MODELING AND FORECASTING EXCHANGE RATE DYNAMICS IN PAKISTAN USING ARCH FAMILY OF MODELS

Transcription

1 Elecronic Journal of Applied Saisical Analysis EJASA (01), Elecron. J. App. Sa. Anal., Vol. 5, Issue 1, 15 9 e-issn , DOI /i v5n1p15 01 Universià del Saleno hp://siba-ese.unile.i/index.php/ejasa/index MODELING AND FORECASTING EXCHANGE RATE DYNAMICS IN PAKISTAN USING ARCH FAMILY OF MODELS Shahla Ramzan *(1), Shumila Ramzan (), Faisal Maqbool Zahid (1) (1) Deparmen of Saisics, Governmen College Universiy Faisalabad, Pakisan () Universiy of Agriculure, Faisalabad, Pakisan Received 14 Ocober 010; Acceped 05 Ocober 011 Available online 6 April 01 Absrac: The main objecive of his paper is o provide an exclusive undersanding abou he heoreical and empirical working of he GARCH class of models as well as o exploi he poenial gains in modeling condiional variance, once i is confirmed ha condiional mean model errors presen ime varying volailiy. Anoher objecive is o search he bes ime series model among auoregressive moving average (ARMA), auoregressive condiional heeroscedasiciy (ARCH), generalized auoregressive condiional heeroscedasiciy (GARCH), and exponenial generalized auoregressive condiional heeroscedasiciy (EGARCH) o give bes predicion of exchange raes. The daa used in presen sudy consiss of monhly exchange raes of Pakisan for he period ranging from July 1981 o May 010 obained from Sae Bank of Pakisan. GARCH (1,) is found o be bes o remove he persisence in volailiy while EGARCH(1,) successfully overcome he leverage effec in he exchange rae reurns under sudy. Keywords: Condiional variance, Exchange raes, GARCH, EGARCH, Volailiy modeling. 1. Inroducion In he era of globalizaion and financial liberalizaion, exchange raes play an imporan role in inernaional rade and finance for a small open economy like Pakisan. This is because movemens in exchange raes affec he profiabiliy of mulinaionals and increase exchange exposure o enerprises and financial insiuions. A sable exchange rae may help venure and financial insiuions in evaluaing he performance of invesmens, financing and hedging, and * shahla_ramzan@yahoo.com 15

2 Modeling and forecasing exchange rae dynamics in Pakisan using arch family of models hus reducing heir operaional risks. Flucuaions in he exchange rae may have a significan impac on he macroeconomic fundamenals such as ineres raes, prices, wages, unemploymen, and he level of oupu. The behavior of exchange raes is of crucial ineres for he policy makers. Recen developmens in financial economerics require he use of quaniaive models ha are able o explain he aiude of invesors no only owards he expeced reurns and risks, bu owards volailiy as well. Hence, marke paricipans should be aware of he need o manage risks associaed wih volailiy, which requires models ha are capable of dealing wih he volailiy of he marke. Due o unexpeced evens like, he non-consan variance in he financial markes, and uncerainies in prices and reurns, financial analyss sared o model and explain he behavior of exchange rae reurns and volailiy using ime series economeric models. One of he mos prominen ools for capuring such changing variance is he Auoregressive Condiional Heeroscedasiciy (ARCH) and Generalized Auoregressive Condiional Heeroscedasiciy (GARCH) models. These models were developed by Engle [1] and exended by Baillie [] and Nelson [3]. Two imporan characerisics wihin financial ime series, he fa ails and volailiy clusering (or volailiy pooling), can be capured by he GARCH family models. Many sudies have been done o model he exchange raes of a counry. Core e al [4] used Bayesian mehods for esimaion and ranking of a se of empirical exchange rae models, and consruced combined forecass based on Bayesian model averaging. Sing [5] invesigaed he GARCH model for a comprehensive se of weighed (expor and rade) and unweighed (official and black) real exchange rae series in India. He found evidence of dimensionally weak and saisically insignifican auoregressive condiional heeroscedasiciy effecs as compared o GARCH effecs in almos all he exchange rae series. Oher sudies include Baillie and Bollerslev [], Poon and Granger [6], Arize e al [7], Fidrmuc and Horváh [8] and Rapach and Srauss [9]. The economy of Pakisan is he 7h larges economy in he world in erms of purchasing and he 45h larges in absolue dollar erms. Pakisan has been caegorized as an emerging marke. There are hree sock exchange markes in Pakisan. The Karachi Sock Exchange (KSE) being he mos liquid and he bigges in erms of marke capializaion and rading volume and been awarded he ile of bes performing emerging sock marke of he world in 00 by Business Week. Jalil and Feridun [10] explained he exchange rae movemens in he Pakisani foreign exchange marke using he marke micro srucure approach, which had no been applied o dae due o he unavailabiliy of high-frequency daa on he order flow for Pakisan. Mohammad [11] empirically assess he effec of Euro-Dollar Exchange rae on chosen macroeconomic variables like real oupu, price level, and money supply for Pakisan. Mohammad [11] applied vecor auoregessive (VAR) approach o find he relaion beween he above and he resuls are eviden of no significan impac of Euro and US dollar depreciaion on he macroeconomic variables of Pakisan. This paper offers insighs on exchange raes in Pakisan and measures he sources of volailiy by using Auoregressive Condiional Heeroscedasiciy (ARCH), Generalized Auoregressive Condiional Heeroscedasiciy (GARCH) and Exponenial General Auoregressive Condiional Heeroscedasiciy (EGARCH) echniques. The res of his paper is caegorized as follows: Secion presens he descripion of he daa used in his sudy and some economeric mehodology used for analysis. Secion 3 presens saisical feaures of he daa and resuls of he analysis wih some discussion. Finally las secion concludes he ex wih some concluding remarks. 16

3 Ramzan, S., Ramzan, S., Zahid, F.M. (01). Elecron. J. App. Sa. Anal., Vol. 5, Issue 1, Maerial and Mehods.1 Daa and Economeric mehodology For convenional analysis and volailiy modeling, he daa used in he presen sudy consiss of monhly average Foreign Exchange raes of Pakisan (Pak rupees per US $). The daa is obained from Sae Bank of Pakisan (SBP) Karachi wih sample period ranging from July 1981 o May 010. Since he official exchange markes remain close on Sunday, here are oal of 347 monhly observaions. To model his daa, differen economeric ime series models like ARMA, ARCH, GARCH, IGARCH and EGARCH are used. A shor descripion of each is given in he following ex.. ARMA (p,q) In an ARMA (p, q) process, here are p auoregressive and q moving average erms. The algebraic represenaion of he saisical model is: y = µ + Φ... ε 1 y 1 + Φ y Φ p y p + ε + θ1ε 1 + θ ε + + θ q q (1) where he inercep parameer µ is relaed o he mean of y and errors ε are assumed o be uncorrelaed random variables, wih E[ ε ] = 0 and var[ ε ] = σ. If his process is saionary, hen i mus have a consan mean µ for all ime periods..3 Auoregressive Condiional Heeroscedasiciy (ARCH) Model The auoregressive condiional heeroscedasiciy (ARCH) model is he firs model of condiional heeroscedasiciy. In economerics, a model feauring auoregressive condiional heeroscedasiciy considers he variance of he curren error erm or innovaion o be a funcion of he acual sizes of he previous ime periods' error erms. Ofen he variance is relaed o he squares of he previous innovaions. Such models ofen called ARCH models were inroduced by Engle [1] and is known as a firs order auoregressive condiional heeroscedasiciy ARCH(1) process. Specifically, le ε denoe he error erms of reurn residuals wih respec o mean process and assume ha ε = σ z where ~ N(0,1 ) and he series σ are modeled by: z iid q... + α qε q = α 0 + αi i i= 1 σ = α + α ε + ε () Here α 0 > 0 and α i o, i > o. An ARCH(q) model can be esimaed using ordinary leas squares..4 Generalized Auoregressive Condiional Heeroscedasiciy (GARCH) Model Unforunaely, volailiy is no an easy phenomenon o predic or forecas. One class of models which have proved successful in forecasing volailiy in many siuaions is he GARCH family 17

4 Modeling and forecasing exchange rae dynamics in Pakisan using arch family of models of models. The Generalized Auoregressive Condiional Heeroscedasiciy (GARCH) models were propounded by Engle [1] and Baillie []. The GARCH (p, q) model is formulaed as: σ = α 0 +α 1 ε α q ε q + β 1 σ β p σ p q = α 0 + α i ε i + β i σ i i=1 p i=1 (3) where p is he order of he GARCH (lagged volailiy) erms, and q is he order of he ARCH (lagged squared-error) erms..5 Inegraed Generalized Auoregressive Condiional Heeroscedasiciy (IGARCH) Model Inegraed Generalized Auoregressive Condiional Heeroscedasiciy IGARCH is a resriced version of he GARCH model, where he sum of he persisen parameers sum up o one, and herefore here is a uni roo in he GARCH process. The condiion for his is: p i= 1 q β i + α i = 1 (4) i= 1.6 Exponenial General Auoregressive Condiional Heeroscedasic (EGARCH) Model The Exponenial General Auoregressive Condiional Heeroscedasiciy (EGARCH) model is anoher popular GARCH model which is inroduced by Nelson [3]. In EGARCH model here is no need for he parameers o follow nonnegaive resricion and anoher imporan feaure of EGARCH model is ha i successfully capure he leverage effec while GARCH model is no o do so. EGARCH ensures ha he variance is always posiive even if he parameers are negaive. The EGARCH (1, 1) could be given as: lnh ε ε α (5) 1 1 = 0 + α1 + φ + β1 lnh 1 h 1 h 1 where φ capures he leverage effec (i.e. asymmeric effec)..7 Forecasing Performance Commonly used measures for comparison of he forecasing performance are Mean Absolue Error (MAE), Roo Mean Square Error (RMSE), Mean Absolue Percenage Error (MAPE), and Theil-U inequaliy ([1]). The seleced GARCH (1, ) model in our sudy is evaluaed by using all hese measures for evaluaing he forecasing performance. The menioned forecas error saisics are defined as: Mean Absolue Error = T k + = T + 1 yˆ h y (6) 18

5 Ramzan, S., Ramzan, S., Zahid, F.M. (01). Elecron. J. App. Sa. Anal., Vol. 5, Issue 1, Roo Mean Squared Error = T k + = T + 1 ( yˆ y ) h (7) These wo forecas error saisics depend on he scale of he dependen variable. These should be used as relaive measures o compare forecass for he same series across differen models. The smaller he error, he beer is he forecasing abiliy of ha model according o a given crierion. The remaining wo saisics are scale invarian. The Theil inequaliy coefficien always lies beween zero and one, where zero indicaes a perfec fi. Mean Absolue Percenage Error = 100 T + k = T + 1 yˆ h y y (8) Theil Inequaliy Coefficien = T + k ( yˆ y ) = T + 1 T + k yˆ = T + 1 h h + T + k y = T + 1 h (9) 3. Resuls and Discussion 3.1 Descripive saisics To assess he disribuional properies of he exchange rae reurn daa, various descripive saisics are repored in Table 1. Table 1. Summary saisics. Mean Sd. Dev. Skewness Kurosis Jarque-Bera Prob.* <0.001 As expeced for a ime series of reurns, he mean is close o zero. The sandard deviaion is also high which indicaes high level of flucuaions of he exchange rae reurns. There is also evidence of posiive skewness, wih long righ ail indicaing ha exchange rae has non symmeric reurns. The hisogram of he series in Figure 1 repors ha he exchange rae reurns are lepokuric or fa ailed because of is large kurosis value. The reurn series of exchange rae is non normal according o he Jarque-Bera es wih p-value less han (see Table 1). So he hypohesis of normaliy of he exchange rae reurns is rejeced. 19

6 Modeling and forecasing exchange rae dynamics in Pakisan using arch family of models Figure 1. Hisogram of exchange rae reurn series. 3. Tess for checking Saionariy I is imporan o confirm wheher he daa is saionary before using he ime series models like Auoregressive Inegraed Moving Average (ARIMA), Auoregressive Condiional Heeroscedasiciy (ARCH), Generalized Auoregressive Condiional Heeroscedasiciy (GARCH) and Exponenial General Auoregressive Condiional Heeroskedasic (EGARCH), which are being used in our sudy. Three imporan mehods of checking saionariy of ime series are graphical analysis, Correlogram and uni roo es exchange rae reurns Index Figure. Time series plo of exchange raes series. 0

7 Ramzan, S., Ramzan, S., Zahid, F.M. (01). Elecron. J. App. Sa. Anal., Vol. 5, Issue 1, In Figure he graphical analysis of exchange rae series shows an upward rend, suggesing ha he mean of he exchange raes has been changing. This perhaps suggess ha he exchange rae series is no saionary. Figure 3. Correlogram of exchange rae series. The Correlogram of he series is shown in Figure 3 which indicaes a paer up o 30 lags. The auocorrelaion coefficien sars wih a very high value of a lag 1 and declines very slowly owards zero. I seems ha he exchange rae series is nonsaionary. Table presens he resuls of Augmened Dickey Fuller (ADF) es and Phillips-Perron (PP) es on exchange rae daa. Since he saisic value for boh ADF and PP ess is greaer han heir corresponding criical values, so we do no rejec he null hypohesis of he presence of uni roo in he series and conclude ha he exchange rae series is nonsaionary. To ransform he nonsaionary exchange rae series we calculae he exchange rae reurns as R = ln( y / y 1) *100 (10) The ime series plo of he ransformed daa ha is exchange rae reurns series is shown in Figure 4. This plo shows ha he mean of he series is now abou consan. So we can assume ha he series is saionary. 1

8 Modeling and forecasing exchange rae dynamics in Pakisan using arch family of models Table. ADF and Phillip-Perron es on exchange rae series. Augmened Dickey-Fuller es saisic -Saisic Prob.* Tes criical values: 1% level % level % level -.57 Phillips-Perron es Adj. -Sa Prob.* Tes criical values: 1% level % level % level -.57 Furhermore, he Correlogram in Figure 5 shows no rend in exchange rae reurns, hence suggesing ha he exchange rae reurn series is saionary. RT Figure 4. Time series plo of exchange rae reurns

9 Ramzan, S., Ramzan, S., Zahid, F.M. (01). Elecron. J. App. Sa. Anal., Vol. 5, Issue 1, Figure 5. Correlogram of exchange rae reurns. In Table 3 he resuls of Augmened Dickey Fuller (ADF) es and Phillips Perron (PP) es show ha he exchange rae reurn series is saionary. The variance is high ha clearly exhibi volailiy clusering, which allows us o carry on furher o apply he ARCH family models. Table 3. ADF and PP es on exchange rae reurn series. Augmened Dickey-Fuller es saisic -Saisic Prob.* <0.01 Tes criical values: Phillips-Perron es Adj. -Sa Prob.* <0.01 Tes criical values:

10 Modeling and forecasing exchange rae dynamics in Pakisan using arch family of models 3.3 Model fiing As he descripive saisics given in Table 1 reflec ha he disribuion of he exchange rae reurn series does no follow a normal disribuion i means volailiy clusering is presen. Suiable economeric modeling echniques are required for our exchange rae reurn series. To sar wih, we model he condiional mean process by auoregressive process AR(1) and moving average MA(1). Boh of hese processes demonsrae high correlaion in residuals. Among differen models applied o he daa, ARMA(1, ) appears o be relaively beer fi on he basis of Akaike crierion, Schwarz crierion, and Durbin Wason saisic. The resuls of ARMA(1, ) are shown in Table 4. Table 4. Resuls of ARMA (1, ) model. Coefficien Sd. Error -Saisic Prob. C <0.01 AR(1) <0.01 MA() R-squared 0.18 Mean dependen var 0.7 Adjused R-squared 0.18 S.D. dependen var 0.57 S.E. of regression 0.51 Akaike info crierion 1.5 Sum squared resid Schwarz crierion 1.55 Log likelihood Hannan-Quinn crier F-saisic 38.4 Durbin-Wason sa.0 Prob(F-saisic) <0.01 The correlogram of ARMA(1, ) residuals in Figure 6 shows ha none of he auocorrelaion and parial auocorrelaion is individually saisically significan. In oher words he correlogram of boh auocorrelaion and parial auocorrelaion give he impression ha he esimaed residuals are purely random. Hence, here is no need o search ou for anoher ARMA model. Once an appropriae ARMA model for condiional mean has been idenified and esimaed, he nex sep is o es wheher he esimaed errors are heeroscedasic or no. For his purpose he correlogram of squared residuals from ARMA(1, ) is presened in Figure 7. I shows high quaniy of auocorrelaion in residuals which sugges o carry on furher o apply he ARCH family of models. The ARCH family of models requires he presence of ARCH effec in he residuals. To es he presence of ARCH effec, we use he Lagrange Muliplier (LM) es for exchange rae reurns series as suggesed by Engle [1]. The resuls of Lagrange Muliplier es are presened in Table 5. The p-value indicaes ha here is evidence of remaining ARCH effec. So, we rejec he null hypohesis of absence of ARCH effec even a 1% level of significance. 4

11 Ramzan, S., Ramzan, S., Zahid, F.M. (01). Elecron. J. App. Sa. Anal., Vol. 5, Issue 1, Figure 6. Correlogram of he residuals of ARMA(1, ) model. Figure 7. Correlogram of square residuals of ARMA (1, ) model. 5

12 Modeling and forecasing exchange rae dynamics in Pakisan using arch family of models Table 5. ARCH LM es on ARMA(1, ) residuals. F-saisic Prob. F(,340) <0.01 Obs*R-squared Prob. Chi-Square() <0.01 Therefore he condiional heeroscedasiciy in esimaed ARMA errors is modeled using GARCH(1, ) specificaion. The resuls are shown in Table 6. The esimaed coefficien of boh condiional mean equaion and condiional variance equaion of GARCH(1, ) model are highly significan. Therefore GARCH(1, ) model can be cied as he suiable model on he basis of AIC and BIC crieria given in Table 6 and correlogram of he squared residuals from GARCH(1, ) in Figure 8. Table 6. Esimaion Resul of GARCH(1, ) model. Coefficien Sd. Error z-saisic Prob. C E <0.001 AR(1) <0.001 MA() <0.001 Variance Equaion C <0.001 RESID(-1)^ GARCH(-1) <0.001 GARCH(-) GED PARAMETER R-squared Mean dependen var Adjused R-squared S.D. dependen var S.E. of regression Akaike info crierion Sum squared resid Schwarz crierion Log likelihood Hannan-Quinn crier F-saisic Durbin-Wason sa.5448 Prob(F-saisic) <0.001 Once he heeroscedasiciy is modeled using appropriae GARCH model, nex we check wheher he GARCH(1, ) has adequaely capured he persisence in volailiy and here is no ARCH effec lef in he residuals from he seleced models. The ARCH LM es is conduced for his purpose. The resuls of LM es given in Table 7 indicae ha he residuals do no show any ARCH effec. Hence, GARCH(1, ) is found o be reasonable o remove he persisence in volailiy. 6

13 Ramzan, S., Ramzan, S., Zahid, F.M. (01). Elecron. J. App. Sa. Anal., Vol. 5, Issue 1, Figure 8. Correlogram of squared residuals from GARCH (1, ) model. Table 7. ARCH LM es on GARCH (1, ) residuals. F-saisic Prob. F(1,339) Obs*R-squared 0.48 Prob. Chi-Square(1) Normaliy es of sandardized squared residuals, however suggess ha residuals are posiively skewed due o he leverage effec. EGARCH is used o es he leverage effec ha successfully capures he asymmery. The resuls on asymmeric condiional variance for exchange rae reurn series are repored in Table 8. The GARCH(1, ) parameer is significan, showing high degree of persisence. Coefficien of he asymmeric funcion is posiive and significanly differen from zero means here is no leverage effec lef. 7

14 Modeling and forecasing exchange rae dynamics in Pakisan using arch family of models Table 8. Esimaion Resul of EGARCH (1, ). Coefficien Sd. Error z-saisic Prob. C <0.001 AR(1) <0.001 MA() <0.001 Variance Equaion C(4) C(5) C(6) <0.001 C(7) <0.001 C(8) GED PARAMETER < Forecas Analysis Forecas performance of he fied GARCH(1, ) model of exchange rae reurns is invesigaed hrough mean absolue error (MAE), roo mean square error (RMSE), mean absolue percen error (MAPE), and Theil inequaliy coefficien. The resuls are shown in Table 9.. Table 9. Forecasing performance of exchange rae reurns. Roo Mean Square Error Mean Absolue Error Mean Abs. Percen Error Theil Inequaliy Coefficien Bias Proporion Variance Proporion Covariance Proporion The value of Theil inequaliy is indicaing he model is a beer fi. The bias proporion and variance proporion are close o zero. The value of covariance proporion is nearly one. We can say ha his model is good for forecasing purpose along wih capuring he volailiy and he leverage effecs. 8

15 Ramzan, S., Ramzan, S., Zahid, F.M. (01). Elecron. J. App. Sa. Anal., Vol. 5, Issue 1, Conclusion This paper focuses on building a model for he exchange rae of Pakisan using ime series mehodology. As he financial ime series like exchange rae may possess volailiy, an aemp is made o model his volailiy using ARCH and GARCH models. Monhly average foreign exchange raes of Pakisan for he period ranging from July 1981 o May 010 are used for his purpose. Firs of all, he saionariy of he exchange rae series is examined using graphical analysis, correlogram and uni roo es which showed he series as nonsaionary. To make he exchange rae series saionary, he exchange raes are ransformed o exchange rae reurns. Then ARMA(1, ) model is fied o he daa. To capure he volailiy, GARCH (1, ) model is used. This model is furher having leverage effec. This leverage effec is capured using an EGARCH model. Finally, he forecas performance is measured using differen measures like MAE, RMSE and MAPE ec. The GARCH family of models capures he volailiy and leverage effec in he exchange rae reurns and provides a model wih fairly good forecasing performance. References [1]. Engle, R.F. (198). Auoregressive Condiional Heeroscedasiciy wih esimaes of variance of Unied Kingdom inflaion. Economerica, 50(4), []. Baillie, R.T., Bollerslev, T. (00). The massage in daily exchange raes: A condiional variance ale. Journal of Business and Economic Saisics, 0(1), [3]. Nelson, D.B. (1991). Condiional heeroscedasiciy in asse reurns: A new approach. Economerica, 59(), [4]. Core, P.D., Sarno, L., Tsiakas, I. (008). An Economic Evaluaion of Empirical Exchange Rae Models. The review of financial sudies, (9), [5]. Singh, T. (00). On he GARCH Esimaes of Exchange Rae Volailiy in India. Applied Economics Leers, 9, [6]. Poon, S.H., Granger, C.W.J. (003). Forecasing financial marke volailiy: A review. Journal of Economic Lieraure, 41(), (6). [7]. Arize, A.C., Osang, T., Sloje, D. J. (006). Exchange-rae volailiy in Lain America and is impac on foreign rade. Inernaional Review of Economics & Finance, 17(1), [8]. Fidrmuc, J., Horváh, R. (007). Volailiy of exchange raes in seleced new EU members: Evidence from daily daa. Economic Sysems, 3(1), [9]. Rapach, D.E., Srauss, J.K. (008). Srucural breaks and GARCH models of exchange rae volailiy. Journal of Applied Economerics, 3(1), [10]. Jalil, A., Feridun, M. (010). Explaining exchange rae movemens: an applicaion of he marke microsrucure approach on he Pakisani foreign exchange marke. The Journal of Developing Areas, 44(1), [11]. Mohammad, S.D. (010). The Euro - Dollar Exchange Raes & Pakisan Macroeconomics Dynamics. European Journal of Scienific Research, 4(1), [1]. Abrosimova, N., Dissanaike, G., Linowski, D. (005). Tesing he weak-form Efficiency of he Russian Sock Marke. hp://papers.ssrn.com/sol3/papers.cfm?absrac_id=3087 This paper is an open access aricle disribued under he erms and condiions of he Creaive Commons Aribuzione - Non commerciale - Non opere derivae 3.0 Ialia License. 9