Application of Statistical Methods of Time-Series for Estimating and Forecasting the Wheat Series in Yemen (Production and Import)

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1 American Journal of Alied Mahemaics 16; 4(3): h:// doi: /j.ajam ISSN: (Prin); ISSN: 33-6X (Online) Mehodology Aricle Alicaion of Saisical Mehods of Time-Series for Esimaing and Forecasing he Whea Series in Yemen (Producion and Imor) Douaik Ahmed 1, Youssfi Elkean, Abdulbakee Kasem 1 The Naional Insiue of Agronomic Research (INRA), Raba, Morocco Dearmen of Mahemaics Faculy of Sciences, Universiy Ibn Tofail, Kenira, Morocco address: ahmed-douaik@yahoo.com (D. Ahmed), elkeani@univ-ibnofail.ac.ma (Y. Elkean), kasemabdulbakee@gmail.com (A. Kasem) To cie his aricle: Douaik Ahmed, Youssfi Elkean, Abdulbakee Kasem. Alicaion of Saisical Mehods of Time-Series for Esimaing and Forecasing he Whea Series in Yemen (Producion and Imor). American Journal of Alied Mahemaics. Vol. 4, No. 3, 16, doi: /j.ajam Received: Aril 4, 16; Acceed: Aril 15, 16; Published: May 4, 16 Absrac: Due o he imorance of he whea cro which reresens 9% of he grain consumed, In his aers, we comared beween he following saisical mehods : Box and Jenkins model, exonenial smoohing models (wih rend and wihou seasonal) and Simle regression for esimaing and forecasing o wo ime series of whea(roducion and imor). We reached o he following resuls: 1. Brown exonenial smoohing model for modeling he imored whea series.. ARIMA (1, 1, 1) model for modeling he roduc whea series. For he whea cro, he raio of roducion o consumion is execed o reach 6.3% in 15 and coninues o decline even u o 5.4% in. This means ha he roblem of food securiy well be worse in Yemen. Keywords: Time Series, Whea Cro, Forecasing, Box and Jenkins, Exonenial Smoohing 1. Inroducion The human needs o know he as in order o redic he fuure o find oimal soluions of many roblems which face humaniy in his cenury. Yemen is one of he Arab counries where local demand for food is growing exonenially. Therefore, i suffers from a huge lack o cover all he oulaion needs of foodsuffs esecially whea which reresens sale food of mos he oulaion. Alhough in recen years he amoun roducion of whea comared wih imored whea reach in 1 o 9%. According o wha has menion above we comered hese saisical mehods of ime series: Box and Jenkins mehodology,exonenial smoohing model nd Simle regression o esimae and forecas he wo whea ime series (imor,roduc) from 1961 o 1 of he Organizaion s sie of Food and Agriculure (FAO) and he Cenral Bureau of Saisics in Yemen. We used hese rogrammes SPSS, EVIEWS and EXCLE.. Theoreical Formulaion.1. Hol and Brown s Exonenial Smoohing Mehod In he case where he series has a rend, we can ado he following redicion formula: yˆ = a b h + h + The values a and b are consanly udaed by he following equaions: and a = α1y + (1 α1)( a 1 + b 1) b = α ( a a + (1 α ) b ) 1 1 This forecas model is known as he model name of HOLT. A secial case of model HOLT, called model BROWN or dual exonenial smoohing is obained when he smoohing

2 15 Douaik Ahmed e al.: Alicaion of Saisical Mehods of Time-Series for Esimaing and Forecasing he Whea Series in Yemen (Producion and Imor) consans α 1 and α are relaed o he same arameer α, by he relaions: α = α( α) e = α 1 α. For hese α wo models, we need o give iniial values a and b o roduce forecass. Thus we ake b which equals he coefficien simle linear regression calculaed on he basis of he firs five values of he series. Thereafer, a is deduced by he relaion: a = y1 b, as he smoohing consans hey are se by he user. In racice ofen gives a value o α beween.1 and.3. [3].. Saionary Process A second rocess is saionary if: Z, E( X ) < Z, E( X ) = m Z, h Z, Cov( X, X ) = γ( h).3. Whie Noise + h Whie noise is a saionary rocess such ha: E( ε ) = m, ( ) =, Var ε σ Cov( ε, ε 1) = γ( h),, h > This noion of whie noise corresonds o he usual assumions on residues in mulile regression. Random variables ε are also called random shocks. we imlicily assumes ha random shocks ε follow a normal disribuion N(, σ ).4. Auocorrelaion The auocorrelaion funcion is he alicaion ρ of Z in R defined by: γ( h) ρ( h) =, h Z γ() ρ ( h) Measuring he correlaion beween X i and i because: Cov( X, X + h) γ( h) γ( h) = =, h Z V( X ) V( X ) γ() γ() γ() + h.5. Auocorrelaion Parial X + h The arial auocorrelaion funcion wih delay k is defined as he arial correlaion coefficien beween X e X he influence of oher variables shifed by k eriods X k, X 1,..., X k + 1 have been wihdrawn..6. Auoregressive Process AR() Le a rocess ( X, Z), X is said auoregressive rocess of order ( AR( )) if X = φ X + φ X φ X + ε 1 1 Where ε, whie noise and φ1, φ,..., φ are consans..7. Moving Average Process (MA(Q)) Le a rocess ( ε, Z), X is said auoregressive rocess of order q ( MA( q )) if X = ε + θ ε + θ ε θ ε 1 1 q q Where ε, whie noise and θ1, θ,..., θ are consans..8. Moving Average Processes Auoregressive A saionary rocess X has an ARMA reresenaion (, q ) Minimum if i saisfies: where Φ( L) X = Θ ( L) ε, Φ ( L) X = X + φ X φ X 1 Θ ( X ) = ε + θ ε θ ε 1 q q φ, θ and L( X ) = X q 1 he olynomials Θ and Φ have heir uer modules sricly roos o 1. Θ and Φ have no common roos. ε = ( ε, Z) is a whie noise of variance V( ε ) = σ.9. Process ARIMA A rocess X = ( X, ) is a rocess ARIMA(, d, q ) [Auoregressive inegraed moving average] if i saisfies an equaion of ye : q i d i ϕ i δ θi ε i=1 i=1 (1 L)(1 L ) X = (1 + L), where δ consan L( X ) = X 1 and ε is whie noise..1. Augmened Dickey Fuller Tes The Augmened Dickey Fuller es ( ADF ) is a uni roo es of he null hyohesis of uni roo (or non saionariy). The ADF es esimaed hree models: X = α X + β X + ε 1 1 j j j=1 X = α + α X + β X + ε 1 1 j j j=1

3 American Journal of Alied Mahemaics 16; 4(3): X = α + α X + β X + δ + ε 1 1 j j j=1 The null hyohesis of ADF es is he uni roo hyohesis of he variable X is he hyohesis H : α 1 =. The ADF es consiss of comaring he esimaed value Suden associaed wih he arameer α 1 o he abulaed values of his saisic. The values abulaed for differen es however abulaed values of Suden es. The criical values of his saisic, ADF denoed in he following, are given by MacKinnon (1996). The null hyohesis H of non-saionary of he ime series is rejeced a he 5% level when he observed value of he Suden s -es is less han he criical value abulaed by MacKinnon (1996) or abs < ADF Box Jenkins Mehodology This is he echnique for selec he mos aroriae ARMA or ARIMA model for a given variable. I comrises four ses: 1. Idenificaion of he model, his involves selecing he mos aroriae lags for he AR and MA ars, as well as selecing if he variable requires firs-differencing o become saionariy. The ACF and PACF are used o idenify he bes model. (Informaion crieria can also be used). Esimaion, his usually involves he use of a leas squares esimaion rocess. 3. Diagnosic esing, which usually is he es for auocorrelaion. If his ar is failed hen he rocess reurns o he idenificaion secion and begins again, usually by he addiion of exra variables. 4. Forecasing, he ARIMA models are aricularly useful for forecasing due o he use of lagged variables. 3. Alicaion 3.1. Grah Series Figure 1. Grah of ime series of whea(roduc and imor)from 1961 o 1. Through he grah figure, we observed a general uward rend over he eriod, his means ha he series is no saionary. 3.. Auocorrelaion and Auocorrelaion Parial Figure. Auocorrelaion of ime series of whea(roduc and imor).

4 17 Douaik Ahmed e al.: Alicaion of Saisical Mehods of Time-Series for Esimaing and Forecasing he Whea Series in Yemen (Producion and Imor) Figure 3. Auocorrelaion arial of ime series of whea(roduc and imor). We examine he auocorrelaion and arial auocorrelaion funcion in figures and 3 we observed ha he esimaed auocorrelaion arameer decreases exonenially owards zero while ha only he firs arial auocorrelaion arameer is no significan. To confirm he revious resuls we execue he Dickey-Fuller es and observed in Figure 4 and 5 ha he series is no saionary. Figure 4. Dickey-Fuller es of ime series of whea roduc.

5 American Journal of Alied Mahemaics 16; 4(3): Figure 5. Dickey-Fuller es of ime series of whea imor. When we execue he firs differences, we noe of figure 6 and 7 ha a saionary series. Figure 6. Dickey-Fuller es of firs differences of whea roduc.

6 19 Douaik Ahmed e al.: Alicaion of Saisical Mehods of Time-Series for Esimaing and Forecasing he Whea Series in Yemen (Producion and Imor) Figure 7. Dickey-Fuller es of firs differences of whea imor. we noe ha he series is saionary. We deduce ha d = 1 in he ARIMA model (, d, q) Idenificaion and Selcion of Model for Whea Producion Series Alhough i aears ha each arial auocorrelaion arameer afer he second arameer is no significanly differen from zero a a =.5 bu he auocorrelaion funcion is gradually decreasing owards zero, his may be sufficien evidence ha he random rocess is AR (1). For ensure we es he following saisical hyohesis: H : ϕ 11 = ; H : ϕ11, 1 1 SE( ϕ 11) = = =,141,,99 Z = ϕ11 /( SE( ϕ 11)) = = 6,5>, n 5,141 we deduce ha he firs arial auocorrelaion arameer is no significanly differen from zero a α =.5. We examining he auocorrelaion arial arameers, we find ha ϕ <.8 for each k =,3,..., ha suors he ossibiliy kk of using he AR (1) and herefore ARIMA (1,1,). For imor whea series we ge he same resuls. And hen we comared beween ARIMA models wih he exonenial smoohing(hol,brown)and simle regression. We ge he following resuls. Figure 8. Cooarison of model.

7 American Journal of Alied Mahemaics 16; 4(3): We ake 4 observaion of he original series and forecas for he nex en years, hen comare beween models by MAPE and choose he bes model. The resuls were as follows: 1. Brown s exonenial smoohing model for redic he series of whea roducion.. The ARIMA (1,1,1) model for redic he series of whea exors Tess of Residues We es he bes model: Grahic residues confidence limis, ACF, PACF Grahic disersion of oins in arallel form residuals around zero ACF, PACF Ljung-Box value is significan If he model realizes he revious ess, we use i o forecas Forecasing Then we use he revious models o calculae he forecas from 11 o and he resuls were as follows: Figure 9. Forecasing of series of he whea (roduc and imor). Figure 1. Grah forecasing of series of he whea (roduc and imor).

8 131 Douaik Ahmed e al.: Alicaion of Saisical Mehods of Time-Series for Esimaing and Forecasing he Whea Series in Yemen (Producion and Imor) 4. Conclusion Whea imors will increase from.9 million onnes in 11 o 4 million onnes in, where he roorion of imors was 9% in 1 and i is exec ha he whea imor roorion will increase o 94% in. whereas, whea roducion will dro by 6% during his decade. References [1] Peer J. Brockwell e Richard A Davis (), Inroducion o Time Series and Forecasing, Sringer. [] Philie Marier, cours Prévision de la demande, Consorium de Recherche Universié Laval. [3] Lahoussaine Baamal, (1), Cours d analyse des Séries Chronologiques. Universié Ibn Tofail,Kénira. [4] E. Oseragova, O. Oserag, The Simle Exonenial Smoohing Model, h:// [5] Rodolhe Palm, (7), Eude des séries chronologiques ar méhodes de lissage, Faculé Universiaire des Sciences agronomiques, Unié de Saisique, Informaique e Mahémaique aliquées, Belgique. [6] Bary Adnan Majed, (), Saisical forecasing mehods-1, Universié du Roi Saoud. [7] Ruey S. Tsay, (5), Analysis of financial ime series, Wiley,Hoboken, New Jersey. [8] Lahoussaine Baamal, (1), Cours de Prévision ar la Méhodologie de Box e Jenkins. Universié Ibn Tofail. [9] IBM, (11), IBM SPSS Forecasing. [1] A.Douaik, (1991), Prévision des rendemens agricoles ar les méhodes de lissage exoneniel. Mémoire ingénieur, Faculé des Sciences Agronomique de Gembloux, Belgique.