ON THE AUTOREGRESSIVE FRACTIONAL UNIT INTEGRATED MOVING AVERAGE (ARFUIMA) PROCESS

Transcription

1 Joural of Susaiable Developme i Africa (Volume 3, No.5, 2) ISSN: Clario Uiversiy of Pesylvaia, Clario, Pesylvaia ON THE AUTOREGRESSIVE FRACTIONAL UNIT INTEGRATED MOVING AVERAGE (ARFUIMA) PROCESS Olarewaju I. Shiu a OlaOluwa S. Yaya Deparme of Saisics, Uiversiy of Ibaa, Nigeria ABSTRACT This paper preses a osaioary fracioal ui iegrae movig average process ha ca moel ime series aa ha are o saioary afer he firs ifferece. This is auoregressive fracioally ui iegrae movig ARFUIMA p q process wih.5 < < 2.5. The classical ui ifferecig of Box-Jekis is average, (,, ) combie wih he semi-parameric approach o esimae he fracioal ifferece parameer. The moel whe applie o quarerly Nigeria gross omesic proucs (GDP) series iicaes ha ARFUIMA moel is beer ha he correspoig auoregressive iegrae movig average (ARIMA) moel whe compare base o iagosic ess a forecas. Keywors: Fracioal ui iegraio, ARFIMA, semi-parameric esimaio. INTRODUCTION I has bee observe ha several ecoomic a fiacial aa have bee moelle o he assumpio ha ifferecig parameer is usually a ieger. Alhough moels obaie from ieger ifferecig have bee fou o be fairly aequae a reliable, however, beer moels wih higher forecas performace ca be obaie if appropriae fracioal ifferece parameer is use. Whe fracioal ifferecig parameer is o-zero, o-saioariy is suspece. This implies here is srog epeece bewee isa observaios. A umber of suies have bee carrie ou o his subjec a hese iclue suies o real aioal prouc (Diebol e al., 989); cosumer a wholesale price (Geweke a Porer-Huak, 982) a sock marke prices (Lo, 99). The moivaio for his paper a preseaio is erive from Faoki, e al., (2) who ivesigae he aual GDP from 98 o 27. The series was fou o be iegrae of orer 2 I ( 2) a auoregressive iegrae movig average, ARIMA (, 2,) was fie usig he usual moel ieificaio a orer eermiaio ools. We are of he opiio ha furher suy coul reveal a possibiliy of he series beig iegrae of a fracioal orer a hece coul lea o beer forecas performace. I is observe ha classical ADF ui roo es may o give coclusive remark o fracioal ifferece bu a es esige for his purpose such as Kwiakowski, Phillips, Schmi a Shi (KPSS) es coul. Deail abou KPSS es ca be fou i Kwiakowski, Phillips, Schmi a Shi (992). Followig Shiu a Yaya (29), a improve moel i erms of parameer esimaes a forecass ca be obaie by moellig he remaiig log memory i a series afer he firs or seco ifferece. This paper herefore proposes a class of 225

2 osaioary (,, ) ARFUIMA p q where.5 < < 2.5 process which caer for series ha may sill be fracioally ifferece afer firs or seco iffereces. The remaiig par of his paper is srucure as follows: Secio 2 preses he moel; Secio 3 iscusses he esimaio meho a forecas evaluaio; Secio 4 preses he aa aalysis resuls a iscussio a Secio 5 coclues he work. THE ARFUIMA PROCESS Le X be ay ime series process. Fracioally Ui Iegrae (FUI) be efie as ( ) y = L X () where X is he osaioary ime series, y is he covariace saioary process a ( ) L is he ifferece operaor wih he fracioal ui ifferece parameer,. Proposiio Suppose a series is osaioary a ca be expresse as ( ) y = L X. If a ADF es of ui roo cofirme ha he series is I( u ), if furher fracioal ifferecig of I( ) for (.5 < < u), he he resulig series of I( ) where = + u is a saioary series. A fracioally iegrae series is iverible a saioary whe is i he ierval (.5.5) < < a osaioary whe >.5 (Sowell, 992). Whe he Augmee Dickey Fuller (ADF) es iicaes saioariy a = or = 2, i implies ha > 2. Whe his siuaio arises, Robiso (995) a Velasco (999) iicae ha he fracioal ifferece parameer ca be obaie from he ifferece series. Therefore, seig = u (2) where u is he ui ifferece parameer, he equaio () ca be re-wrie as: y ( ) L X u+ ( L) X ( ) u L ( L) X where u is he ui ifferece parameer (, 2,... ) = (3) = (4) = (5) u = a is he fracioal ifferece parameer (.5 < < ). From (3), aiiviy of ifferece parameers hols a biomial heorem applie o he ifferece operaor resuls io complex fucio. Usig he Biomial expasio, he fracioal ifferece operaor i equaio () becomes, ( ) k k L = ( ) L k = k 226

3 u+ = k = k = ( ) k k L, ( u ) k ( u ) ( k ) Γ + + k L (6) k = Γ + Γ + The above expressio will hol uer he assumpio ha he fracioally ifferece series is sill saioary i he.5 < < u+. The process i (4) ca he be re-wrie as, ierval ( ) y ( u ) k ( u ) ( k ) Γ + + = X (7) j k = Γ + Γ + Whe u =, y Whe u = 2, y ( ) k ( ) ( k ) Γ + + = X (8) Γ + Γ + j k = ( 2 ) k ( 2 ) ( k ) Γ + + = X (9) Γ + Γ + j k = Thus u ca ake values,2,3,4,..., however i rarely ges beyo 2. METHODOLOGY We cosier esimaio of fracioal ifferece parameer, i he ARFUIMA( p,, ) q moel usig he frequecy omai approach escribe i Geweke a Porer-Huak (983) (GPH) a applie i Yaya a Shiu (2). The efiiio use i (4) above sill applies as ifferecig a aig back meho (Velasco, 25). The osaioary series, X is he ifferece, u imes i orer o guaraee ha he rue is i.5 < < u accorig o proposiio. Oce he value of is eermie, y% = ( uˆ ˆ ) j ˆ ( u ˆ ) ( j ) ( ( uˆ ˆ ) k) ˆ ( u ) k Γ + + j j= = Γ + Γ + Γ y is approximae by usig X ( ˆ ) ( ) ˆ = uˆ + ˆ i, X k () k = Γ + Γ

4 Forecass Evaluaio Evaluaio of forecas performace of he moels will be juge o Meese a Rogoff (988), MR saisic. The crierio uses he raio of he roo mea square preicio error (RMSPE) of oe of he moels o he moel o he give base moel o check he saisical sigificace. The MR saisic efie as: MR = s UV 2 j= uv 2 2 j j is asympoically ormally isribue wih mea zero a variace oe where is he umber of forecass geerae, u a v are rasforme fucios of forecas errors of he wo moels; s UV is he sample covariace of he meas of U a V, approximae by suv ( u j u)( v j v) = j= where u = u j a j = () v = v j wih j = u j = e j e2 j a v j = e j + e2 j i which j e 2, i =, 2 is he h j forecas error of he moel i a is he umber of forecass. The ull hypohesis of MR saisic is cov (U, V) =. Whe MR > Z α, ull hypohesis is 2 rejece. This implies ha forecas accuracy i he firs moel is sigificaly beer ha he seco moel. The es is mos reliable whe is large. RESULTS AND DISCUSSION Quarerly Nigeria Gross Domesic Proucs (GDP) aa were use o illusrae he propose moel. The aa spa from 96 o 28 wih 96 aa pois. Eve hough Faoki, e al., (2) use aual aa o GDP (98-27) i.e. 28 aa pois. Prelimiary aalysis give i Table below shows ha GDP is posiively skewe (Skewess=2.5249) a heavy aile (Kurosis= ). Table : Descripive aalysis o GDP series Mea Meia Maximum Miimum S. Dev. Skewess Kurosis JB Prob The ime plo presee i Figure shows ha here has bee asroomical icrease i he Nigeria GDP series. 228

5 Figure. Time Plo of Quarerly Nigeria Gross Domesic Prouc i Millio Naira (96-28) Quarerly GDP The aa will be moelle i wo scearios. Firs he aa will be moelle as ARIMA(p,, q) a secoly as ARFUIMA(p,, q) moel. The heir forecas performace will be examie wih a view o eermiig he beer moel o he basis of he RMS forecas error. Table 2: Saioariy Tess o GDP Series Origial Series Firs Differece Seco Differece Tes ADF KPSS ADF KPSS ADF KPSS Saisic % % (.) (.9725) (.).463 % From Table 2, he hypohesis of ui roo of he augmee Dickey Fuller (ADF) es is o sigifica a level a firs ifferece series of he Nigeria GDP a 5% level bu afer seco ifferece, he series is saioary which implies ha Nigeria GDP is I ( 2) series. The seco ifferece series is furher subjece o KPSS es of log memory a a 5% level of sigificace, ull hypohesis of series saioariy is accepe agai he aleraive of log memory. Figure 2. Plo of Seco Differece of Gross Domesic Prouc Series Seco Differece of GDP 229

6 The GDP series aais saioariy afer seco ifferece, I( u = 2). Usig he moel ieificaio ools of ACF a PACF a oher iagosic ools, he mos appropriae moel for he aa ARIMA (4,2,) moel wih miimum AIC of as show below. Table 3: Esimaio of Parameers of Subse of ARIMA (4, 2, ) Moel Esimaors Coefficie Saar Error -probabiliy û 2 ˆ φ ˆ φ ˆ φ ˆ φ ( ) ( ).98( ).86367( ).66294( ) L 2 X = L 2 X L X 2 L X 3 L X 4 +ε (.) (.) (.) (.26) Log-likelihoo Skewess.4838 AIC Excess Kurosis 2. Variace of Resiuals E+9 Normaliy es [.]** Pormaeau es [.2]** ARCH es [.76]** The oliear esimaio wih PcGive versio. followig he GPH approach esimae =.7539 wih saar error of.62. The value of is wihi he cofiece ierval of [ ,.87389]. The esimae fracioal ui iegrae parameer is obaie usig (2) as =.7539 a he resulig series is assume o be i he saioary fracioally iegrae rage accorig o proposiio. Rescale Saisics (RS) approach employe o he seco ui ifferece I( u = 2) series of he GDP aa compue as -.35, which implies =

7 The esimae ARFUIMA (3,.7539, ) moel is presee below: Table 4: Esimaio of Parameers of ARFUIMA (3,.7539) Moel Esimaors Coefficie Saar Error -probabiliy û ˆ ˆ φ ˆ φ ˆ φ 2 ˆ φ ( L) X ( L) X.967( L) X ( L).7539 = X 3 +ε (.) (.) (.) (.) Log-likelihoo Skewess.24 AIC Excess Kurosis.66 Variace of Resiuals E+9 Normaliy es [.]** Pormaeau es [.7] ARCH es [.]** Table 5: Forecass of ARFUIMA moel base o Moifie GPH esimaio Approach of Fracioal Differece Parameer,. Horizo ARIMA ARFUIMA Forecass (i Millios S. Error (i Millios Cofiece Ierval (i Millios Naira) Forecass (i Millios S. Error (i Millios Cofiece Ierval (i Millios Naira) Naira) Naira) Naira) Naira) 29Q [62892, ] [653744, ] 29Q [63939, ] [ , ] 29Q [6968, ] [ , 77753] 29Q [ , 74455] [ , 72588] 2Q [ , ] [67626, 78298] 2Q [724527, ] [ , ] 2Q [ , 82564] [724526, ] 2Q [ , ] [74282, ] 2Q [ , ] [ , 89383] Usig he Meese a Rogoff (MR) (988), who evelope he MR-saisic as reviewe i Secio 4: H : Forecas Errors are he same (Moels are ieical) H : Forecas Errors are o he same (Moels are o ieical) 23

8 Sice he compue MR=2.86 is greaer ha Z.25 =.96, we rejec he ull hypohesis of equaliy of forecas errors for he wo moels. Thus, he esimae moels are o ieical. The ARFUIMA (,2.27,) is cosiere cosierig he smaller MAD of 57. agais of ARIMA (,2,) moel. CONCLUSION We have cosiere i he paper auoregressive fracioally ui iegrae movig average a (ARFUIMA) moel. Is performace was compare wih he ARIMA(p,,q) moel i erms of moel aequacy a forecas performace. The ARFUIMA moel performe beer whe applie o GDP aa ha he ARIMA moel as iicae by smaller mea square error of forecas. REFERENCES Diebol, F. X. a Ruebusch, G. D. (989). Log Memory a Persisece i Aggregae Oupu", Joural of Moeary Ecoomics, 24: Geweke, J. a Poer-Huak, S. (983): The Esimaio a Applicaio of Log memory Time series moels, Joural of Time Series Aalysis, 4: Faoki, O., Ugochukwu, M. a Abass, O.(2). A Applicaio of ARIMA Moel o he Nigeria Gross Domesic Prouc (GDP). Ieraioal Joural of Saisics a Sysems, 5(): Lo, A.W. (99). Log erm memory i Sock marke prices", Ecoomerica 59: Kwiakowski, D., Phillips, P. C.B., Schmi, P, a Shi, Y (992), "Tesig he Null Hypohesis of Saioariy Agais he Aleraive of a Ui Roo," Joural of Ecoomerics, 54: Hassler, U. (993). Frakioal iegriere Prozesse i er Okoomerie, Frakfura m Mai: Haag & Herche. Meese, R. A. a Rogoff, K. (988). Was Is Real? The Exchage Rae-Ieres Differeial Relaio over he Moer Floaig Rae Perio. Joural of Fiace. 43: Robiso, P.M. (995). Log-Perioogram Regressio for Time Series wih log Rage Depeece". Aals of Saisics 23: Sowell, F. (992). Maximum Likelihoo Esimaio of Saioary uivariae Fracioally Iegrae ime Series Moels, Joural of Ecoomerics 53: Shiu, O.I. a Yaya, O.S. (29). Measurig Forecas Performace of ARMA a ARFIMA Moels: A Applicaio o US Dollar/UK Pou Foreig Exchage Rae. Scieific Research 32: Europea Joural of Velasco, C. (999) No-saioary log-perioogram regressio. Joural of Ecoomerics, 9: Velasco, C. (25). Semiparameric Esimaio of Log-memory Moels. Deparme of Ecoomics Uiversia Carlos III e Mari, Spai. Yaya, O.S. a Shiu, O.I. (2). Log Memory a Esimaio of Memory Parameers: Nigeria a US Iflaio Raes. Ieraioal Joural of Physical Scieces, 2(3): 2-3. ABOUT THE AUTHORS: Olarewaju I. Shiu a OlaOluwa S. Yaya: Deparme of Saisics, Uiversiy of Ibaa, Nigeria 232