ESTIMATING STOCK MARKET VOLATILITY USING ASYMMETRIC GARCH MODELS. Dima Alberg, Haim Shalit and Rami Yosef. Discussion Paper No

Transcription

1 ESTIMATING STOCK MARKET VOLATILITY USING ASYMMETRIC GARCH MODELS Dima Alberg, Haim Shali and Rami Yosef Discussion Paper No Sepember 006 Monaser Cener for Economic Research Ben-Gurion Universiy of he Negev P.O. Box 653 Beer Sheva, Israel Fax: Tel:

2 Sepember, 006 Esimaing Sock Marke Volailiy Using Asymmeric GARCH Models. DIMA ALBERG Deparmen of Economics, Ben-Gurion Universiy of he Negev, Beer Sheva, 8405 Israel HAIM SHALIT Deparmen of Economics, Ben-Gurion Universiy of he Negev, Beer Sheva, 8405 Israel RAMI YOSEF Deparmen of Business Adminisraion, Ben-Gurion Universiy of he Negev, Beer Sheva, 8405 Israel

3 Absrac A comprehensive empirical analysis of he reurn and condiional variance of Tel Aviv Sock Exchange (TASE) indices is performed using GARCH models. The predicion performance of hese condiional changing variance models is compared o newer asymmeric GJR and APARCH models. We also quanify he day-of-he-week effec and he leverage effec and es for asymmeric volailiy. Our resuls show ha he EGARCH model using a skewed Suden- disribuion is he mos successful in forecasing he TASE indices. Keywords: GARCH, Leverage Effec, Day-of- Week Effec, Marke Volailiy.

4 Esimaing Sock Marke Volailiy Using Asymmeric Changing Variance. Inroducion Models Volailiy clusering and lepokurosis are common observaions in financial ime series (Mandelbro (963)). Anoher phenomenon ofen encounered is he socalled leverage effec (Black (976)), which occurs when sock prices changes are negaively correlaed wih changes in volailiy. Observaions of his ype in financial ime series have led o he use of various changing variance models. In his seminal paper, Engle (98) proposed o model ime-varying condiional variance wih Auo-Regressive Condiional Heeroskedasiciy (ARCH) processes using lagged disurbances. Early empirical evidence shows ha a high ARCH order is needed o capure he dynamic behavior of condiional variance. The Generalized ARCH (GARCH) model of Bollerslev (986) fulfills his requiremen as i is based on an infinie ARCH specificaion which reduces he number of esimaed parameers from infiniy o wo. Boh models capure volailiy clusering and lepokurosis, bu as heir disribuion is symmeric, hey fail o model he leverage effec. To address his problem, many nonlinear exensions of GARCH have been proposed, such as he Exponenial GARCH (EGARCH) model by Nelson (99), he so-called GJR model by Glosen, Jagannahan, and Runkle (993) and he Asymmeric Power ARCH (APARCH) model by Ding, Granger, and Engle (993). Anoher problem encounered when using GARCH models is ha hey do no always fully embrace he hick ails propery of high frequency financial imesseries. To overcome his drawback Bollerslev (987), Baillie and Bollerslev (989), Kaiser (996) and Beine, Lauren, and Lecour (000) have used he Suden's - disribuion. Similarly o capure skewness Liu and Brorsen (995) use an asymmeric sable densiy. Bu he variance of such a disribuion rarely exiss. For modelling boh skewness and kurosis Fernandez and Seel (998) used he skewed Suden's -disribuion ha was laer exended o he GARCH framework by Lamber and Lauren (000, 00). Forecasing condiional variance wih asymmeric GARCH models has been comprehensively sudied by Pagan and Schwer (990), Brailsford and Faff (996), Franses, Neele, and Van Dijk (998) and Loudon, Wa, and Yadav (000). A 3

5 comparison of normal densiy wih non-normal ones was made by Hsieh (989), Baillie and Bollerslev (989), Peers (000), Lamber and Lauren (00). The purpose of his paper is o characerize a volailiy model by is abiliy o forecas and capure commonly held sylized facs abou condiional volailiy, such as persisence of volailiy, mean revering behavior, and asymmeric impacs of negaive versus posiive reurn innovaions. We invesigae he forecasing performance of GARCH, EGARCH, GJR and APARCH models ogeher wih he differen densiy funcions: normal disribuion, Suden's -disribuion, and asymmeric Suden's - disribuion. We also compare beween symmeric and asymmeric disribuions using he hree differen densiy funcions. We forecas wo major Tel-Aviv Sock Exchange (TASE) indices: TA00 and TA5. To compare he resuls, we use several sandard performance measuremens. Our resuls sugges ha one can improve overall esimaion by using he asymmeric GARCH model wih fa-ailed densiies for measuring condiional variance. Moreover, we find ha he asymmeric EGARCH model is a beer predicor han he asymmeric GARCH, GJR and APARCH models. The paper is srucured as follows. Secion presens he daa. In Secion 3, we presen he mehodology and he GARCH models used in he paper. In Secion 4, we describe he esimaion procedures and presen he forecasing resuls. 4

6 . Daa The daa consis of 3058 daily observaions of he TA5 index from 0/0/99 o 3/5/005 and 9 daily observaions of he TA00 index from 0/07/997 o3/5/005. To esimae and forecas hese indices, we creaed many calibraing programs 3 and use G@RCH.0 by Lauren and Peers (00), a package whose purpose is o esimae and forecas GARCH models and many of is exensions. The code wrien by Doornik (999) in he Ox programming language provides a dialog-oriened inerface wih feaures ha are no available in sandard economeric sofware. Parameers were esimaed using he QML echnique by Bollerslev and Wooldridge (99). The opimizaion algorihm used is he Broyden-Flecher- Goldfarb-Shanno (BFGS) quasi-newon mehod. The TA5 Index is a value-weighed index of 5 socks raded on he Tel Aviv Sock Exchange (TASE). The TA00 Index is a value- weighed index of 00 socks raded on he TASE. 3 Coded wih Visual Basic Applicaions 5

7 3. Mehodology Early empirical evidence has shown ha o capure condiional variance dynamics one needs o selec a high ARCH order. The Bollerslev (986) Generalized ARCH (GARCH) model, which is based on infinie ARCH specificaions, allows us o reduce he number of esimaed parameers by imposing non-linear resricions. The GARCH (p, q) model expresses he variance as: σ = w+ q i= α ε i i + p j= β σ j j Using he lag operaor L, he variance becomes: ( L) ε β( L) σ σ = w+ α + q i j wih α ( L) = α i L and β ( ) = β L j. i= p L j= If all he roos of he polynomial β ( L) = 0 lie ouside he uni circle, we have: σ = w [ β( L) ] + α( L) [ β( L) ] ε. This may be envisaged as an ARCH ( ) process since he condiional variance depends linearly on all previous squared residuals. As such, he condiional variance of y 4 can become larger han he uncondiional variance. Then, if pas realizaions of ε are larger han σ i is given by: σ E ( ε ) w = q p β α i i= j= Like ARCH, some resricions are needed o ensure ha σ is posiive for all. Bollerslev (986) shows ha imposing w> 0, α i 0 ( for i=, K, q) and β 0 ( for j =, K, p) is sufficien for he condiional variance o be posiive. j To capure he asymmery observed in he daa, a new class of ARCH models was inroduced: he GJR, he exponenial GARCH, and he EGARCH (p,q), models : ln q p ( σ ) = a0 + ( ai z i + γ i z i) + b j ln( σ j), i= i = j= j where z ε σ i i y = E Ω + ε, where 4 Mean equaion: ( y ) Ω is he informaion se a ime - 6

8 The parameers allow us o capure he asymmeric effecs. For example, if γ = 0 a posiive surpriseε > 0 has he same effec on volailiy han a negaive surpriseε < 0. The presence of a leverage effec can be invesigaed by esing he hypohesis haγ < 0. Engle's (98) ARCH model uses he normal disribuion of normalized residuals z. Bollerslev (987), on he oher hand, proposed a sandardized Suden's - disribuion wih v > degrees of freedom whose densiy is given by: where Γ ( v) = 0 e x x v D ( z ; v) dx = Γ Γ( ( v+ ) / ) ( v / ) π( v ) z + v ( v+ ) is he gamma funcion and v is he parameer measuring he ail hickness. The Suden's -disribuion is symmeric around mean zero. For v > 4, he condiional kurosis equals 3(v )/(v 4), which exceeds he normal value of 3. The common mehodology for esimaing ARCH is by maximum likelihood assuming i.i.d. innovaions. For ( z ; v) Suden's -disribuion is given by: D, he log-likelihood funcion of { ( θ) }, y for he T v+ v z LT( { y} ; θ) = TlnΓ ln lnπ( v ) = v where θ is he vecor of parameers o be esimaed for he condiional mean, he condiional variance and, he densiy funcion. When ( ) ln( σ ) + ( + ) ln+ v, v we have a normal disribuion, so ha he lower v is, he faer are he ails. Recenly, Lamber and Lauren (000, 00) exended he skewed Suden's -disribuion proposed by Fernandez and Seel (998) o he GARCH framework. Using D( z ; v), he loglikelihood funcion of { ( θ) } L T y for he skewed Suden's -disribuion is given by: ({ y }; θ) = T lnγ ln ln( π( v ) ) + ln + ln( s) T = ln v+ v sz + m I ( σ ) + ( + v) ln+ ξ, v ξ + ξ 7

9 8 where ξ is he asymmery parameer, v he degree of freedom of he disribuion and,, m s and v v v m s m z if s m z if I + = Γ + Γ = < = ξ ξ ξ ξ π (See Lamber and Lauren (00) for more deails.) Maximum likelihood esimaes of parameers are usually obained using he BFGS numerical maximizaion procedure. In our work insead, we use he quasimaximum likelihood esimaor (QMLE). According o Bollerslev and Wooldridge (99) his esimaor is generally consisen, has a normal limiing disribuion, and provides asympoic sandard errors ha are valid under non-normaliy.

10 4. Esimaion Resuls 4. Descripive Saisics and he Saionariy Consrain To obain a saionary series, we use reurns r ( log( P) log( P )) = 00 where P is he closing value of he index a dae. The samples for TA5 and TA00 have means of and 0.045; sandard deviaions of.4895 and.398; skewness of 0.06 and ; and kurosis of and The sample kurosis is greaer han 3, meaning ha reurn disribuions have excess kurosis for boh indices. Excess skewness is also observed, leading o high Jarque-Bera saisics indicaing non-normaliy. Table : Descripive Saisics for Logarihm Differences [ ln( P ) ln( P )] 00 Index Average Min. Max. Sd. Dev. Kurosis Skewness Jarque Bera Sa. TA TA As daily sock reurns may be correlaed wih he day-of-he-week effec, we avoid he poenial calendar effec on he volailiy analysis by filering he daily means and variances using he following wo regressions: () r = α SUN + α MON + α TUE + α WED + α THU + δ, ˆ () ( r r) = β SUN + β MON + β TUE + β WED + β THU + ε, where SUN, MON, TUE,WED and THU are he dummy variables for Sunday, Monday, Tuesday, Wednesday and Thursday; and rˆ is he ordinary leas squares (OLS) fied value of r from regression () a dae. Table : Regression Coefficiens for Day-of-he-Week Effec TA5 Regression Sunday Monday Tuesday Wednesday Thursday Mean () 0.** S.E Variance () 3.4**.45 **.9**.9**.** S.E

11 Table 3: Regression Coefficiens for Day-of-he-Week Effec TA00 Regression Sunday Monday Tuesday Wednesday Thursday Mean () 0.78** S.E Variance ().73** 0.878**.658**.757**.66** S.E The OLS esimaes of he wo regressions on Tables and 3 show ha he TA5 and TA00 indices have significanly posiive daily means on Sunday and significan daily variaions for Sunday hrough Thursday 5. y To eliminae he daily effecs, we sandardize he daily reurns using ( r rˆ ) / ηˆ =, where ηˆ is he fied value of ( ) r rˆ from regression () a dae. We now subsiue he original daily reurn r wih y and referred o i as he daily reurn a dae. Table 4: Descripive Saisics for "Sandardized" Reurns y Index Average Min. Max. Sd.Dev. Kurosis Skewness Jarque Bera Sa. TA TA Reurns y are hus normalized o zero mean and uni variance. The sample skewness and kurosis of y s are and 3.58; and for he wo indices. 4. Choosing a Volailiy Model For he TA5 index, convergence could no be reached wih he EGARCH model and a Suden's -disribuion. Therefore we urn o he oher hree models where all asymmeric coefficiens are significan a sandard levels. Moreover, he Akaike informaion crieria (AIC) and he log-likelihood values indicae ha he EGARCH, APARCH or GJR models beer esimae he series han radiional GARCH. These models are esimaed by he approximae quasi-maximum likelihood esimaor assuming normal, Suden- or skewed Suden- errors. Noe ha i is quie 5 * and ** - means significance a 5% and % levels, respecively. 0

12 eviden ha he recursive evaluaion of maximum likelihood is condiional on unobserved values and herefore he esimaion canno be considered o be perfecly exac. To solve he problem of unobserved values, we se hese quaniies o heir uncondiional expeced values. When we analyzed he densiies we found ha he wo Suden's -disribuions (symmeric and skewed) clearly ouperform he normal disribuion. Indeed, he loglikelihood funcion increases when using he skewed Suden's -disribuion, leading o AIC crieria of.70 and.730 for he normal densiy versus.665 and.697 for he non normal densiies, for he TA 00 and he TA 5 respecively. Table 5: Comparison beween he Models for he TA00 6 TA00 Normal Suden's Skewed APARCH APARCH EGARCH Q(0) Q (0) P(50) Prob[] Prob[] AIC Log-Lik The skewed Suden's -disribuion shows resuls ha are superior o he symmeric Suden- disribuion when modeling he TA 5 and TA00. A possible explanaion for his resul is ha, if skewness is significan in boh series, is magniude will be inferior in boh indices. I may herefore be necessary o add wo asymmeric parameers (asymmeric GARCH + asymmeric disribuion). 6 In Tables 5 and 6 Q(0) and Q (0) are respecively he Box-Pierce saisics a lag 0 of he sandardized and squared sandardized residuals. P(50) is he Pearson goodness-of-fi wih 50 cells. AIC is he Akaike informaion crierion. Log-Lik is he log-likelihood value.

13 Table 6: Comparison beween he Models for he TA5 TA5 Normal Suden's Skewed GJR APARCH GJR APARCH EGARCH Q(0) Q (0) P(50) Prob[] Prob[] AIC Log-Lik All he models describing he dynamics of he firs wo momens of he series are shown by Box-Pierce saisics for residuals and squared residuals. All are nonsignifican a he 5% level. The saionary consrains are observed for every model and for every densiy. The values (ranging from 0.83 o 0.98) sugges long persisence of he volailiy for he indices.

14 4.3 Forecasing The forecasing abiliy of GARCH models has been comprehensively discussed by Poon and Granger (00). However Andersen and Bollerslev (997) poined ou ha he squared daily reurns may no be he proper measure o assess he forecasing performance of he differen GARCH models for condiional variance. Thus, we consider he following five measures o assess forecasing abiliy:. Mean squared error (MSE):. Median squared error (MedSE): S + MSE = h+ h = S ( ) ˆ σ σ ( e )), where e = ( ˆ σ σ ) and [ S S h] MedSE = Inv( f, 3. Mean absolue error (MAE): MAE = h+ Med + S + h = S ˆ σ σ 4. Adjused mean absolue percenage error (AMAPE): S + h AMAPE ˆ σ σ = h+ = ˆ σ + σ S, where h is he number of lead seps, S is he sample size, variance and σ is he acual variance. 5. Theil's inequaliy coefficien (TIC) 7 : ˆ σ is he forecased TIC = h+ h+ S+ h = S S+ h = S ( yˆ y ) h+ S+ h ( yˆ ) ( y ) = S The forecasing abiliy is repored by ranking he differen models wih respec o he five measures. This is done in ables 7 and 8 where we compare he disribuions for he TA 5 and TA 00 indexes. 7 The Theil inequaliy coefficien is a scale invarian measure ha lies always beween zero and one, where zero indicaes a perfec fi. 3

15 Table 7: Forecasing Analysis for he TA5 Index: Comparing beween Densiies TA5 GARCH EGARCH GJR APARCH Suden- Skewed- Skewed- Suden- Skewed- Skewed- MSE() MSE() MedSE() MedSE() MAE() MAE() RMSE() RMSE() AMAPE() TIC() TIC() ()- Mean Equaion, ()-Variance Equaion For he TA 5 index shown on Table 7 he resuls suppor he use of he asymmeric EGARCH model. For mos measures in he variance equaion, he EGARCH model ouperforms he APARCH model. The GARCH model provides much less saisfacory resuls and he GJR model provides he poores forecass. For he TA 00 index shown on Table 8, he EGARCH model gives beer forecass han he GARCH model while he APARCH and GJR models give he poores forecass. The skewed Suden- disribuion is he mos successful in forecasing he TA00 condiional variance, conrary o he TA5 where he resuls conflic. Therefore we were unable o draw a general conclusion. The skewed Suden- disribuion seems o be he bes for forecasing series showing higher skewness. In fac Lamber and Lauren (00) found ha he skewed Suden- densiy is more appropriae for modeling he NASDAQ index han symmeric densiies. 4

16 Table 8: Forecasing Analysis for he TA00 Index: Comparing beween Densiies TA00 GARCH EGARCH GJR APARCH Skewed- Skewed- Suden- Suden- MSE() MSE() MedSE() MedSE() MAE() MAE() RMSE() RMSE() AMAPE() TIC() TIC() ()- Mean Equaion, ()-Variance Equaion 5. Conclusion We compared he forecasing performance of several GARCH models using differen disribuions for wo Tel Aviv sock index reurns. We found ha he EGARCH skewed Suden- model is he mos promising for characerizing he dynamic behavior of hese reurns as i reflecs heir underlying process in erms of serial correlaion, asymmeric volailiy clusering, and lepokuric innovaion. The resuls also show ha asymmeric GARCH models improve he forecasing performance. Among he esed models, he EGARCH skewed Suden- model ouperformed GARGH, GJR and APARCH models. This resul furher implies ha he EGARCH model migh be more useful han he oher hree models when applying risk managemen sraegies for Tel Aviv sock index reurns. 5

17 References Baillie, R.T. and T. Bollerslev (989): Common Sochasic Trends in a Sysem of Exchange Raes, Journal of Moneary Economics, 44, Black, F. (976): Sudies of Sock Marke Volailiy Changes, Proceedings of he American Saisical Associaion, Business and Economic Saisics Secion, Bollerslev, T. (986): Generalized Auoregressive Condiional Heeroskedasiciy, Journal of Economerics, 3, Bollerslev, T. (987): A Condiionally Heeroskedasic Time Series Model for Speculaive Prices and Raes of Reurn, Review of Economics and Saisics, 69, Bollerslev, T., and J. Wooldridge (99): Quasi-Maximum Likelihood Esimaion Inference in Dynamic Models wih Time-varying Covariance, Economeric Theory,, Brailsford, T., and R. Faff (996): An Evaluaion of Volailiy Forecasing Techniques, Journal of Banking and Finance, 0, Ding, Z., Granger and R. Engle (993): A Long Memory Propery of Sock Reurns and a New Model, Journal of Empirical Finance,, Doornik, J. A. (999): An Objec Oriened Marix Programming Language Timberlake Consulan Ld., 3rd ed. Engle, R. (98): Auoregressive Condiional Heeroskedasiciy wih Esimaes of he Variance of Unied Kingdom Inflaion, Economerica, 50, Engle, R., and T. Bollerslev (986): Modeling he Persisence of Condiional Variances, Economeric Reviews, 5, 50. 6

18 Fernandez, C., and M. Seel (998): On Bayesian Modeling of Fa Tails and Skewness, Journal of he American Saisical Associaion, 93, Glosen, L., R. Jagannahan, and D. Runkle (993): On he Relaion beween Expeced Reurn on Socks, Journal of Finance, 48, Jarque, C., and A. Bera (987): A Tes for Normaliy of Observaions and Regression Residuals, Inernaional Saisical Review, 55, Lauren, S., and J.-P. Peers (00): G@RCH.0 : An Ox Package for Esimaing and Forecasing Various ARCH Models, Proceedings 8h Forecasing Financial Markes, London, May 00. Lamber, P., and S. Lauren (000): Modelling Skewness Dynamics in Series of Financial Daa, Discussion Paper, Insiu de Saisique, Louvain-la-Neuve. Lamber, P., and S. Lauren (00): Modelling Financial Time Series Using GARCH-Type Models and a Skewed Suden Densiy, Mimeo, Universié de Liège. Mandelbro, B. (963): The Variaion of Cerain Speculaive Prices, Journal of Business, 36, Nelson, D. (99): Condiional Heeroskedasiciy in Asse Reurns: a New Approach, Economerica, 59, Palm, (996): GARCH Models of Volailiy, Handbook of Saisics, Vol.4, Pagan, A. and Schwer, (990): Alernaive Models for Condiional Sock Volailiy, Journal of Economerics, 45, Peers, J. (000): Developmen of a Package in he Ox Environmen for GARCH Models and Two Empirical Applicaions, Mimeo, Universié de Liège. 7