Box-Jenkins Modelling of Nigerian Stock Prices Data

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1 Greener Journal of Science Engineering and Technological Research ISSN: Vol. (), pp , Sepember 0. Research Aricle Box-Jenkins Modelling of Nigerian Sock Prices Daa Ee Harrison Euk*, Barholomew Uchendu, Ephraim Okon Udo Deparmen of Mahemaics/Compuer Science, Rivers Sae Universiy of Science and Technology, Nigeria ABSTRACT *Corresponding Auhor s eeuk@yahoo.com Nigerian sock prices daa is modelled by Box-Jenkins approach and he use of auomaic model selecion crieria: Akaike Informaion crierion (AIC), Schwarz Informaion Crierion (SIC), R. I is inferred ha he mos adequae model is auoregressive inegraed moving average of orders, and 3(ARIMA (,,3)). Forecass are obained on he basis of he model. Key Words: Sock prices, ARIMA modelling, AIC, SIC, Nigeria. INTRODUCTION A ime series is defined as a se of daa colleced sequenially in ime. I has he propery ha neighbouring values are correlaed. This endency is called auocorrelaion. A ime series is said o be saionary if i has a consan mean and variance. Moreover he auocorrelaion is a funcion of he lag separaing he correlaed values called he auocorrelaion funcion (ACF). A saionary ime series { } 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... ()... p p = ε βε β ε q q Or (B) = β(b)ε () where {ε } is a sequence of random variables wih zero mean and consan variance, called a whie noise process, and he i s and β j s consans; (B) = B B... p B p and β(b) = β B β B... β q B q and B is he backward shif operaor defined by B k = -k. If p=0, model () becomes a moving average model of order q (designaed MA(q)). If, however, q=0 i becomes an auoregressive process of order p (designaed AR(p)). An AR(p) model of order p may be defined as a model whereby a curren value of he ime series depends on he immediae pas p values: -, -,..., -p. On he oher hand an MA(q) model of order q is such ha he curren value is a linear combinaion of immediae pas values of he whie noise process: ε, ε,..., ε q. Apar from saionariy, inveribiliy is anoher imporan requiremen for a ime series. I refers o he propery whereby he covariance srucure of he eries is unique (Priesley, 98). Moreover i allows for meaningful associaion of curren evens wih he pas hisory of he series (Box and Jenkins, 976). An AR(p) model may be more specifically wrien as p - p -... pp -p = ε Then he sequence of he las coefficiens{ ii } is called he parial auocorrelaion funcion(pacf) of { }. The ACF of an MA(q) model cus off afer lag q whereas ha of an AR(p) model is a combinaion of sinusoidals dying off slowly. On he oher hand he PACF of an MA(q) model dies off slowly whereas ha of an AR(p) model cus off afer lag p. AR and MA models are known o have some dualiy properies. These include:. A finie order AR model is equivalen o an infinie order MA model.. A finie order MA model is equivalen o an infinie order AR model. 3. The ACF of an AR model exhibis he same behaviour as he PACF of an MA model. 3

2 Greener Journal of Science Engineering and Technological Research ISSN: Vol. (), pp , Sepember The PACF of an AR model exhibis he same behaviour as he ACF of an MA model. 5. An AR model is always inverible bu is saionary if (B) = 0 has zeros ouside he uni circle. 6. An MA model is always saionary bu is inverible if β(b) = 0 has zeros ouside he uni circle. Parameric parsimony consideraion in model building enails preference for he mixed ARMA fi o eiher he pure AR or he pure MA fi. Saionariy and inveribiliy condiions for model () or () are ha he equaions (B) = 0 and β(b) = 0 should have roos ouside he uni circle respecively. Ofen, in pracice, a ime series is non-saionary. Box and Jenkins [] proposed ha differencing of an appropriae daa could render a non-saionary series { } saionary. Le degree of differencing necessary for saionariy be d. Such a series { } may be modelled as ( ) d = β(b)ε (3) where = B and in which case (B) = = 0 shall have uni roos d imes. Then differencing o degree d renders he series saionary. The model (3) is said o be an auoregressive inegraed moving average model of orders p, d and q and designaed ARIMA(p, d, q). The purpose of his paper is o fi an ARIMA model o Nigerian sock prices. MATERIALS AND METHODS The daa for his work are monhly sock prices daa from January987 o December 006 obained from Nigerian Sock Exchange Office, Por Harcour, Nigeria. Deerminaion of he differencing order d: Preliminary analysis of ime series involves he ime-plo and he correlogram. A saionary ime series exhibis no rend and he degree of variabiliy is invarian wih ime. In addiion he covariance is a funcion of he ime lag. The ime plo of a saionary ime series shows no change in he mean level as well as he variance over ime. The auocorrelaion funcion should decay fas o zero. Tes for saionariy: The ACF of a non-saionary ime series sars high and declines slowly. Moreover o es for saionariy we shall be using he Augmened Dickey-Fuller (ADF) es. This involves esing for b= agains b < in = a b - ε. The sofware Eviews 3. ha we shall use has faciliy for he ADF es also. Deerminaion of he orders p and q: As already menioned above, an AR(p) model has a PACF ha runcaes a lag p and an MA(q)) has an ACF ha runcaes a lag q. In pracice are he nonsignificance limis for boh funcions. We shall explore he range of models ARMA(a,b), 0 a p, 0 b q for an opimum one. To do his we shall use he auomaic model deerminaion crieria AIC and SIC ( e.g. Akaike(970), Euk (987, 988), and Schwarz(978)) defined by: AIC p d q n ~ ( ) = ln σ p SIC( p d q) nln ~ = σ d q p d q ( p d q) ( p d q)ln( n) / n where is he maximum likelihood esimae of he residual variance when he model has k parameers.the opimum model corresponds o he minimum of he crieria wihin he explored range. Model Esimaion: The involvemen of he whie noise erms in an ARIMA model enails a nonlinear ieraive process in he esimaion of he parameers, i s and β j s. An opimizaion crierion like leas error of sum of squares, maximum likelihood or maximum enropy is used. An iniial esimae is usually used. Each ieraion is expeced o be an improvemen of he las one unil he esimae converges o an opimal one. However,for pure AR and pure MA models linear opimizaion echniques exis (See for example Box and Jenkins (976), Oyeunji (985)). There are aemps o adop linear mehods o esimae ARMA models (See for example, Euk(987, 988, 996)). 33

3 Greener Journal of Science Engineering and Technological Research ISSN: Vol. (), pp , Sepember 0. Diagnosic Checking: The model ha is fied o he daa should be esed for goodness-of-fi. The auomaic order deerminaion crieria AIC and SIC are hemselves diagnosic checking ools. Furher checking can be done by he analysis of he residuals of he model. If he model is correc, he residuals would be uncorrelaed and would follow a normal disribuion wih mean zero and consan variance. RESULTS AND DISCUSSION The ime plo of he original series NSP in Figure, he correlogram of Figure and he ADF es of Table clearly depic non-saionariy. Differencing he series once yields a saionary process, DNSP; he ime plo is in Figure 3, he correlogram in Figure 4 and he ADF es in Table. We noe ha in his able he dependen variable is he second difference SNSP of he original series. From fig. 4, he ACF cus off a lag 5 and PACF a lag 4. Exploring he range of models {ARMA(p,q): 0 p 4, 0 q 5} which are saionary and inverible wih posiive R for he opimal on he basis of AIC and SIC yields an ARMA(, 3). The model esimaion is summarized in Table 3. FIGURE : CORRELOGRAM OF NSP 34

4 Greener Journal of Science Engineering and Technological Research ISSN: Vol. (), pp , Sepember 0. TABLE : ADF TEST FOR NSP 35

5 Greener Journal of Science Engineering and Technological Research ISSN: Vol. (), pp , Sepember 0. FIGURE 4: CORRELOGRAM OF DNSP TABLE : ADF TEST ON DNSP 36

6 Greener Journal of Science Engineering and Technological Research ISSN: Vol. (), pp , Sepember 0. TABLE 3: MODEL ESTIMATION The chosen model as summarized in Table 3 is ARIMA(,, 3) and is given by DNSP = DNSP DNSP ε ε ε -3 ε (±0.093) (± ) (± ) (±0.0495) (± ) Clearly non-linear echniques used by Eviews 3. involved an ieraive process ha converged afer hiry one ieraions. We observe ha only he firs and hird MA coefficiens are no significan, each being less han wice is sandard error. The roos of (B) = 0 and β(b) = 0 all lie ouside he uni circle indicaing saionariy and inveribiliy respecively. Besides he residual plo of Fig. 5 confirms ha he residuals follow he normal disribuion wih zero (acually 0.) mean. FIGURE 5: HISTOGRAM OF MODEL RESIDUALS Forecasing: An ARIMA(,, 3) is of he form 37

7 Greener Journal of Science Engineering and Technological Research ISSN: Vol. (), pp , Sepember 0. = ε 3 3 ε = ( ) ( ) = A ime k, k = ( ) k ( ) k k 3 βε k β ε k β 3ε k3 ε k Taking condiional expecaions a ime, ˆ ˆ () () = ( ) ( ) 3 ˆ = ( ) () ( ) 3 ˆ (3) = ( ˆ ˆ ) () ( ) () β3 ˆ ( k) = ( ) ˆ ( k ) ( ) ˆ ( k ) ˆ ( k 3), k 4 where ˆ ( k) is he k-poin ahead forecas. Tha is he forecas of k given he series up o. ε Table 6. Forecass TIME RESIDUALS DNSP NSP Ocober 006 November 006 December January 007 February 007 March 007 April CONCLUSION: We have fied an adequae ARIMA (,,3) model o Nigerian Sock Prices. Tha means ha he firs differences DNSP follow an ARMA (,3) model. On he basis of he model, we have made some forecass. REFERENCES Akaike, H. (970). Saisical Predicor Idenificaion. Annals of he Insiue of Saisical Mahemaics, Volume : pp Box, G. E. P. And Jenkins, G. M. (976). Time Series Analysis, Forecasing and Conrol, Holden-Day, San Francisco. Euk, E. H. (987). On he Selecion of Auoregressive Moving Average Models. An unpublished Ph. D. Thesis, Deparmen of Saisics, Universiy of Ibadan, Nigeria. Euk, E. H. (988). On Auoregressive Model Idenificaion. Journal of Official Saisics, Volume 4, No. ; pp Euk, E. H. (996). An Auoregressive Inegraed Moving Average (ARIMA) Simulaion Model: A Case Sudy. Discovery and Innovaion, Volume 0, Nos & :pp Oyeunji, O. B. (985). Inverse Auocorrelaions and Moving Average Time Series Modelling. Journal of Official Saisics, Volume : pp Priesley, M. B. (98). Specral Analysis and Time Sreies. Academic Press, London. Schwarz, G. (978). Esimaing he dimension of a model. Annals of Saisics, Volume 6: pp