MPRA Munich Personal RePEc Archive The seasonal KPSS Tes: some exensions and furher resuls Ghassen El Monasser Ecole supérieure de commerce de Tunis 10. March 014 Online a hp://mpra.ub.uni-muenchen.de/5490/ MPRA Paper No. 5490, posed. April 014 07:0 UTC
The seasonal KPSS Tes: some exensions and furher resuls Ghassen El Monasser * Universi of Mannouba École supérieure de commerce de Tunis Deparmen of quaniaive mehods March 014 Absrac The lieraure disinguishes finie sample sudies of seasonal saionari quie less inensel han i shows for seasonal uni roo ess. Therefore, he use of boh pes of ess for beer exploring ime series dnamics is seldom noiced in he relaive sudies on such a opic. Recenl, Lhagen (00) inroduced for quarerl daa he seasonal KPSS es which null hpohesis is no seasonal uni roos. In he same manner, as mos uni roo limi heor, he asmpoic heor of he seasonal KPSS es depends on wheher he daa has been filered b a preliminar regression. More specificall, one ma proceed o he exracion of deerminisic componens such as he mean and rend from he daa before esing. In his paper, I ook accoun of de-rending on he seasonal KPSS es. A skech of is limi heor was provided in his case. Also, I sudied in finie sample he behaviour of he es for monhl ime series. This could enrich our knowledge abou i since i was as I menioned above earl inroduced for quarerl daa. Overall, he obained resuls showed ha he seasonal KPSS es preserved is good size and power properies. Furhermore, like he es of Kwiakowski e al. [KPSS] (199), he nonparameric correcions of residual variances ma smooh he wide variaions of he seasonal KPSS empirical sizes. Kewords: KPSS es, deerminisic seasonali, Brownian moion, LM es JEL classificaion: C Time series models * E-mail: g.elmonasser@gmail.com 1
1. Inroducion Nowadas, he use of seasonall unadjused daa is on he increase. Behind ha, he inference disorion and he derimenal informaion loss in dnamic models could be caused b seasonal adjusmen. In his respec, Ghsels and Perron (199) showed ha seasonal adjusmen filers can seriousl affec seasonal uni roos. Furhermore, researchers, in heir ques o reveal as much ime series informaion, have shown grea ineres in suding heir unobserved componens. Especiall, several auhors have shown ha he seasonal and cclical componens are linked; see, iner alia, Canova and Ghsels (1994). Tha s wh he ssemaic eliminaion of he seasonal componen can generae non-rigorous deducions. However, having decided no o eliminae his componen, he quesion ha comes immediael afer: wha model should be given o seasonali? The lieraure has considered several differen seasonali models. The firs approach is o model seasonali as a deerminisic componen; see Barsk and Miron (1989). The second approach is o consider seasonali as a deerminisic variable wihin is saionar sochasic model; see Canova (199). Finall, he hird approach is o consider seasonali as sochasic. In his approach, he auhors concern is he developmen of seasonal uni roo ess. The es of Hlleberg e al. [HEGY] (1990) is now he preeminen seasonal uni roo es, wih is asmpoic orhogonali as a ke proper allowing is generalizaion a an observaional frequenc. The subsequen rejecion of heir null hpohesis implies a srong resul ha he series exhibis a saionar seasonal paern, bu heir es is found o suffer from he problem of low power in moderae sample sizes. In agreemen wih wha was found in he convenional case, Hlleberg (1995) suggesed he join use of seasonal uni roo and saionari ess. Lieraure was relaivel small in seasonal saionari ess. One can refer o he ess of Canova and Hansen (1995) and Caner (1998). The difference beween he wo ess lies a he correcion of he error erm when he sandard assumpions, which i should verif in regression analsis, do no appl. The firs es used a non-parameric correcion like he es of Kwiakowski e al. [KPSS] (199) and he second used a parameric correcion. Likewise, Lhagen (00) proposed anoher version of he KPSS es in he seasonal conex which resuled in a frequenc-based es. More explicil, Lhagen (00) esed he hpohesis of level saionari agains a single seasonal uni roo. Thus, his es can be ermed seasonal KPSS es. I was shown b Khedhiri and El Monasser (010) wih a Mone Carlo mehod ha he seasonal KPSS es is robus o he magniude and he number of addiive ouliers. Furhermore, he
saisical resuls obained cas an overall good performance of he finie-sample properies of he es. Khédhiri and El Monasser (01) provided a represenaion of he seasonal KPSS es in he ime domain and esablished is asmpoic heor. This represenaion allows he generalizaion of he es s asmpoic heor when he basic equaion incorporaes oher addiional dnamics. This paper differs from Khédhiri and El Monasser (01) in ha i akes ino accoun he presence of a linear rend in he basic equaion of he seasonal KPSS es and he monhl observaional frequenc. In doing so, I will conduc a Mone Carlo analsis o sud he es s properies of size and power in such circumsances. In addiion, a skech of is asmpoic heor is provided in he presence of a linear rend. The ouline of his paper is as follows. In secion, I inroduce some preliminaries of he seasonal KPSS. In secion, I conduc a Mone Carlo simulaion sud o assess he finie sample properies of he es in erms of is size and power performance when including a linear rend in is basic equaions. Also, I consider in his sud he effec of he observaional frequenc on he es properies. To his aim, I involve monhl daa. The las secion concludes.. Preliminaries on he Seasonal KPSS Tes Le be a ime series observed quarerl. Since he goal is o es for he presence of negaive uni roo, i would be suiable o use he appropriae filer in order o isolae he effecs of oher uni roos in he series. Therefore, he es will be applied o he ransformed series: (1 L L L ), where L is he lag operaor. This ransformaion is obained from he (1) 4 seasonal difference filer 1 L (1 L)(1 L)(1 L ) (1 L )(1 L ). Nex, one es he uni roo of 1 in he series (1) x r u, 1,..., T, (1) where 4 T 4N, x ai Di and he shorhand noaion D i ( i, 4[( 1) / 4]) and also i1 where [.] denoes he larges ineger funcion and ( i, j) is he Kronecker s funcion.
The erm u is zero mean weakl dependen process wih auocovariogram h E( uuh ) and a sricl posiive long run variance. The componen r is drawn from he following process: u r r 1 v, () where v is zero mean weakl process wih variance and long run variance 0. v v The ransformaion needed o carr ou he seasonal KPSS es for complex uni roos given b he following variable, i is (1 L ). () The es of such complex uni roos is based on he regression, ( x c e, () ) where x e is zero mean weakl dependen process wih long run variance 0 and 4 i1 b D i i. The componen c is given b c, (4) c where is anoher zero mean weakl dependen process wih variance long run variance. e and sricl posiive Adding he deerminisic erms in (1) and () is ver imporan because i allows he seasonal KPSS es o include deerminisic seasonali. The esing procedure follows in wo seps: Firs, he uni roo of -1 is esed, and hen he complex roos are esed where heir null hpohesis will be specified hereafer. The seasonal KPSS es is a Lagrange Muliplier-based es. Hence, he null hpohesis of a roo 0 v equals o -1 is H : 0. Under his null hpohesis, (1) is wrien as: (1) x u, (5) where he series is rend saionar afer seasonal mean correcion. Under he alernaive hpohesis H : 0, 1 v (1) has a uni roo corresponding o Nquis frequenc. 4
Le u ~ be he residual series obained from leas squares regression applied o equaion (5), 1,,...,T. Following Breiung and Franses (1998, eq. (18), p. 09), and Busei and Harve (00, eq. (8), p. 4) and Talor (00, eq. (8), p. 05), we replace he long-run variance b an esimae of ( imes) he specrum a he observed frequenc in order o deal wih uncondiional heeroscedasici and serial correlaion. This nonparameric esimaion of he long-run variance is a useful soluion o he nuisance parameer problem (Talor, 00). Thus, for he Nquis frequenc, his nonparameric esimaion is wrien as follows: u where he weigh funcion as T l T ~ u, k k 1 k 1 k 1 1 ~ 1 ( l) T u T w( k, l)( u~ u~ )cos( ), () k w ( k, l) 1 and l is a lag runcaion parameer such ha l l 1 T and l o( n 1/ ). Now from equaion (), I choose a Barle kernel following Newe and Wes (1987). I should be noed ha Andrews (1991) showed ha such a runcaion lag can produce good resuls in pracice, as also shown in KPSS (199). Similarl, he null hpohesis of he es regarding complex uni roos is given b H : 0. Under his null hpohesis, wrien as follows: 0 () is ( x e (7) ) Using he residuals e ~ obained from he leas squares regression of equaion (7), he Barle kernel esimaor of e can be compued as follows: ~ e, T l T 1 ~ 1 ( l) T e T w( k, l)( e~ e~ k )cos( k) (8) 1 k1 k1 Define he parial sums ~ i j S e u~ j j1 and ~ i j P e e~ j1. I follows ha he es saisics for uni roo of -1 is given b: ( ) T ~ ~ S S 1 1 ~ (9) T ( l) u, This saisic ma be wrien for he complex uni roos, as 5
T ~ ~ P P 1 1 ~, (10) T ( l) e, where S ~ and P ~ are he conjugae numbers of S ~ and P ~, respecivel. ( ) Khédhiri and El Monasser (01) have shown under H : 0, V ( r) dr where 0 v 1 d 0 V (r) is a sandard Brownian bridge, d denoes weak convergence in probabili and R I r [0,1]. However, for H 0 : 0, he auhors have shown ha d [ V ( ) V ( ) ] d 0 1 1 R I where V ( ) and V ( ) are wo independen sandard Brownian bridges and [0,1 ]. Remark 1: Asmpoicall has he firs level Cramer-von Mises disribuion ( CvM 1 ) under he null hpohesis while he limi heor of ( / ) was shown as a funcion of a generalized Cramer-von Mises wih wo degrees of freedom. Specificall, he asmpoic heor of his 1 saisic is as follows: d CvM 1(). The reader can refer o Anderson and Darling (195) for his pe of disribuions. The criical values of he seasonal KPSS es wih seasonal dummies can be compued from Nblom (1989) or from Canova and Hansen (1995). These criical values are also shown in Table 1 of Khédhiri and El Monasser (01). Remark : I can be shown ha he seasonal frequenc has no effec on he asmpoic disribuion of es saisics. In oher words, ma reain he same limi disribuion as above and he saisic associaed wih he complex uni roos in quesion has he same limi disribuion as ( ). Onl he se of seasonal uni roos will change and i ma no include he uni roo which corresponds o he Nquis frequenc, i.e. when he periodici is odd. Remark : Recall ha if here is a ime rend in he regression of he sandard KPSS es, he parial sum of residuals from a firs order polnomial regression weakl converges o a second level Brownian bridge denoed B where, as in McNeill (1978),
1 B 1 ( r) W ( r) rw (1) r(1 r) W (1) W ( s) ds 0, (11) wih W (.) being a sandard Wiener process or Brownian moion. Then he es saisic follows he so-called second level Cramer von Mises disribuion; see Harve (005). However, his resul canno be generalized o seasonal KPSS es. Indeed, he saisics follows he so called zero level Cramer von Mises noed CvM 0 ; see Harve (005). Specificall, 1 ( ) d W ( r) dr. (1) 0 Meanwhile, when he deerminisic componen is represened b onl a rend in Eq. (), i can be shown ha 1 d CvM 0 (). (1) The criical values of he seasonal KPSS es in his case can be obained from Nblom (1989, Table 1) and he are shown in Table 1. Table 1: Criical values of he seasonal KPSS es in he case of firs order polnomial rend 1% 5% 10% Roo -1.787 1.5 1.19 Roos i 1.945 1.10 1.01 Even hough onl a consan is included in he Eqs. (1) and (), hese criical values are sill appropriae. These findings show indeed ha he generalizaion of he asmpoic resuls of he sandard KPSS es should no be done in an auomaic wa, bu raher i is advisable o ake some serious reflecion o esablish equivalen resuls for he seasonal KPSS es.. The Mone Carlo Analsis To evaluae he size performance of he seasonal KPSS saisic in presence of firs order linear rend, I conduc Mone Carlo simulaion experimens wih seasonal roos of a quarerl process. The daa generaing process (DGP) for he negaive uni roo is 7
x r, 1,..., T, (14a) where b: x onl represens a firs order linear rend and he auoregressive process r is given r r v, (14b) 1 The error erms v are normall disribued wih zero mean and uni variance. The DGP for complex uni roos is given b: x c, 1,..., T, (15a) where x onl includes a firs order linear rend and he process c is given b: c, (15b) c are normall disribued wih zero mean and uni variance. I choose alernaive values of 1, 0.8, 0., 0, 0., 0.8 and I onl consider he 5% nominal size. The bandwidh values chosen in our experimens are given b: l 0 0, 4 4 l ineger ( /100) 1/ 4 T and 1 4 l ineger ( /100) 1/ 1 T. I use 0000 replicaions and all he simulaion experimens were carried ou wih Malab programs. The corresponding resuls are summarized in Table. Table : Rejecion frequencies for he seasonal KPSS es wih a firs order polnomial rend for seasonal quarerl uni roos, significance level: 5% (size and power) (1) T l 0 l 4 l 1 l 0 l 4 l1 ( i) -1 80 0.949 0.718 0.17 0.9774 0.909 0.8 00 0.99 0.8498 0.5981 0.9987 0.974 0.818-0.9 80 0.7894 0.415 0.110 0.91 0.7575 0.99 00 0.898 0.5981 0.1481 0.97 0.794 0.841-0. 80 0.114 0.054 0.010 0.18 0.078 0.04 00 0.1158 0.0577 0.098 0.141 0.071 0.040 0 80 0.0514 0.098 0.017 0.0510 0.098 0.017 00 0.05 0.047 0.09 0.0479 0.045 0.01 0. 80 0.0181 0.09 0.0145 0.01 0.004 0.011 00 0.014 0.08 0.0 0.0100 0.055 0.04 0.9 80 0.00 0.005 0.00 0.00 0.0019 0.0007 00 0.00 0.000 0.0074 0.00 0.00 0.0007 8
Wha has been observed in Table of Khédhiri and El Monasser (01) sill be seen from Table. Indeed, he size of he es increases wih decreasing values of. Similarl, he sample size does no noiceabl affec he es s size no markedl improved wih he non-parameric correcions (l4 ) and (l 1 ). To see he effec of observaional frequenc on he seasonal KPSS es in finie samples, monhl periodici is aken ino accoun. I onl consider a deerminisic seasonali. More specificall, I assume ha he deerminisic componen is represened b 1 seasonal dumm 11 variables. Remember ha seasonal uni roos exhibied b he filer S( L) (1 L L... L ) i corresponding o he seasonal frequencies i, i 1,,.... For size experimens, I 1 assume a paricular value of he null hpohesis specifing an i.i.d. process as a daa generaing process and corresponding o 0 in (14b) and (15b) for he quarerl case. For power experimens, I suppose ha he process r, for he seasonal frequencies oher han he Nquis one, is oulined b: r i 1 cos r r, i 1,,...5. (1) However, when he process shows a uni roo corresponding o he Nquis frequenc, i will be generaed b: r r 1 v (17) The considered sample sizes are T=40 e T=00 which displa he same number of ears as in able. As menioned above, he criical values of he es are obained from Table 1 of Khédhiri and El Monasser (01) where he firs line corresponds o he uni roo -1 and he second one o complex uni roos. 9
Table : The size of he seasonal KPSS es for monhl daa, significance level 5% T=40 T=00 0.058 0.05 5 5 5 ( l1) ( l1) ( l1) ( l1) ( l1) 0.055 0.0491 0.047 0.0470 0.0581 0.054 0.051 0.0500 0.0457 0.0471 0.0590 0.057 0.054 0.0499 0.047 0.049 0.059 0.05 0.058 0.0505 0.0471 0.048 0.058 0.0515 0.059 0.050 0.0454 0.0474 ( ) 0.058 0.050 ( ) 0.058 0.0500 ( ) ( l1) 0.0444 0.0459 10
Table 4: The power of he seasonal KPSS es for monhl daa, significance level 5% T=40 T=00 1 1 5 5 5 ( l1) ( l1) ( l1) ( l1) ( l1) 0.9988 0.9998 0.949 0.9944 1 1 0.9990 1 0.95 0.9949 1 1 0.9987 1 0.9517 0.994 1 1 0.998 1 0.9515 0.994 1 1 0.998 1 0.9450 0.994 ( ) 1 1 ( ) 0.948 0.999 ( ) ( l1) 0.7744 0.9194 11
According o Table, all empirical rejecion frequencies approach he heoreical significance level of 5%. This shows indeed an excellen empirical size no subjec o an disorion. Also, he increase of sample size mosl resuls in a sligh decrease in size. Table 4 shows again ha he seasonal KPSS es for monhl daa preserves is good power properies. In his regard, a reducion of power corresponding o he roo of -1 and he funcion l1 is noable bu no surprising. Indeed he value 0.7744 ha appears in he las box of he firs column of Table 4 is ver close o he values provided b KPSS (199, Table 4) for he convenional uni roo. This similari is due o he mirror effec siuaion ha occurs beween he uni roos a frequencies zero and. 4. Conclusion As poined b Hlleberg (1995), he mos imporan reserve agains he seasonal uni roo es was ha he null hpohesis of a uni roo a he seasonal frequencies is problemaic because seasonal uni roo allows more variaion in he seasonal paern ha is acuall observed. So if he daa generaing process (DGP) is a seasonal uni-roo- process, winer ma become summer. Anoher limiaion going along wih he firs one is manifesed b he fac ha he HEGY es, like he Dicke-Fuller es, has low power agains reasonable alernaives and ha he exisence of moving average erms wih roos close o he uni circle impl ha he power is almos equal o he size. Even hough here were some recommendaions o handle such siuaions, ineres has been graned for he consrucion of ess wih beer properies han he exising ones eiher agains similar or differen alernaives or for differen esablished assumpions. In his research specrum, one ma refer o he ess of Canova and Hansen (1995) and Lhagen (00) adoping a ver similar framework. In his paper, I sudied he finie sample properies of he second one in presence of a linear rend and also b considering a monhl periodici. The effec of changing observaional frequencies should be sudied since his es was earl se for quarerl daa. The boom line of his Mone Carlo sud is ha he seasonal KPSS es preserves good size and power properies boh in including a linear rend and considering monhl ime series. Moreover, is empirical rejecion frequencies ofen approximae nominal sizes when using he nonparameric correcions of he residual variances. The exension of he seasonal KPSS o a vecor of ime series is a fuure avenue of research. In ha framework, one can examine if a se of daa exhibi a common deerminisic seasonali. This exension would be analogous o ha which Nblom and Harve (000) made o he KPSS es. 1
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