A New Unit Root Test against Asymmetric ESTAR Nonlinearity with Smooth Breaks

Transcription

1 Iran. Econ. Rev. Vol., No., 08. pp. 5-6 A New Uni Roo es agains Asymmeric ESAR Nonlineariy wih Smooh Breaks Omid Ranjbar*, sangyao Chang, Zahra (Mila) Elmi 3, Chien-Chiang Lee 4 Received: December 7, 06 Acceped: April 9, 07 Absrac his paper proposes a new uni roo es agains he alernaive of symmeric or asymmeric exponenial smooh ransiion auoregressive (AESAR) nonlineariy ha accouns for muliple smooh breaks. We provide small sample properies which indicae he es saisics have good empirical size and power. Also, we compared small sample properies of he es saisics wih Chrisopoulos and Leon-Ledesma (00) es. he resuls indicae ha our uni roo es approach is superior o he es mehod of Chrisopoulos and Leon- Ledesma (00) for boh ransiion parameers (i.e. slow and fas speed), and he es power increases along wih he frequency. We apply our es saisics for examining he real ineres rae pariy hypohesis among OECD counries. Keywords: Uni Roo, Asymmery, ESAR, Smooh Breaks, Real Ineres Rae Pariy. JEL Classificaions: C, G5.. Inroducion his paper develops a new uni roo es o allow smooh breaks in he deerminisic componens and asymmeric nonlinear adjusmen. We also exend he Sollis (009) asymmeric exponenial smooh ransiion auoregressive (AESAR) nonlinear uni roo wih smooh breaks by means of a Fourier funcion. he paper is o se ou as follows. Secion inroduces new uni roo es and is consrucion of. Iran and rade Promoion Organizaion, Allameh abaaba'i Universiy, ehran, Iran (Corresponding Auhor: omid.ranjbar6@gmail.com).. Deparmen of Finance, Feng Chia Universiy, aichung, aiwan (ychang@mail.fcu.edu.w). 3. Faculy of Economics, Universiy of Mazandaran, Babolsar, Iran (z.elmi@umz.ac.ir). 4. Deparmen of Finance, Naional Sun Ya-sen Universiy, Kaohsiung, aiwan (cclee@cm.nsysu.edu.w).

2 5/ A New Uni Roo es agains Asymmeric ESAR criical values. Secion 3 analyzes he properies of small samples. Secion 4 provides an empirical applicaion aiming a he real ineres rae pariy hypohesis (RIRPH), and final secion concludes briefly.. Uni Roo es and AESAR Nonlineariy wih Muliple Smooh Breaks Suppose ha a series { } follows he daa generaing process (DGP) as y y (), () where ( ) Z n k n k, k sin( ) k, k k cos( ), () is a ime-varying deerminisic componen. In order o obain a global approximaion from he smooh ransiion and unknown number, and o equip deerminisic componens wih breaks, we follow Gallan (98) approach wih employing he Fourier approximaion and puing boh erms of n k n k k k sin( ) and k k cos( k ) ino he model. he reason o selec sin( ) and k cos( ) in he model is based on he fac ha a Fourier expression is capable of approximaing absoluely inegrable funcions o any desired degree of accuracy. Where k,, and are he number of frequencies of he Fourier funcion, sample size, and a rend erm, respecively, and π Z is an opional exogenous regressor which consiss of eiher a consan or a consan wih rend erm; n denoes he number of frequencies conained in he approximaion, and i saisfies n.

3 Iran. Econ. Rev. Vol., No., 08 /53 he esimaion of equaion () involves wo parameers choice - he choice of n and he choice of k. As noed by Becker e al. (004), i is reasonable o resric n= since join null hypohesis of s is rejeced for one frequency (i.e., 0 ), and ime invariance hypohesis, k, k is also rejeced. Similarly, Enders and Lee (0) noed ha he resricion n= is useful o save he degrees of freedom and prevens over-fiing problem. Hence we re-specify equaion () as follows: k k y Z sin( ) cos( ) () where, ] measures he ampliude and displacemen of he [ frequency componen. Paricularly he sandard linear specificaion is a special case of equaion () while seing 0. here mus be a leas one of he boh frequency componens exised if a srucural break is appeared. Becker e al. (004) uilize his propery of equaion () o develop a more powerful es o deec srucural breaks under an unknown form han Bai and Perron (003) es. In deermining an opimal k, we se he maximum of k equal o 5. For any K=k, we esimae equaion () employing ordinary leas squares (OLS) mehod and save he sum of squared residuals (SSR). Frequency k* is seing as opimum frequency a he minimum of SSR. Wih above assumpion and respec o he deerminisic componens, we es he following null hypohesis: H :, u (3) 0 where u is assumed o be an I(0) process wih zero mean. o es he null hypohesis, we follow Chrisopoulos and Leon-Ledesma (00) o calculae he saisic via hree seps shown in following. Firs sep: we se a maximum k equals o 5, and hen find ou opimal frequency of k* by employing he mehodology described above. We compue he OLS residuals as ha:

4 54/ A New Uni Roo es agains Asymmeric ESAR ˆ y ˆ() (4) ˆ k k ˆ () Z ˆ sin( ) ˆ cos( ) Second sep: a uni roo on he OLS residuals given from equaion (4) is esed by using AESAR model. he AESAR model combines exponenial funcion and logisic funcion ogeher, and i assumes he ˆ as a ransiion variable o saisfy ha (5) ˆ (, ˆ ){S (, ˆ ) ( S (, ˆ )) } ˆ ; ~ iid(0, ) (, ˆ ) exp( ( ˆ )) 0 S (, ˆ ) [ exp( ( ˆ ))] 0 o es he uni roo null hypohesis, unlike he alernaive hypohesis of globally saionary symmeric or asymmeric ESAR nonlineariy wih a uni roo cenral regime, we follow Sollis (009) model o es he null hypohesis H 0 : 0 in equaion (5) and propose wo aylor approximaions, in which one is for exponenial funcion (, ˆ ) around 0, and he oher is for logisic funcion S, ˆ ). We propose he following model: ( l 3 4 i i i ˆ ˆ ˆ ˆ. (6) l We include erms i ˆ i o avoid auocorrelaion wih he error i erm. he null hypohesis : 0 in equaion (5) becomes o es: H0 H 0 : 0. (7) As noed by Sollis (009), i is no allowed o calculae he sandard criical values o es null hypohesis H 0 : 0 in equaion (5). Hence, his paper ries o compue he finie-sample criical values via

5 Iran. Econ. Rev. Vol., No., 08 /55 wo alernaive models. he firs model is only assumed an inercep erm in equaion () (i.e., Z= []). he second model is equipped an inercep erm wih rend erm (i.e., Z= [,]). Wih seing k beween o 5 and sample sizes included 00, 00, 300, and 500 observaions, we compue asympoic criical values by Mone Carlo simulaion based on random walk model wih pseudo-iid N(0,) random number and 00,000 replicaions. he resuls of criical values are presened in able. Sample size able : Asympoic Criical Values Model wih inercep Model wih inercep and rend k= k= k=3 k=4 k=5 k= k= k=3 k=4 k=5 0% =00 5% % % =00 5% % % =300 5% % % =500 5% % Noe: Criical values were calculaed using Mone Carlo experimen based on 00,000 replicaions. When he null of uni roo is rejeced agains he alernaive of saionary symmeric or asymmeric ESAR nonlineariy, we are able o es he null hypohesis of symmeric ESAR nonlineariy by esing H 0 agains H 0 wih a sandard F-es (-es, 0 : : or Lagrange Muliplier (LM) es). hird sep: when he null hypohesis of uni roo is rejeced in he second sep, we furher es wheher he nonlinear componen in equaion () is absen. Following Becker e al. (006), we calculae he F-es saisic:

6 56/ A New Uni Roo es agains Asymmeric ESAR (SSR resriced SSR unresriced (k )) / F(k ), SSR unresriced (k ) / q (8) where SSR unresriced indicaes he SSR wih full regression equaion () and SSR resriced is he SSR wihou s from shor regression under he cases of wih and wihou nonlinear componen, respecively. Due o he presence of nuisance parameer, he F-es saisic has no sandard disribuion. We hereby employ he criical values abulaed in Becker e al. (006). 3. Finie-Sample Size and Power Properies 3. Finie-Sample Size Aiming a he es wih a finie-sample size, we consider a following DGP: * * k k y Z sin( ) cos( ) (9),,..., where ~ iid (0, ), {0., }, k * {,,3 }, sample size {00,300}, nominal size is 5%, and 0,000 replicaions. he values 0. and for and are relaed o an almos linear and nonlinear process respecively. he resuls are presened in able, and all repored sizes for hese DGPs are close o 5%. Model Model wih inercep Model wih inercep and rend able : Empirical Size of he es Saisic Parameer =00 =300 k= k= k=3 k= k= k= Noe: Nominal size is a he 5% level. Number of replicaions is 0,000.

7 Iran. Econ. Rev. Vol., No., 08 / Finie-Sample Power o invesigae he finie-sample power properies from he uni roo es, unlike globally saionary process, we consider a following Fourier-AESAR model as DGP: * * k k y Z sin( ) cos( ) (0) o o (,o ){S (,o ) ( S (,o )) }o ; ~ iid(0, ) (,o ) exp( (o )) 0 S (,o ) [ exp( (o ))] 0 where all combinaions of he parameers and frequencies values are specified o be ha { 0.05, 0., 0.3}, { 0.05, 0., 0.3, 0.7, 0.9, }, {0.,},, {0., }, k * {,,3}, ω ~ iid(0,), and 300. he values 0. and wih respec o η indicae respecively slow and fas ransiion. he small corresponds o symmeric ESAR nonlineariy, and he small linking wih a big generaes he large degree of asymmeric ESAR nonlineariy. In addiion, we compare our es resuls wih he finie-sample power wih he uni roo es developed by Chrisopoulos and Leon- Ledesma (00). We herefore calculae he criical values wih he uni roo es of Chrisopoulos and Leon-Ledesma (00) for boh models, in which one equipped only wih inercep, and anoher conains boh inercep and rend erms. he resuls are presened in able 3. Clearly, he resuls indicae (a) our es saisic resul is much beer han he es model of Chrisopoulos and Leon-Ledesma (00) in all cases, especially for he model wih inercep and rend erms. (b) When he degree of asymmery is large, our uni roo es is becoming more powerful (almos 0% and 3%, respecively) han he model wih boh inercep and rend in Chrisopoulos and Leon- Ledesma (00). Ineresingly, we find ha boh uni roo ess have equal power when symmeric ESAR nonlineariy exiss, and i hereby appears no subsanive loss in power if our uni roo es is subsiued wih he es of Chrisopoulos and Leon-Ledesma (00).

8 58/ A New Uni Roo es agains Asymmeric ESAR In addiion, he resuls also show ha our uni roo es approach is superior o he es mehod of Chrisopoulos and Leon-Ledesma (00) for boh ransiion parameers (i.e. slow and fas speed), and he es power increases along wih he frequency. 4. Empirical Applicaion In his secion, we employ our es saisics o re-es he RIRPH among OECD counries. We exend our daa se, compared wih he examinaion of Rapach and Wohar (004), over he period from 960Q4 o 0Q3. he nominal ineres rae and CPI variables are colleced from OECD Economic Indicaors. In order o es he RIRPH, we firs calculae real ineres rae and furher use he U.S as benchmark counry o compue he real ineres rae differenials series according o he following equaion (): f f (i ) (i ) RID () where i and raes, respecively, and denoe domesic nominal ineres rae and inflaion f i and f indicae respecively he U.S nominal ineres rae and inflaion rae. In nex sep, we es he uni roo hypohesis aiming a he RID series wih our es saisic when muliple smooh breaks are allowed in inercep model. he empirical resuls are presened in he able 4. he significan F saisic in he fourh column indicaes ha boh sine and cosine erms should be included in he esimaed model for all counries. Obviously, he join null hypohesis of uni roo es is rejeced for all counries excep for Canada. Also, he resuls in column five show ha he null of symmeric adjusmen for he RID is rejeced in Belgium, Neherlands, New Zealand, Norway, and UK.

9

10 * k able 3: Empirical Power Comparison a he 5% Nominal Level Sollis-Fourier model KSS-Fourier model (Chrisopoulos and Leon-Ledesma (00)) Consan model Consan & rend model Consan model Consan & rend model

11 * k Iran. Econ. Rev. Vol., No., 08 / able 3: Empirical Power Comparison a he 5% Nominal Level Sollis-Fourier model KSS-Fourier model (Chrisopoulos and Leon-Ledesma (00)) Consan model Consan & rend model Consan model Consan & rend model Noe: Empirical power were calculaed using Mone Carlo experimen based on 0,000 replicaions and =300.

12

13 6/ A New Uni Roo es agains Asymmeric ESAR able 4: Empirical Resuls for RIRPH Counry Opimum frequency Opimum lag F-es H saisic 0: 0 H 0 : 0 Ausralia * Belgium ** 0.968*** Canada Denmark * France *** Ireland *** Ialy *** Neherlands **.0*** New Zealand ** 7.300* Norway ** 0.478*** Swizerland *** Unied Kingdom ** 9.765*** Noe: he opimum lag order seleced based on he recursive -saisic. ***, ** and * indicae he null hypohesis is rejeced a he %, 5% and 0% levels, respecively. 5. Conclusion In his paper, we generalize he Sollis (009) AESAR nonlinear uni roo es wih allowing muliple smooh emporary breaks by calculaing means in Fourier funcion. he simulaion resuls of Mone Carlo sudies also show ha our empirical es approach is more reliable and is es saisic is much powerful. Lasly, we furher employ our es o examine he real ineres rae pariy hypohesis over OECD counries and find ha he RID series are shor-lived, nonlinear mean reversing wih muliple smooh breaks, and some adjusmens deviaed from equilibrium are asymmerical. References Bai, J., & Perron, P. (003). Compuaion and Analysis of Muliple Srucural Change Models. Journal of Applied Economerics, 8,.

14 6/ A New Uni Roo es agains Asymmeric ESAR Becker, R., Enders, W., & Lee, J. (004). A General es for ime Dependence in Parameers. Journal of Applied Economerics, 9, Becker, R., Enders, W., & Lee, J. (006). A Saionary es in he Presence of an Unknown Number of Smooh Breaks. Journal of ime Series Analysis, 7, Chrisopoulos, D. K., & Leon-Ledesma, M. A. (00). Smooh Breaks and Non-Linear Mean Reversion: Pos-Breon Woods Real Exchange Raes. Journal of Inernaional Money and Finance, 9, Enders, W., & Lee, J. (0). A Uni Roo es Using a Fourier Series o Approximae Smooh Breaks. Oxford Bullein of Economics and Saisics, 74(4), Gallan, R. (98). On he Basis in Flexible Funcional Form and an Essenially Unbiased Form: he Flexible Fourier Form. Journal of Economerics, 5, 353. Rapach, D. E., & Wohar, M. E. (004). he Persisence in Inernaional Real Ineres Raes. Inernaional Journal of Finance and Economics, 9, Sollis, R. (009). A Simple Uni Roo es agains Asymmeric SAR Nonlineariy wih an Applicaion o Real Exchange Raes in Nordic Counries. Economic Modelling, 6, 8 5.