Appled Mahemacal Scences Vol. 008 no. 8 363-379 Tesng he Null Hypohess of no Conegraon agans Seasonal Fraconal Conegraon L.A. Gl-Alana Unversdad de Navarra Faculad de Cencas Economcas Edfco Bbloeca Enrada Ese E-31080 Pamplona Span alana@unav.es Absrac In hs arcle we propose a procedure for esng he null hypohess of no conegraon agans he alernave of seasonal fraconal conegraon. I s a wosep procedure based on he unvarae ess of Robnson 1994. Fne-sample crcal values are compued and he power properes of he ess are examned. The ess are also exended o allow seasonally fraconally conegraed alernaves a each of he seasonal frequences separaely. An emprcal applcaon llusrang he use of he ess s also carred ou a he end of he arcle. Mahemacs Subec Classfcaon: 6P0; 91B70 Keywords: Seasonal fraconal conegraon; Long memory; Seasonaly. 1. Inroducon Modellng he seasonal componen of macroeconomc me seres s a maer ha sll remans conroversal. Seasonal dummy varables were nally employed bu hey were shown o be napproprae n many cases especally f he seasonal componen changes or evolves over me. Followng he un roo approach na lly developed by Box & Jenkns 1970 and wdely used afer he semnal paper by Nelson & Plosser 198 seasonal un roo models became popular and many es sascs of hs ype were developed by Dckey Hasza & Fuller DHF 1984; Hylleberg Engle Granger & Yoo HEGY 1990; Canova & Hansen 1995 and ohers. Seasonal un roo models were laer exended o allow for oher ypes of long memory behavour n parcular allowng for a fraconal degree of negraon see e.g. Porer-Hudak 1990; Ray 1993; Suclffe 1994 and more recenly Gl-Alana & Robnson 001. The dea behnd he concep of seasonal fraconal negraon s ha he number of seasonal dfferences requred
364 L. A. Gl-Alana o oban saonary mgh no necessarly be an neger bu a real value. Thus assumng ha s s he number of me perods whn a year he seasonal polynomal 1 - L s d can be expressed n erms of s Bnomal expanson such ha for all real d s d d s s d d 1 s 1 L 1 L 1 d L + L... 0 and hgher he d s he hgher s he level of assocaon beween he observaons far apar n me. The concep of seasonal fraconal conegraon has hardly been nvesgaed. For he purpose of he presen paper we say ha a gven vecor Y s seasonally fraconally conegraed f: a all s componens y are seasonally fraconally negraed of he same order say d.e. 1 L s d y u for all where u s an I0 process defned as n Secon and b here s a leas one lnear combnaon of hese componens whch s seasonally fraconally negraed of order b wh b < d. Oher more complex defnons of seasonal fraconal conegraon allow us o consder dfferen orders of negraon for each of he ndvdual seres. However Robnson & Marnucc 001 show ha n a bvarae conex a necessary condon for conegraon s ha boh ndvdual seres share he same order of negraon. In he conex of fraconal processes hs assumpon may appear unrealsc because of he connuy on he real lne for he orders of negraon. However n emprcal work here mgh be cases when even hough he orders of negraon of boh seres are fraconal and dfferen he ess are unable o reec he un roo null d 1. In such cases we can proceed furher wh conegraon analyss. In hs arcle we propose a wo-sep procedure for esng he null hypohess of no conegraon agans he alernave of seasonal fraconal conegraon whch s based on he unvarae ess of Robnson 1994. The oulne of he paper s as follows: Secon frsly descrbes he ess of Robnson 1994. Then he wo-sep procedure agans seasonally fraconally conegraed alernaves s presened. Secon 3 gves fne-sample crcal values of he new ess and he esng procedure s exended o he case of fraconally conegraed alernaves for each of he frequences separaely. The power properes of he ess agans dfferen fraconal alernaves are examned n Secon 4. Secon 5 conans an emprcal applcaon and fnally Secon 6 offers some concludng remarks.. The ess of Robnson 1994 and seasonal fraconal conegraon We frsly descrbe a verson of he unvarae ess of Robnson 1994 for esng seasonally fraconally negraed hypoheses. Assume ha y s he observed me seres 1 T and consder he followng model s d 1 L y u 1... 1
Seasonal fraconal conegraon 365 where L S s he seasonal lag operaor L S y y -s ; d s a real number and u s an I0 process defned as a covarance saonary process wh specral densy funcon ha s posve and fne a any frequency on he nerval [0 π]. Clearly f d 0 n 1 y u and a weakly auocorrelaed y s allowed for. However f d > 0 y s defned as a long memory process also called srongly dependen and so-named because of he srong assocaon n he seasonal srucure beween observaons far apar n me. If d 0 0.5 y s covarance saonary havng auocovarances whch decay much more slowly han hose of a seasonal ARMA process n fac so slowly as o be non-summable; f d 0.5 y s nonsaonary and as d ncreases beyond 0.5 can be vewed as becomng more nonsaonary n he sense for example ha he varance of he paral sums ncreases n magnude. Noe ha he varance of he paral sums s OT d+1 so ha saonary mples d > 0.5. See Hoskng 1981. Few emprcal applcaons can be found based on seasonal fraconal models. The noon of fraconal Gaussan nose wh seasonaly was nally suggesed by Abrahams & Dempser 1979 and Jonas 1981 and exended n a Bayesan framework by Carln Dempser & Jonas 1985 and Carln & Dempser 1989. Porer-Hudak 1990 appled a seasonally fraconally negraed model o quarerly U.S. moneary aggregaes wh he concluson ha a fraconal model could be more approprae han sandard ARIMAs. The advanages of seasonally fraconally negraed models for forecasng are llusraed n Ray 1993 and Suclffe 1994 and anoher recen emprcal applcaon can be found n Gl- Alana & Robnson 001. In general we wan o es he null hypohess: Ho : d do for a gven real number d o and he es sasc proposed by Robnson 1994 whch s based on he Lagrange Mulpler LM prncple s gven by: 1/ T â rˆ 3 Â ˆ σ where * * ˆ π 1 π 1 π σ σ ˆ τ g ; τˆ I ; â ψ g ; τˆ I ; T T T 1 * * * * Â ˆ ' ˆ ˆ ' ˆ T ψ ψ ε ε ψ ε ε ψ log sn + log cos + log cos ; εˆ log g ; τˆ. τ s I s he perodogram of û where û 1 L do y and he funcon g above s a known funcon comng from he specral densy of u f ; τ σ / π g ; τ evaluaed a ˆ τ arg mn * σ τ where T * s a τ Τ compac subse of he R q Eucldean space. Fnally he summaon on * n he
366 L. A. Gl-Alana above expressons s over M where M {: -π < < π ρ l - 1 ρ l + 1 l 1 s} such ha ρ l 0 π/ -π/ and π are he dsnc poles of ψ on -π π]. Noe ha hese ess are purely paramerc and herefore hey requre specfc modellng assumpons abou he shor memory specfcaon of u. Thus f u s whe nose g 1 and hus ε ˆ 0 and f u s an AR process of form φlu ε g φe - wh σ Vε so ha he AR coeffcens are a funcon of τ. Under he null hypohess Robnson 1994 showed ha under ceran regulary condons rˆ d N01 as T. 4 These condons are very mld regardng echncal assumpons on ψ whch are sasfed by model 1. Thus an approxmae one-sded es of agans H a : d > d o wll be gven by he rule: Reec H o f rˆ > z α where he probably ha a normal varae exceeds z α s α and conversely a es of agans H a : d < d o wll be gven by he rule: Reec H o f rˆ < -z α. As hese rules ndcae we are n a classcal large sample esng suaon for he reasons spel ou n Robnson 1994 who also showed ha he ess are effcen n he Pman sense.e. ha agans local alernaves of form: H a : d d o + δ T -1/ for δ 0 rˆ has an asympoc dsrbuon gven by a normal dsrbuon wh varance 1 and mean ha canno when u s Gaussan be exceeded n absolue value by any rval regular sasc. The es sasc presened us above was used n Gl-Alana & Robnson 001 o sudy he seasonal quarerly srucure of he UK and Japanese consumpon and ncome. Oher versons of Robnson s 1994 ess based on annual seasonal monhly and cyclcal models were suded n Gl-Alana & Robnson 1997 and Gl-Alana 1999 001 respecvely. Ooms 1997 also proposed ess based on seasonal fraconal models. They are Wald ess and hus requre effcen esmaes of he fraconal dfferencng parameers. He used a modfed perodogram regresson esmaon procedure due o Hassler 1994. Also Hosoya 1997 esablshed he lm heory for long memory processes wh he sngulares no resrced a he zero frequency and proposed a se of quas log-lkelhood sascs o be appled n raw me seres. Unlke hese mehods he ess of Robnson 1994 do no requre esmaon of he long memory parameers snce he dfferenced seres have shor memory under he null. Nex we nroduce a esng procedure based on rˆ n 3 for esng he null hypohess of no conegraon agans he alernave of seasonal fraconal conegraon. For smplcy we consder a bvarae sysem of wo me seres y 1 and y ha mgh be seasonally fraconally conegraed. In hs bvarae conex a necessary condon for conegraon s ha boh seres mus have he same degree of negraon say d o. Thus n he frs sep we can use Robnson s 1994 unvarae ess descrbed above o es he order of negraon of each of he ndvdual seres and f boh are seasonally negraed of he same order say d o 1 we can go furher and es he degree of negraon of he resduals from he conegrang regresson. There also exs mulvarae versons of he ess of Robnson 1994 for smulaneously esng he degree of negraon of he ndvdual seres e.g. Gl-Alana 003a. Ths procedure however has only been
Seasonal fraconal conegraon 367 developed for non-seasonal cases and he exenson o he seasonal case s sll n progress. I mgh be argued ha he use of Robnson's 1994 ess on he ndvdual seres s no adequae snce he wo seres may be dependen. In general hs s a problem ha s faced by all unvarae procedures. Noe however ha hs s he same approach as he one used by Engle & Granger 1987 n her classcal paper on conegraon and also by Cheung & La 1993 and Dueker & Sarz 1998 when esng for conegraon a he long run frequency. A problem occurs here as he resduals are no acually observed bu obaned from mnmzng he resdual varance of he conegrang regresson and hus a bas mgh appear n favour of saonary resduals. Noe ha hs problem s smlar o he one noced by Engle & Granger 1987 when esng conegraon a he long run or zero frequency wh he ess of Fuller 1976 and Dckey & Fuller 1979. See also Phllps & Oulars 1991 and Kremers Ercsson & Dolado 199. In order o solve hs problem fne-sample crcal values of he ess wll be compued n he nex secon. We can consder he model: s d 1 L e v 1... where e are he OLS resduals from he conegraon regresson of y 1 on y or vceversa and v s I0 and es H o agans he alernave: Ha : d < do. 5 Noe ha f we canno reec H o on he esmaed resduals above we wll fnd evdence of no conegraon snce he resduals wll be negraed of he same order as he ndvdual seres. On he oher hand reecons of H o agans 5 d < d o wll suppor fraconal conegraon snce he esmaed resduals wll be negraed of a smaller order han ha of he ndvdual seres. 3. Fne-sample crcal values and exensons of he ess Table 1 repors fne-sample crcal values of he ess of Robnson 1994 for esng he null hypohess of no conegraon agans seasonal fraconal conegraon. We use Mone Carlo smulaons based on 50000 replcaons for sample szes T 48 96 144 and 19 assumng ha he rue sysem consss of wo quarerly Id seasonal processes of he form: 4 1 L do y ε 1... 1 6 wh Gaussan ndependen whe nose dsurbances ha are no conegraed and ake values of d o rangng from 0.6 hrough 1.5 wh 0.1 ncremens. We have concenraed on values of d > 0.5 snce mos macroeconomc me seres are nonsaonary hough he resuls based on d < 0.5 are no subsanally dfferen from hose repored n he ables. We can see ha he fne-sample crcal values are smaller han hose from he normal dsrbuon whch s conssen wh he earler argumen ha when esng H o agans 5 he use of he sandard crcal values wll resul n he conegrang ess reecng he null hypohess of no conegraon oo ofen. We can also noe ha hese crcal values are smlar around d o and as we
368 L. A. Gl-Alana ncrease he number of observaons hey approxmae he values from he normal dsrbuon. The seasonal srucure descrbed n he precedng secon may be oo resrcve n he sense ha mposes he same degree of negraon a each of he frequences of he process. Noe for example ha he polynomal 1 L 4 can be facored as 1 - L 1 + L 1 + L conanng four roos of modulus uny: one a he long run or zero frequency one a wo cycles per year correspondng o he frequency π and wo complex pars a one cycle per year correspondng o frequences π/ and -π/. Thus he seasonal process 6 mposes he same degree of negraon d o a each of hese frequences. Nex we consder he possbly of wo me seres beng conegraed a a sngle frequency.e. followng he same srucure a a gven frequency for a gven value d o. We examne he possbly of her beng conegraed eher a he zero frequency or alernavely a he seasonal frequences π or π/ -π/.
Seasonal fraconal conegraon 369 TABLE 1 Fne-sample crcal values of he ess of Robnson 1994 for esng he null hypohess of no conegraon agans seasonal fraconal conegraon T 48 P. / d o 0.50 0.60 0.70 0.80 0.90 1.00 1.10 1.0 1.30 1.40 1.50 0.1 % -3.16-3.16-3.0 -.1-3.1-3.17-3.19-3.0-3.0-3.0-3.19 0.5 % -.94 -.93 -.93 -.94 -.95 -.95 -.95 -.95 -.95 -.95 -.95 1 % -.81 -.81 -.8 -.8 -.84 -.84 -.84 -.83 -.84 -.83 -.84 % -.68 -.69 -.70 -.71 -.71 -.71 -.71 -.70 -.70 -.70 -.70.5 % -.64 -.65 -.66 -.66 -.66 -.66 -.66 -.66 -.66 -.66 -.66 5 % -.49 -.50 -.50 -.51 -.51 -.51 -.51 -.51 -.51 -.51 -.51 10 % -.30 -.31 -.33 -.3 -.3 -.3 -.3 -.3 -.3 -.3 -.3 T 96 P. / d o 0.50 0.60 0.70 0.80 0.90 1.00 1.10 1.0 1.30 1.40 1.50 0.1 % -.94 -.94 -.93 -.93 -.93 -.93 -.93 -.93 -.93 -.93 -.93 0.5 % -.66 -.68 -.68 -.67 -.67 -.67 -.66 -.66 -.66 -.66 -.66 1 % -.55 -.56 -.56 -.56 -.56 -.55 -.56 -.56 -.55 -.55 -.55 % -.40 -.40 -.40 -.40 -.40 -.40 -.40 -.40 -.40 -.39 -.39.5 % -.35 -.34 -.35 -.35 -.35 -.35 -.35 -.35 -.34 -.34 -.34 5 % -.19 -.19 -.19 -.18 -.18 -.18 -.18 -.18 -.17 -.17 -.17 10 % -1.98-1.99-1.99-1.98-1.98-1.98-1.97-1.97-1.97-1.97-1.97 T 144 P. / d o 0.50 0.60 0.70 0.80 0.90 1.00 1.10 1.0 1.30 1.40 1.50 0.1 % -.80 -.8 -.84 -.84 -.8 -.83 -.83 -.81 -.80 -.78 -.78 0.5 % -.53 -.53 -.5 -.5 -.51 -.51 -.5 -.51 -.51 -.51 -.51 1 % -.39 -.39 -.41 -.41 -.41 -.40 -.41 -.40 -.41 -.41 -.40 % -.4 -.5 -.5 -.5 -.5 -.4 -.4 -.4 -.4 -.4 -.4.5 % -.0 -.0 -.0 -.0 -.19 -.18 -.19 -.19 -.19 -.19 -.19 5 % -.0 -.03 -.03 -.03 -.03 -.03 -.0 -.03 -.0 -.0 -.0 10 % -1.8-1.8-1.8-1.83-1.8-1.8-1.81-1.81-1.81-1.81-1.81 T 19 P. / d o 0.50 0.60 0.70 0.80 0.90 1.00 1.10 1.0 1.30 1.40 1.50 0.1 % -.77 -.77 -.76 -.75 -.7 -.7 -.73 -.73 -.74 -.74 -.74 0.5 % -.5 -.51 -.53 -.51 -.49 -.48 -.49 -.49 -.49 -.50 -.51 1 % -.34 -.35 -.35 -.34 -.35 -.36 -.36 -.35 -.35 -.34 -.34 % -.17 -.17 -.18 -.18 -.18 -.17 -.18 -.18 -.18 -.18 -.18.5 % -.1 -.1 -.13 -.13 -.13 -.13 -.13 -.13 -.1 -.13 -.1 5 % -1.94-1.95-1.95-1.95-1.94-1.94-1.94-1.94-1.94-1.94-1.93 10 % -1.71-1.71-1.71-1.71-1.71-1.71-1.71-1.70-1.70-1.70-1.70 50000 replcaons were used n each case. The procedure s exacly he same as before. Once we have shown ha boh seres are negraed of he same order d o a a gven frequency we es H o agans 5 wh he one-sded ess of Robnson 1994 n he model
370 L. A. Gl-Alana... 1 v e L 1 d 7 f we focus on he long run or zero frequency or alernavely n he models... 1 v e L 1 d + 8 or... 1 v e L 1 d + 9 f we concenrae on he seasonal frequences π or π/ -π/ respecvely. Based on 7 8 and 9 he es sascs wll be gven by rˆ 1 and 3 respecvely where 3 1 ˆ â Â T rˆ 1/ σ 10 wh τ ψ π * 1 I ˆ ; g T â ˆ ' ˆ ˆ ' T Â * * 1 * * ψ ε ε ε ε ψ ψ ; cos log ; cos log ; sn log 3 1 ψ ψ ψ and g ˆ ˆ ε σ and I as below 3 bu for he new resduals obaned from 7 8 and 9. Fne-sample crcal values of he new versons of he ess were also compued and he resuls are gven n Table.
Seasonal fraconal conegraon 371 TABLE Fne-sample crcal values of he ess of Robnson 1994 for esng he null hypohess of no conegraon agans seasonal fraconal conegraon a a gven frequency T 48 ρl P./ d o 0.50 0.60 0.70 0.80 0.90 1.00 1.10 1.0 1.30 1.40 1.50 1 - L d 1 % -.56 -.57 -.56 -.55 -.55 -.53 -.5 -.53 -.53 -.51 -.50 5 % -.10 -.11 -.11 -.11 -.11 -.10 -.09 -.08 -.08 -.08 -.06 1 + L d 1 % -.49 -.51 -.5 -.5 -.51 -.51 -.51 -.51 -.50 -.48 -.47 5 % -.03 -.05 -.05 -.05 -.04 -.04 -.04 -.03 -.03 -.03 -.0 1 + L d 1 % -.96 -.97 -.97 -.97 -.97 -.98 -.97 -.98 -.90 -.96 -.96 5 % -.5 -.53 -.54 -.54 -.53 -.5 -.53 -.53 -.53 -.53 -.53 T 96 ρl P./ d o 0.50 0.60 0.70 0.80 0.90 1.00 1.10 1.0 1.30 1.40 1.50 1 - L d 1 % -.50 -.50 -.49 -.48 -.48 -.48 -.47 -.46 -.45 -.43 -.44 5 % -.03 -.04 -.03 -.04 -.0 -.01 -.00-1.99-1.99-1.99-1.98 1 + L d 1 % -.44 -.45 -.47 -.47 -.47 -.47 -.48 -.46 -.46 -.45 -.46 5 % -1.98-1.98-1.99-1.99-1.99-1.98-1.97-1.97-1.97-1.96-1.96 1 + L d 1 % -.83 -.84 -.85 -.86 -.85 -.83 -.8 -.8 -.83 -.83 -.83 5 % -.31 -.3 -.33 -.3 -.3 -.3 -.3 -.3 -.3 -.3 -.33 T 144 ρl P./ d o 0.50 0.60 0.70 0.80 0.90 1.00 1.10 1.0 1.30 1.40 1.50 1 - L d 1 % -.46 -.46 -.47 -.47 -.45 -.4 -.41 -.41 -.4 -.41 -.41 5 % -1.95-1.96-1.95-1.94-1.94-1.94-1.94-1.93-1.93-1.9-1.9 1 + L d 1 % -.46 -.47 -.45 -.44 -.4 -.40 -.39 -.39 -.38 -.38 -.37 5 % -1.94-1.95-1.96-1.94-1.9-1.9-1.91-1.90-1.89-1.89-1.89 1 + L d 1 % -.76 -.77 -.77 -.77 -.76 -.76 -.76 -.78 -.78 -.76 -.77 5 % -.3 -.3 -.3 -.3 -.3 -. -. -. -.1 -.1 -.1 T 19 ρl P./ d o 0.50 0.60 0.70 0.80 0.90 1.00 1.10 1.0 1.30 1.40 1.50 1 - L d 1 % -.43 -.43 -.44 -.44 -.43 -.41 -.40 -.41 -.41 -.40 -.39 5 % -1.93-1.94-1.94 1.93-1.93-1.9-1.90-1.89-1.88-1.88-1.88 1 + L d 1 % -.45 -.46 -.46 -.45 -.43 -.41 -.39 -.41 -.40 -.37 -.47 5 % -1.93-1.93-1.94-1.94-1.94-1.91-1.91-1.89-1.88-1.89-1.95 1 + L d 1 % -.71 -.7 -.71 -.71 -.70 -.70 -.70 -.71 -.71 -.71 -.71 5 % -.16 -.17 -.17 -.17 -.16 -.15 -.15 -.15 -.14 -.15 -.15 50000 replcaons were used n each case. Smlarly o Table 1 we see ha all he crcal values are smaller han hose from he normal dsrbuon wh slgh dfferences n some cases across d o. When ncreasng he sample szes hey become hgher bu even wh T 19 hey are sll below hose correspondng o he normal dsrbuon. These resuls renforce he argumen ha he use of he sandard crcal values when esng conegraon wh he ess of Robnson 1994 wll lead o reec he null
37 L. A. Gl-Alana hypohess of no conegraon more ofen han expeced suggesng ha fnesample crcal values should be employed. 4. The power of he ess agans fraconal alernaves In hs secon we examne he power properes of Robnson s 1994 ess agans fraconally conegraed alernaves and consder a bvarae sysem where y 1 and y are gven by: y + y u 1... where nally and y 1 1 1 y u + 1... 11 4 1 L u1 ε1 1... 1 4 d 1 L u ε 1... 13 wh he nnovaons ε 1 and ε generaed as ndependen sandard normal varaes. Thus f d 1 n 13 he wo seres are quarerly I1 and nonconegraed whle d < 1 wll mply ha y 1 and y are seasonally fraconally conegraed and 11 wll be he conegrang relaonshp. We also consder cases where he roo occurs a a sngle frequency ha s u 1 and u are generaed as 1 L u ε 1... or alernavely or 1 1 1 d L u ε 1... 14 1 + L u1 ε1 1... d 1 + L u ε 1... 15 1 + L u1 ε1 d 1... 1 + L u ε 1.... 16 Agan n all hese cases f d 1 n 14 16 y 1 and y wll be nonconegraed and f 0 < d < 1 boh seres wll be fraconally conegraed wh he roos occurrng a zero a π and a π/ -π/ respecvely. Table 3 repors he reecon frequences of rˆ n 3 and 10 wh d 0 0.10 0.90; T 48 96 144 and 19 and nomnal szes of 5% and 1% based on 50000 replcaons. We see ha he reecon frequences consderably mprove as d becomes smaller and also as we ncrease he number of observaons. These values are relavely hgh n all cases f d 0.60 and T 144 or 19. Sarng wh he case of four seasonal roos we observe ha f T 96 he reecon probables are hgher han 0.50 for all cases wh d 0.60 even a he 1% sgnfcance level. Lookng a he resuls for he ndvdual frequences he reecon probables are also relavely hgh especally f he sample sze s 144 or 19. Thus for example f T 19 and d o 0.70 he reecon frequences assocaed o he frequences 0
Seasonal fraconal conegraon 373 π and π/ -π/ are respecvely 0.959 0.954 and 0.964 a he 5% level. Smlar expermens were also carred ou based on auocorrelaed dsurbances. Fnesample crcal values were compued and he power properes examned. If he roos of he AR MA polynomals are close o he un roo crcle he resuls are poor. However f hey are far away from 1 hey are smlar o hose repored here he reecon probables beng relavely hgh for d 0.6 and T 144. TABLE 3 Reecon frequences of he ess of Robnson 1994 agans fraconal conegraon ρl T Sz/ d 0.90 0.80 0.70 0.60 0.50 0.40 0.30 0.0 0.10 0.00 48 1% 0.01 0.049 0.106 0.197 0.39 0.471 0.619 0.735 0.85 0.886 5% 0.088 0.159 0.87 0.437 0.600 0.733 0.844 0.913 0.95 0.974 1% 0.037 0.138 0.35 0.650 0.870 0.961 0.99 0.998 0.999 1.000 96 1 L 4 d 5% 0.136 0.349 0.635 0.866 0.968 0.994 0.999 1.000 1.000 1.000 144 1% 0.057 0.67 0.659 0.918 0.991 0.999 0.999 1.000 1.000 1.000 5% 0.186 0.53 0.859 0.981 0.999 1.000 1.000 1.000 1.000 1.000 19 1% 0.068 0.381 0.8 0.985 0.999 1.000 1.000 1.000 1.000 1.000 5% 0.6 0.670 0.947 0.997 1.000 1.000 1.000 1.000 1.000 1.000 ρl T Sz/ d 0.90 0.80 0.70 0.60 0.50 0.40 0.30 0.0 0.10 0.00 48 1% 0.05 0.07 0.176 0.351 0.549 0.741 0.867 0.940 0.970 0.988 5% 0.108 0.31 0.40 0.630 0.815 0.94 0.975 0.991 0.997 0.999 1 L d 96 1% 0.045 0.199 0.506 0.81 0.96 0.994 0.999 1.000 1.000 1.000 5% 0.151 0.43 0.761 0.939 0.99 0.999 1.000 1.000 1.000 1.000 144 1% 0.084 0.377 0.794 0.968 0.999 1.000 1.000 1.000 1.000 1.000 5% 0.33 0.60 0.917 0.989 0.999 1.000 1.000 1.000 1.000 1.000 19 1% 0.11 0.535 0.908 0.99 0.999 1.000 1.000 1.000 1.000 1.000 5% 0.79 0.741 0.959 0.997 0.999 1.000 1.000 1.000 1.000 1.000 ρl T Sz/ d 0.90 0.80 0.70 0.60 0.50 0.40 0.30 0.0 0.10 0.00 48 1% 0.01 0.064 0.153 0.317 0.54 0.717 0.848 0.94 0.964 0.981 5% 0.103 0.15 0.394 0.69 0.807 0.94 0.969 0.990 0.996 0.998 1% 0.045 0.177 0.49 0.804 0.95 0.993 0.999 1.000 1.000 1.000 96 1 + L d 5% 0.159 0.43 0.753 0.938 0.99 0.999 1.000 1.000 1.000 1.000 144 1% 0.079 0.374 0.771 0.965 0.997 0.999 1.000 1.000 1.000 1.000 5% 0.1 0.61 0.90 0.989 0.999 1.000 1.000 1.000 1.000 1.000 19 1% 0.104 0.508 0.893 0.990 0.999 1.000 1.000 1.000 1.000 1.000 5% 0.68 0.719 0.954 0.996 0.999 1.000 1.000 1.000 1.000 1.000 ρl T Sz/ d 0.90 0.80 0.70 0.60 0.50 0.40 0.30 0.0 0.10 0.00 48 1% 0.06 0.058 0.140 0.7 0.455 0.65 0.78 0.875 0.938 0.964 5% 0.097 0.00 0.359 0.551 0.735 0.86 0.940 0.973 0.991 0.995 1% 0.039 0.163 0.446 0.753 0.99 0.989 0.998 0.999 0.999 1.000 96 1 + L d 5% 0.145 0.395 0.713 0.97 0.987 0.999 1.000 1.000 1.000 1.000 144 1% 0.064 0.315 0.76 0.950 0.996 0.999 1.000 1.000 1.000 1.000 5% 0.00 0.594 0.901 0.988 0.999 1.000 1.000 1.000 1.000 1.000 19 1% 0.09 0.484 0.890 0.991 1.000 1.000 1.000 1.000 1.000 1.000 5% 0.69 0.736 0.964 0.997 1.000 1.000 1.000 1.000 1.000 1.000
374 L. A. Gl-Alana 5. An emprcal llusraon We analyse he quarerly UK and Japanese consumpon and ncome seres ha were used by Hylleberg e al. 1990 and Hylleberg Engle Granger & Lee 1993. For he UK he daa are he log consumpon expendure on non-durables and he log personal dsposable ncome from 1955.1 o 1984.4 and for Japan he log of oal real consumpon and he log of real dsposable ncome from 1961.1 o 1987.4 n 1980 prces. These daa were also used by Gl-Alana & Robnson 001 o es he presence of un and fraconal roos n unvarae conexs. The resuls from ha paper for he case of esng un roos wh whe nose dsurbances are summarzed n Table 4. We see ha n boh counres he un roo null hypohess canno be reeced for any seres. Ths s found when we mpose four un roos smulaneously.e. model 6 wh d o 1 bu also when each of he roos s consdered separaely. TABLE 4 Tesng he null hypohess of a un roo H o : d 1 wh he ess of Robnson 1994 on he ndvdual seres Counry Model UNITED KINGDOM JAPAN Consumpon Income Consumpon Income 1 - L 4 d x u -1.00-1.00-1.0-1.05 1 - L d x u -0.30-0.31-0.37-1.07 1 + L d x u -0.9-1.09-0.98-1.06 1 + L d x u -1.19-1.51-1.1-1.4 and n bold: Non-reecon value of a un ro a he 95% sgnfcance level. The resuls n hs able have been aken from Gl-Alana and Robnson 001. Nex we look a he possbly of boh seres consumpon and ncome beng conegraed. The resulng OLS regressons were c 1.1 + 0.87 y y 1.171 + 1.14 c 0.114 0.011 and 0.145 0.011 for he UK and c.389 + 0.901 y 0.100 0.017 for Japan. 0 and y.193 + 1.065 c 0.115 0.017 0
Seasonal fraconal conegraon 375 TABLE 5 Tesng he null hypohess of no conegraon d 1 agans seasonal fraconal conegraon d < 1 wh he ess of Robnson Counry Model UNITED KINGDOM JAPAN Cons. / Inc. Inc. / Cons. Cons. / Inc. Inc. / Cons. 1 - L 4 d x u -.58 -.55 -.11-1.9 1 - L d x u -4.69-4.67-5.37-5.38 1 + L d x u -4.56-4.7-4.87-4.41 1 + L d x u -6.71-6.77-6.93-6.95 and n bold: Reecon values of he null hypohess of no conegraon agans fraconal conegraon a he 95% sgnfcance level. Table 5 repors values of he ess of Robnson 1994 esng he null hypohess of no conegraon agans seasonal fraconal conegraon frs mposng he same order of negraon a all frequences and hen esng each frequency separaely. Tha s we calculae rˆ gven by 3 and 10 esng H o agans 5 wh d o 1 frsly n 4 d 1 L e v 1... 17 and hen n 7 9. Sarng wh he case of four seasonal roos.e. 17 we see ha he null hypohess of no conegraon s clearly reeced for he UK. However hs hypohess canno be reeced for Japan even a he 10% sgnfcance level. If we look a he resuls for each of he frequences separaely.e. 7 9 we observe ha n boh counres all cases lead o reecons of he null n favour of conegraon. The resuls for Japan mgh seem surprsng snce we fnd evdence of conegraon a 0 π and π/ 3π/ when esng hese frequences separaely bu we canno reec he null hypohess of no conegraon when hese frequences are esed ogeher. Ths may be explaned by he fac ha all he es sascs oulned n hs secon have been evaluaed usng whe nose dsurbances and hus he lack of reecon n he case of Japan when esng all roos smulaneously mgh reflec he poenally auocorrelaed srucure underlyng he I0 dsurbances n he esmaed resduals of he conegrang regressons. 6. Conclusons In hs paper we have presened a procedure for esng he null hypohess of no conegraon agans seasonal fraconal conegraon. I s a wo-sep procedure based on he unvarae ess of Robnson 1994. Inally we es he order of negraon of he ndvdual seres and f all of hem have he same degree of negraon we proceed o esng he order of negraon of he esmaed
376 L. A. Gl-Alana resduals from he conegrang regressons. A smlar procedure was proposed by Gl-Alana 003b n non-seasonal conexs. We frs examned he case of processes wh he same degree of negraon a all frequences.e. a zero and he seasonal ones. Then he procedure was exended o he case of seasonal fraconal conegraon a each of he frequences separaely. Fne-sample crcal values of he ess were compued and several Mone Carlo expermens were conduced n order o examne he power properes of he new ess. The resuls ndcae ha he ess behave relavely well agans fraconal alernaves especally f he sample sze s large. The ess were appled o he UK and Japanese consumpon and ncome seres and was found ha boh seres may be fraconally conegraed a each of he frequences separaely. However when esng agans seasonal conegraon a all frequences smulaneously he null hypohess of no conegraon was reeced for he UK bu no for Japan. The presen sudy can be exended n several ways. Frs he same mehodology can be employed allowng for more han wo varables and also for weakly auocorrelaed dsurbances when esng boh he ndvdual seres and he esmaed resduals from he conegrang regressons. However n boh cases fne-sample crcal values should be compued. ARMA srucures for he I0 dsurbances have been wdely used by appled researchers; however her mplemenaon n he conex of long memory processes s sll n s nfancy and he processes descrbed here can be vewed as compeng wh he ARMA models n modellng he degree of assocaon beween he observaons. Also he ess of Robnson 1994 allow us o consder deermnsc regressors lke an nercep a lnear rend and/or seasonal dummes. However once agan he ncluson of hese deermnsc componens changes he emprcal dsrbuon of he ess. Oher mehods of esmang and esng he fraconal dfferencng parameers based on paramerc or sem-paramerc procedures e.g. Robnson 1995ab; Ooms 1997; Hosoya 1997; Slvapulle 001; ec. may also be appled n he second sep of hs procedure. An example s he approach due o Cheung & La 1993 for he case of he long run or zero frequency whch uses he logperodogram esmaon procedure of Geweke and Porer-Hudak 1983. In nonseasonal conexs oher more elaborae echnques on fraconal conegraon esmang and esng he fraconal dfferencng parameers along wh he coeffcens of he conegrang regresson are beng developed by Robnson and hs coauhors Robnson and Marnucc 001 Robnson and Yama 00 Robnson and Hualde 003. Fnally a more general procedure for smulaneously esng seasonal fraconal conegraon under he null hypohess n a smlar way o Johansen s 1988 procedure for non-seasonal conexs would also be desrable. Work n all hese drecons s now n progress.
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