Modeling the Nigerian Inflation Rates Using Periodogram and Fourier Series Analysis

Transcription

1 CBN Joural of Applied Saisics Vol. 4 No.2 (December, 2013) 51 Modelig he Nigeria Iflaio Raes Usig Periodogram ad Fourier Series Aalysis 1 Chukwuemeka O. Omekara, Emmauel J. Ekpeyog ad Michael P. Ekeree This work cosiders he applicaio of Periodogram ad Fourier Series Aalysis o model all-iems mohly iflaio raes i Nigeria from 2003 o The mai objecives are o ideify iflaio cycles, fi a suiable model o he daa ad make forecass of fuure values. To achieve hese objecives, mohly all-iems iflaio raes for he period were obaied from he Ceral Bak of Nigeria (CBN) websie. Periodogram ad Fourier series mehods of aalysis are used o aalyze he daa. Based o he aalysis, i was foud ha iflaio cycle wihi he period was fify oe (51) mohs, which coicides wih he wo admiisraios wihi he period. Furher, appropriae sigifica Fourier series model comprisig he red, seasoal ad error compoes is fied o he daa ad his model is furher used o make forecas of he iflaio raes for hiree mohs. These forecass compare favourably wih he acual values for he hiree mohs. Key words: Fourier series aalysis, periodogram, frequecy domai, ime series, forecasig. JEL Subjec Classificaio: E31, E Iroducio Iflaio is a persise rise i he geeral price levels of goods ad services i a ecoomy over a period of ime. Iflaio rae has bee regarded as oe of he major ecoomic idicaors i ay coury. Accordig o Olauji e al. (2010), iflaio is udeiably oe of he leadig ad mos dyamic macroecoomic issues cofroig almos all ecoomies of he world. Is dyamism has made i a imperaive issue o be cosidered. Is imporace o he ecoomic growh of a coury makes may researchers ad ecoomiss o apply various ime series ad ecoomeric models o forecas or model iflaio raes of couries. These models iclude Auoregressive Iegraed Movig Average (ARIMA) Models ad Seasoal Auoregressive Iegraed Movig Average (SARIMA) Models all i he ime 1 Deparme of Saisics, Michael Okpara Uiversiy of Agriculure, Umudike, Abia Sae, Nigeria.

2 52 Modelig he Nigeria Iflaio Raes Usig Periodogram ad Fourier Series Aalysis Omekara e al. domai, error correcio, VAR ad oher ecoomeric models, bu o much have bee doe usig he Frequecy domai models. This paper explores he frequecy domai approach o model iflaio raes as a ime series, usig periodogram ad Fourier series aalysis mehods, because of heir simple way of modelig seasoaliy ad elimiaig sigifica peaks wihou re-esimaig he model. Periodogram aalysis eables oe o ideify cycles or periods i series. We also seek o use his model o make forecass of fuure values. Gary (1995) aalyzed he domia facors ifluecig iflaio i Nigeria usig a error correcio model based o moey marke equilibrium codiios. I his aalysis, he foud ha moeary expasio drive maily by expasioary fiscal policies, explais o a large exe he iflaioary process i Nigeria. Some ecoomeric models have bee used o describe iflaio raes, bu hey are resricive i heir heoreical formulaios ad ofe do o icorporae he dyamic srucure of he daa ad have edecies o iflic improper resricios ad specificaios o he srucural variables (Saz, 2011). Odusaya ad Aada (2010) deermied he dyamic ad simulaeous ierrelaioship bewee iflaio ad is deermias growh rae of Gross Domesic Produc (GDP), growh rae of moey supply (M 2 ), fiscal defici, exchage rae (U.S dollar o Naira), imporace ad ieres raes, usig ecoomeric ime series model. Olauji e al. (2010) examied he facors affecig iflaio i Nigeria usig coiegraio ad descripive saisics. They observed ha here were variaios i he red paer of iflaio raes ad some variables cosidered were sigifica i deermiig iflaio i Nigeria. These variables iclude aual oal impor, aual cosumer price idex for food, aual agriculural oupu, ieres rae, aual goverme expediure, exchage rae ad aual crude oil expor. Mordi e al. (2007), i heir sudy of he bes models o use i forecasig iflaio raes i Nigeria ideified areas of fuure research o iflaio dyamics o iclude re-ideifyig ARIMA models, specifyig ad esimaig VAR models ad esimaig a P-Sar model, amogs ohers ha ca be used o forecas iflaio wih miimum mea square error.

3 CBN Joural of Applied Saisics Vol. 4 No.2 (December, 2013) 53 Socko ad Glassma (1987) coduced a comparaive sudy o hree differe iflaio processes amely raioal expecaios model, moearis model ad he expecaio augmeed Philips curve ha are based o ecoomic heory relaioships ha explai ad form iflaio. These processes were compared o oe aoher, uilizig heir ou-of-sample forecas performace o a eigh-quarer horizo ad i addiio a simple Auoregressive Iegraed Movig Average (ARIMA) model was used as a bech mark o subsaiae he heoreical validiy of he ecoomeric models. Their fidigs showed ha he ARIMA model ouperformed boh he raioal expecaios model ad he moearis model ad was foud o perform jus as good as he Philips curve i all specificaios. They cocluded ha despie all heoreical effors o explai causes of iflaio, a simple ARIMA model of iflaio ured i such a respecable forecas performace relaive o he heoreically based specificaios. A reaso why simple ime series models ed o ouperform heir heoreical couerpars lies i he resricive aure of ecoomeric models wih heir improper resricio ad specificaios o srucural variables. The absece of resricio i he ARIMA model gives i he ecessary flexibiliy o capure dyamic properies ad hus sigifica advaage i shor-ru forecasig (Saz, 2011). Ecouraged by hese empirical resuls o he superioriy of ARIMA models, Saz (2011) applied Seasoal Auoregressive Iegraed Movig Average (SARIMA) model o forecas he Turkish iflaio. Logius (2004) examied he ifluece of he major deermias of iflaio wih a paricular focus o he role of exchage rae policy of Tazaia from 1986 o He discovered ha he parallel exchage rae had a sroger ifluece o iflaio. Oher works o modelig iflaio raes are see i he works of Faukasi (2003), Euge e al. (2007) ad Tidiae (2011). Havig kow he advaages of usig Fourier Series Aalysis o model periodic daa, Ekpeyog ad Omekara (2007) modeled he mea mohly emperaure of Uyo Meropolis usig Fourier Series Mehod. Also Ekpeyog (2008) modeled raifall daa usig Pseudo-addiive Fourier series model, which he modified from Fourier Series Model. I he same vei, we seek o use Fourier Series Aalysis ad Periodogram aalysis o ivesigae he properies of iflaio raes i Nigeria ad also model he raes so as o make good forecas of i.

4 54 Modelig he Nigeria Iflaio Raes Usig Periodogram ad Fourier Series Aalysis Omekara e al. 2.2 Mehod of Aalysis The mai mehods of aalysis of his work are periodogram ad Fourier series aalysis. If he series is largely iflueced by seasoaliy or periodiciy, oe ca immediaely ispec ad guess he period or he frequecy of he series; bu if he period of he series cao be guessed accuraely, i calls for he cosrucio of a periodogram o deermie he period or frequecy of he series. I his work, we seek o model iflaio raes of Nigeria usig hese mehods. Sice iflaio is affeced by oher facors, periodogram aalysis becomes ecessary o deermie he iflaio cycle of he series for he period uder sudy. The daa for his work are All-iems iflaio raes, from 2003 o 2011, which are obaied from he Ceral Bak of Nigeria websie ( I cosrucig he periodogram, ime series is viewed i he frequecydomai view poi as oe ha cosiss of sie ad cosie waves a differe frequecies. X T N i1 aicos i bi Sii Z (1) Where : T is he red equaio X is iflaio rae a ime ω i is he agular frequecy measured i radias Z is he error erm a i, b i are he coefficies I shor, he series cosiss of he red, seasoal ad error compoes. The series is firs dereded ad he ordiary leas squares esimaes of he parameers obaied o he dereded series as:

5 CBN Joural of Applied Saisics Vol. 4 No.2 (December, 2013) 55 Sice { { Ad N sii 0 (7) 1 N cosi 0 for1 i N, where i 2f 2 i 1 The above resuls are obaied from complex variables as orhogoaliy ad idepede properies of Fourier Series or siusoidal models. I case of ime series wih eve umber of observaios N = 2q, q = N/ 2, he same defiiios are applicable excep for aˆ bˆ q q 1 N 0 N i0 1 X (9) Therefore he Periodogram for a ime series ΔX wih N = 2q + 1 (odd umber of observaios) is defied as he fucio of iesiies I(f i ) a frequecy f i as: 2 2 a b, i 1, 2, q N ( fi ) i i..., (10) 2 where f i is he h harmoic of frequecy N 1 ad 0 < fi < 0.5 for eve umber of observaios, (f 0.5 ) = 2 Na q (11)

6 56 Modelig he Nigeria Iflaio Raes Usig Periodogram ad Fourier Series Aalysis Omekara e al. The plo of he iesiies agais he frequecies (f 1 ) or periods periodogram. 1 f i is he The periodogram measures he ampliude of a ime series for all possible frequecies ad waveleghs. I ca be ierpreed as he amou of he oal series sums of squares ha is explaied by specific frequecies. The period or frequecy of he series is ideified as ha wih he larges iesiy, (f i ). The frequecy would he be used as he Fourier frequecy f i o obai he parameer esimaes of he model. I is oed here ha he period obaied from he periodogram would give he iflaio cycle. The period or frequecy obaied is ow used as Fourier frequecy o fi a Fourier series model o he iflaio raes daa. The combiaio of mehods of esimaig he compoes of ime series gives a geeral model of Fourier series aalysis used i forecasig ime series. The geeral Fourier series model is give by: [ ] The esimaed model for forecasig ime series is give by: [ ] where Xˆ = esimaed values of iflaio Tˆ = esimaed red equaio ˆ, ˆ ( i 1,..., k) = parameer esimaes i i i 2f i k = highes harmoic of The highes harmoic, k i Fourier series aalysis model is he umber of observaios per seaso divided by wo (2) for a eve umber of observaios ad ( 1)/2 for a odd umber of observaios (Priesly, 1981).

7 CBN Joural of Applied Saisics Vol. 4 No.2 (December, 2013) 57 The red is firs isolaed or removed by fiig he liear red model or quadraic red or eve he overall mea of he daa, usig he mehod of leas squares. The red equaio is fied based o is sigificace i he model. Thus he red equaio icludes: X a a a a 2 p p (14) Where The red, afer esimaig, is ow removed from series ad he dereded series used o esimae seasoal variaio. The sie ad cosie fucios i equaio (13) give he esimaed model for he seasoaliy of he model. I is give by: [ ] where The above equaio is cas as a muliple liear regressio o obai he esimaes of i ad i. To esimae Z, we firs of all deermie if he residual values are radom. This ca be doe by assessig he auocorrelaio fucio of he residual. If he residual or error compoe is o radom, a firs order auoregressive model ca be fied o he error values as: Z 1 (16) Z Where = he auoregressive coefficie = he purely radom process (whie oise) The esimaed equaio is give as:

8 58 Modelig he Nigeria Iflaio Raes Usig Periodogram ad Fourier Series Aalysis ˆ ˆZ 1 Omekara e al. Z (17) These esimaios are doe usig MINITAB. A combiaio of hese hree compoes gives he geeral Fourier series model used o esimae Iflaio raes. This model is he used o esimae iflaio raes of Nigeria ad make forecass for fuure values of iflaio raes. I esig for he adequacy of he forecass, Theil Iequaliy Coefficie is used. The Theil coefficie is give as: U 1 i1 ( X X ) 1 1 i 2 2 Xi Xi i1 i1 i 2 (18) Where X i ad X i are he acual ad esimaed iflaio raes respecively; is he umber of observaios. The closer he value of U is o zero he beer he forecass. 3.0 Daa Aalysis A clear assessme of he graphical represeaio of origial series shows ha here is a exisece of red, cyclical ad seasoal variaios, bu he cycle or period of he series cao be exacly asceraied (see Figure 1). I addiio he ormal probabiliy plo of he series i Figure 2 idicaes ha he series is o ormal or sable. I is herefore impora o rasform he daa for sabiliy ad ormaliy. A square roo rasformaio was herefore appropriae as show i Figure 3. Figure 1: Plo of Origial Series(All-iems Iflaio Rae)

9 CBN Joural of Applied Saisics Vol. 4 No.2 (December, 2013) 59 Figure 2: Normal P-P Plo of he Series Figure 3: Normal P-P Plo of he rasformed Series 3.2 Periodogram Aalysis The iesiies I(f i ) a various frequecies are obaied usig equaios (2), (3) ad (10) as show i Table 1. The plos of hese iesiies agais he frequecies ad periods are give i figures 4 ad 5. From figures 4 ad 5, i is observed ha he frequecy ad period wih he larges iesiy are ad 51 mohs respecively. This is also show i Table 1 of his work. Hece he period or cycle for he daa is 51 mohs. These are ow used o fi he Fourier series model o he daa.

10 60 Modelig he Nigeria Iflaio Raes Usig Periodogram ad Fourier Series Aalysis Omekara e al. Table 1: Esimaes, Frequecies, Period ad Iesiies a b a**2 b**2 I(f) f PERIOD a b a**2 b**2 I(f) f PERIOD Figure 4: Plo of Iesiies agais Frequecy

11 CBN Joural of Applied Saisics Vol. 4 No.2 (December, 2013) 61 Figure 5: Plo of Iesiies agais Periods 3.3 Fiig he Geeral Fourier Series Model The red (T ) is esimaed as: T a a 0 1 Is aalysis is give i Table 2 wih a red equaio give as: Tˆ Table 2: Tes for Sigificace of he Parameer Esimaes i he Tred Model Predicor Coef SDev T P Cosa T S = R-Sq = 2.9% R-Sq(adj) = 2.0% From Table 2, i is see ha he parameer esimae a o oly is sigifica i he red equaio a 5% level of sigificace. Therefore T = The seasoal compoe is he esimaed from he dereded series as:

12 62 Modelig he Nigeria Iflaio Raes Usig Periodogram ad Fourier Series Aalysis Omekara e al k ad X Cosi Sii i i i1 The parameers i Table3. s a ' i ad are obaied by mehod of leas squares as show ' s i Table 3: Tes for Sigificace of he Parameer Esimaes i he Seasoal Compoe Predicor Coef SDev T P Predicor Coef SDev T P Nocosa Nocosa Cosw si13w Siw cos14w cos2w si14w si2w cos15w cos3w si15w si3w cos16w cos4w si16w si4w cos17w cos5w si17w si5w cos18w cos6w si18w si6w cos19w cos7w si19w si7w cos20w cos8w si20w si8w cos21w cos9w si21w si9w cos22w cos10w si22w si10w cos23w cos11w si23w si11w cos24w cos12w si24w si12w cos25w cos13w si25w S = From Table3, i is observed ha he parameer esimaes ha are sigifica i he model are:, ˆ ad ˆ. Therefore, he esimaed seasoal model is ˆ1 2 3 SQRT X cos cos 2 si

13 CBN Joural of Applied Saisics Vol. 4 No.2 (December, 2013) 63 SQRT X cos cos si 3 (19) The esimaes of he seasoal compoe are obaied usig equaio (19). I assessig he auocorrelaio ad parial auocorrelaio fucio of he error compoe, i was foud ha he error was o radom. The behaviour of he auocorrelaio ad parial auocorrelaio fucio sugges a auoregressive model of order oe. Table 4 shows he es for sigificace of he model. Table 4: Tes for Sigificace of Parameer Esimae of he Error Compoe Type Coef SDev T AR Number of observaios: 102 From Table 4, he parameer esimae i he error compoe is sigifica i he model. Hece Z 0.903Z 1 (20) The geeral model for he series, which cosiss of he esimaed red, seasoal ad error compoe, is give as: SQRTXˆ cos cos si Z 1 (21) The model is ow used o esimae iflaio raes. (See able 5). The plos of he origial ad esimaed series of boh he acual ad rasformed values give i Table 5 are show i figures 6 ad 7, ad hey show ha he model fis well o he daa. Figure 6: Plo of Acuals ad Esimaes of he Trasformed Series

14 64 Modelig he Nigeria Iflaio Raes Usig Periodogram ad Fourier Series Aalysis Omekara e al. Table 5: Acuals ad Esimaes of Nigeria All-iems Iflaio Raes ( ) ESTSQRTX ESTX X SQRTX ESTSQRTXESTX X SQRTX ESTSQRTXESTX X SQRTX Figure 7: Plo of Acuals ad Esimaes of he Origial Series

15 CBN Joural of Applied Saisics Vol. 4 No.2 (December, 2013) Forecass for he Fuure Raes Forecass from Sepember 2011 o Sepember 2012 are made usig he esimaed geeral model give i equaio (22). The values obaied from equaio (22) are squared o give he values i able 6: Table6: Forecass for Fuure Periods FORCAST ACTUALS The plo of he Acual Series ad he Forecass is give i Figure 8: Figure 8: Plo of Acual Series ad Forecass I assessig he adequacy of he forecas, Theil Iequaliy Coefficie is calculaed as:

16 66 Modelig he Nigeria Iflaio Raes Usig Periodogram ad Fourier Series Aalysis ( Xi Xi) , Xi , Xi i 1 i1 i1 Omekara e al. U Sice U= 0.07 is very close o zero, he he model ca favourably be used o forecas he fuure values of iflaio raes. 4.0 Discussio The periodogram aalysis reveals ha here exis boh shor erm ad log erm cycles wihi he period uder sudy. The log erm cycle is 51 mohs while he shor erm cycle is approximaely 20 mohs. This is kow by checkig he secod larges iesiy i able 1. This goes a log way o buress he fac ha iflaio is iflueced by cyclical or periodic variaio. I ca also be deduced from he periodogram he relaioship bewee he iflaio cycle ad he various goverme admiisraios wihi he period. Uder his period of sudy, i is kow ha wo admiisraios exised, amely Obasajo s ad Yaradua/Joaha s period. The iflaio cycle relaes o hese wo admiisraios. The firs fify-oe mohs represe Preside Olusegu Obasajo s period while he secod cycle represes Yaradua/Joaha s period. From he foregoig, i ca be deduced ha ecoomic policies, aciviies, implemeaio of policies, budges, ec. of goverme admiisraio ifluece iflaio raes i Nigeria. Mos ecoomiss have cofirmed ha a expasioary budgeary provisio amog oher facors helps o icrease iflaio raes (Dodge, 2011). A clear look a he origial plo of he series shows ha durig Obasajo s period, here were higher iflaio raes ha he Yaradua/Joaha s period. This may be as a resul of icessa supplemeary budges raised ad he iabiliy of he admiisraio o follow budgeary provisios i he implemeaio of he budge. Bu he laer admiisraio records lower iflaio raes, which may be as a resul of reduced passage of supplemeary budges ad implemeaio of some reforms i he ecoomic secors.

17 CBN Joural of Applied Saisics Vol. 4 No.2 (December, 2013) Coclusio The periodogram aalysis has ideified a major iflaio rae cycle for he period uder sudy o be fify oe (51) mohs ad he former series aalysis has esablished a former series model for he all-iems iflaio raes o be SQRTXˆ cos cos si Z 1 This model is used o make good forecass of iflaio raes. Therefore Fourier series models ca also be used o model iflaio raes because of is advaage of ideifyig iflaio cycles i addiio o esablishig a suiable model for he series. Refereces Dodge, E.R. (2011): Expasioary ad Coracioary Fiscal Policy Review for AP Ecoomics. Available a Ekpeyog, E. J. (2008): Pseudo Addiive (Mixed) Fourier Series Model of Time Series: Asia Joural of Mahemaics ad Saisics 1(1): Ekpeyog, E. J. ad C. O. Omekara, (2007): Applicaio of Fourier Series Aalysis o Modelig Temperaure Daa of Uyo Meropolis: Global Joural of Mahemaical Scieces, 7(1), Euge F. ad Cypria, S. (2007): A muliple Regressio model for Iflaio raes i Romaia I he elarged EU. Available Olie a: hp://mpra.ub.ui-mueche.de/11473/ Faukasi B. (2003): Deermias of Iflaio i Nigeria: A Empirical Aalysis. Ieraioal Joural of Humaiies ad Social Sciece, 1(18). Available Olie a: Gary G. M. (1995): The Mai Deermias of Iflaio i Nigeria. IMF Saff Papers.Ieraioal Moeary Fud, 42(2), Logius R. (2004): Exchage raes Regimes ad Iflaio i Tazaia.AERC Research Paper 38, Africa Ecoomic Research Cosorium, Nairobi.

18 68 Modelig he Nigeria Iflaio Raes Usig Periodogram ad Fourier Series Aalysis Omekara e al. Mordi, C.N.O, Essie, E.A, Adeuga, A.O, Omaukwe, P.N, Oougbo, M.C, Oguade, A.A, Abeg, M.O, Ajao, O.M (2007): The Dyamics of Iflaioi Nigeria: Mai Repor. Occasioal Paper No. 2. Research ad Saisics Deparme. Ceral Bak of Nigeria, Abuja. Odusaya, I. A., ad Aada, A. A. M (2010): Aalysis of Iflaio ad is Deermias i Nigeria: Pakisa Joural of Social Scieces 7(2): Olauji, G. B, Omoesho, O. A, Ayide, O. E, ad Ayide, K (2010): Deermias of Iflaio i Nigeria: A Co-iegraio Approach. Joi 3rd Africa Associaio of Agriculural Ecoomiss (AAAE) ad 48 h Agriculural Ecoomiss Associaio of Souh Africa (AEASA) Coferece, Cape Tow, Souh Africa, Sepember Priesly, M. B. (1981): Specral Aalysis of Time Series: Academic Press; Lodo. Saz, Gokha (2011): The Efficiecy of SARIMA Models for Forecasig Iflaio raes i Developig Couries: The Case for Turkey: Ieraioal Research Joural ad Fiace ad Ecoomics, (62), Socko, D. J. ad Glassma J. E. (1987): A Evaluaio of he Forecas Performace of Aleraive Models of Iflaio; The Review of Ecoomics ad Saisics 69(1): Tidiae, K. (2011): Modelig Iflaio i Chad. IMF Workig Paper. Ieraioal Moeary Fud.