Research & Reviews: Journal of Statistics and Mathematical Sciences

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1 Research & Reviews: Journal of Saisics and Mahemaical Sciences A Transfer Funcion-Auoregressive Noise Model of Naira Exchange Raes for Us Dollar and Swiss Franc Iwok IA* Deparmen of Mahemaics/Saisics, Universiy of Por-Harcour, P.M.B.533, Por-Harcour, Rivers Sae, Nigeria Research Aricle Received dae: 8//05 Acceped dae: 7/0/06 Published dae: 3/0/06 *For Correspondence Iwok IA, Deparmen of Mahemaics/ Saisics, Universiy of Por-Harcour, P.M.B.533, Por-Harcour, Rivers Sae, Nigeria, Tel: ibywok@yahoo.com Keywords: Transfer funcion, Pre-whiening, Cross correlaion, Raional polynomial, Noise model, Saionariy. ABSTRACT This work aims a modelling he Naira exchange raes for US Dollars X and Swiss Franc Y using a ransfer funcion (TF) echnique. Daa for he wo currencies were obained from he Cenral Bank of Nigeria (for a period of 53 years). Afer obaining saionariy of he wo series using appropriae ransformaion, he inpu series x was pre-whiened o remove he spurious correlaion effec. The oupu series y was also pre-whiened and he cross correlaion beween he pre-whiened inpu (α ) and oupu (β ) was examined. From he behavior of he cross correlaion, raional polynomial represenaion ω( B) of he dynamic ransfer funcion was obained. The esimaed δ B noise ( ˆn ) was found o be auo correlaed. Thus, he noise was modelled separaely using Box and Jenkins auoregressive (AR) mehod. This provided he missing componen of he TF model which was used o fi he overall model. The resuled model underwen a diagnosic check and was found o be appropriae. Hence forecas was generaed. INTRODUCTION Basically, no counry is compleely economically independen. Every counry mus depend on one or several counries for her economic growh. All naions are iner-dependen, because hey have limied resources and have o rade wih each oher o saisfy heir wans. This implies ha he analysis of economic variables such as gross domesic produc (GDP), inflaion, exchange raes ec. are of paramoun imporance o any naion s growh. Unforunaely in mos African counries, less aenion is paid o he sudy and monioring of he rise and falls of hese essenial and sensiive variables. A volaile or inappropriae exchange rae has been a major hindrance o he growh of many African counries which Ghana is inclusive Appiah and Adeunde []. Exchange raes vary largely according o exen and naure of each counry s exposure o rade and global financial markes. In some sable economies, exchange raes of currencies compee favorably while he developing economies appear o be influenced heavily by he developed ones. This is because, he less developed counries receives less aenion concerning exchange rae compared o he developed or indusrialized economies []. The currencies of many sub-saharan African counries faced depreciaions wih respec o US Dollar a he onse of he global financial crisis. According o Laifa, e al. [3], five counries: Ghana, Kenya, Nigeria, Uganda and Zambia depreciaed by a leas 0% beween June 008 and March 009; while Tanzania and Rwanda depreciaed by0% or less during same period. According o Jeffrey [4], Argenina being he vicim of he wors marke financial crisis in 00; experienced problems in he lae 990 s and he siuaion became severe because of is link wih a paricular currency, he US Dollar. The Dollar gained value over oher major currencies beginning in he mid-995, and caused he marke for Argenina s imporan agriculural expor producs (whea, mea and soya beans) o decline sharply. Thus, he decline in he prices of hese commodiies expressed in Dollar was paricularly dramaic. All hese pu ogeher, led o sharp increase in he raio of deb o expors. RRJSMS Volume Issue June, 06 39

2 Some earlier sudies sugges ha insabiliy in exchange rae has he poenial o affec a counry s economic performance. In Nigeria, he value of Naira has depreciaed significanly compared wih oher currencies due o poor economic managemen, poliical sysem and oher unknown facors. However, a close observaion of he Naira exchange rae for US Dollar and Swiss Franc shows ha he wo currencies are highly relaed. Thus, sudying he behavior and relaionship of hese wo currencies in relaion o Naira can provide useful informaion for he economy of he wo Counries (USA and Swizer Land) and Nigeria a large. This work herefore, inends o build a model ha esablishes he relaionship beween he wo currencies (US Dollar and Swiss Francc) in relaion o Nigeria s currency (Naira). The model is called he ransfer funcion model. Transfer funcion model (TFM) is a saisical model describing he relaionship beween an oupu variable and one or more inpu variables [5]. TFM can be a single equaion or muliple equaion model, which he laer could be referred o as a simulaneous ransfer funcion (STF) model [6-8]. Some auhors will prefer o disinguish he wo models as single-inpu and muliple-inpu ransfer funcion models. In mos applicaions, linear equaion is used o describe he relaionship resuling from he disribued-lag model. Elkhem, e al. [9] considered he naure of crude oil as a mixure of hydrocarbons wih differen boiling emperaures. Conrol became essenial for he fracionaion column o keep producs a he limiaions. The paper revealed he idenificaion of ransfer funcion for relevan differen developed conrol sraegy and relevan ransfer funcion were idenified using MATLAB Sofware. A Work on forecasing foreign exchange by Znaczko [0] invesigaed a ransfer funcion model wih muliple variables. The sudy revealed ha he bes forecas of foreign exchange raes depends on curren and pas prices. Kannan and Farook [] fied a ransfer funcion model o global warming eniies such as amospheric emperaure, and amospheric CO emissions. A srong relaionship was demonsraed beween annual amospheric CO emissions and presen/ pas amospheric emperaure. Pracically, he oupu is no a deerminisic funcion of he inpu. Transfer funcion model is quie differen from he ARIMA model. The laer is a univariae ime series model while he ransfer funcion model is a mulivariae ime series model. The ARIMA model relaes he series only o is pas, bu beyond his indicaion, he ransfer funcion model will relae he series o oher ime series []. Due o he close relaionship wih he regression models, he ransfer funcion models are also referred o as dynamic regression models [3]. The ransfer funcion models can use more han one explanaory variable, bu he explanaory variables mus be linearly independen of each oher [4]. Transfer funcion approach as a mehod of daa analysis, has been in use for some decades. Virually every field of sudy has demonsraed he imporance and relevance of he ransfer funcion approach especially in is abiliy o enhance forecas based on relaing differen series. In a sable economic rend, such model can enable a researcher or planner o predic favorable chances an indigenous economy sands amids higher ones in he global marke. In ime series analysis, he Box-Jenkins mehod describes he ransfer funcion model as a model ha predics fuure values of a ime series (oupu series) from pas values of same series and one or more relaed series (inpu series). In his work, however, we inend o apply he Box-Jenkins mehod and he pre-whiening approach o build a ransfer funcion- AR noise model of Naira exchange raes for he US Dollar and Swiss Franc. In our mehod, he leading indicaor is idenified and he noise componen is modeled separaely and added o he dynamic relaionship beween he inpu and oupu series. The resuling noise is found o saisfy is basic assumpions [ ε NIID(0, á ) ] hus showing ha he model is adequae. Le X and Y be wo ime series. Saionariy METHODOLOGY The ime series X is said o be saionary if he saisical properies are essenially consan hrough ime. In oher words, E( X ) = µ and Var( X ) = σ. In his work, saionariy is obain by differencing. Tha is, x= X X. Pre-Whiening Pre-whiening approach involves finding auoregressive inegraed moving average (ARIMA) model for X ha yields whie noise residuals while he funcional relaionship beween he X and Y is sill mainained. For his work, an ARIMA (p,d,o) model is idenified and we have x =φ x + + φ x + α p p Rewriing he equaion, he residuals or he pre-whiened inpu α ˆ = x { φ x + } is obained as The y is subsiued for x o ge he filered equaion for he pre-whiened oupu as: RRJSMS Volume Issue June, 06 40

3 β ˆ = y { φ y + +φ y } p p By pre-whiening, cross correlaion accuraely reflecs he srucure of he impulse funcion [5]. Here, α and β represen he pre-whiened inpu and oupu series respecively. Cross Correlaion The cross correlaion for lag k is given as Cxy r ( xy) = SS x y n k C = x x y y k = 0,, ( + ) xy k n = Wher x and y are he sample means of x and y, s and s are he sample sandard deviaions respecively. y Transfer Funcion Model The dynamic relaionship beween inpu series Y and oupu series X is usually approximaed by a linear ransfer funcion y = vx 0 + vx + + n, = v B x + n ' Where vs j are he ransfer funcion weighs and is a noise erm. Esimaes of he ransfer funcion weighs can be calculaed from ráâ ( js ) â v ˆ j =, sα j= 0,,k ; (3.3) Where s α and s β are he esimaed sandard deviaions of he pre-whiened inpu and he oupu daa respecively. is he esimaed cross correlaion beween he pre-whiened inpu and oupu. Insignifican weighs are rounded off o zero. B denoes m he backshif operaor such ha B X = X m, The funcion v (B) deermines he impac of inpu X on y ; and ( B) ( B) ω b v( B) = B δ Thus, can be represened as ω δ( B) ( B) x ( B) b BB y = x + n ω = + n δ b According o Box and Jenkins [6], he parameers in v (B) are esimaed by he equaion δ ( B) v( B ) = ω( B) Explicily, i can be expressed as, ( δ δ r )( ) = ( ω0 ω ωs ) B B v vb vb B B B r s b The parameers r, b, s, represen he order of he numeraor polynomial, he delay parameer ha indicaes he ime lag unil inpu affecs he oupu also called dead-ime or delay ime, and order of he denominaor polynomial respecively. The Auoregressive (AR)-Noise Model In he model idenificaion, v (B) parameers are obained and fied; and he noise erm n is added. In his concep, however, i is believed ha he noise n could be serially correlaed; hus violaing is assumpion. The noise is hen modeled separaely using Box and Jenkins mehod as: ε φ ( Bn ) =ε n = φ ( B) Where, φ ( B) = ( φ p B φb φpb ) and is he auoregressive order of n and ф i s are he auoregressive parameers. Thus he resuling model is: (3.5) RRJSMS Volume Issue June, 06 4

4 y ε = v( B) x + and is found o be serially uncorrelaed as expeced. φ ( B) Daa Analysis and Resuls The analysis of he daa available for his hesis was carried ou using Miniab sofware. Daa used was secondary daa wih fify hree observaions each (960-03). We inend o build a ransfer funcion -AR noise model which could be used for fuure forecas of he naira exchange rae for he wo currencies. Le X denoes he Naira exchange rae of US Dollar and Y, he Naira exchange rae of Swiss Franc In he language of our sudy, X is he inpu series and Y he oupu series. Raw Daa Plo Figure (appendix) demonsraes he combined raw daa plo for X and Y Time Series Plo of X, Y Variable X Y Daa Index Figure. Raw daa plo of and. From hese plos, i is observed ha he series are no saionary. This demanded for an appropriae ransformaion o make hem saionary. Correlaion of he Two Series The correlaion beween X and Y gives This value indicaes a srong posiive relaionship beween he wo series. This is also eviden in Figure. Differencing X and Y were boh differenced once o obain saionariy. Model Idenificaion for he Inpu Series X An ARIMA (,,0) was idenified for he X resuling in he model: ( 0.086B)( B) X =α Pre-Whiening of X and Y From he above model, he pre-whiened X is: α ˆ = X X 0.086X X and ha of Y β ˆ = Y Y 0.086Y Y Cross Correlaion Funcion for α and β The cross correlaion of he pre-whiened series r αβ are seen in Figure below. Since he r αβ a lag and are no saisically significan, he weighs v 0 and v are sysemaically equal o zero. Thus from he figure; b=, s= and r=. In addiion, since rαβ ( k) are no saisically differen from zero; i means he presen values of are relaed o he pas values of X and no oherwise. Hence, X is he leading indicaor. The four ransfer funcion significan weighs are shown in Table below wih heir respecive cross correlaions and lags. Idenificaion of he Transfer Funcion Parameers We had earlier expressed in mehodology ha:. δ ( B) v( B ) = ω( B) Tha is, ( δ δ r )( ) = ( ω0 ω ωs ) B B v vb vb B B B Subsiuing b=, s= and r=, we have r s b RRJSMS Volume Issue June, 06 4

5 X XX XX XX XXX XXXX XX X XX XXX XXXXX XXXXXX XXXXX XXX XX XX XXX XXX 0.03 XX XXXXXXXXXXXXXXXXXXX XXXXXXXXXXXXXXXXXXXXXX XXXXXXXXXXXXXXXX XXXXXXXXXXXXXX XXXX XXX XXXXXX XXXX XXX XX X XX XXX XX X XXX Figure. Cross Correlaion funcion (CCF) for, Table. Cross Correlaion and Transfer Funcion Weigh. Lag Cross Correlaion Weigh Vˆj ( δ δ ) j = ( ω0 ω) B B vb B j j= 0 j j+ J+ 3 vb j vb j vb j 0B B j= 0 j= 0 j= 0 δ δ = ω ω Solving he above equaion, we have δ =0.757, δ =0.5750, ω 0 =0.5750; ω 0 = , B B v( B) = 0.757B B Esimaion of he Noise n The noise is esimaed as n ˆ = y ŷ = y ( δ ˆ yˆ +δ ˆ yˆ +δ ˆ yˆ +ωˆ x ω ˆ x ωˆ x ) r r 0 b b s b s ( ˆ ˆ x 0.74x ) = y 0.75y y Model Idenificaion for he Noise Series n Auocorrelaion and parial auocorrelaion funcion for n : The auocorrelaion funcion decays exponenially o 0, while he parial auocorrelaion funcion cu off afer lag (Figures 3 and 4). Thus he idenified model is ARIMA (,0,0) which is equivalen o AR() (Table ). RRJSMS Volume Issue June, 06 43

6 Auocorrelaion Funcion for n Auocorrelaion Lag Figure 3. Auocorrelaion funcion for n.. Parial auocorrelaion funcion for n Parial Auocorrelaion Lag Figure 4. Parial auocorrelaion funcion for n. Table. The parameer esimaes for he model. Parameers Esimae p-value ω ω δ δ φ Wriing he model explicily, we have φ Bn =ε p So ha φ Bn =ε and n = ε φb The Transfer Funcion-AR Noise Model Combining he wo models, we have y = v B x + n δ( B) b ω BB y = x + n ω ωb y = x + ε 0 b δb δb φ B The final ransfer funcion-ar noise model is obained as; ω ωb y = x + ε 0 δb δb φb Where x = X X and y = Y Y RRJSMS Volume Issue June, 06 44

7 DIAGNOSIS To check for model adequacy, he following diagnosic check was applied. Auocorrelaion Funcion for he Residual ε The auocorrelaion funcion of he residual erm ε (Figure 5) below demonsraes no paricular paern or spikes. No auocorrelaion is significan. This indicaes ha he residual follows a whie noise process. Hence he fied model is adequae. Auocorrelaion Funcion for e Auocorrelaion Lag Forecasing Figure 5. Parial auocorrelaion funcion for ε. Having esed he adequacy of he ransfer funcion AR noise model and was found o be appropriae; nex is o generae forecass. The forecass of he nex en years are presened in Table 3. Table 3. Forecass from period 54 a 95 percen limis. Period Forecas Lower Upper , SUMMARY/CONCLUSION This work provides a soluion o a dynamic relaionship where he residual fails o follow a whie noise process or violae is usual assumpions. The work modeled he residual obained from fiing he ransfer funcion (TF) model separaely. The ARIMA model obained from he TF noise is added o he major componen of he model resuling in ransfer funcion auoregressivenoise model. Daa from naira exchange rae of US Dollars and Swiss Franc (970-03) was used o demonsrae he workabiliy of he model. The model was found o be appropriae and forecass for he Naira exchange of Swiss Franc were generaed using he model. REFERENCES. Appiah ST and Adeunde IA. Forecasing exchange rae beween he Ghana Cedi and he US Dollar using ime series analysis. African Journal of Basic applied sciences.0;3: Kwame-Poku O. Exchange rae volailiy in LDCs: Some findings from Ghanaian, Mozambican and Tanzanian markes. PhD Thesis Submied o Economics Sudies Deparmen, College of Ars and Social Science, Universiy of Dundee, 00;pp: Laifa NB, e al. Impac of he global financial crisis on exchange raes and policies in Sub Saharan Africian. Inernaional moneary fund, African deparmenal paper HG398, B Jeffrey FA. proposed moneary regime for small commodiy exporers: Peg he expor price PEP. Inernaional Finance Paper Par Tsay RS. Analysis of Financial Time Series, nd Ediion. A Wiley-Inerscience publicaion. John Wiley & Sons, Inc. New York RRJSMS Volume Issue June, 06 45

8 6. Liu LM, e al. 986; The SCA saisical sysem: Reference manual for forecasing and Time series Analysis. Scienific Compuing Associaes, Illinois Liu LM. Sales forecasing using muli-equaion Transfer Funcion Model. Journal of Forecasing. 987;6: Liu LM. Use of Linear Transfer Funcion Analysis in Economeric Time Series Modeling. Saisisca Simica. 99;: Elkhem SG and Karama B. Transfer funcion and uning of crude disillaion uni conroller a Kharoum refinery-sudan. Journal of applied and indusrial Sciences.04;: Znaczko TM. forecasing foreign exchange raes; A hesis in Economics and Finance, Sae Universiy of New York, Buffalo. 03,. Kannan S and Farook AJ. Transfer Funcion Modeling for global warning. 03;: Ook BW and Suharono. Developmen of rainfall forecasing model in Indonesia by using ASTAR, Transfer funcion and ARIMA mehods. European journal of scienific research. 009;38: Pankraz A. Forecasing wih univariae Box and Jenkins models. Conceps and cases, John Wiley, New York Jha GK. Advanced forecasing models using SAS sofware, IARI, Pusa New Delhi Box GEP, e al. Time Series Analysis, Forecasing and Conrol, Pearson educaion, Delhi Box GEP and Jenkins G. Time Series Analysis, Forecasing and Conrol, Holden-Day San Francisco RRJSMS Volume Issue June, 06 46