Imo Udo Moffat Department of Mathematics/Statistics, University of Uyo, Nigeria

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1 Inernaional Journals of Advanced Research in Compuer Science and Sofware Engineering ISSN: (Volume-7, Issue-8) a Research Aricle Augus 2017 Applicaion of Inerruped Time Series Modelling o Prime Moor Spiri Disribuion in Nigeria Ee Harrison Euk Deparmen of Mahemaics, Rivers Sae Universiy, Nigeria DOI: /ijarcsse/V7I7/01706 Imo Udo Moffa Deparmen of Mahemaics/Saisics, Universiy of Uyo, Nigeria Azubuike Samuel Agbam Deparmen of Banking and Finance, Rivers Sae Universiy, Nigeria Absrac An inspecion of he ime-plo of monhly Prime Moor Spiri (PMS) disribuion in Nigeria from 2009 o 2015 reveals an abrup jump in January 2013 wih he series coninuing a ha level ill Clearly he rend of he series was inerruped in January 2013 and i is believed ha his perurbaion was due o he deregulaion of he downsream secor of he crude oil indusry. A -es comparison of he pre- and he pos-inervenion means is highly significan (p < ) indicaing he impac of he inervenion. A model of he ARIMA family was o be fied o he pre-inervenion daa which were observed o have a downward rend and be non-saionary. Differencing once rendered i saionary. An adequae ARIMA(2,1,0) model was fied o he original pre-inervenion series. Posinervenion forecass were obained on he basis of his model. These forecass were subraced from heir respecive pos-inervenion counerpar observaions. These differences were modelled o obain he ransfer funcion of he inervenion. The resulan inervenion model closely fis he pos-inervenion daa and may be used o explain and conrol he siuaion. Keywords Prime moor spiri, inervenion, arima modelling, Nigeria, disribuion I. INTRODUCTION A commodiy consumed on a consan basis and exensively in Nigeria is Prime Moor Spiri (PMS) oherwise known as gasoline or perol. I is used for elecriciy generaion for auomobiles and oher uses. I is a major componen of crude oil which is drilled in commercial quaniies in he counry which is one of he major producers in he world. Par of he consumed PMS comes from he refineries in he refined form and some oher par comes as impor. This work is a sudy of he disribuion of he commodiy. I has been observed ha he disribuion of he produc beween 2009 and 2015 is such ha here was an abrup rise in January 2013 probably due o he deregulaion of he downsream secor of he peroleum indusry of he counry in January In paricular i is he aim of his sudy o propose an inervenion model o explain he impac of his inervenion o he monhly disribuion of he produc. The approach adoped is he Box-Tiao [1] echnique for which auoregressive inegraed moving average (ARIMA) modelling is done and from which an inervenion ransfer funcion is obained. This approach has been exensively and successfully used o model cases of inervenions. For insance, Guerard [2] demonsraed he superioriy of his approach o he applicaion of random walk wih drif in he forecasing a quarerly earnings per share series. A comparaive analysis of ARIMA ransfer funcions has been made by Helfensein [3]. Sridharan [4] used inervenion analysis o show ha aboliion of parole in January 1995 had he effec of reducing crime raes. Ray e al. [5] have found he performance of ARIMA inervenion o be superior o ha of he convenional ARIMA modelling for forecasing yields of coon a Gujara, Maharashra in India. By he use of his approach he esablishmen of Federal Road Safey Corps in Nigeria has been shown o have he impac of reducing he number of accidens on Nigerian roads emporarily by Oreko e al. [6]. Bell e al. [7] evaluaed he impac of healhcare inervenion on he smoking habis of some pregnan women and observed ha i produced he expeced effec of increased quiing of he habi by he pregnan women on delivery. Euk and Vicor-Edema [8] proposed an inervenion model for monhly Naira/Euro exchange raes. This is o menion a few cases. Secion I of his work is an inroducion o he paper. Secion II discusses he maerials and mehods used in his work. Secion III discusses he resuls of daa analysis and Secion IV is he conclusion. Afer he References here is an Appendix which is a lising of he daa analysed in his research work. II. MATERIALS AND METHODS A. Daa Daa for his work are monhly Nigerian PMS disribuion daa from January 2009 o December 2015 from he websie of he Nigerian Naional Peroleum Corporaion (NNPC) hp://nnpcgroup.com/. They are expressed in meric onnes and are lised ou in he appendix of his paper. B. Inerruped Time Series Modelling A ime series { } is said o follow an auoregressive moving average of order p and q denoed by ARMA(p,q) if All Righs Reserved Page 46

2 Euk e al., Inernaional Journal of Advanced Research in Compuer Science and Sofware Engineering7(8) ISSN(E): , ISSN(P): , DOI: /ijarcsse/V7I7/01706, pp p p q q (1)... where { } is a whie noise process and he s and s are consans chosen such ha he model is boh saionary and inverible. In erms of backward shif noaion model (1) may be pu as A(L) = B(L) (2) p where A(L) is he auoregressive (AR) operaor defined by A( L) 1 L L 2... L 1 2 P and B(L) is he q moving average (MA) operaor defined by B( L) 1 L L 2... L k 1 2 q where L k. If he series { } is replaced by is d h difference, {(1-L) d } in (1) or (2), { } is said o follow an auoregressive inegraed moving average model of order p, d and q denoed by ARIMA(p,d,q). Suppose ha he ime series { } experiences an inervenion a = T. Box and Tiao [1] propose ha he pre-inervenion daa be modelled wih an adequae ARIMA model. Suppose i is an ARIMA(p,d,q) model. In pracice he model (1) or (2) is esimaed saring wih he model dimensions represened by p, d and q. The differencing order d is pu a 0 firs, and saionariy esed by he Augmened Dickey Fuller (ADF) es. If he series is adjudged saionary hen d=0. If no, d=1 and if he series is saionary, d=1 bu if no, he process coninues. The AR order p is deermined by he cu-off lag of he auocorrelaion funcion ACF and he MA order q by he cu-off poin of he parial auocorrelaion funcion PACF. Esimaion of he s and s is done by he leas squares procedure. Le pos-inervenion forecass F be made on he basis of he model and suppose Z = F, T. Then according o Pennsylvania Sae Universiy [9] C(1)*(1 C(2)^ ( T 1)) Z (3) (1 C(2)) The overall inervenion model is given by B( L) Y I d Z A( L) where I = 0, < T and I = 1, T. C. Compuer Package The compuer sofware used in his work is Eviews 7. This package uses he leas square procedure for esimaion purposes. III. RESULTS AND DISCUSSION Figure 1 shows ha he monhly disribuion of PMS experienced an inervenion in January This is probably due o he deregulaion of he downsream secor of he Nigerian crude oil indusry in January The inervenion poin is January 2013 one year afer he deregulaion was pu in place by he hen Presiden, Goodluck Ebele Jonahan. The ime plo of he pre-inervenion disribuion in Figure 2 shows a downward rend. Figures 3 and 4 are he hisograms of he pre- and pos-inervenion daa respecively. A suden s -es of heir mean difference is saisically significan wih p < This is an indicaion of he impac of he inervenion of he disribuion of he commodiy. However, applicaion of -es in his siuaion is no ideal as is applicaion is in violaion of he independence requiremens of he daa because hey consiue a ime series. (4) Figure 1: PMS Disribuion in Nigeria All Righs Reserved Page 47

3 Euk e al., Inernaional Journal of Advanced Research in Compuer Science and Sofware Engineering7(8) ISSN(E): , ISSN(P): , DOI: /ijarcsse/V7I7/01706, pp Figure 2: Pre-inervenion PMS Disribuion Figure3: Hisogram of Pre-inervenion PMS disribuion Figure 4: Hisogram of Pos-inervenion PMS disribuion All Righs Reserved Page 48

4 Euk e al., Inernaional Journal of Advanced Research in Compuer Science and Sofware Engineering7(8) ISSN(E): , ISSN(P): , DOI: /ijarcsse/V7I7/01706, pp A beer approach is o fi an inervenion model. Firsly an ARIMA model has o be fied o he pre-inervenion series. Wih a es saisic value of and wih respecive 1%, 5% and 10% criical values of -3.68, and -2.60, he ADF Tes adjudges he pre-inervenion series as non-saionary. The series was differenced and he difference, wih an ADF es saisic value of -8.00, was pronounced as saionary. The correlogram of he differences are given in Figure 5. The firs auocorrelaion and he firs wo parial auocorrelaions are he only saisically significan correlaions. One of he mos adequae models on AIC grounds is he ARIMA(2,1,0) model esimaed in Table 1 which is given by (5) where { } is he monhly disribuion of PMS and = 1-L. Adequacy of model (5) is eviden from Figures 6 and 7 which show ha he residuals are uncorrelaed and normally disribued respecively. Forecass on he basis of model (5) for he pos-inervenion period were obained. Le F be he forecas for ime, 49. Then Z = F, 49 was modelled using equaion (3). The inervenion ransfer funcion is as esimaed in Table 2 as Z *( ) ( ) (6) where 49. The overall inervenion model is given by Y F I Z (7) where F and I 2 = 0, if < 49 and I = 1, if 49. A comparison of he inervenion ( L L ) forecass and pos-inervenion observaions in Figure 8 shows a close agreemen. This corroboraes he -es conclusion of a significan impac of he deregulaion policy. Figure 5: Correlogram of he differences of he pre-inervenion daa TABLE 1: ESTIMATION OF THE ARIMA(2,1,0) PRE-INTERVENTION MODEL Dependen Variable: (1-L) Variable Coefficien Sandard Error -Saisic Probabiliy AR(1) AR(2) R-squared AIC Invered AR Roos i i All Righs Reserved Page 49

5 Euk e al., Inernaional Journal of Advanced Research in Compuer Science and Sofware Engineering7(8) ISSN(E): , ISSN(P): , DOI: /ijarcsse/V7I7/01706, pp Figure 6: Correlogram of he residuals of he ARIMA(2,1,0) pre-inervenion model Figure 7: Hisogram of he residuals of he ARIMA(2,1,0) pre-inervenion model TABLE 2: Esimaion of he Inervenion Transfer Funcion Dependen Variable: Z Coefficien Sandard Error -Saisic Probabiliy C(1) C(2) R-squared AIC Figure 8: Comparison of he Inervenion Model and Pos-inervenion daa All Righs Reserved Page 50

6 Euk e al., Inernaional Journal of Advanced Research in Compuer Science and Sofware Engineering7(8) ISSN(E): , ISSN(P): , DOI: /ijarcsse/V7I7/01706, pp IV. CONCLUSIONS I may be concluded ha PMS disribuion has been significanly affeced by he deregulaion of he downsream secor of he Nigerian crude oil indusry. An adequae inervenion model in his respec is given by equaion (7). This migh be used o explain and conrol his phenomenon. REFERENCES [1] G. E. P. Box and G. C. Tiao, Inervenion Analysis wih applicaions o economic and environmenal problems, Journal of American Saisical Associaion, vol. 70, pp , [2] J. B. Guerard, Auomaic ime series modeling, inervenion analysis, and effecive forecasing, Journal of Saisical Compuaion and Simulaion., vol. 34, no. 1, pp , [3] U. Helfensein, The use of ransfer funcion models, inervenion analysis and relaed ime series mehods in epidemiology, Inernaional Journal of Epidemiology., vol. 20, no. 3, pp , [4] S. Sridharan, Inervenion ime series analysis of crime raes: he impac of senence reforms in Virginia, hps://papers.inbergen.nl/03040.pdf accessed 26 July [5] M. Ray, V. Ramasubramanian, A. Kumar and A. Rai, Applicaion of ime series inervenion modeling for modeling and forecasing coon yield, Saisics and Applicaions, vol. 12, nos. 1 & 2, pp , [6] B. U. Oreko, C. C. Nwobi-Okoye, S. Okiy and A. C. Igboanugo, Modelling he impac of inervenion measures on oal acciden cases in Nigeria using Box-Jenkins mehodology: A case sudy of Federal Road Safey Commission, Cogen Engineering, vol. 4, no. 1., [7] R. Bell, S. V. Gliniaia, Z. Waal, A. Close, E. Moloney, S. Jones e al. Evaluaion of a complex healhcare inervenion o increase smoking cessaion in pregnan women: inerruped ime series analysis wih economic evaluaion, accessed on 26 July [8] E. H. Euk and U. A. Vicor-Edema, Box-Tiao inervenion modeling of monhly EUR-NGN exchange raes due o Nigerian economic recession, Journal of Science and Engineering Research, vol. 4, no. 3, pp , [9] The Pennsylvania Sae Universiy, Welcome o STAT 510! Applied Time Series Analysis. Deparmen os Saisics Online Program. accessed 9 h November APPENDI Daa Year January February March April May June July Augus Sepember Ocober November December All Righs Reserved Page 51