A Forecasting Model for Monthly Nigeria Treasury Bill Rates by Box-Jenkins Techniques

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1 Inernaional Journal of Life Science and Engineering Vol. 1, No. 1, 2015, pp hp:// A Forecasing Model for Monhly Nigeria Treasury Bill Raes by Box-Jenkins Techniques Ee Harrison Euk 1, *, Azubuike Samuel Agbam 1, Barholomew Anuriobi Uchendu 2 1 Deparmen of Mahemaics/Compuer Science, Rivers Sae Universiy of Science and Technology, Por Harcour, Nigeria 2 Deparmen of Mahemaics/Saisics, Federal Polyechic, Nekede, Imo Sae, Nigeria Absrac A realisaion, TBR, of Nigeria Treasury Bill Raes from January 2006 o December 2014 is analysed by seasonal ARIMA mehods. The ime plo of he realisaion reveals an overall downward rend from 2006 o 2009 followed by an overall upward rend up o Twelve-monhly differencing of TBR yields he series SDTBR which has an overall upward rend. Nonseasonal differencing of SDTBR yields he series DSDTBR wih an overall horizonal rend and no clear seasonaliy. By he Augmened Dickey-Fuller Uni Roo Tes boh TBR and SDTBR are adjudged non-saionary whereas DSDTBR is adjudged saionary. The correlogram of DSDTBR has a negaive significan spike in he auocorrelaion funcion a lag 12, an indicaion of seasonaliy of period 12 monhs and he presence of a seasonal moving average componen of order one. By a novel proposal credied o Suharono, iniially he (0, 1, 1)x(0, 1, 1) 12 SARIMA model is fied. The non-significance of he lag 13 coefficien suggess he addiive SARIMA model wih significan coefficiens a lags 1 and 12. This model is fied and has been shown o be he more adequae model and may be used o forecas fuure Nigeria Treasury Bill Raes. Keywords Nigeria Treasury Bill Monhly Raes, SARIMA Models, Addiive Models, Time Series, ARIMA Models Received: March 17, 2015 / Acceped: April 1, 2015 / Published online: April 2, 2015 The Auhors. Published by American Insiue of Science. This Open Access aricle is under he CC BY-NC license. hp://creaivecommons.org/licenses/by-nc/4.0/ 1. Inroducion Many economic and financial ime series are known o be seasonal as well as volaile. For insance, crude oil prices, inflaion raes, foreign exchange raes, crude oil domesic producion are a few such series. Seasonal ime series may be modelled using seasonal auoregressive moving average (SARIMA) echniques. Nigeria Treasury Bill Raes consiue such a ime series. The purpose of his wrie-up is o model hese raes by SARIMA mehods. Nigeria Treasury Bill is a shor-erm deb insrumen given ou by he Federal Governmen of Nigeria hrough he Cenral Bank of Nigeria (CBN) o provide shor-erm funding for he governmen (Cashcraf Asse Managemen Limied, 2013). I is risk-free and re-discounable hrough licensed Money Marke Dealers (Balogun, 2013). The raes are deermined by he CBN according o marke realiies. Furher descripions of his faciliy include he following. I is a shor-erm negoiable bill of exchange, used by governmen o fund naional borrowing requiremens, quoed for purchase and sale in he secondary marke on an annual percenage yield o mauriy and issued a a discoun. Is feaures furher include being a zero coupon rae, ha is, no ineres is paid during he enure of he bill, being issued in fixed enures of one-fold, wo-fold and four-fold muliples of 91 days. Primary dealers and invesmen may underwrie i * Corresponding auhor: address: eeuk@yahoo.com (E. H. Euk), eeheuk@gmail.com (E. H. Euk), azubuikesamuelagbam@yahoo.com (A. S. Agbam), uchendubarholomew@yahoo.com (B. A. Uchendu)

2 Inernaional Journal of Life Science and Engineering Vol. 1, No. 1, 2015, pp and he ineres is paid upfron. Benefis accruable include he fac ha hey are regarded as liquid asses for he purpose of liquidiy raio calculaion and ineres received on i is no axable. A few examples of such use of he SARIMA mehodology o model economic and financial daa include he following. Saz(2011) highlighed he comparaive advanage of using SARIMA models in erms of parsimony and he efficiency of daa modelling. Euk e al. (2012) modelled Nigerian monhly inflaion raes by such mehods oo. Ou e al. (2014) fied a SARIMA(1, 1, 1)x(0, 0, 1) 12 model o Nigeria s inflaion raes and used i o forecas he series up o Luo e al. (2013) fied a SARIMA(1, 0, 1)x(1, 1, 1) 12 model o cucumber marke prices in China. Dan e al. (2014) fied a SARIMA(1, 1, 1)x(0, 0, 1) 12 model o malaria moraliy in Imo Sae. Lee e al. (2008) were able o observe he superioriy of he SARIMA mehod over wo oher mehods, a neural-based mehod and a Bayesian based one. Eni and Adesola (2013) fied a SARIMA(1, 1, 0)x(1, 0, 1) o predic he fuure volume of raffic from Cross Lines Limied, a ranspor Company. Osarunwense (2013) fied a SARIMA(0, 0, 0)x(2, 1, 0) model o quarerly rainfall daa in Por Harcour. Teye-Okuu (2011) fied a SARIMA(1, 1,1,1)x(0,0,1) 12 and used i o forecas Ghanaian inflaion. Monhly precipiaion in Moun Kenya region has been modelled as a SARIMA (1,0,1)x(1,0,0) 12 by Kibunja e al. (2014). Where seasonal series are involved hey have been observed o exhibi comparaive advanage (see for insance Lee e al. (2008)). I is noeworhy ha mos fied SARIMA models are muliplicaive as defined by Box and Jenkins (1976). For insance all he above-menioned works applied muliplicaive models. Box and Jenkins (1976) however considered he possibiliy of a non-muliplicaive such model. They did no define an addiive SARIMA model. I is Suharono (2011) and Suharono and Lee (2011) ha defined a subse, muliplicaive and addiive SARIMA model. Using airline daa Suharono observed ha he subse model oudid he addiive one. However wih he ouris arrival o Bali daa, he addiive model displayed some supremacy over he subse model. Suharono and Lee (2011) observed ha he addiive model oudid he ohers in erms forecasing he number of ouris arrivals a Bali. Boh works aimed a proposing a new algorihm for SARIMA fiing. This algorihm shall be considered in deail in he mehodology secion. The SARIMA models fied by Euk e al. (2013) and Euk e al.(2014) o monhly Nigerian Naira-CFA Franc Exchange Raes and Monhly Nigerian Savings Deposi Raes respecively urned ou o be addiive. 2. Mehodology The daa for his work are monhly Nigeria Treasury Bill Raes from January 2006 o December 2014 rerievable from he Money Marke Indicaors, Daa and Saisics publicaion of he CBN in is websie Sarima Models A saionary ime series {X } is said o follow an auoregressive moving average model of orders p and q, designaed ARMA(p, q), if i saisfies he following difference equaion X α X α X... α X p p = ε + β ε + β ε β ε q q where {ε } is a whie noise process and he α s and β s are consans such ha he model is boh saionary and inverible. Model (1) may be wrien as (1) A(L)X = B(L)ε (2) where A(L) = 1 - α 1 L - α 2 L α p L p and B(L) = 1 + β 1 L + β 2 L β q L q and L is he backward shif operaor defined by L k X = X -k. Mos real life ime series are non-saionary. Box and Jenkins(1976) proposed ha differencing o a sufficien order could render a non-saionary ime series saionary. Suppose he series {X } is non-saionary. Le d be he minimum order such ha he d h difference of X, namely, d X, is saionary where = 1 - L. Saionariy may be esed using Augmened Dickey-Fuller(ADF) Uni Roo Tes. If he series { d X } follows an ARMA(p, q) he original series {X } is said o follow an auoregressive inegraed moving average model of orders p, d and q designaed ARIMA(p, d, q). For a seasonal ime series of order s, Box and Jenkins(1976) proposed ha {X } be modelled by A( L) Θ s d D s Φ ( L ) s X = B( L) ( L ) (3) where he series mus have been subjeced o seasonal differencing D imes and non-seasonal differencing d imes, s = 1 L s, being he seasonal differencing operaor. Moreover Φ(L) and Θ(L) are he seasonal auoregressive and moving average operaors respecively. These seasonal operaors are polynomials in L. Suppose ha Φ(L) = 1 + φ 1 L + φ 2 L φ P L P and Θ(L) = 1 + θ 1 L + θ 2 L θ Q L Q hen he ime series {X } is said o follow a muliplicaive seasonal auoregressive inegraed moving average model of orders p, d, q, P, D, Q and s, designaed (p, d, q)x(p, D, Q) s SARIMA model. ε

3 22 Ee Harrison Euk e al.: A Forecasing Model for Monhly Nigeria Treasury Bill Raes by Box-Jenkins Techniques Suharono(2011), using moving average (MA) symbolism, defines a subse SARIMA model as d D s X = ε + β ε β ε β ε (4) s s + s+ 1 s 1 where β s+1 β 1 β s. Oherwise, i is a muliplicaive SARIMA model. If β s+1 = 0, he model (4) is said o be an addiive SARIMA model. He goes on o propose he following se of seps for SARIMA fiing: 1) Fi a subse SARIMA model. 2) Find ou if β s = 0. If so, he model is addiive. If no, find ou if he model is muliplicaive. If no, he model is subse. low crierion value is an indicaion of relaive model adequacy. 3. Resuls 2.2. Model Esimaion In order o fi he model (3) he orders p, d, q, P, D, Q and s mus firs be deermined. Ofen seasonaliy is no eviden from he ime plo. Afer due differencing a spike in he auocorrelaion funcion (ACF) and ofen in he parial auocorrelaion funcion (PACF) indicaes seasonaliy of period equal o he corresponding lag s. In order o avoid undue model complexiy i has been advised ha d + D < 3. Mos ofen i is allowed ha d= D=1. The non-seasonal auoregressive order p is esimaed by he non-seasonal cuoff lag of he PACF and q by he non-seasonal cu-off lag of he auocorrelaion funcion ACF. The seasonal orders P and Q are respecively deermined by he seasonal cu-off lags of he PACF and ACF. The parameers of he model are invariably esimaed by non-linear opimizaion echniques because of he involvemen of iems of he whie noise process in he model. An iniial esimae of he parameers is usually made and on is basis furher esimaes are obained ieraively, each esimae expeced o be an improvemen on is predecessor unil opimaliy is aained depending on he degree of accuracy required. The opimaliy crierion employed is usually he leas squares crierion, he maximum likelihood crierion, he maximum enropy crierion, ec. Afer model fiing analysis of he residuals is usually done wih a view o asceraining he adequacy of he model. This is called diagnosic checking. Assuming model adequacy he residuals should have zero mean, be uncorrelaed and follow a Gaussian disribuion. In his wrie-up all analyical work is done using he Eviews 7 sofware which uses he leas squares approach o esimaion. Besides Eviews 7 displays values of hree informaion crieria on he basis of which model comparison could be done. The informaion crieria are Akaike Informaion crierion (AIC) (Akaike, 1977), Schwarz informaion crierion (Schwarz, 1978) and Hannan- Quinn informaion crierion (Hannan and Quinn, 1979). A Figure 1. TBR Figure 2. SDTBR Figure 3. DSDTBR The ime plo of he realizaion TBR in Figure 1 reveals an overall negaive rend beween 2006 and 2009, followed by an overall posiive rend from 2010 onwards. Regular seasonaliy is no obvious. However an inspecion of he daa reveals ha yearly minimums occur in May, July, December, February, March, July, December, February and Sepember respecively. Tha means ha six of he nine minimums lie

4 Inernaional Journal of Life Science and Engineering Vol. 1, No. 1, 2015, pp beween February and July of he same year. Similarly, five of he nine maximums lie in he remaining par of he year. This is an indicaion of a 12-monhly seasonaliy. This jusifies a SARIMA modelling approach. A welve-monh differencing of his original series yields he series SDTBR which has an overall slighly upward rend wih no observable regular seasonaliy as eviden from Figure 2. A non-seasonal differencing resuls in he series DSDTBR which shows no rend on he overall and no clear seasonaliy (See Figure 3). The ADF uni roo es saisic for TBR, SDTBR and DSDTBR is equal o -1.9, -2.4 and -8.6 respecively. Wih he 1%, 5% and 10% criical values equal o -3.5, -2.9 and -2.6 respecively, he ADF es adjudges boh TBR and SDTBR non-saionary bu DSDTBR saionary. In he correlogram of DSDTBR in Figure 4, here is a negaive significan spike a lag 12 in he ACF. This shows ha he series is seasonal of period 12 monhs. Moreover, his is an indicaion of he presence of a seasonal MA componen of order one. Adoping Suharono s (2011) se of seps, a (0, 1, 1)x(0, 1, 1) 12 SARIMA model fi yields he model (5) as summarized in Table 1: 2) The uncorrelaedness of he residuals as seen from heir correlogram of Figure 6. None of he 36 auocorrelaions is significan (i.e. ouside of he range ±2/ n, where n is he sample size). 4. Conclusion I may be concluded ha Nigeria Treasury Bill Raes follow he addiive SARIMA model (6). Tha means ha a curren value of he ime series depends on he pas value of a monh ago and ha of a year ago of shocks or is error erms. This model has been shown o be adequae and could be used for he forecasing of Nigeria reasury Bill raes. However effors should be made o explore he possibiliy of models ha beer accoun for he variabiliy in he ime series. X = ε ε ε + ε ( ± ) ( ± ) ( ± ) (5) Clearly he lag 13 coefficien is non-significan. Hence he model is addiive which is re-esimaed as summarized in Table 2 as X = ε ε + ε 12 ( ± ) ( ± ) (6) which is clearly superior o model (5) on he grounds of he minimum of he informaion crieria: Akaike informaion crierion, Schwarz informaion crierion and Hannan-Quinn informaion crierion. The adequacy of model (6) is eviden from 1) The close agreemen of he fied model and he acual model (See Figure 5). Figure 4. Correlogram of Dsdbr Table 1. (0, 1, 1)X(0, 1, 1) 12 Sarima Model Esimaion Dependen Variable: DSDTBR Mehod: Leas Squares Variable Coefficien Sd. Error -Saisic Prob. MA(1) MA(12) MA(13) Akaike info crierion Schwarz crierion Hannan-Quinn crier Invered MA Roos ±.99,.86±.50i,.49±.86i, ±.99i, -0.13, -.50±.86i, -.86±.50i

5 24 Ee Harrison Euk e al.: A Forecasing Model for Monhly Nigeria Treasury Bill Raes by Box-Jenkins Techniques Table 2. Addiive Sarima Model Esimaion Dependen Variable: DSDTBR Mehod: Leas Squares Variable Coefficien Sd. Error -Saisic Prob. MA(1) MA(12) Akaike info crierion Schwarz crierion Hannan-Quinn crier [2] Balogun, M. (2013). Undersanding he Nigerian Treasury Bills Marke, g-he-nigerian-reasury-bills-marke.hml [3] Box, G. E. P. and Jenkins, G. M. (1976). Time Series Analysis, Forecasing and Conrol, Holden-Day, San Francisco. [4] Cashcraf Asse Managemen Limied (2013). Treasury Bills, icle&id=3448&iemid=157 Figure Residual ---- Acual ---- Fied [5] Dan, E. D., Jude, O., Idochi, O. (2014). Modelling and Forecasing Malaria Moraliy rae Using SARIMA Models (A Case Sudy of Aboh Mbaise General Hospial, Imo Sae, Nigeria), Science Journal of Applied Mahemaics and Saisics, 2(1), [6] Eni, D. and Adesola, A. W. (2013). Sarima Modelling of Passenger Flow a Cross Line Limied, Nigeria. Journal of Emerging Trends in Economics and Managemen Sciences (JETEMS), 4(4), [7] Euk, E. H., Aboko, I. S., Vicor-Edema, U., Dimkpa, M. Y. (2014). An Addiive Seasonal Box-Jenkins model for Nigerian monhly savings deposi raes. Issues in Business Msanagemen and Economics, 2(3), [8] Euk, E. H., Asuquo, A. and Essi, I. D. (2012). Seasonal Box- Jenkins Model for Nigerian Inflaion Rae Series, Journal of Mahemaical Research, 4(4), [9] Euk, E. H., Wokoma, D. S. A. and Moffa, I. U. (2013). Addiive Sarima Modelling of Monhly Nigerian Naira-CFA Franc Exchange Raes. European Journal of Saisics and Probabiliy, 1(1), [10] Hannan, E. J. and Quinn, B. G. (1979). The Deerminaion of he Order of an Auoregression. Journal of he Royal Saisical Sociey, Series B, 41, pp [11] Kibunja, H. W., Kihoro, J. M., Orwa, G. O., Yodah, W. O. (2014). Forecasing Precipiaion Using SARIMA model: A Case Sudy of M. Kenya Region. Mahemaical Theory and Modeling, 4(11), [12] Lee, K. J., Chi, A.Y., Yoo, S. Jin, J. J. (2008). Forecasing Korean Sock Price Index (KOSPI) using back propagaion neural nework model, Bayesian Chio s model, and SARIMA Model. Allied Academies Inernaional Conference, Figure 6. Correlogram of he Residuals of he Addiive Sarima Model References [1] Akaike, H. (1977). On Enropy Maximizaion Principle: Proceedings of he Symposium on Applied Saisics (P. R. Krishnaiah, ed.) Amserdam, Norh-Holland. [13] Luo, C. S., Zhou, L. Y. and Wei, O. F. (2013). Applicaion of SARIMA Model in Cucumber Price Forecas, Applied Mechanics and Maerials, , [14] Osarunwense, O. (2013). Applicabiliy of Box-Jenkins SARIMA model in rainfall Forecasing: A Case sudy of Por Harcour Souh Souh Nigeria. Canadian Journal on Compuing in Mahemaics, Naural sciences, Engineering and Medicine, 4(1), 1-4.

6 Inernaional Journal of Life Science and Engineering Vol. 1, No. 1, 2015, pp [15] Ou, A. O., Osuji, G. A., Opara, J. Mbachu, H. I., Iheagwara, A. I. (2014). Applicaion of Sarima Models in Modelling and Forecasing Nigeria s Inflaion Raes. American Journal of Applied Mahemaics and Saisics, 2(1), [16] Saz, G. (2011). The Efficacy of SARIMA Models for Forecasing Inflaion Raes in Developing Counries: The case for Turkey. Inernaional Research Journal of Finance and Economics, 62, [17] Schwarz, G. (1978). Esimaing he Dimension of a model. Annals of Saisics, 6, pp [19] Suharono and Lee, M. H. (2011). Forecasing of Touris Arrivals Using Subse, Muliplicaive or Addiive Seasonal ARIMA Model, Maemaika, 27(2), [20] Teye-Okuu, E.M.K. (2011). Modelling and Forecasing Ghana s inflaion raes using Sarima models. A Thesis submied o he Deparmen of Mahemaics, Kwame Nkrumah Universiy of Science and Technology in parial fulfilmen of he requiremens for he award of M.Sc. in Indusrial Mahemaics. [18] Suharono (2011). Time Series Forecasing by using Auoregressive Inegraed moving Average: Subse, Muliplicaive or addiive Model. Journal of Mahemaics and Saisics, 7(1),