Estimates and Forecasts of GARCH Model under Misspecified Probability Distributions: A Monte Carlo Simulation Approach

Transcription

1 Journal of Modern Applied Saisical Mehods Volue 3 Issue Aricle 8-04 Esiaes and Forecass of GARCH Model under Misspecified Probabiliy Disribuions: A Mone Carlo Siulaion Approach OlaOluwa S. Yaya Universiy of Ibadan, Ibadan, Nigeria, os.yaya@ui.edu.ng Olusanya E. Olubusoye Universiy of Ibadan, Ibadan, Nigeria, busoye00@yahoo.co Oluwadare O. Ojo Federal Universiy of Technology Akure, Akure, Nigeria, daruu08075@yahoo.co Follow his and addiional works a: hp://digialcoons.wayne.edu/jas Par of he Applied Saisics Coons, Social and Behavioral Sciences Coons, and he Saisical Theory Coons Recoended Ciaion Yaya, OlaOluwa S.; Olubusoye, Olusanya E.; and Ojo, Oluwadare O. (04) "Esiaes and Forecass of GARCH Model under Misspecified Probabiliy Disribuions: A Mone Carlo Siulaion Approach," Journal of Modern Applied Saisical Mehods: Vol. 3 : Iss., Aricle 8. DOI: 0.37/jas/ Available a: hp://digialcoons.wayne.edu/jas/vol3/iss/8 This Regular Aricle is brough o you for free and open access by he Open Access Journals a DigialCoons@WayneSae. I has been acceped for inclusion in Journal of Modern Applied Saisical Mehods by an auhorized edior of DigialCoons@WayneSae.

2 Journal of Modern Applied Saisical Mehods Noveber 04, Vol. 3, No., Copyrigh 04 JMASM, Inc. ISSN Esiaes and Forecass of GARCH Model under Misspecified Probabiliy Disribuions: A Mone Carlo Siulaion Approach OlaOluwa S. Yaya Universiy of Ibadan Ibadan, Nigeria Olusanya E. Olubusoye Universiy of Ibadan Ibadan, Nigeria Oluwadare O. Ojo Federal Universiy of Technology Akure Akure, Nigeria The effec of isspecificaion of correc sapling probabiliy disribuion of Generalized Auoregressive Condiionally Heeroscedasic (GARCH) processes is considered. The hree assued disribuions are he noral, Suden, and generalized error disribuions. The GARCH process is sapled using one of he disribuions and he odel is esiaed based on he hree disribuions in each saple. Paraeer esiaes and forecas perforance are used o judge he esiaed odel for perforance. The AR-GARCH- GED perfored beer on he hree assued disribuions; even, when Suden disribuion is assued, AR-GARCH-Suden does no perfor as he bes odel. Keywords: specificaion Generalized Error Disribuion, forecass, GARCH, isspecificaion, Inroducion Since he inroducion of Generalized Auoregressive Condiional Heeroscedasic (GARCH) odel of Bollerslev (986), housands of aricles have been published applying he odel on financial series. The odel capures volailiy in he arke, and is disribuional specificaion akes i special aong oher nonlinear ie series odels. The GARCH process exiss on he assupion of Noral, Suden, and Generalized Error Disribuions (GED). The Noral disribuion is he usual assupion in any ie series esiaion, bu due o he fac ha he disribuion of GARCH process is lepokuric, Noral disribuion was found o be in appropriae OlaOluwa Yaya and Olusanya Olubusoye are par of he acadeic saff in he Deparen of Saisics, Universiy of Ibadan. Eail he a os.yaya@ui.edu.ng. and oe.olubusoye@ui.edu.ng. O. O. Ojo is a lecurer in he Deparen of Saisics, Federal Universiy of Technology, Akure. Eail hi a daruu08075@yahoo.co. 479

3 ESTIMATES OF GARCH MODEL in capuring he ail behavior of he series. Bollerslev (987) herefore proposed Suden disribuion o capure he long ail behavior of he process. Nelson (99) proposed he GED disribuion. Apar fro he real applicaions of GARCH odels on financial series, here is need o sudy he effec of isspecifying he GARCH disribuional assupions during esiaion. Aricles are very scarce along his line of hough. Wang (00) affirs ha spurious and inefficien inference is expeced when pure GARCH odels are isspecified. This as well ay affec he Quasi Maxiu Likelihood Esiaes (QMLEs) of he isspecified odel. The QMLE of pure GARCH(,) odels indicaes ha he ARCH paraeer is sall, GARCH paraeer is close o uniy and he su of boh paraeers approaches uniy as he sapling frequency increases (Engle and Bollerslev, 986; Bollerslev and Engle, 993; Baillie, Bollerslev and Mikkelsen, 996; Ding and Granger, 996; Andersen and Bollerslev, 997, and Engle and Paon, 00.) This fac is refleced in he Inegraed GARCH (IGARCH) of Engle and Bollerslev (986). A ore recen paper by Jensen and Lange (00) shows ha in a GARCH (,) odel, he esiaes of ˆ and ˆ end o zero and uniy respecively as he sapling frequency increased, which is an IGARCH effec. This IGARCH effec is known for pure-garch processes. In a linear AR-GARCH or nonlinear AR-GARCH processes, IGARCH effec is no plausible. The presen work considers AR-GARCH process, and herefore IGARCH effec ay no be expeced. As ail disribuion of he GARCH odel is capured using he hree disribuions, and paraeers esiaed adjus accordingly, forecass perforances of he odel are affeced. Exensive Mone Carlo siulaion was perfored on he GARCH odel using he hree disribuions. The GARCH (,) odel The GARCH (,) odel proposed in Bollerslev (986) is () w where ε are he reurns series of he financial asse; σ is he volailiy a ie and z gives he assued disribuion. The paraeers, α and β are condiioned as w > 0, α 0, β 0, and α + β < in order o ensure saionariy of he whole process (Bollerslev, 986). This condiion is esablish by defining 480

4 YAYA ET AL. k z where z N(0,). Using his in () resuls in he Auoregressive Moving Average (ARMA) represenaion w k k () where k is serially uncorrelaed wih ean zero. Saionariy of he process is hen ensured when he roos of α() β() = (α + β) = 0 lie ouside he uni circle and his is no condiioned on ie as i is easured direcly fro he paraeers of he odel. Hence i is expeced ha (3) for exisence of covariance saionary process. For he saionary process, he finie uncondiional variance of ε is given by w (4) Kurosis of GARCH (,) odel 3 For any GARCH (p,q) process, E(z) = 0 and Var(z) =. 4 and E z is he skewness E z gives he easure of skewness. Because he ephasis is on ail behaviour of GARCH residuals, he expression for he uncondiional kurosis is nex derived. 4 E exis, hen i suffices o wrie Assuing ha E and E z 4 k 3; because E 0; z E E E z because z and and z are independen 48

5 ESTIMATES OF GARCH MODEL Then, squaring GARCH (,) odel, w, gives w w w. 4 4 Taking expecaion of he resuling expansion, as well as applying he properies oulined above E w k z 4 4 Using he relaion E kz 3 E E 4 where kz is he excess kurosis of z, hen w kz 3 k z 4 E Using he forula k 3 for excess kurosis and wih he fac ha E w E fro he properies above, k w w kz 3 k z k 3 z kz 3 3 (5) 48

6 YAYA ET AL. wih norally disribued innovaions z, kz = 0 and k (6) 6 wih non-norally disribued innovaions z, as in Suden and GED, Var z E z and k z 4 E z 4 E z 0, hen E z k z z 6 z k k k kz (7) In hese wo cases, i is observed ha is iporan in deerining he ail behavior of, because once 0, k 0. Hence, k kz for he non-norally disribued case and i iplies he siilariy of he ail behaviors of boh and z Disribuional Assupions and Esiaion For GARCH odels, he uncondiional disribuions are always non noral, and his gives faer ails han he noral disribuion. In pracice, z is assued o follow he noral disribuion or non-noral disribuions. These non-noral disribuions have been proved o perfor well in odeling he faer ails (lepokuriciy) observed in GARCH residuals. The non-noral disribuions are he Suden disribuion proposed in Bollerslev (987) and Generalized Error Disribuion (GED) by Nelson (99) The sandardized Noral disribuion is f z exp, z z (8) wih he log likelihood funcion N Lz N log z (9) 483

7 ESTIMATES OF GARCH MODEL where N is he saple size. The sandardized Suden disribuion proposed in Bollerslev (987) is given as v / z f z, v v / v / v, z v (0) This disribuion is syeric around zero as i is observed in is specificaion wih v >. A v =, he Suden reduces o Cauchy disribuion. A < v 4, is condiional kurosis is less han 3, which eans ha he resuling ail effec is noral. For v > 4, he kurosis becoes 3(v )(v 4), which is greaer han 3, hence he ail effec becoes non-noral disribuion. As v, he disribuion converges o noral disribuion. The log likelihood funcion of Suden disribuion is hen siplified as v v/ N z Lz, v N log v log v / The sandardized GED proposed in Nelson (99) is given as () v / 3/ 3 v 3 f z, v v exp z v v v v v/ () where < z < and v > 0. The GED reduces o he sandard noral disribuion a v = 4. A 0 < v <, he disribuion has hicker ail han he noral disribuion, for exaple, a v = he disribuion becoes a double exponenial (Laplace) disribuion. A v >, he disribuion of z has hinner ails han he noral disribuion, for exaple, as v ends o infiniy, z reduces o a unifor disribuion on he inerval 3, 3. The log likelihood of his disribuion is hen expanded as / 3/ 3 v 3 f z, v v exp z v v v v v/ (3) 484

8 YAYA ET AL. These likelihood funcions are hen esiaed using he nuerical derivaives based on he fac ha GARCH odels lack closed for esiaion. Bernd, Hall, Hall and Hausan (BHHH) algorih of Bernd, e al (974) is hen used. This algorih is ered Gauss-Newon in general Nonlinear Leas Squares (NLS) and BHHH in MLE esiaion. Unlike soe oher derivaives, i uses only firs derivaives of he likelihood funcion and copues a se of paraeer values as i i i N L z,. L z,. L z,. i i N. ' (4) where L(z,.) is he likelihood funcion. The iniial paraeer se is given as ψ (0) and he paraeer se which axiize he likelihood funcion is denoed as ψ (i+). The esiaion of GARCH (, ) odel wih Suden disribuion and GED follow he usual Quasi Maxiu Likelihood Esiaion (QMLE) because noraliy assupion is violaed in hese cases. Misspecificaion of disribuion of GARCH odel could lead o saionariy and explosion of he series in soe poins. Though sandard errors will be i consisen; he QML esiaors are generally closed o he exac ML esiaor ˆ i for syeric GARCH disribuion. For non-syeric condiional disribuions, boh he asypoic and finie saple loss in efficiency are quie large and paraeric esiaion approach are no applicable in his regard (Mills and Markellos, 008). Forecass Evaluaion Forecas evaluaion crieria considered are he Roo Mean Squares Forecas Error (RMSFE), Mean Absolue Error (MAE), Mean Absolue Percenage Forecas Error (MAPFE) and Theil Inequaliy of Theil (96;966). The MSFE is defined as where MSFE ˆ (5) ˆ is he prediced in-saple condiional variances, and his depends on he scale of he variance series,. The square roo of MSFE is he RMSFE 485

9 ESTIMATES OF GARCH MODEL RMSFE ˆ (6) The MAFE and MAPFE are obained by aking he absolue differences of he prediced condiional volailiies and he observed volailiies as MAFE ˆ (7) ˆ MAPFE 00 (8) The Theil inequaliy is given as TI ˆ ˆ (9) The inequaliy coefficien is ie invarian and always lies beween 0 and uniy. The saller hese forecas evaluaion crieria, he beer he candidae odel represen well he daa. Mone Carlo Siulaions The Mone Carlo experien is se up using he AR() GARCH(,) DGP y y, 0.60, (0) wih he error disribuion ε = σz where z is assued o follow Noral, Suden and GED disribuions. The paraeers of he AR() and GARCH(,) odels are se wihin he saionary region in order o avoid probles daa explosion. The saple sizes N are varied as 000, 4000 and 6000 wih in-saple forecass generaed as 5% of he daa lengh. The resuls are hen presened as Scenarios 486

10 YAYA ET AL. o 3 in Tables 6 below. Each Scenario gives resuls for paraeer esiaion, volailiy, excess kurosis and forecass evaluaions crieria. Scenario : When he rue Disribuion is Noral Tables and presen he resuls when he GARCH processes are siulaed based on Noral disribuion assupion, and hese processes are used o esiae he GARCH process based on Suden, GED and he sae Noral disribuion. The resuls in Table show ha he AR-GARCH paraeer esiaes, easures of volailiy and kurosis are no consisen wih saple sizes. Boh he AR and GARCH paraeer esiaes copued for Suden disribuion have larger biases in copared wih ha of Noral and GED disribuions, even hough excess kurosis of he AR-GARCH-Suden odel is he salles. Volailiy of he AR- GARCH-Suden odel is also observed o be higher han ha of he Noral and GED disribuions. The excess kurosis of he AR-GARCH-Noral odel was expeced o be he salles because he series is sapled fro Noral disribuion bu his was no he case. Looking a he resuls of he in-saple forecass realized fro he AR- GARCH odels as given in Table, he AR-GARCH-Noral and AR-GARCH- GED odel perfor beer han AR-GARCH-Suden odel on forecass as given by he iniu values of he RMSPE and Theil inequaliy coefficiens. The AR- GARCH-GED is expeced o realize beer forecass han AR-GARCH-Noral odel. Table. Model Paraeer, Volailiy and Kurosis when GARCH processes are siulaed based on Noral disribuion assupion Assued Disribuion Noral ˆ Saple 0 (0.500) ˆ (0.5000) ŵ (0.000) ˆ (0.500) ˆ (0.6000) Persisence (0.8500) Volailiy Exc. Kurosis Suden GED

11 ESTIMATES OF GARCH MODEL Table. Forecas evaluaion esiaes when GARCH processes are siulaed based on Noral disribuion assupion Assued Disribuion Noral Saple RMSPE MAFE MAPFE Theil Suden GED Scenario : When he rue Disribuion is Suden Tables 3 and 4 presen he resuls when he rue GARCH disribuion follows Suden. Here, he disincions in he GARCH esiaes can only be ade using he persisence and uncondiional volailiy easures. The AR-GARCH-Suden odel sill presens salles persisence and highes volailiy. The excess kurosis of he AR-GARCH-Suden odel is he salles followed by ha of AR- GARCH-Noral odel. Table 3. Model Paraeer, Volailiy and Kurosis when he rue GARCH disribuion follows Suden Assued Disribuion Noral ˆ Saple 0 (0.500) ˆ (0.5000) ŵ (0.000) ˆ (0.500) ˆ (0.6000) Persisence (0.8500) Volailiy Exc. Kurosis Suden GED

12 YAYA ET AL. Table 4. Forecas evaluaion esiaes when he rue GARCH disribuion follows Suden Assued Disribuion Noral Saple RMSPE MAFE MAPFE Theil Suden GED In ers of forecass, he AR-GARCH-Suden odel is he wors, even hough he DGP is realized fro he sae probabiliy disribuion. The forecas perforances of AR-GARCH-Noral and AR-GARCH-GED see no differen fro each oher as indicaed by he forecas evaluaion esiaes. Scenario 3: When he rue Disribuion is GED Table 5 and 6 presen he resuls when he rue GARCH disribuion is GED. In Table 5, in he AR() esiaes, he esiaes for he consan ˆ 0 are all consisen wih saple sizes when he hree probabiliy disribuions are assued. The auoregressive paraeers ˆ are no consisen wih saple sizes. The GARCH paraeer esiaes copued for Suden disribuion are he sae o ha of Table and 3 while he AR() paraeer are differen in he wo resuls. The Suden disribuion assupion of GARCH odel sill presens odel esiaes wih highes volailiy bu wih lowes persisence of his volailiy. Misspecifying GED for Suden disribuion here also caused he excess kurosis o be negaive in AR-GARCH-Suden odel and his is a very spurious case. The forecas evaluaion resuls of he odel esiaes follow in In Table 6. Saring wih he AR() GARCH(,) Suden odel, he odel is he wors in ers of forecass because i presens he highes RMSPE, MAPE, MAPFE and Theil inequaliy coefficien. The bes odel is AR() GARCH(,) GED odel, and his is expeced because he DGP assued GED iniially. The perforance of 489

13 ESTIMATES OF GARCH MODEL AR() GARCH(,) Noral in ers of forecas is very close o ha of AR() GARCH(,) GED odel. Table 5. Model Paraeer, Volailiy and Kurosis when he rue GARCH disribuion is GED Assued Disribuion Noral ˆ Saple 0 (0.500) ˆ (0.5000) ŵ (0.000) ˆ (0.500) ˆ (0.6000) Persisence (0.8500) Volailiy Exc. Kurosis Suden GED Table 6. Forecas evaluaion esiaes when he rue GARCH disribuion is GED Assued Disribuion Noral Saple RMSPE MAFE MAPFE Theil Suden GED

14 YAYA ET AL. Conclusion The isspecificaion of GARCH probabiliy disribuion funcions were considered. These are he Noral, Suden and Generalized Error Disribuions (GED). The esiaion convergence ie varied based on he disribuion and he se saple sizes. When a Noral disribuion was assued, he AR GARCH GED seeed o perfor arginally beer han AR GARCH Noral odel in ers of forecass as revealed in he esiaes of he Theil inequaliy. Though, he AR GARCH Noral was he bes odel here in ers of paraeer esiaes, and his was expeced because he DGP assued Noral disribuion iniially. Wih he assupion of Suden disribuion in he DGP, he forecas perforance of he odels copued wih Noral disribuion and GED reduced and hese sill presened beer odels han he corresponding AR GARCH Suden odel. Siilar resuls were obained when he DGP assued GED. I was also observed ha all he resuls obained, paricularly he paraeer esiaes were no consisen wih saple sizes. These are expeced because volailiy cae ino play. In epirical odeling research like his, ineres should eiher lie in he behavior of he volailiy assuing a probabiliy disribuion which will give us he bes volailiy easureen or in he forecass. The bes GARCH odel ay no acually produce he bes forecas esiaes and probabiliy disribuions have effec on he ail disribuion of he innovaions. This work can be replicaed using higher order of he odel, and in ha case, ore sophisicaed sofware is recoended for he siulaion in order o avoid convergence probles. References Andersen, T. G., & Bollerslev, T. (997). Inraday periodiciy and volailiy persisence in financial arkes. Journal of Epirical Finance, 4, Baillie, R.T., Bollerslev, T., & Mikkelsen, H. O. (996). Fracional inegraed generalized auoregressive condiional heeroscedasiciy, Journal of Econoerics, 74, Bernd, E., Hall, B., Hall, R., & Hausan, J. (974). Esiaion and inference in nonlinear srucural odels. Annals of Econoic and Social Measureen, 3, Bollerslev, T. (986). Generalized Auoregressive Condiional Heeroscedasiciy. Journal of Econoerics, 3,

15 ESTIMATES OF GARCH MODEL Bollerslev, T. (987). A condiional heeroscedasic ie series odel for speculaive prices and raes of reurn. Review of Econoics and Saisics, 69, Bollerslev, T., & Engle, R. F. (993). Coon persisence in condiional variances. Econoerica, 6, Ding, Z., & Granger, C. W. J. (996), Modelling volailiy persisence of speculaive reurns: a new approach. Journal of Econoerics, 73, Engle, R. F., & Bollerslev, T. (986). Modeling he persisence of condiional variances. Econoeric Reviews, 5, 50. Engle, R. F., & Paon, A. J. (00). Wha good is a volailiy odel? Quaniaive Finance,, Jensen, A. T., & Lange, T. (00). On convergence of he QMLE for isspecified GARCH odels. Journal of Tie Series Econoerics, (), -3. Mills, T. C., & Markellos, R. (008). The Econoeric Modelling of Financial Tie Series (3rd ed.). Cabridge, UK: Cabridge Universiy Press. Nelson, D. B. (99). Condiional heeroscedasiciy in asse reurns: A new approach. Econoerica, 59, Theil, H. (96). Econoic forecass and policy. Aserda: Norh- Holland. Theil, H. (966). Applied Econoic forecasing. Aserda: Norh-Holland Wang, Y. (00), Asypoic nonequivalence of GARCH odels and diffusions. Annals of Saisics, 30(3),