2012, TexRoad Publicaion ISSN 2090-4304 Journal of Basic and Applied Scienific Research www.exroad.com Arima Fi o Nigerian Unemploymen Daa Ee Harrison ETUK 1, Barholomew UCHENDU 2, Uyodhu VICTOR-EDEMA 3 1 Deparmen of Mahemaics/Compuer Science, Rivers Sae Universiy of Science and Technology, Nigeria 2 Deparmen of Mahemaics/Saisics, Federal Polyechnic, Nekede, Imo Sae, Nigeria 3 Deparmen of Mahemaics/Saisics, Rivers Sae Universiy of Educaion, Nigeria ABSTRACT Nigerian unemploymen daa is modelled by Box-Jenkins approach and he use of auomaic model selecion crieria Akaike Informaion crierion (AIC) and Schwarz Informaion Crierion (SIC). I is inferred ha he mos adequae model is auoregressive inegraed moving average of orders 1, 2 and 1(ARIMA(1,2,). Forecass are obained on he basis of he model. KEY WORDS: Unemploymen daa, 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 or... 2... 1 1 2 2 p p 1 1 2 q q ( (B) = (B) (2) where { } is a sequence of random variables wih zero mean and consan variance, called a whie noise process, and he i s and j s consans; (B) = 1 + 1 B + 2 B 2 +... + p B p and (B) = 1 + 1 B + 2 B 2 +... + 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: -1, -2,..., -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: 1, 2,..., 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 [7]. Moreover i allows for meaningful associaion of curren evens wih he pas hisory of he series [2]. An AR(p) model may be more specifically wrien as + p1-1 + p2-2 +... + pp -p = 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: 1. A finie order AR model is equivalen o an infinie order MA model. 2. 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. 4. 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 (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 (2) are ha he equaions (B) = 0 and (B) = 0 should have roos ouside he uni circle respecively. *Corresponding Auhor: Ee Harrison ETUK, Deparmen of Mahemaics/Compuer Science, Rivers Sae Universiy of Science and Technology, Nigeria. Email: eeuk@yahoo.com 5964
ETUK e al., 2012 Ofen, in pracice, a ime series is non-saionary. Box and Jenkins [2] 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 (1 + α B ) d = (B) (3) where = 1 B and in which case (B) = (1 + α 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 monhly unemploymen rae daa of Nigeria. MATERIALS AND METHODS The daa for his work are monhly unemploymen rae daa from 1999 o 2008 obainable from quarerly absracs of he Cenral Bank of Nigeria. Unemploymen rae in his conex is he percenage of he workforce ha are wihou jobs. 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=1 agains b < 1 in = a + b -1 +. The sofware Eviews 3.1 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 ±2 n 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. [1], [3], [4] and[ 8]) defined by: AIC p d q n ~ 2 ( ) ln p d q 2( p d q) SIC( p d q) nln ~ 2 ( p d q)ln( n) / n p d q where σ 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 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 [2],[6]). There are aemps o adop linear mehods o esimae ARMA models (See for example, [3], [4], [5]). 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 NUMP in Fig.1and he correlogram of Figure 2 clearly depic nonsaionariy. Differencing he series once yields a sill non-saionary process, DNUMP; he ADF es of Table 1 confirms he non-saionary naure. This necessiaed second order differencing. The ADF es of Table 2 aess o he saionary naure of he second differences SNUMP. We noe ha in his able he dependen variable is he hird difference TNUMP of he original series. From fig. 4, he ACF cus off a lag 5 and he PACF a lag 4. Exploring he range of models {ARMA(p,q): 0 p 4, 0 q 5}for he opimal on he basis of AIC and SIC yields an ARMA(1, as summarized in Table 3. 5965
FIG. 2. CORRELOGRAM OF NUMP 5966
ETUK e al., 2012 TABLE 1: Augmened Dickey Fuller Tes on DNUMP ADF Tes Saisic -1.400118 1% Criical Value* -3.4885 5% Criical Value -2.8868 10% Criical Value -2.5801 *MacKinnon criical values for rejecion of hypohesis of a uni roo. Augmened Dickey-Fuller Tes Equaion Dependen Variable: SDNUMP Mehod: Leas Squares Dae: 12/24/11 Time: 17:16 Sample(adjused): 1999:07 2008:12 Included observaions: 114 afer adjusing endpoins Variable Coefficien Sd. Error -Saisic Prob. D((-) -0.222120 0.158644-1.400118 0.1643 D((-,2) -0.830500 0.161518-5.141845 0.0000 D((-2),2) -0.519311 0.166142-3.125705 0.0023 D((-3),2) -0.276272 0.147540-1.872524 0.0638 D((-4),2) -0.243273 0.099133-2.454007 0.0157 C 0.012374 0.012392 0.998626 0.3202 R-squared 0.596501 Mean dependen var 0.002632 Adjused R-squared 0.577821 S.D. dependen var 0.197981 S.E. of regression 0.128639 Akaike info crierion -1.212419 Sum squared resid 1.787181 Schwarz crierion -1.068408 Log likelihood 75.10788 F-saisic 31.93177 Durbin-Wason sa 2.014234 Prob(F-saisic) 0.000000 TABLE 2: Augmened Dickey Fuller Tes on SDNUMP ADF Tes Saisic -7.831665 1% Criical Value* -3.4890 5% Criical Value -2.8870 10% Criical Value -2.5802 *MacKinnon criical values for rejecion of hypohesis of a uni roo. Augmened Dickey-Fuller Tes Equaion Dependen Variable: TNUMP Mehod: Leas Squares Dae: 12/24/11 Time: 17:10 Sample(adjused): 1999:08 2008:12 Included observaions: 113 afer adjusing endpoins Variable Coefficien Sd. Error -Saisic Prob. D((-,2) -3.783728 0.483132-7.831665 0.0000 D((-,3) 1.730148 0.424541 4.075343 0.0001 D((-2),3) 1.016548 0.331170 3.069566 0.0027 D((-3),3) 0.555296 0.215904 2.571960 0.0115 D((-4),3) 0.130074 0.097086 1.339781 0.1832 C 0.009594 0.012215 0.785434 0.4339 R-squared 0.878016 Mean dependen var 0.002655 Adjused R-squared 0.872316 S.D. dependen var 0.361905 S.E. of regression 0.129319 Akaike info crierion -1.201433 Sum squared resid 1.789405 Schwarz crierion -1.056616 Log likelihood 73.88096 F-saisic 154.0331 Durbin-Wason sa 2.003449 Prob(F-saisic) 0.000000 FIG. 4: CORRELOGRAM OF SNUMP 5967
Table 3. Comparison of models wihin he range of exploraion using AIC and SIC p q AIC SIC 0 1-0.997-0.974 0 2-1.234-1.187 0 3-1.223-1.153 0 4-1.116-1.022 0 5-1.208-1.091 1 0-0.999-0.975 1 1-1.250-1.203 1 2-1.235-1.164 1 3-1.224-1.129 1 4-1.215-1.097 1 5-1.150-1.009 2 0-1.174-1.127 2 1-1.231-1.159 2 2-1.220-1.126 2 3-1.234-1.116 2 4-1.217-1.075 2 5-1.215-1.049 3 0-1.158-1.086 3 1-1.212-1.117 3 2-1.225-1.105 3 3-1.208-1.065 3 4-1.194-1.027 3 5-1.190-0.999 4 0-1.225-1.129 4 1-1.229-1.109 4 2-1.211-1.067 4 3-1.238-1.070 4 4-1.222-1.030 4 5-1.218-1.002 TABLE 4: Model Esimaion Dependen Variable: SDNUMP Mehod: Leas Squares Dae: 12/24/11 Time: 18:32 Sample(adjused): 1999:04 2008:12 Included observaions: 117 afer adjusing endpoins Convergence achieved afer 6 ieraions Backcas: 1999:03 Variable Coefficien Sd. Error -Saisic Prob. AR( -0.388391 0.099793-3.891961 0.0002 MA( -0.716548 0.077618-9.231771 0.0000 R-squared 0.587306 Mean dependen var 0.005128 Adjused R-squared 0.583717 S.D. dependen var 0.199069 S.E. of regression 0.128440 Akaike info crierion -1.249770 Sum squared resid 1.897123 Schwarz crierion -1.202554 Log likelihood 75.11156 F-saisic 163.6567 Durbin-Wason sa 1.966983 Prob(F-saisic) 0.000000 Invered AR Roos -.39 Invered MA Roos.72 The chosen model as summarized in Table 4 is ARIMA(1, 2, and is given by SDNUMP = -0.388391SDNUMP -1 0.716548-1 + ( 0.099793) ( 0.077618) Clearly non-linear echniques used by Eviews 3.1 involved an ieraive process ha converged afer six ieraions. We observe ha he coefficiens are boh highly significan, each being more han wice is sandard error. The roos of (B) = 0 and (B) = 0 are -2.56 and 1.39, boh ouside he uni circle indicaing saionariy and inveribiliy respecively. Besides he residual plo of Fig. 5 confirms ha he residuals follow he normal 5968
ETUK e al., 2012 disribuion wih zero (acually 0.0 mean. The kurosis is 2.8 which compares favourably wih he normal disribuion sandard of 3. 0. Forecasing: An ARIMA(1, 2, model may be wrien as 2 This ranslaes ino 2 1 1 1 1 2 FIG. 5. HISTOGRAM OF THE MODEL RESIDUALS 1 2 1( 1 2 2 3) 1 1 Tha is, ( 1 2) 1 2 1 3 1 1 A ime +k, he model may be wrien as k ( 1 2) k 1 k 2 1 k 3 1 k 1 Taking condiional expecaions a ime, we have ˆ ( ( 1 2) 1 1 2 1 ˆ ˆ (2) ( 1 2) ( 1 1 ˆ (3) ( 1 2) ˆ (2) ˆ ( 1 ˆ ( k) ( 2) ˆ ( k ) ˆ ( k 2) ˆ ( k 3), ˆ 1 1 1 k where ( k) is he k-sep ahead forecas. Tha is he forecas of +k. TABLE 6. Forecass Ocober 2008 November 2008 December 2008 January 2009 February 2009 March 2009 Residuals SNUMP DNUMP NUMP 0.14626 0.4 0.4 6.6 0.06016-0.2 0.2 6.8 0.16543 0.2 0.4 7.2 0.33015 0.56511 0.75936 0.73015 1.29526 2.05462 4 7.9 9.2 11.3 5969
Conclusion We have successfully fied an ARIMA(1, 2, model o Nigerian monhly unemploymen daa. This means ha he second differences SNUMP follow an ARMA(1, model. Is adequacy has been esablished and on is basis we have made forecass. REFERENCES 1. Akaike, H. (1970). Saisical Predicor Idenificaion. Annals of he Insiue of Saisical Mahemaics, Volume 22: pp. 203 217. 2. Box, G. E. P. And Jenkins, G. M. (1976). Time Series Analysis, Forecasing and Conrol, Holden-Day, San Francisco. 3. Euk, E. H. (1987). On he Selecion of Auoregressive Moving Average Models. An unpublished Ph. D. Thesis, Deparmen of Saisics, Universiy of Ibadan, Nigeria. 4. Euk, E. H. (1988). On Auoregressive Model Idenificaion. Journal of Official Saisics, Volume 4, No. 2; pp. 113 124. 5. Euk, E. H. (1996). An Auoregressive Inegraed Moving Average(ARIMA) Simulaion Model: A Case Sudy. Discovery and Innovaion, Volume 10, Nos 1 & 2:pp. 23 26. 6. Oyeunji, O. B. (1985). Inverse Auocorrelaions and Moving Average Time Series Modelling. Journal of Official Saisics, Volume 1: pp. 315 322. 7. Priesley, M. B. (198. Specral Analysis and Time Sreies. Academic Press, London. 8. Schwarz, G. (1978). Esimaing he dimension of a model. Annals of Saisics, Volume 6: pp. 461 464. 5970