Evaluation of GARCH model Adequacy in forecasting Non-linear economic time series data

Transcription

1 Journal of Comuaons & Modellng, vol.3, no., 03, -0 ISSN: (rn), (onlne) Scenress Ld, 03 Evaluaon of GARCH model Adequacy n forecasng Non-lnear economc me seres daa M.O. Aknunde, P.M. Kgos and D.K. Shangodoyn 3,* Absrac To dae n leraure, GARCH model has been descrbed no suable for non-lnear foregn exchange seres and herefore hs aer rooses an Augmened GARCH model ha could caure boh lnear and non-lnear behavor of daa. The roeres of hs new model s derved and found o have a mnmum varance comared wh GARCH model. We emloy he use of Brock-Decher- Schenkman (BDS) es sasc o confrm he suably of GARCH model on he daa; he new mehodology roosed s llusraed wh foregn exchange rae daa from Grea Bran (Pound) and Boswana (Pula) agans Uned Saes of Amerca (Dollar). Dearmen of Sascs, Unversy of Boswana, Boswana, Gaborone. Dearmen of Sascs, Unversy of Boswana, Boswana, Gaborone. 3 Dearmen of Sascs, Unversy of Boswana, Boswana, Gaborone. * Corresondng auhor. Arcle Info: Receved : December 4, 0. Revsed : January 9, 03 Publshed onlne : June 0, 03

2 Evaluaon of GARCH model Adequacy n forecasng... Keywords: GARCH models, Augmened GARCH models, Brock-Decher- Schenkman (BDS) es, B-lnear models, foregn exchange daa Inroducon The auoregressve condonal heeroscedascy model (ARCH), nroduced by Engle (98) and s generalzaon GARCH, nroduced by Bollerslev (986) have been wdely aled o model volaly n fnancal me seres. These models have been useful because hey are convenen reresenaon of he erssence of varance over me dese he lack of sascal and economc heory jusfcaon (Hall e al., 989). Several sudes have nvesgaed he adequacy of GARCH model n fnancal me seres. Claudo and Jean (0) used GARCH o model sock marke ndces and concluded ha he model fals o caure he sascal srucure of he marke reurns seres for all he counres economes nvesgaed. Lm e.al.(005) emloyed he Hnch ormaneau bcorrelaon es o deermne he adequacy of GARCH model for egh Asan sock markes. They conclude ha hs model canno rovde an adequae characerzaon for he underlyng marke ndces. Brooks and Hnch (998), Lew, e.al.(003) and Lm e.al (004) have suded he behavor of exchange raes daa usng GARCH models, was concluded ha hese models could no caure adequaely he sascal roeres of non-lneary resen n he seres. Besdes hese fndngs, olcal and fnancal nsably ha arses from erod o erod n mos counres roduces esodc non-lneares n he foregn exchange markes ndces (Bonlla e.al. 006 and Romero-Meza e.al. 007), f he rocedure ulzed n he analyss of foregn exchange s no adequae may jeoardze forecasng effcacy and lead o dsoron of nference made. I herefore may be of neres o examne he sascal roeres of modfed

3 M.O. Aknunde, P.M. Kgos and D.K. Shangodoyn 3 GARCH model and s suably n he resence of non-lneares behavor of exchange rae daa. Ths aer examnes he sascal roeres of augmened GARCH model; he augmenaon s erformed usng B-lnear funcon o caure he nsably of he non-lneary n he daa se. We analycally comare he new model wh convenonal GARCH model usng he model varance. The Brock-Decher- Schenkman es (BDS) s aled o es he adequacy of GARCH model on he seres used. Guglemo e.al (005) have ulzed hs es sasc o deermne he adequacy of GARCH models for caurng non-lneary n daa se. The rocedure nvolves subjecng he sandardzed resduals of he fed GARCH models o BDS under he null hyohess of GARCH suffcen characerzaon of he seres. If he BDS es rejecs he null hyohess usng arorae crcal values, hen he fed GARCH model s assumes o be nadequaely characerzed he daa. Monhly daa used n hs aer covered he erod of January 975 o December 00 (444 monhs). The behavors of he seres examned are as shown n fgures a o b. Tes for saonary was carred ou usng Augmened Dckey-Fuller es and un roo es were erformed. The remanng ar of hs aer s organzed as follows: secon covers he secfcaon of augmened GARCH models, effcency of AGM, esmaon of he arameers of augmened GARCH model (AGM), roeres of derved esmaors of AGM, secon 3, emrcal llusraon, denfcaon of non-lneary saus of he seres wh BDS es, denfcaon of saonary condon of seres, esmaon of classcal GARCH and augmened GARCH models secon 4 emrcal comarson of models and concluson.

4 4 Evaluaon of GARCH model Adequacy n forecasng... Secfcaon of augmened GARCH models Leraure has shown ha fnancal me seres daa resen volaly cluserng effecs, and hs volaly occurs nermenly. To ake care of hs suaon researchers make use of a condonal varance model, where he varance of he errors s allowed o change over me n an auoregressve condonal heeroskedascy framework. Followng Bollerslev (986), he GARCH (, q ) model can be reresened n he followng form: { } Le y ( ) be he me seres of an exchange rae reurn, hen y = σε ( ) q = 0 + y + j j = j= () σ α α βσ where α0 > 0, α 0 and nnovaon sequence { ε} s ndeenden and = dencally dsrbued ( d ) wh ( ) E ε 0 =0 and E ( ε 0 ) =. The man dea s ha σ, he condonal varance of y gven nformaon avalable u o me has an auoregressve srucure and s osvely correlaed o s own recen as and o recen values of he squared reurn, erssen, large (small) values of y. Ths caures he dea of volaly beng y are lkely o be followed by large (small) values. The GARCH model formulaon caures he fac ha volaly s changng n me. The change corresonds o a weghed average among he long erm average varance, he volaly n he revous erod, and he fed varance n he revous erod as well. The model descrbed n equaon () s used o arameerze fnancal me seres and n arcular foregn exchange. An augmened GARCH model s an exenson of he GARCH model as ool for modelng fnancal me seres. I allows us o caure asymmeres n he condonal mean and varance of fnancal and economc me seres by means of neracons beween as shocks and volales. The blnear GARCH models

5 M.O. Aknunde, P.M. Kgos and D.K. Shangodoyn 5 ake no accoun varaons beween he ndeenden varables as well as covaraons beween he varables. Ths s very moran n he sudy of fnancal marke daa where he covarance beween ndeenden varables may lay a sgnfcan role n deermnng marke volaly. We use AGM because we dscovered ha s modelng s daa drven as we augmen he model y o hs error erm and observe seres. The ncluson of blnear rocess o equaon () wll caure he non-lnear behavor ar of y, blnear akes no accoun he varaon whn ndeenden varables as well as co-varaons beween he varable. On he oher hand Augmened-GARCH models (AGM) allow us o caure asymmeres n he condonal varance of fnancal and economc-me seres by means of neracons beween as shocks and volales; hus we osulae an augmened GARCH ( AGM ) as: y j j = j= q = σε + τ y ε () To nvesgae he rooron of () we consder s mean and varance as follows: mean of y s derved usng Ey ( ) σ E( ε ) τ Ey ( ε ) as = + j j = j= q { } = E y 0 j σε τj, =j = To derve he varance of ( ) ( ( )) Var( y ) = E y E y Consder an alernave reresenaon y from he convenonal exresson gven as: ( ) Z = y σ = σ ε (3) (4) y = σ + Z,

6 6 Evaluaon of GARCH model Adequacy n forecasng... where Z s a marngale dfferences wh mean zero q 0 y j j Z = j= = α + α + βσ + q 0 y j j jzk j Z = j= j= = α + α + βσ β + If we denoe max ( q, ) can be wren as: In oher words =, α = 0 for > and β j = 0 for j P q = α0+ ( α+ β) β j j + = j= y y Z Z > q, hen he above y s an ARMA rocess wh marngale dfference nnovaons. Usng saonary,.e. E( y ) = E( y ) s now easy o oban, he uncondonal varance q ( ) = α0 + ( α + β j) ( ) β j ( j) + ( ) E y E y E Z E Z = j= reduces o ( ) ( j) α0 + E y α + β, ( ) E y = α = + 0 ( α β j) = Also usng equaon we have y = σε + τ y ε I ( ) = ( σε ) + τ ( ) E y E E y ( ) E y E ( σ ) α ( τ ) ( α + β j) ( τ ) = = 0 (5a) (5b)

7 M.O. Aknunde, P.M. Kgos and D.K. Shangodoyn 7 Usng (3) and (5a and 5b) n (4) gves Var( y ) α0 + ( α β ) j = l= α0 + l= ( α β) ( τ ) (6) = σε τ, = j. 4 where ( ). Effcency of AGM To comare he effcency of he AGM wh GARCH, we relae he varances of AGM o ha of classcal GARCH as follows: The varance of AGM was derved as: n Le T and T be wo esmaors of a aramerc funcon k( θ); θ R ; s he Eucldan sace. The effcency of T relave o T s defned as: If for all et { },, T, { / T} et T { } { } MSE T = MSE T θ s more effcen han T, oherwse T s more effcen hant. If T and T are unbased esmaors of k ( θ ), he effcency of T relave o T s he rao of ( ) o ( ) V T V T are unbased esmaors, Then he effcency of AGM relave o GM usng eguaon (6) and (7) s as follows: Var y 0 ( α + β) AGM = + ( ( ) ) ( ( GM ) ) Var y α = 0 0 ( ( α β )) = = = ξ α α ( α + β )

8 8 Evaluaon of GARCH model Adequacy n forecasng... ( ) ( α + β) where ξ =. α he 0 I can be seen ha f ξ >, hen AGM s more effcen han GM; besdes s osve and relaces he varance of AGM comared wh ha of GM. We shall look a emrcal mlcaons of hese quanes laer.. Esmaon of he arameers of augmened GARCH model (AGM) To esmae he arameers of he models n equaon (), a wo sage echnque s suggesed as follows. The reduced form of equaon () s: y = τ j zj + v (7) ( j) In marx form Y τ z v E vv = 0 j. = +, where we assume v N( 0, σ j ) and Y = τ z+ v (8) Now, a he frs sage we aly he mehod of MLE o oban arameers of () and he second sage gven ndeendence of he arameers n model (), we aly OLS o he reduce form (8), hus we have: τˆ = ZZ ZY (9) and ( ) τ = τ + = τ [ ˆ] ( ) E E ZZ Z Z V ( )( ) ( ) ( ) ( ) ( ) Var( ˆ τ) = E ˆ τ τ ˆ τ τ = E ZZ ZVVZ ZZ = σ ZZ The esmaes n (9) are unbased and conssen and usual es of hyohess can be underaken o asceran her sgnfcance.

9 M.O. Aknunde, P.M. Kgos and D.K. Shangodoyn 9.3 Proeres of derved esmaors of AGM We evaluae he roeres of he derved esmaor of AGM n hs secon based on some basc roeres of sascal esmaor..3. Lneary and unbased roeres of leas-squares esmaors From equaon (0), we have zy ˆ τ = = z z Such ha k =. Ths shows ha ˆ τ s a lnear esmaor because s a lnear z funcon of Y ; acually s a weghed average of Y wh weghs. The assumons on weghs k, are () ky z and k are assumed o be non-sochasc () k = 0 () k ( ) = z, and (0) k servng as he (v) kz =. These assumons can be drecly verfed from he defnon of nsance, k ; for Snce for a gven samle mean value, s always zero. z k = = z z z s known = 0, snce z Now subsue Y = τ+ τz + u no (0) o oban ( ) z., sum devaon from he ˆ τ = k τ + τ z + u = τ k + τ kz + ku = τ + ku ()

10 0 Evaluaon of GARCH model Adequacy n forecasng... Now akng he execaons of () on boh sdes and nong ha k, beng non-sochasc, can be reaed as consans, we oban Snce ( ) 0 ( ) ke( u) E ˆ τ τ = τ. = + E u = by OLS assumon. Therefore, ˆ τ s an unbased esmaor of ˆ τ. Lkewse can be roved ha ˆ τ s also an unbased esmaor of ˆ τ..3. Mnmum-varance roery of leas-squares esmaors of AGM I was shown ha he leas-squares ˆ τ s lnear as well as unbased (hs holds for ˆ τ also). To show ha hese esmaors also have mnmum varance n he class of all lnear unbased esmaors, consder he leas squares esmaor τˆ gvng as ˆ ky τ = where k weghs. where z z z = = ( z z) z. Ths shows ha ˆ τ s a weghed average of he Y' s, wh k servng as he Le us defne an alernave lnear esmaor of ˆ τ as τ = wy w are weghs, no necessarly equal k. Now, ( τ ) = ( ) = ( τ + τ ) = τ + τ E we Y w z w wz. Therefore for ˆ τ o be unbased, we mus have w = 0 and wz =. Also we may wre where ( τ) = wy = w ( Y) = σ w, var var var

11 M.O. Aknunde, P.M. Kgos and D.K. Shangodoyn ( ) var ( u ) var Y = = σ σ w = + z z z z z z z z w ( z ) = σ w + σ + σ z z z z () = σ w + σ z z Equaon () reduces o var ( τ ) σ = = var ˆ τ z By equaons (0) hrough (3) we have shown ha he derved model esmaors of AGM arameers sasfy he convenonal roeres of esmaors vs-à-vs unbasedness, mnmum varance and bes lnear unbased esmaors (BLUE). (3) 3 Emrcal llusraon The exchange rae daa colleced for Grea Bran and Reublc of Boswana akng Uned Saes of Amerca as bass for comarsm s ulzed for he emrcal llusraon of our roosed mehodology. The sascal ackage for he daa analyss n hs aer s E-vews. The analyss resened here focused on monhly exchange rae, of wo economes, vz-a vz develoed economy reresened by Grea Bran and develong economy reresened by Reublc of Boswana, he currences are denomnaed n Brsh Pound and Boswana Pula agans Uned Saes of Amerca Dollar.

12 Evaluaon of GARCH model Adequacy n forecasng Idenfcaon of non-lneary saus of he seres wh BDS es The currences exchange raes were analyzed hrough he use of E-vew and he hyohess was accordngly se as follows: H 0 : H : GARCH model s a suffcen characerzaon of seres H 0 s no rue In Table he null hyohess ha he GARCH model s a suffcen characerzaon of seres are rejeced, onng o he fac ha hs resul agreed wh Claudo A.B and Jean S (0), Chrs B and Hnch M.J. (0), Claudo A.B e.al (008), Kang-ng lm, e al (005), Chrs B and Hnch M.J (999) jus o menon he few ha GARCH s no adequae for fnancal me seres daa. Table : BDS es sasc values Seres BDS Sasc Sd. Error z-sasc Normal Prob. Boosra Prob. Pound Pula Idenfcaon of a saonary condon of he seres The lne grah of all he seres (fgures a andb) ndcaes he nonsaonary of he seres, snce volale values are evden and hese do no flucuae around a consan mean. We hus examne he frs dfferences of he seres (Fgures a and b) snce has no erssen rend and s values flucuae around a consan mean of zero.

13 M.O. Aknunde, P.M. Kgos and D.K. Shangodoyn 3.8 POUND Fgure a: Lne grah of he leveled exchange rae of Dollar/ula PULA Fgure b: Lne grah of he leveled exchange rae of Dollar/ula 500 POUND Fgure a: Lne grah of he frs dfference exchange rae of Dollar o Nara PULA Fgure b: Lne grah of he frs dfference of exchange rae of Dollar/ula

14 4 Evaluaon of GARCH model Adequacy n forecasng... The saonary condon of he seres can be formally verfed by usng un roo es (URT) for he leveled and frs dfferences of he seres. We es for a un roo usng he augmened Dckey-Fuller (ADF) sasc. A level all he seres are no saonary bu a frs dfference all seres are saonary as shown n Tables (a) and (b) below. Table a: Un Roo Tes Ouu for he leveled for he Seres Seres ADF-Tes sasc Crcal value Macknnon rob (5%) Pound Pula Table b: Un Roo Tes Ouu for he frs dfference for he Seres Seres ADF-Tes sasc Crcal value Macknnon rob (5%) Pound Pula Esmaon of classcal GARCH model To generae arameer esmaes for he GARCH model, we used E-vew o analyzed dfferenced daa for he sudy as follows: Each of he currency vz-a-vz Pound and Pula were ndvdually analysed. Based on ables 3a and 3b he esmaed GARC(,) model are obaned for boh Pound and Pula as follows: y = σε POUND/ US () where σ and ε are obanable from he fed model:

15 M.O. Aknunde, P.M. Kgos and D.K. Shangodoyn 5 y = y + ε POUND/ US () and (4) y σ = = σε PULA/ US () ε ( σ ) where σ and ε are obanable from he fed model: y = y + ε PUKA/ US () and (5) σ = The ouus of he resul are as follows: ε ( σ ) Table 3a: GARCH model esmaes for ound Deenden Varable: POUND Mehod: ML - ARCH (Marquard) - Normal dsrbuon GARCH = C() + C(3)*RESID(-)^ + C(4)*GARCH(-) Varable Coeffcen Sd. Error z-sasc Prob. DATE E Varance Equaon C E RESID(-)^ GARCH(-) R-squared Mean deenden var.4498 Adjused R-squared S.D. deenden var S.E. of regresson Akake nfo creron Sum squared resd Schwarz creron Log lkelhood Hannan-Qunn crer Durbn-Wason sa

16 6 Evaluaon of GARCH model Adequacy n forecasng... Table 3b: GARCH model esmae for ula Deenden Varable: PULA Mehod: ML - ARCH (Marquard) - Normal dsrbuon GARCH = C() + C(3)*RESID(-)^ + C(4)*GARCH(-) Coeffcen Sd. Error z-sasc Prob. DATE E Varance Equaon C RESID(-)^ GARCH(-) R-squared Mean deenden var Adjused R-squared S.D. deenden var.576 S.E. of regresson Akake nfo creron Sum squared resd Schwarz creron Log lkelhood Durbn-Wason sa Esmaon of augmened GARCH model Esmaon of arameers here was done here n wo sages as he sandard devaon obaned from classcal GARCH was used o oban he arameers of augmened GARCH models. The reduced form n equaon (0) was esmaed by makng use of Blnear (,) he reason for he choce of blnear (,) was due o he fac ha few arameers make he models o be arsmonous; from where ses of daa were generaed and OLS aled and he followng resuls were obaned for he wo seres (Pound and Pula foregn exchange wh resec o Dollar).

17 M.O. Aknunde, P.M. Kgos and D.K. Shangodoyn 7 Table 4a: Augmened GARCH model for ound Deenden Varable: y σε =ACMINFIT(POUND) Mehod: Leas Squares Dae: 0/09/ Tme: 4:56 Samle: 975M0 0M Included observaons: 444 τ y ε ACMINFIT =C()* Coeffcen Sd. Error -Sasc Prob. C() R-squared Mean deenden var Adjused R-squared S.D. deenden var S.E. of regresson Akake nfo creron Sum squared resd Schwarz creron Log lkelhood Hannan-Qunn crer Durbn-Wason sa Table 4a: Augmened GARCH model for ula Mehod: Leas Squares Dae: 0/09/ Tme: 5:8 Samle: 444 Included observaons: 444 τ y ε ACMINFIT =C()* Coeffcen Sd. Error -Sasc Prob. C() R-squared Mean deenden var Adjused R-squared S.D. deenden var.459 S.E. of regresson Akake nfo creron

18 8 Evaluaon of GARCH model Adequacy n forecasng... Sum squared resd.7477 Schwarz creron Log lkelhood Durbn-Wason sa By usng he values generaed n Table 4a he AGM fed s y = σε y ε ( 0.38) wh varance of he model Also usng he values generaed n 4b, he AGM fed s wh varance of he model y = σε y ε ( ) 4 Emrcal comarson of models and concluson Table 5 summarzed he resuls obaned for he varances of boh classcal GARCH models (GM) and augmened GARCH models (AGM), hs wll ceranly enable us o arecae he effcency of he new model. The mlcaon of hs s ha he augmened GARCH models (AGM) s more effcen han GARCH model (GM) and hs acually asser he suerory of he new model. Forecasng exchange rae s radonally mlemened usng GARCH model, he shorcomng of hs model s ha daa analyzed ofen exhb some non-lneary ha hs model canno caured as shown when he BDS was used o analyze he daa. For s nably o caure he non-lnear comonens of he seres, he model was augmened usng B-lnear and hs roduced a beer resul han he classcal GARCH model n erm of her varances. For nsance, he varances of classcal GARCH model for Pound and Pula are and.444 resecvely whle Augmened-GARCH gave for ound and.66 for Pula n ha order. The suerory of hs model les on he varance reducon. The mlcaon of hs resul s ha Augmened-GARCH can be used o forecas foregn exchange n

19 M.O. Aknunde, P.M. Kgos and D.K. Shangodoyn 9 hese wo counres more accuraely and wll gve a desre resul more han classcal GARCH model. Table 5: Varances and relave effcences of GM and AGM SERIES G.M A.G.M R.E. POUND PULA From he fed model we have he followng able on he varance and relave effcences comued and he suerory of AGM over GM s evden. References [] T.G. Andersen and T. Bollerslev, Answerng he skecs: Yes, sandard volaly models do rovde accurae forecass, Inernaonal Economc Revew, 39(4), (998), [] Anderson, Choosng lag lenghs n nonlnear dynamc models, Monash Economercs and Busness Sascs, Workng Paers, /0, Monash Unversy, Dearmen of Economercs and Busness Sascs, (Dec. 00). [3] R. Balle and T. Bollerslev, The message n daly exchange raes: A condonal varance ale, Journal of Busness and Economc Sascs, 7(3), (989), [4] T. Bollerslev, Generalzed auoregressve condonal heeroskedascy, Journal of Economercs, 3, (986), [5] C. Bonlla, R. Romero-Meza and M.J. Hnch, Esodc nonlneares n he Lan Amercan sock marke ndces, Aled Economcs Leers, 3, (006),

20 0 Evaluaon of GARCH model Adequacy n forecasng... [6] C. Brooks, Tesng for non-lneary n daly serlng exchanges raes, Aled Fnancal Economcs, 6, (996), [7] C. Brooks and M. Hnch, Esodc nonsaonary n exchange raes, Aled Economcs Leers, 5, (998), [8] C. Brooks and Melvn J. Hnch, Bcorrelaons and cross- bcorrelaons as non-lneay and ools for exchange raes forecasng, (00). [9] C. Bonlla, R. Romero-Meza and M. Hnch, Esodc nonlneares n he Lan Amercan Sock Marke ndces, Aled Economcs Leers, 3, (005), [0] C. Brun, G. Dullo and G. Koch, Blnear Sysems: An ealng Class of Nearly Lnear Sysem n Theory and Alcaon, IEEE Trans. Auo Conrol, Ac-9, (974), [] R.F. Engle, Auoregressve condonal heeroscedascy wh esmaes of he varance of Uned Kngdom nflaons, Economercal, 50, (98), [] Lew, e. al, The nadequacy of lnear auoregressve models for real exchange raes: emrcal evdence from Asan economes, Aled Economcs, 35, (003), [3] K.P. Lm, M.J. Hnch and V. Lew, Adequacy of GARCH models for ASEAN exchange raes reurn seres, Inernaonal Journal of Busness and Socey, 5, (004), 7-3. [4] K.P. Lm, M.J. Hnch and V. Lew, Sascal nadequacy of GARCH models for Asan sock markes: evdence and mlcaons, Inernaonal Journal of Emergng Marke Fnance, 4, (005), [5] C. Sarca, GARCH (,) as good a model as he Nobel rze accolades would mly?, Economcs, Workng Paer, 0405, Archve Econ WPA, Unversy of Gohenburg, Sweden, (004).