Forecasting the Twelve Month Treasury Bill Rates in Sri Lanka: Box Jenkins Approach

Transcription

1 IOSR Journal of Economics and Finance (IOSR-JEF) e-issn: , p-issn: Volume, Issue (May. Jun. 203), PP Forecasing he Twelve Monh Treasury Bill Raes in Sri Lanka: Box Jenkins Approach D.M.K.N. Senevirana, Mao Shuhua 2 ( School of Economics, Wuhan Universiy of Technology, Wuhan, P.R. China, Deparmen of Inerdisciplinary Sudies, Faculy of Engineering, Universiy of Ruhuna, Sri Lanka) ( 2 Deparmen of Saisics, School of Sciences, Wuhan Universiy of Technology, Wuhan, P.R. China, ) Absrac : In his sudy, univariae ime series Auoregressive Inegraed Moving Average (ARIMA) model is used o forecas governmen welve monh Treasury bill raes in Sri Lanka over he period June, 2008 o June, 203. Box Jenkins mehodology is mainly used o build four models and differen diagnosic ess and crieria were applied o selec he appropriae model. The accuracy of he forecased values is compared wih Mean Squared Error (MSE) and Mean Absolue Error (MAE). The empirical resuls reveal ha he bes ARIMA model for he welve monh reasury bill raes is ARIMA (,,2). The obained model was used o forecas nex five weeks period and he resuls showed ha he slow decay of he T-bill raes. The decreasing of he ineres raes implies ha he increasing he considerable demand for he governmen T-bills. Therefore he findings of his sudy have been given some impression o he invesors for planning heir fuure invesmens. Keywords - ARIMA models, Box Jenkins, Forecasing, Treasury bill raes I. INTRODUCTION The Financial marke is a place where issues he securiies o he public. Basically here are wo ypes of governmen securiies namely Treasury Bills and Treasury Bonds. Among hem, Treasury bills can be known as shor erm, highly markeable, liquid and low-risk deb securiies. Moreover, T-bills make a funcional relaion beween he public and he governmen. Generally, governmen raises heir necessary capial requiremens by selling heir bills o he public. On he oher hand public gives heir conribuion hrough invesing on bills. The mauriy periods of T-bills are differen. They are varying weekly such as 4, 3, 26 and 52. The public can purchase T-bills direcly from he reasury or aucion and he secondary marke as well as can be sold a any ime in he secondary marke afer purchasing [8]. Though he risky siuaion always incorporaes wih securiies while invesing, T-bills are caegorized as low risk asses compared o he ohers since hey are issued under he governmen auhoriy. Due o hese reasons he governmen T-bills are highly markeable han he oher invesmens. Normally he public can purchase T-bills wih a discoun hrough a compeiive bidding process. Discouns represens he ineres, which would be differen based on he mauriy. This ineres will increase when he prices go down and will decrease when he prices rise. The flucuaion of he T-bill raes is affeced by several facors such as supply and demand of T-bills, inflaion, economic condiions and moneary policy. Normally here is a significan demand for governmen T-bills due o he shor erm period and low risk asses. When he demand for T-bills goes up, he governmen reduces he discoun since he number of available invesors are high. The governmen drops a supply of T-bills when here is a budge surplus in he counry. As well as T-bill raes go up due o he poor invesmen condiions would arise during he period of inflaion. On he oher hand T-bill raes go up and down when he counry is in a recession and boom respecively. Therefore he volailiy of he ineres or discouns of T-bills is a very imporan condiion for boh he public and he governmen o undersand he economic behavior of he counry. People are more ineresed in invesing heir capials in governmen T-bills raher han he oher securiies. Aenive on ineres raes and low risk asses can be considered as significan facors for increasing invesmens.therefore if i is possible o give an impression abou he volailiy of he T-bill raes, i will help o invesors as well as he sabiliy of he counry s economy. Hence he main objecive of his sudy is o find an appropriae Auoregressive Inegraed Moving Average (ARIMA) model o forecas governmen T-bill raes. For his purpose governmen welve monh reasury bill raes (by week) in Sri Lanka are considered over he period from June, 2008 o June 203. Secondary daa are obained hrough he published daa from Cenral bank of Sri Lanka. The nex par of he sudy is organized as follows. Secion II gives he background of he lieraure based on ARIMA model. Mehodology explains in secion III. The empirical resuls are given in secion IV. Secion V presens he conclusion of his sudy. 42 Page

2 Forecasing he Twelve Monh Treasury Bill Raes in Sri Lanka: Box Jenkins Approach II. LITERATURE REVIEW Differen ype of mehodologies have been developed o overcome his problem. Among hem, many researchers used ARIMA models for forecasing resuls. In 203, Paul e al used ARIMA model for forecasing average daily share price index of pharmaceuical companies in Bangladesh. In heir sudy, hey found he bes fied ARIMA model afer considering he differen ype of facors such as Akaike Informaion, correced Akaike informaion, Schwarz informaion, mean absolue percen error, roo mean square error and absolue mean error. Their empirical resuls indicaed ha he ARIMA (2,,2) model is he bes for forecasing he average daily share price indices []. Chujai e al conduced a research o find a model for forecasing he elecriciy consumpion in a household. As a main objecive of heir sudy, he mos suiable forecasing mehod was fied. The analysis resuls suggesed ha, wo forecasing mehods called auoregressive inegraed moving average and auoregressive moving average (ARMA) are mos suiable for forecasing fuure resuls. For his purpose hey applied Box Jenkins mehod and idenified he bes suiable model is ARIMA for monhly and quarerly as forecasing periods. On he oher hand, hey showed ha he ARMA model is suiable for forecasing based on daily and weekly periods [2]. Inflaion rae in Nigeria was forecased by Olajide e al based on he Box Jenkins approach. Yearly daa from 96 o 200 was used. According o he empirical resuls, hey suggesed ha he ARIMA (,,) model is he mos adequae for he inflaion rae. Based on he suggesed model, hey prediced he inflaion rae a 6.27% in he year 200 [3]. Kumar e al presened a sudy o build a univariae ime series model o forecas he expors of indusrial goods from Punjab. They followed he Box Jenkins mehodology and used several crieria o check he validiy of he model. Finally hey observed ha he ARIMA(2,d,) is he opimal for forecasing he variable expors [4]. According o he lieraure, anoher sudy carried ou by Chen e al (2008) based on ARIMA model. They used his model o forecas shor-erm propery crime for one ciy of China. In his sudy hey compared ARIMA model wih oher wo exponenial smoohing models and showed ha he bes fi is given by ARIMA model [5]. Al-Sahib sudied he predicabiliy of he Amman Sock Exchange(ASE) based on ARIMA model over a period of seven days. Differen diagnosic ess used o perform he bes fied model and showed ha he seleced model is suiable for forecasing on ASE [6]. Anoher research based on ARIMA model has done by Nochai e al. They invesigaed o find a model o forecas hree ypes of oil palm price in Thailand such as Farm price, Wholesale price and Pure oil price. Non-seasonal Box Jenkins mehodology is used and hree models are found based on he minimum of mean absolue percenage error (MAPE). Finally hey developed model for hree ypes of palm oil price and found ha models ARIMA (2,,0) for he farm price, ARIMA (,0,) for he wholesale price, and ARIMA (3,0,0) for he pure oil price [7]. III. DATA & METHODOLOGY The sudy is carried ou based on weekly daa of Twelve monh Treasury bill raes as secondary daa, which have been colleced from Cenral Bank of Sri Lanka, over he period from June, 2008 o June 203. The objec of his sudy is o find he appropriae ARIMA model o forecas welve monh reasury bill raes in Sri Lanka. For his purpose non-seasonal Box Jenkins approach is used o find he bes fied ARIMA model and he accuracy of he forecasing values are checked by comparing residuals. The seps of he suggesed model and is forecasing can be explained in he following seps. Deermining wheher he ime series is saionary or no is a very imporan concep before making any inferences in ime series analysis. Therefore Augmened Dickey Fuller (ADF) and Phillips-Person (PP) ess has been used o check he saionariy of he daa series. There are several mehods can be applied o fi a ime series model. Among hem, Auoregressive Inegraed moving average (ARIMA) model is used on he saionary daa in his sudy. 3.. Auoregressive Inegraed Moving Average (ARIMA) Model ARIMA models are exensions of an ARMA process by he inegraed (I) par and can be obained by combining he AR(p) and MA(q) models and defined in (). d d 2 ( B) Y ( B)( B) Y ( B) Z, { Z} ~ WN(0, ) () Where p and q are orders of AR and MA models, (B) and (B) are polynomials of order p and q respecively. d, non negaive ineger is he number of differences. B is he backward shif operaor [9]. Esimaing an ARIMA model was firs approached by Box and Jenkins (976) and according o heir mehodology, i follows hree seps as Idenificaion, Esimaion, and Diagnosic Checking [0]. The hree seps can be summarized in he following secions. 43 Page

3 Forecasing he Twelve Monh Treasury Bill Raes in Sri Lanka: Box Jenkins Approach 3.2. Model Idenificaion : Box Jenkins Approach Model idenificaion is he deerminaion of he order of he model, basically which can idenify based on sample auocorrelaion (ACF) and sample parial auocorrelaion (PACF) plos [0]. The Table explained he way of deermining he ARIMA model using sample ACF and sample PACF [9]. Table : Properies of he ACF and PACF of AR, MA and ARMA Series Model ACF PACF AR (p) Tails off Cu off afer lag p MA (q) Cu off afer lag q Tails off ARMA (p, q) Tails off Tails off 3.3. Esimaion of he Model Parameers Afer idenifying he possible ARIMA models, he maximum likelihood mehod is used o esimae he model parameers [0] Diagnosic Checking The nex sep is o selec he bes model among all he idenified models. For his, residual diagnosics and informaion crieria as AICC and BIC were used o check he adequacy. Under he residual diagnosics, Ljung-Box Q saisic is used o check wheher he residuals are random or no [0]. The corresponding null and alernaive hypohesis can be wrien as follows. H : 0 The model does no exhibi lack of fi H : The model exhibis lack of fi The es saisic Q is displayed in (2). Q n( n 2) m k rˆ 2 k n k rˆ is he esimaed auocorrelaion of he series a lag k, m is he number of lags includes in he ; where k es, and n is he number of residuals. The conclusion is considered based on he p-value associaed wih he Q saisic. If p-value <, hen i implies he adequacy of he model []. Moreover correced Akaike s and Bayesian informaion crieria can be used o selec he suiable model which has he lowes AICC and BIC values. On he oher hand ACF and PACF plos of residuals can be used o check he randomness of he residuals Forecasing Finally, Forecasing is done for all he seleced models and calculaed mean squared error (MSE) and mean absolue error (MAE) for all he models. The mos accurae forecasing among he models is seleced by considering he lowes value of he MSE and MAE [0]. Then he bes fied value has been used o forecas welve monh reasury bill raes for he nex Five weeks. IV. RESULTS AND DISCUSSION 4.. Saionariy As a firs sep, behavioral analysis was carried ou based on Time series plo and ACF plo. Time series plo of Fig. shows ha, nonexisence of significan rend and seasonal componens of daa in our experimen period. The ACF plo of Fig. shows he srong and slowly decaying auocorrelaions. Moreover, i implies ha he non saionariy of he level daa. (2) Figure : ime series plo and ACF plo of daa (by week) 44 Page

4 Forecasing he Twelve Monh Treasury Bill Raes in Sri Lanka: Box Jenkins Approach Fig.2 shows he sample ACF and PACF of he firs difference series. Clearly, i shows ha he significan auocorrelaion a lag and ohers are no significan. I suggess ha he firs difference series is saionary. Figure 2: Sample ACF and PACF of he Firs difference series Table 2: Uni roo es resuls Tes Level Firs Difference ADF (0.2735) (0.0000) PP (0.3007) (0.0000) The ADF and PP es resuls show ha he series is non saionary a levels and saionary a he firs difference. The resuls coincide wih he Time series plo, ACF plos and uni roo resuls. So, resuls suggesed ha he series is saionary a he firs difference Model Idenificaion According o he plos and uni roo ess resuls, level daa are no saionary and he series is saionary a he firs difference. Therefore ARIMA model can be proposed for he firs difference series. Based on he ACF and PACF plos in Fig.2, i shows ha he auocorrelaion of he firs lag in he wo plos is significan. Therefore Four possible models can be suggesed for he difference series. They are ARIMA (,,0), ARIMA (0,,), ARIMA (0,,2) and ARIMA(,,2) Model Esimaion In his sudy Maximum likelihood mehod is used o esimae he model parameers and he resuls are displayed in Table 3. Table 3: Esimaed model parameers Model Coefficiens S.E ARIMA(,,0) ARIMA(0,,) ARIMA(0,,2) (for MA()) (for MA(2)) ARMA(,,2) (for AR()) (for MA()) (for MA(2)) According o he maximum likelihood esimaion, Four models can be explained in (3), (4), (5), and (6). ARIMA (,,0) Y Y Z or Y Y ( Y Y ) Z (3) ARIMA (0,,) 0 2 Y Z Z or Y Y Z Z (4) ARIMA (0,,2) Y Z Z 0. Z or Y Y Z Z Z 2 ARIMA (,,2) (5) 45 Page

5 Forecasing he Twelve Monh Treasury Bill Raes in Sri Lanka: Box Jenkins Approach Y or Y Y Y Z Z Z ( Y Y 2) Z Z Z 2 (6) 4.4. Model Selecion The values for AICC, BIC, Ljung-Box saisic, and probabiliy of he seleced Four models are given in Table 4. Table 4: Comparaive resuls of models Model AICC BIC Ljung-Box Saisic P-value ARMA (,,0) E E ARMA (0,,) E E ARMA (0,,2) E E ARMA (,,2) E E Compared wih all he resuls in Table 4, he minimum values for boh AICC and BIC are indicaed o he ARIMA (,,2) model. As well as he minimum sandard error is also given by he same model. On he oher hand, he null hypohesis do no rejec a he 0.05 level of significance of he ARIMA (,,2) model based on he Ljung-Box saisic. By comparing all hese es values in Table 4, i can be suggesed ha he ARIMA (,,2) is he mos appropriae model for represening he Twelve monh reasury bill raes Model Forecasing Since he accuracy of he forecasing values is measured by comparing he mean square error (MSE) and mean absolue error (MAE), residuals are obained for all he proposed ARIMA models. Then he MSE and MAE are calculaed and displayed in Table 5. Table 5: Forecasing resuls of he seleced models Model MSE MAE ARIMA (,,0) ARIMA (0,,) ARIMA (0,,2) ARIMA (,,2) The minimum values for MSE and MAE are given under he model ARIMA (,,2), i suggess ha he accurae forecasing values can be obained using he ARIMA (,,2) model. Then he forecasing values are shown in Fig.3 and Table 6. Figure 3: Weekly Time Series Forecasing for Treasury Bill Raes (welve monh) The forecased values for nex five weeks from June 203 for welve monh T-bill raes are shown in Table Page

6 Forecasing he Twelve Monh Treasury Bill Raes in Sri Lanka: Box Jenkins Approach Table 6: Forecased welve monhs T-bill raes since June, 203 Sep Predicion Sqr (MSE) Approximae 95 Percen Predicion Bounds Lower Upper The forecasing resuls suggesed ha, here will be a slow decreasing of he welve monh reasury bill raes in he nex five weeks. V. CONCLUSION This sudy was carried ou o find an appropriae ARIMA model o forecas Twelve monh governmen T-bill raes in Sri Lanka. For his purpose Box Jenkins mehodology was used and ARIMA (,,2) model was seleced as he mos suiable model based on MSE and MAE. ARIMA (,,2) forecasing model used o forecas he nex five weeks T-bill raes and i does no show significan volailiy. However, here is a very slow decay can be seen. Therefore mainly invesors can plan heir fuure invesmens by considering he fuure behavior of he T- bill raes. On he oher hand decreasing of he ineres raes implies ha he increasing he considerable demand for he governmen T-bills. REFERENCES [] J.C. Paul, S. Hoque, and M.M. Rahman, Selecion of Bes ARIMA Model for Forecasing Average Daily Share Price Index of Pharmaceuical Companies in Bangladesh: A case Sudy on Square Pharmaceuical Ld, Global Journal of Managemen and Business Research, 3(3), 203. [2] P. Chujai, N. Kerdprasop, and K. Kerdprasop, Time Series Analysis of Household Elecric Consumpion wih ARIMA and ARMA Models, Proc. IMECS Conf., Hong Kong, 203. [3] J.T. Olajide, O.A. Ayansola, M.T. Odusina, and I.F. Oyenuga, Forecasing he Inflaion Rae in Nigeria: Box Jenkins Approach, IOSR Journal of Mahemaics (IOSR-JM), 3(5), 202, 5-9. [4] G. Kumar, and S. Gupa, Forecasing Expors of Indusrial Goods from Punjab An Applicaion of Univariae ARIMA Model, Annals of he Universiy of Perosani, 0(4), 200, [5] P. Chen, H. Yuan, and X. Shu, Forecasing Crime Using he ARIMA Model, 5h IEEE Conf. on Fuzzy Sysems and Knowledge Discovery, [6] M. Al-Shiab, The Predicabiliy of he Amman Sock Exchange using he Univariae Auoregressive Inegraed Moving Average (ARIMA) Model, Journal of Economic & Adminisraive Sciences, 22(2), [7] R. Nochai, and T. Nochai, ARIMA Model for Forecasing Oil Palm Price, Proc. 2nd IMT-GT Regional Conf. on Mahemaics, Saisics and Applicaions, Universiy Sains Malaysia, Penang, [8] Z. Bodie, A. Kane, and A.J. Marcus, Invesmens (Inc and China Machine Press: McGraw-Hill Companies, 202). [9] MAS328, Time series analysis (Universiy London: School of Mahemaical Sciences, 2006). [0] C. Brooks, Inroducory economerics or finance (New York, USA: Cambridge Universiy Press, 2008). [] G.S. Maddala, Inroducion o economerics (Wes Sussex, England: John Wiley & Sons Ld, 2002). 47 Page