Available online Journal of Scientific and Engineering Research, 2017, 4(8): Research Article

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1 Available online , 4(8): Research Aricle ISSN: CODEN(USA): JSERBR Modeling he Exchange rae long-range dependence of some world emerging markes Sanusi Alhaji Jibrin 1 *, Abdulhameed A. Osi 1, Aremu T. Adeyemi 2, Samaila Mohammed 2, Umar A. Zubair 3, Ahmed S. Saidu 3 * 1 Deparmen of Saisics, Kano Universiy of Science and Technology, Wudil, Nigeria 2 Deparmen of Mahemaics and Saisics, NuhuBamalli Polyechnic Zaria, Kaduna 3 Deparmen of Saisics, Cenral Bank of Nigeria Absrac This research examined over en years Chinese Yuan (CNY), Indian Rupees (INR), Nigerian Naira (NGN) and Malaysia Ringgis (MYR) daily o he U.S Dollar exchange rae using he condiional mean models namely: he Auoregressive Inegraed Moving Average (ARIMA) and he Auoregressive Fracional Inegraed Moving Average (ARFIMA) models. The bes candidae modelswere seleced using Akaike Informaion Crieria (AIC). Approaches used for esing and esimaing long memory parameers are he rescaled range ess [1-3] and Local While Esimaor developed by Robinson [4]. The whie noise, serial correlaion and he heeroscedasiciy es was carried ou. Uni roos ess confirmed he nonsaionary of all he four series while he inconsisency resuls obained from he long memory parameers esimaes guided he use of wo modeling approaches. The findings revealed ha he ARIMA model is he bes o sudy CNY, INR and MYR o he U.S Dollar exchange rae while ARFIMA mehod is he suiable o model he NGN exchange rae. Keywords Modeling, Exchange rae 1. Inroducion The increasing use of large ime series daa has iniiaed some compeiive research work. Among oher procedures ha can be used o analyze a large daa se is he long memory (LM) analysis. The behavior of businesses, economies, raffic flow and he walk of a drunkard are few simple examples of daa wih long-range dependence. LM represens a degree of dependence beween observaions or a siuaion where some figures persiss or occurred in high frequency in a given daa. The LM was firs discovered in physical science daa by Hurs (1951) and his effor was complimened by a lo of gian sride in he economies and financial daa [5]. LM processes are saionary processes whose Auocorrelaion Funcions (ACF) decayed more slowly han a shor memory process. Because he ACF die ou slowly, he LM process displayed a ype of long range dependence. The shor memory process, ACF decayed o zero a a geomeric rae while ha of LM decayed a he hyperbolic rae [3]. Baillie and Beran inroduced simple explanaions o LM processes and also developed models o sudy he series characerized by his aribue [6-7]. Long range-dependence of wo oil prices was sudied and prediced for he possibiliy for a crude oil price o decline drasically in 2014 [8]. The research esimaed wo long memory models, he ARFIMA (1, 0.47, 2) and ARFIMA (2, 0.09, 0) using Wes Texas Inermediae (WTI) and Bren series respecively. Nezhad e al examined he fracional inegraion in Organizaion of Peroleum Exporing Counries (OPEC) prices and furher confirmed ha i is necessary o model he price using a fracional inegraion model [9]. The price in he opion marke is dominaed by long memory characerisics and a fracional coinegraed relaionship [10]. The fracional inegraion es may discriminae beween spurious long memory and a rue fracional inegraion parameer [11]. Jibrin e al sudied he impac of Floaing exchange rae regime inroduced by he Nigerian governmen in 2016 using ARIMA 360

2 Jibrin S e al, 2017, 4(8): inervenion approach. The naira value winessed a persisen depreciaion agains he Unied Saes dollar. The research findings furher revealed ha he currency and he economy will be affeced by he devaluaion for a long ime excep criical economics measures are aken by he auhoriies concerned [12]. I is paramoun o noe ha considerable cauions are necessary whenever he nonparameric and semiparameric memory esimaion procedures are used o explore he degree of long memory specifically when he series displayed he nonsaionary, deerminisic and liner rend. Phillips and Shimosu emphasized on he need o ake precauions when using LWE o deermine he long-range dependence of a series ha has rending aribues and deerminisic rends. These characerisics of he ime series someime led o varying LM parameer esimae. Occasionally, for a paricular series, some LM esing and esimaion procedureconverge o uniy indicaing ha he series is inegraed of order one, I(1), while ohers said is fracional inegraed, I(0<d<1). For I(1), i is clearly ha he Auoregressive Inegraed Moving Average (ARIMA) model could be suiable o sudy he series while if i is I(0<d<1), hen Auoregressive Fracionally Inegraed Moving Average (ARFIMA) model ha is appropriae [13]. In view of hese, he crucial issue for he analysis of daa is making a decision on he bes modeling approach. The use of single mehodology (for example, he use of ARIMA or SARIMA or ARFIMA model) o sudy a series someimes can lead o erroneous conclusions. Lieraure in he pas indicaed ha he behaviors of he ime series are no easily or graphically idenified unless various formal ess are explored [8]. The wo popular semiparameric LM procedures are he log-periodogram regression discussed [14] and Gaussian semiparameric esimae also known as Local While Esimae (LWE) proposed [15]. This research is moivaed by he sudies of [13, 17]. In heir separae researches, hey discussed he inconsisency of he various long memory esimaion mehods/ess. Robinson [4] and Velasco [17] boh shows asympoic normaliy, consisency and oherwise of he LWE for d(-0.5,0.5) and d[(-0.5,1),(-0.5,0.75)] respecively. This research wan o deermine he bes condiional mean model ha can be used o effecively sudy he exchange rae of some emerging world markes ha exhibied long-range dependence. The remaining par paper consiss of maerial and mehods used in secion 2, resuls and discussion in secion 3 while he concluding remarks is discussed in secion Maerials and Mehods 2.1. The daase The daa for his research were obained from he websie of he Quandle daabase and include CNY, INR, NGN and MYR o he U.S dollar daily exchange rae for he period 01/01/2007 o 31/05/ Mehodology Numerous real lifeime series is nonsaionary. ARIMA model was inroduced for such a ime series [18] while Box Jenkins proposed ha differencing up o an order d could render hose series o be saionary [19]. Researchers conribued in designing models o sudy ime series wih long range dependence [6-7, 14, 20-21]. A daa is called a fracionally inegraed process if i can be represened as: The expansion of (1) produced he following equaion: d 1, (1) d d1 d (2) 2! A general form of an ARFIMA model can be presened by d 1, for0 d 0.5 (3) where he parameer d is a fracional value, 2 mean 0 and variance. The ( B) and ( B) X is he original daa a he ime, is normally disribued wih, represens AR and MA polynomials wih lag B, respecively. Furhermore, an ARIMA model can also be represened in a similar manner as he ARFIMA model in (3) if he "difference parameer", d, is allowed o ake ineger values. The bes candidae for he ARIMA and ARFIMA model can be seleced using Akaike Informaion Crieria (AIC). The saionariy ess employed in his sudy are 361

3 Jibrin S e al, 2017, 4(8): [22] and [23]. Approaches used for esing and esimaing long memory parameers are he rescaled range ess [1-3] and Local While Esimaor suggesed by Kunsch [15] and laer developed by Robinson [4]. The whie noise, serial correlaion and he heeroscedasiciy es was carried ou using he residual normaliy es, he Pormaneau es and Auoregressive Condiional Heeroscedasiciy Lagrange Muliplier (ARCH-LM) es respecively Sofware used The sofware used is Greel version1.9.4 and G@RCH 7.0 and OxMarics version Analysis and Resuls USD Exc. Rae To Chinese Yuan USD Exc Rae To 1 Indian Rupees USD Exc Rae To 1 Nigeria Naira USD Exc Rae To 1 Malaysia Ring Figure 1: A ime plos of daily CNY, INR, NGN and MYR o he U.S dollar exchange rae The CNY, INR, NGN and MYRo he U.S dollar daily exchange rae are displayed in Figure 1. Three of he graphs exhibied deerminisic rends, namely: Chinese Yuan, Indian Rupees and Malaysian Ringgis. The Nigeria Naira exchange rae is dominaed by sable movemens and regimes swiching. Overall, he four series are no saionary. Table 1: The long memory ess, he saionary es and long memory esimaes Currencies HM/RS Lo/RS KPSS SP GPH LWE USD/CNY USD/INR USD/NGN USD/MYR Hurs-Mandelbro Rescaled Range =HM/RS, Lo Rescaled Range= Lo/RS, Geweke Porer-Hudak=GPH, Local While Esimaor= LWE, Kwiakowski Phillips Schmid and Shin=KPSS and Schmid Phillips =SP. 362

4 Jibrin S e al, 2017, 4(8): The Table 1 displayed above conains he LM rescaled range ess, uni roo ess, and long memory parameer esimaes. Boh he Hurs-Mandelbro and Lo rescaled range ess saisic rejeced he null hypohesis of no auocorrelaion and long-erm dependence respecively a 5% significance levels for all he four exchange rae series. In addiion, he wo uni roo ess confirmed he nonsaionary of he four series. However, he GPH and LWE esimaes converges o uniy and shows ha he CNY, INR, MYR and NGN are respecively inegraed of order one, I(1). Therefore, he ARIMA model is a candidae model o sudy he series. Anoher poin is ha GPH and LWE also indicaed ha NGN and CNY, INR, MYR respecively are of inegraed order I(0<d<1) and herefore can be sudy using he ARFIMA model. Table 2: ARIMA candidae models for CNY, INR, NGN and MYR Currencies ARIMA(p,1,q) AIC Norm. of Resi. Porm. Tes ARCH-LM es USD/CNY ARIMA(1,1,2) (0,1.0160) 89.36(0.0104) (0.0000) USD/INR ARIMA(2,1,0) (0,1.0277) 92.72(0.0893) (0.0000) USD/NGN ARIMA(0,1,2) (0,7.7774) 1.84(0.9855) (0.0000) USD/MYR ARIMA(0,1,2) (0,1.0243) 80.93(0.0000) (0.0000) Table 3: ARFIMA candidae models for CNY, INR, NGN and MYR Currencies ARFIMA(p, fd, q) AIC Norm. of Resi. Porm. Tes ARCH-LM es USD/CNY ARFIMA(2, 0.941, 0) (0.0036,0.2254) 80.96(0.0000) (0.0000) USD/INR ARFIMA(1, 0.941, 1) (0.0149,0.1609) 77.99(0.0000) (0.0000) USD/NGN ARFIMA(0, 0.806, 2) (0.0002,1.2881) 10.11(0.2574) (0.0000) USD/MYR ARFIMA(1, 0.936, 2) (0.0000,0.0917) 64.59(0.0000) (0.0000) fd=fracional difference. The ARIMA and ARFIMA candidae s models for he four currency exchange rae ogeher wih he accuracy measure and various residual ess are displayed in he Table 2 and Table 3 respecively. Seven differen models were idenified for all he four series and he bes model was seleced based on he minimum AIC and residual analysis. A comparison beween he resuls of ARIMA and ARFIMA modeling shows ha ARIMAis he bes model o sudy CNY, INR and MYR series because hey all possesses minimum AIC, whie noise residuals and less serial correlaion while ARFIMA approach is he suiable o model he NGN exchange rae. 4. Conclusion The sudy examined he CNY, INR, NGN and MYR o he U.S Dollar daily exchange rae. The inconsisency resuls obained from long memory ess and parameer esimaes guided he use of wo modeling approaches, he ARIMA and ARFIMA models. Afer models idenified and esimaion, he bes model was seleced based on he minimum AIC and residual analysis ess resuls. The ARIMA model is he bes o sudy CNY, INR and MYR o he U.S Dollar daily exchange rae due o minimum AIC, whie noise and less serial correlaion in he residuals while ARFIMA mehod is he suiable o model he NGN exchange rae. One of he assumpions of condiional mean models is he homoscedasic residuals and he wo models, ARIMA and ARFIMA considered here violae such assumpion. The ARCH-LM es resuls displayed in he Table 3 and 4 for he idenified and esimaed models show he presence of heeroscedasiciy in he model s residuals. Therefore, furher research can be conduced ou using LM condiional volailiy models o accoun for he effecs of some sylized facs. References [1]. Mandelbro, B. B. (1972): Saisical mehodology for non periodic cycles: From he covariance o R/S analysis. Annals of Economic and Social Measuremen, 1: [2]. Lo, A.W. (1991): Long erm memory in Sock marke prices, Economerica, 59: [3]. Box, G. E. P., Jenkins, G. M. and Reinsel, G. C. (2008): Time Series Analysis: Forecasing and Conrol. 4 h ed. Hoboken, NJ: Wiley. 363

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