o be caused by he breakdown in he assumpions (under which he PPP is valid) due o facors such as ransacion coss, axaion, imperfec compeiion, foreign exchange marke inervenions, and he differenial composiion of commodiy baskes and hence he price indices across counries. Subsequenly, an alernaive framework for he empirical analysis of real exchange raes, which allows for fricion in he commodiy rade, has emerged. Dumas (1992) and Sercu, Uppal and Van Hulle (1995) developed equilibrium models of exchange rae deerminaion in he presence of ransacion coss, hence providing empirical evidence of real exchange rae adjusmen being a non-linear ESTAR meanrevering process. See secion 3.1 for more deails. The objecive of his paper is o develop a new sequenial procedure for esing nonsaionariy hypoheses agains local and global ESTAR non-linear saionary alernaives. These ess will be applied o Indonesian real exchange raes agains is major rading parners - U.S., Japan and Singapore. The Indonesian governmen aggressive inervenion in he foreign exchange markes has caused five noable srucural breaks in he real exchanges raes in he sample period we chose o sudy. By incorporaing dummy variables o capure hese breaks, he proposed ess will be exended and applied o all hree real exchange raes, hus esablishing heir global nonlinear saionary properies. Tesing for a non-saionary process agains he ESTAR saionary alernaive is no new. Kapeanios e al. (2003) developed a es for he null hypohesis of non-saionary series agains saionary non-linear alernaives, ignoring he problem of he presence of nuisance parameers under he null hypohesis of non-saionariy. See secion 3.2 for more deails. Furher, he criical values abulaed in his paper are inappropriae o use 2

4 in he empirical invesigaion o be carried ou in his paper, due o he five srucural breaks in Indonesian real exchange raes. This paper proposes a es saisic for he null of non-saionariy agains he local ESTAR saionary alernaives. In his esing, he smoohing and hreshold parameers governing he ESTAR process become unidenified under he null. Earlier sudies have linearised his process o overcome he problem of nuisance parameers, whereas our sudy akes he presence of such parameers ino accoun in developing he esing procedure. Furher, we also develop a esing procedure agains he global ESTAR nonsaionariy. Kapeanios e al. (2003) esimaed he power of heir proposed es agains a global process, and his paper, on he oher hand, will propose a es saisic agains global saionary alernaive hypohesis. Moreover, o dae, ess developed o esablish he saionariy properies of real exchange raes have been applied mainly o OCED counries, wih exchange raes largely being marke-deermined. We found hese ess o be inapplicable direcly, for example, o some Asian real exchange raes, as aggressive governmen inervenions caused significan impacs on hese raes. In his sudy, we exend he proposed ess o accommodae he presence of srucural breaks caused by such inervenions. This paper is organised as follows: Secion 2 briefly oulines he ESTAR model. Secion 3 oulines he model wih linear and non-linear ESTAR componens, jusifying he use of such a model for capuring real exchange rae behaviour. Secion 4 specifies he hypoheses relaed o local and global non-linear saionary processes and proposes es saisics for sequenial esing of local and global ESTAR saionary alernaives. Secion 4 discusses a simulaion experimen conduced o esimae he criical values of he disribuions, and assess he small sample properies of he proposed ess, followed by a discussion of heir resuls. Secion 5 describes he daa series used in his sudy. Secion 6 applies he ess proposed in his paper o Indonesian real exchange raes. Some concluding remarks are made in Secion 7. 3

5 2. Exponenial smooh ransiion auoregressive (ESTAR) models We consider a STAR model of order p for variable, given as: y = +Φ + +Θ +. (1) y ϕ0 x ( λ0 x) G( s; γ, c) u where x ( y y 2,..., y ), Φ = ( ϕ1, ϕ2,..., ϕ p ) and Θ = ( θ1, θ2,..., θ p ) are = 1, p unknown parameer vecors and u ~ ni.. d(0, σ ). G(.) is a ransiion funcion which is coninuous, and bounded by zero and one. s is a ransiion variable, c is he hreshold 2 parameer and γ is he smoohing parameer. The ransiion variable s may be a single sochasic variable, for example, an elemen of x, a linear combinaion of sochasic variables or an exogenous variable. In many empirical sudies employing STAR-ype models, he ransiion variable s is generally assumed o be a lagged endogenous variable, ha is, s = for an ineger d (delay parameer) > 0 (Teräsvira, 1994 and y d Franses and Van Dijk, 2000). The STAR model can be inerpreed as a regime swiching model wih wo regimes, associaed wih wo exreme values of he ransiion funcion, G( s ; γ, c) =0 and G( s ; γ, c) =1, wih he ransiion from one regime o he oher being gradual raher han abrup as in TAR models. In his paper, he exponenial smooh ransiion (EST) funcion, which is exensively being used in applied economic analysis of ime series such as real exchange raes and real ineres raes, will be sudied. Is funcional form is given as, 2 [ ( s ) ] G( s ; γ, c) = 1 exp γ c, γ > 0 (2) where γ measures he speed of ransiion from one regime o anoher and c represens he locaion for hreshold value for s. Equaion (1) combined wih (2) yields he Exponenial STAR (ESTAR) model. The ransiion funcion of ESTAR is symmeric and U-shaped around values of he smoohing parameer c. Figure 1 shows he shapes of he ransiion funcion for some γ. I can be shown ha as γ 0, he funcion 4

6 G(.) goes o zero and (2) reduces o a linear model. On he oher hand, as γ, G(.) converges o one and (2) reduces o a linear model wih a differen se of parameer values. I is noiceable in he graph presened in Figure 1 ha he ransiion becomes faser as he values of oher is abrup and he model becomes TAR. γ increases, and for large γ, he ransiion from one regime o Figure: Examples of he exponenial funcion G ( s ; γ, c) as given in (3) for various value of γ and hreshold c = Gamma=1 Gamma=2.5 Gamma=25 3. Modelling non-linear adjusmens We consider an ESTAR process of order p for y given as: 2 { ( ) } p p = π j j + ( π j j) 1 exp γ + j= 1 j= 1 y y y s c u (3) 2 where is a saionary and ergodic process wih zero mean, u ~ n. i. d(0, σ ), and y γ > 0, which deermines he speed of he ransiion process beween wo exreme 5

7 regimes. The middle regime corresponds o s = c and (3) becomes a linear AR( p) model: p = π 1 + π j j + u (4) j= 2 y y y The ouer regime corresponds o s = ±, and (3) becomes he following linear AR( p) model, = ( π + π ) 1 + p ( π j + π j) j + u j= 2 y y y wih π = π and π π 1 = 1. Now, assuming ha he higher order erms in AR(p) are zero, we re-wrie he model (3) as: ( ) 2 y = δ y 1+ δ y 1 (1 exp[( γ s c ]) + where δ = 1 π and u (5) δ = π. The crucial parameers in (5) are δ and δ which deermine wheher or no he small and large deviaions respecively are saionary processes. The presence of ransacion coss suggess ha here exiss a region of no rading, and herefore, for small deviaions from he equilibrium, he real exchange rae may be non-saionary, following eiher a random walk or an explosive process. Ouside his region (ha is, for large deviaions), on he oher hand, he possibiliy of inernaional arbirage can bring he series close o he equilibrium. See Killian and Taylor (2001) for an alernaive explanaion based on, among ohers, heerogenous raders and noise raders. For his reason, he small deviaions are modelled as linear, whereas he large deviaions as non-linear ESTAR, hese being capured precisely by (5). While δ 0 is possible, he condiions ha δ < 0 and δ + δ < 0 mus be saisfied for he process y o be globally saionary. See Michael, Nobay and Peel (1997) for more discussion on his. Using simulaion sudies, i has been shown ha he ADF-ype ess have no power agains he non-linear ESTAR model. See, for example, Michael, Nobay and Peel (1997) and Taylor, Peel and Sarno (2001). 6

8 As has been argued before, we will focus on anoher issue relaed o esing for nonsaionariy of Indonesian real exchange raes. From our experience in empirical applicaions, we find ha he ess and he models discussed in his paper so far and elsewhere are no direcly applicable o analyse Asian real exchange raes, because he operaion of hese financial markes is differen from hose of OECD markes. In he former, exchange rae movemens are managed by governmen inervenions in he foreign exchange markes frequenly, while in he laer, hey are largely marke deermined. These inervenions can have significan impacs on real exchange raes, resuling in srucural breaks in he series. In he presence of such breaks in he series, he es saisics o be developed in he conex of model (5) will be exended by inegraing hree ideas - local and global saionary hypoheses and he presence of srucural breaks. See secion 3.2 for more deails Hypoheses for esing agains local and global non-linear saionary processes and he esing procedures Recall he model (5), 1 1 ( ) y = δ y + δ y (1 exp[ γ s c ]) + 2 u We will develop procedures for esing wo ses of hypoheses: One is, seing δ = 0, he null hypohesis H 0 L : δ = 0 is esed agains he saionary alernaive hypohesis H1 L : δ < 0, which lends a mehodology for esing agains he local nonlinear saionary process. Second is, he null hypohesis H 0 G : δ + δ 0 is esed agains H 1 G : δ + δ < 0 which lends a mehodology for esing agains he global saionary process. In wha follows, we shall propose he es saisics., (i) Under H OL, he parameers γ and c become unidenified. In he presence of such nuisance parameers, a -es for H δ = 0 agains 0 L : H1 L : δ 0 < based on ˆ ( γ, c) δ 0 = - which is he convenional -es bu a funcion of nuisance parameers γ and c. We overcome his problem by exploiing Davies (1987) idea and compuing he larges 7

9 possible value of -saisics over he space spanned by γ and c. The Davies approach is o define he es saisic as: ˆ sup ˆ sup ˆ (, c δ = γ ) = sup, where L, δ = 0 L, δ = 0 ˆ se.( δ ) space spanned by ( γ, c). ˆ δ is a funcion of he parameer In such cases, simulaion mehods are generally used o compue he criical values of he disribuion of he es saisic. To improve he power properies of he ess, parameer space needs o be resriced, and he previous empirical sudies and sylised facs relaed o ESTAR model can provide some guidance owards his end. (ii) In order o es H δ + δ agains H δ + δ < 0, we compue he F G,(1,T-k) 0 G : 0 saisic, and hence he saisic G, δ+ δ = 0 1 G :, and is esimae has a non-sandard - disribuion and is criical values will be generaed using simulaions. Recall ha y in (5) was assumed o have zero mean. In order o accommodae sochasic processes x wih nonzero mean and/or linear deerminisic rend, he processes should be demeaned as x = µ + y and/or derended as x = µ + β + y, and he resulan residuals y be used in ESTAR model (5). Furher, as has been argued before, here are five srucural breaks presen in he Indonesian real exchange raes, over he sample period o be sudied in his paper. The dummy variables defined in (6) below will capure hese breaks. Clearly, in he presence of srucural breaks, he disribuion of es saisics - supˆ L derended series., δ = 0 and G, δ+ δ = 0 - will no be he same as for he demeaned and/or D 1 1 if < TB1( TB1 = 1983 : M 4) = 0 oherwise 1 if TB1 < TB2 ( TB2 = 1986 : M 9) D2 = (6) 0 oherwise 8

10 D D D if TB2 < TB3( TB3 = 1997 : M 8) = 0 oherwise 1 if TB3 < TB4 ( TB4 = 1998 : M 9) = 0 oherwise if > TB = 0 oherwise 1 4 In he presence of hese breaks in he series, he appropriae criical values of he es saisics will be generaed via simulaions. The dummy-filered series y is generaed as follows: 5 x = µ + β+ φid i + y i= 1 and used in model (5). In order o esablish wheher y is non-saionary or global nonlinear ESTAR saionary, a sequenial esing is done as follows: (i) Tes he null H 0L: δ = 0 agains δ < using he - saisic for which H 1L: 0 sup L, δ = 0 he criical values are abulaed in Table 1. If he null is no rejeced, conclude ha he series under invesigaion is non-saionary. On he oher hand, if he null is rejeced, hen conclude ha he series is locally nonlinear saionary, and proceed o es he following: (ii) Tes H 0 G : δ + δ 0agains H1 G : δ + δ < 0 using he one-sided saisic for G, δ+ δ =0 which he criical values are abulaed in Table 2. According o he resuls of a limied simulaion sudy carried ou in his paper, his es appears o be similar. If he null is no rejeced conclude ha he series is non-saionary, and on he oher hand, if he null is rejeced conclude ha he series is globally ESTAR non-linear saionary process. On he selecion of ransiion variable s : our experience showed ha y -d is a more appropriae proxy for s, han y -d used by Kapeanios e al. (2003). However, o simplify 9

11 he simulaion sudy we assume ha d = 1. In fac, d can be chosen from he se {1, 2,.,d max } based on an informaion crierion such as AIC or a sequence of hypoheses ess. We do no also ake he presence of serial correlaion coefficien ino accoun in developing hese ess. We believe his issue can be handled as ha in applying he ADF es. 4. A simulaion sudy and he resuls The simulaion sudy is conduced in a number of sages: (i) Calculae 1, 5 and 10 per cen criical values of he disribuion of he es saisic supˆ L, δ = 0 for he models A, B and C. Since his saisic is a funcion of γ and c, i has been compued over he parameer space γ = [0.1, 5.0], c = [c L, c U ], where c L and c U are he minimum and maximum values of sored y -1, wih 10 per cen of he op and he boom of hese values rimmed. (ii) Calculae 1, 5 and 10 per cen criical values of he disribuion of he es saisic supˆ L, δ = 0 over he parameer space defined in (i) above for he hree models D, E and F, wih five dummy variables presen. The sages (iii) and (iv) involve esimaing he criical values of he disribuion of G, δ δ + =0 saisic wihou and wih he srucural breaks presen in he variable x. Furher, in his secion, we will ouline he experimenal design used in he simulaion sudy and discuss he resuls. 4.1 Experimenal Design The criical values were compued using 20,000 simulaions for he sample of sizes T=100, 300 and The sizes and he powers (he rejecion frequencies under he null and alernaive hypoheses respecively) were esimaed using 2000 and 1000 simulaions respecively, only for he sample of sizes T=100 and 300. Le x be he ime 10

12 series o be sudied, his series is filered for various deerminisic variables using he following models: x = µ + β+ φ D + 5 i i y (7) i= 1 where µ is he mean, is he ime rend and D i, i=1,,5 are he dummy variables, capuring he srucural breaks noed in Indonesian real exchange rae series. Model A: Model (7) wih µ = β = φ i ( i = 1,..,5) = 0 Model B: Model (7) wih µ 0 & β = φ i ( i = 1,..,5) = 0 Model C: Model (7) wih µ 0, β 0 & φ i ( i = 1,..,5) = 0 The models D, E and F are he same as hose A, B and C respecively, wih φ 0 for i = 1,..,5. i The filered series y is used in he model (5). The criical values for he sages (i) and (ii) above are repored in Table 1, while hose for he sages (iii) and (iv) are repored in Table 2. These criical values were used o compue he rejecion probabiliies under he null (sizes) and under he alernaives (powers) of he ess proposed in his paper. The seleced parameer ses for hese calculaions are defined as: δ = { }, δ ={ }, γ ={ } and c ={ }. 4.2 Simulaion resuls Table 1 repors he 1, 5 and 10 per cen criical values of he disribuion of he supˆ L,δ =0 saisic for models A, B and C for he sample sizes T = 100, 300 and The disribuion of his saisic appears o be much wider for T = 100 han for he oher wo sample sizes. The abulaed criical values for T=300 do no noably differ from hose for T=1000, and herefore, we can say ha he sample size 300 is large enough for he asympoic resuls o hold. Clearly, he disribuions become wider for he models filered wih deerminisic variables, in paricular wih five dummy variables, han hose for he models filered only wih mean and/or deerminisic rend. The criical values of 11

13 he G, δ+ δ =0 -saisic were abulaed in Table 2. Alhough he asympoic resuls are no ye derived heoreically, our experience wih he simulaion experimen conduced in his sudy shows ha he disribuion of ˆ G, δ+ δ = 0 - saisic, o some exen, depends on he parameer values of γ and c. The rejecion raes (sizes) of hese ess under he null hypoheses are reasonably close o he seleced nominal level of five per cen. The rejecion raes (powers) of hese ess under he corresponding alernaive hypoheses are always higher han he sizes of he ess, and hey end o increase as he parameer values and/or he sample size increase as expeced. 5. Daa Series The bilaeral exchange raes beween Indonesian Rupiah agains U.S-Dollar, Japanese Yen, and Singaporean Dollar were colleced from he IMF s Inernaional Financial Saisics CD-ROM for he period January 1979 o June The CPI series were also colleced from IMF Inernaional Financial Saisics CD-ROM. The domesic price index is he Indonesian CPI and foreign price indices are he U.S, Japan and Singapore CPI series. The relaive price is defined as he raio of domesic price o foreign price. The CPI and nominal exchange rae series are used in naural logarihms. Indonesia had a fixed nominal exchange rae regime unil Ocober In November 1978, wih 48 percen rupiah devaluaion, he Indonesian governmen changed he exchange rae regime o a managed-floaing one wih infrequen large devaluaions, and concurrenly pegged he rupiah o a baske of currencies of Indonesia s major rading parners. A 28 per cen exchange rae devaluaion was experienced by Indonesian economy in March 1983, followed by a 31 per cen exchange rae devaluaion in Sepember The managed-floaing sysem was susained unil a major financial crisis in July Following his crisis, he Indonesian rupiah was floaed freely from Augus 1997 onwards and he governmen inervenion ceased. 12

14 The hree derended real exchange rae series - Indonesia-U.S, Indonesia-Japan and Indonesia-Singapore - are ploed in Figures 1, 2 and 3 respecively. I can be seen ha here are wo jumps in he 80 s due o devaluaion in March 1983 and Sepember The big jump in he 90 s is due o he Asian financial crisis in The derended series are furher filered for five srucural breaks and ploed in Figures 4, 5 and 6 respecively. Employing sandard ADF es, and hreshold uni roo ess of Enders and Granger (1998), Lesari, Kim and Silvapulle (2003) found real exchange rae series o be non-saionary. Closely analysing he behaviour of he series, he small deviaions from he long-run equilibrium are found o be persisen, while he large ones o be revering back o he mean very fas. Furher, hese deviaions are generally symmeric around he mean. 6. Applicaion of he esing procedures agains local and global saionary alernaives The non-linear ESTAR model (5) was fied o all hree real exchange rae series, and he ess developed for esing agains local and global saionary process were applied. The es resuls are repored in Table 7 along wih he model parameer esimaes. We firs examine he resuls for he real exchange raes derended as in models B and C. An analysis of he local saionariy es resuls indicaes ha he es saisics (in absolue values) are larger han he corresponding five per cen criical values repored in Table 1. However, alhough saisically insignifican, he esimaed value of he parameer δ is greaer han zero in he lower regime in all cases. These resuls imply ha all hree series are locally saionary, while hey are globally non-saionary as indicaed by insignifican values of he ˆG, - saisic. + = δ δ 0 sup ˆ L,δ =0 Differen resuls emerge for he real exchange series filered for srucural breaks. The esimaed value of he parameer δ is less han zero, alhough saisically insignifican, in he lower regime for all cases. However, he finding ha ˆG, δ+ δ = 0 - saisic larger han he five per cen criical value for all hree real exchange raes indicaes ha hese hree 13

15 series are globally saionary. Furher, Indonesia-U.S and Indonesia-Japan real exchange raes appear o have a slow ransiion from lower o upper regimes as indicaed by small values of γ (0.70 and 0.37 respecively), while he Indonesia-Singapore exchange rae was governed by a large value of γ (2.30). As has been discussed before, previous sudies found many Asian real exchange raes o be non-saionary because hese sudies largely ignored he governmen inervenions in he financial markes and he subsequen srucural breaks in exchange raes. Conrary o hese findings, our sudy provides concree evidence o esablish he saionary properies of Indonesian exchange raes relaive o hree counries U.S., Japan and Singapore. 7. Conclusion This paper develops a new sequenial esing procedure o invesigae saionary properies of real exchange raes. This involves (i) developing a esing procedure, in he presence of nuisance parameers, agains a local non-linear exponenial smooh ransiion auoregressive (ESTAR) saionary process; (ii) developing a es agains a global ESTAR non-linear saionary process; and (iii) exending hese wo ess by incorporaing dummy variables which capure srucural breaks in he series - ino he daa generaing process. In his sequenial esing, he null hypohesis of non-saionary series agains a local ESTAR non-linear saionary alernaive process is esed. If he null is rejeced, hen, esing agains a global ESTAR non-linear saionary alernaive process follows. The ess developed in his paper are applied o real exchange raes of Indonesia agains is major rading parners - U.S., Japan and Singapore. There are five noable srucural breaks in hese real exchanges raes due o Indonesian governmen inervenions in he foreign exchange markes. This paper inegraes hree ideas - local saionariy, global saionariy and srucural breaks - in developing ess o esablish global saionary properies of real exchange raes. Applicaion of he sequenial esing procedure proposed in his paper provides concree evidence in esablishing global non-linear 14

16 saionariy of all hree real exchange raes. These ess will have much wider applicaions han hose provided in he paper. Figure 1: Demeaned and derended Indonesia-U.S. real exchange rae for he original daa EQU Figure 2: Demeaned and derended Indonesia-Japan real exchange rae for he original daa EQJ

17 Figure 3: Demeaned and derended Indonesia-Singapore real exchange rae for he original daa EQS Figure 4: Indonesia-U.S real exchange raes for he dummy filered daa DQU 16

18 Figure 5: Indonesia-Japan real exchange raes for he dummy filered daa DQJ Figure 6: Indonesia-Singapore real exchange raes for he dummy filered daa DQS 17

19 Table 1: Criical values of he esimaed sup ˆ agains locally non-linear saionary process L,δ =0 es saisic for a uni roo y = δ y + δ y (1 exp[( γ s c ]) + u wih δ = 0. Model: ( ) Hypoheses: he null H 0L: 0 δ = is esed agains H 1L: δ < 0 ˆ Tes saisics: sup ˆ sup ˆ (, c δ = γ ) = sup, where ˆ δ is a funcion of he L, δ = 0 L, δ = 0 ˆ se.( δ ) parameer space spanned by ( γ, c). Sample size Models used o filer he series Perceniles of he disribuion of supˆ L, δ = 0 1% 5% 10% 100 Model A Model B Model C Model A Model B Model C Model A Model B Model C Model D Model E Model F Noe: Model (): 5 x = µ + β+ φid i + y i= 1 Model A: Model () wih µ = β = φ i ( i = 1,..,5) = 0 Model B: Model () wih µ 0 & β = φ i ( i = 1,..,5) = 0 Model C: Model () wih µ 0, β 0 & φ i ( i = 1,..,5) = 0 Models D, E and F are equivalen o (respecively) A, B and C wih φ ( i = 1,..,5) 0. i 18

20 Table 2: Criical values of one-sided -es for nonsaionary agains global nonlinear saionary y δ y δ y (1 exp[( γ s c ]) = Model: ( ) 2 Hypoheses: he null hypohesis Tes saisic: saisic. G, δ+ δ = 0 H 0 G : δ δ 0 u wih δ 0. + is esed agains H1 G : δ + δ < 0. Sample size Deerminisic variables Perceniles of he disribuion of. G, δ+ δ = % 5% 10% Model A Model B Model C Model A Model B Model C Model A Model B Model C Model D Model E Model F See he noes for Table 1. The perceniles of he disribuion of he es saisic were calculaed as funcions of of he parameer space spanned by ( γ, c). 19

21 Table 3: Rejecion raes of supˆ L alernaives, δ = 0 es agains local nonlinear saionary y = δ y + δ y (1 exp[( γ s c ]) + u wih δ = 0. Model: ( ) Hypoheses: he null H 0L: δ = 0is esed agains H 1L: 0 Tes saisic: supˆ L, δ = 0 δ <. δ =-0.1 δ = -0.5 δ = -1 T=100 γ= γ= γ= c=0 Model A Model B Model C T=100 c= 0.3 Model A Model B Model C T=300 c= 0 Model A Model B Model C T=300 c= 0.3 Model A Model B Model C See he foonoe for Table 1. 20

22 Table 4: Rejecion raes of G, δ+ δ = 0 for esing global nonlinear saionariy es) y = δ y + δ y (1 exp[( γ s c ]) + u wih δ 0. Model: ( ) Hypoheses: he null hypohesis Tes saisic:. G, δ+ δ = 0 H 0 G : δ δ 0 + is esed agains H1 G : δ + δ < 0. δ =-0.3 & δ=0.1 δ = -0.5 & δ=0.2 δ = -0.7 & δ=0.3 T=100 γ= γ= γ= c=0 Model A Model B Model C T=100 c= 0.3 Model A Model B Model C T=300 c= 0 Model A Model B Model C T=300 c= 0.3 Model A Model B Model C

23 Table 5: Powers of sup ˆ he presence of srucural breaks L,δ =0 Model: ( ) es agains local nonlinear saionary alernaives in y = δ y + δ y (1 exp[( γ s c ]) + u wih δ = 0. Hypoheses: he null H : δ = 0is esed agains H : δ < 0 0L δ =-0.1 δ = -0.5 δ = -1 γ= γ= γ= T=300 c= 0 Model D Model E Model F c= 0.3 Model D Model E Model F L Table 6: Powers of G, δ+ δ =0 he presence of srucural breaks Model: ( ) es agains global nonlinear saionary alernaives in y = δ y + δ y (1 exp[( γ s c ]) + u wih δ 0. Hypoheses: he null hypohesis H 0 G : δ + δ 0 is esed agains H1 G : δ + δ < 0. δ =-0.3 & δ=0.1 δ = -0.5 & δ=0.2 δ = -0.7 & δ=0.3 γ= γ= γ= T=300 c= 0 Model A Model B Model C c= 0.3 Model A Model B Model C Dummy filered daa x = µ + β+ φid i + y 5 i= 1 22

24 Table 7: The resuls of uni roos Indonesian real exchange raes Model: ( ) 2 y = δ y + δ y (1 exp[( γ s c ]) + α y + u 1 1 j j j= 1 Hypoheses: he null H 0L: δ = 0is esed agains H 1L: δ < 0; es saisic: supˆ L, δ = 0 γ, c p Hypoheses: he null hypohesis H0 G : δ + δ 0 is esed agains H1 G : δ + δ < 0; es saisic:. ˆG, δ+ δ = 0. T=300 (approximaely) ˆ δ Model B: Indon-US (0.102) Indon-Jap (0.122) Indon-Sing (0.099) ˆ δ (0.232) (0.283) (0.349) ˆ γ ĉ ˆd supˆ L, δ 0 γ, c (0.311) (0.151) (0.520) ˆG, + = 0 p = δ δ Model C: Indon-US (0.112) Indon-Jap (0.128) Indon-Sing (0.123) (0.215) (0.235) (0.320) (0.254) (0.113) (0.488) Model E: Indon-US (0.070) Indon-Jap (0.055) Indon-Sing (0.090) (0.288) (0.264) (0.446) (0.255) (0.159) (0.499) Model F: Indon-US (0.101) Indon-Jap (0.099) Indon-Sing (0.079) (0.382) (0.481) (0.349) (0.100) (0.157) (0.601) Noe: The saisic sup ˆ was compued assuming δ =0. The ransiion variable s = y L,δ =0 -d. 23

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