Seasonal asymmetric persistence in volatility: an extension of GARCH models

Transcription

1 Seasonal asymmeric persisence in volailiy: an exension of GARCH models Virginie TERRAZA CREA, universiy of Luxembourg Absrac In his paper, we sudy non-linear dynamics in he CAC 40 sock index. Our empirical resuls, sugges combining seasonaliy, persisence and asymmeric effecs o model he condiional volailiy. We observe ha seasonaliy can have an asymmeric impac on he volailiy. In paricular, we show ha negaive shocks observed on Mondays have a greaer impac on he volailiy han he oher days. Then we consruc a seasonal asymmeric GARCH model. I consiss o add seasonal erms in he variance equaion of a GJR-GARCH (1,1) model. Keywords: non lineariy, condiional volailiy, asymmery, seasonal processes, GJR-GARCH model. 1 Inroducion Mandelbro [19] and Fama [10] boh repored evidence ha large (small) changes in he price are ofen followed by oher large (small) changes. This auocorrelaion of he volailiy of reurns was modeled by Engle [9] wihin he framework of ARCH (Auoregressive condiional Heeroskedasiciy) processes exended o GARCH models (Generalized Auoregressive Condiionally Heeroskedasiciic) by Bollerslev [4]. Differen sudies have revealed ha he ARCH and GARCH processes are unsuiable o ake ino accoun effecs of asymmery ofen noiced on he condiional volailiy of sock reurns. I seems ha he condiional volailiy reacs more a he announcemens of bad news. In

2 paricular, Black [3] observes he exisence of a negaive correlaion beween he curren reurn and he fuure volailiy. Volailiy asymmery may be capured using a GJR GARCH (1,1) model inroduced by Glosen, Jagannahan, Runkel, [14]. In his model he condiional volailiy depend on he sign and on he ampliude of he pas esimaion errors. In a general case, ARCH models explain a par of he lepokuric effec noiced in financial series, bu no a all. During he pas decade, some sudies have shown ha big flucuaions could be inheren o he marke srucure. Numerous researches concerning he microsrucure of he markes have been developed like weekend effecs and oher anomalies. In paricular, he day of he week effec has been sudied in a number of papers: French [13] Hamon and Jacquilla [15]. In hese papers, Monday reurns are found o be negaive while he reurns on Friday ended o be higher han he oher days. No only do he average reurns on Monday end o differ, Bessembinder and Herzel [] show ha reurns on Mondays are posiively correlaed wih Fridays reurns while reurns on Tuesdays are negaively correlaed wih Mondays reurns. Then, hese auhors propose a periodic auoregressive model (PAR) in heir empirical sudies. Addiionally, here is evidence ha he volailiy vary wih he day of he week, see Foser and Viswanahan [11]. To ake ino accoun hese laer empirical observaions, Bollerslev and Ghysels [5] use a periodic GARCH model (PGARCH). Franses and Paap [1] observe posiive auocorrelaion on Monday and day of he week variaion in he persisence of volailiy. Then, hey combine he PAR model for he reurns wih he PGARCH model for he volailiy. In his paper, we observe ha seasonaliy can have an asymmeric impac in he condiional variance equaion. In our empirical sudy, we show ha negaive shocks observed on Mondays have a greaer impac on he volailiy han he oher days. Then we propose an asymmeric seasonal GARCH process o model asymmeric and seasonal effecs joinly. We sudy he seasonal effec boh in he reurns and he volailiy in he case of he CAC 40 sock index series from 1987 o 00. The paper is organized as follows. Firs, we give some saisics for he reurns of he CAC 40. Preliminary resuls are menioned. Then, we presen mehodology and empirical resuls. The paper finalizes wih some conclusions. Daa and Saisical analysis The daa used are he daily index series (CAC 40) of he French Sock Exchange during he period 09/14/ /01/00 (390 observaions). The Phillips Perron (PP) [1] uni roo es shows ha one uni roo exiss in he CAC 40 series (he PP value is , which is greaer han he criical value a 5%). We ake he log difference of he value of he index so as o conver he daa ino coninuously compounded reurns. The PP value for his series is now , which is less han he criical value a 5%. Some summary saisics on he reurns are presened in able 1.

3 Table 1: summary saisics of CAC40 reurns Average Sandard Errors skewness kurosis Jarque bera LB( 30 )* LB ( 30 )* (-9.635)** (58.93)** (5.99)** ** he criical values are compared wih 1.96 ; *The Ljung Box es is compared wih (9) As he able 1 shows, he index has a small posiive average reurn. The daily variance is The skewness coefficien indicaes ha he reurns disribuion is subsanially negaively skewed. Furhermore, he excess of kurosis gives evidence of a srong probabiliy of negaive exreme reurns for he index CAC 40. The conclusion is ha he assumpion of normaliy for he reurns index is rejeced. Auocorrelaion is revealed applying he saisics of Ljung Box [18] calculaed wih 30 lags LB (30) o he reurn and he squared of reurns. This es is a firs indicaion on he presence of a srong heeroscedasiciy and on a linear or nonlinear srucure in he series of index reurns. To comfor his resul, nonlineariy ess are applied using he rouine proposed by Ashley and Paerson [1]. Afer prewhiening he daa, we rouinely boosrap he significance levels, as well as compuing hem based on asympoic heory. We draw 1000 T samples a random from he empirical disribuion of he observed T- sample of daa. The Brock, Decher, Scheinkman (BDS,[6]), McLeod-Li [0], Engle [9] and Tsay [4] ess are implemened in Toolki, a Windows-based compuer program presened in Ashley and Paerson [1]. The hypohesis of non-lineariy is acceped if he hresholds of probabiliy are lower han Resuls of he ess are presened in able. Table : Non lineariy ess on he reurns Tess McLeod-Li ( L=4) Engle ( P=5 ) Tsay ( K=5 ) BDS (M=,3,4) ( =0.5,1,) Boosrap Asympoic All of he ess appear o have high power o deec non-lineariy in he daa. We conclude in favour of non-linear srucures bu we canno specify wha kind of non linear process can be used o model reurns series. Tess of Time Reversibiliy (TR) can complemen he exising ess. In paricular, he TR es of Chen Chou and Kuan [7] (he CCK es) is powerful agains asymmery in volailiy while he BDS es is no. In effec, ime series ha exhibi asymmeric behaviours are ypically ime irreversible. When is

4 ime reversible, i can be shown ha for each k = 1,, he disribuion of - -k is symmeric (abou he origin). If his symmeric condiion fails, here is some asymmeric dependence beween and -k. In view of his propery, non-linear ime series are ime irreversible in general. In able 3 we repor he saisics of he CCK es. We consider = 0.5 and 1 and we ake k = 1,,3,4,5 as he empirical applicaions of he auhors. Table 3: CCK es of he daily reurns TR es (C exp, k) k = 0.5 = * -1.98* -3.89* -4.65* * -3.79* * -3.83* * significance a 5% level The CCK ess are significan in all cases excep for k =5 and bea = 0.5. The resuls indicae ha he daa are ime irreversible and ake a firs indicaion on he poenial asymmery in he reurns series. The applicaion of hese differen ess has permied o show he presence of non-lineariy in he series. However i can be possible ha oher effecs explain he srucure of he reurns like deerminisic evens. Some auhors have shown heir exisence in he mean and volailiy characerisics, and have sudied he effecs of seasonaliy observed in he reurns. To es if a weekend effec exiss in he average reurns of he CAC 40 during our period of observaions, we use he regression beween he index reurns and he days of he week. Table 4 confirms he exisence of a Monday effec for he CAC 40 reurns, and a seasonal effec on Tuesday. Table 4: seasonaliies in CAC 40 reurns Monday Tuesday Wednesday Thursday Friday R ² Reurns ( - saisic) *Significance a 5 % level * (-1.96) * (.1) -.97E-05 (-0.06) (1.14) (0.93) Then, o characerize he mean equaion, we consruc an auoregressive seasonal model. This model consiss o add he seasonal dummies in an auoregressive

5 process. We sugges an AR(3) and an MA(1) processes o ake ino accoun he auocorrelaion in he index reurns. We obain he following equaion: r r D D ,,, iid normal (0, ) (1) wih D 1, and D, being dummies for Monday and Tuesday. Table 5 gives he resuls of he esimaion. Table 5: Seasonal auoregressive model Reurn - saisic LB ( 30 )* *The Ljung Box es is compared wih he value equal Even if he effecs of seasonaliy are no very imporan, (see he R saisic in able 4), his model can be acceped since he hypohesis of auocorrelaion is rejeced by he Ljung Box es applied on he residuals. 3 Mehodology and empirical resuls Lile Work has ever been devoed o linking he weekend effec wih heeroscedasiciy and /or o a seasonal behaviour of marke volailiy. Mos sudies ha consider weekend effec for he reurns assume ha he volailiy does no vary wih he day of he week. As Franses and Paap [1] have suggesed, i seems imporan o ake accoun of boh feaures joinly. This weekend effec on volailiy can be explained by he fac ha here is a concenraion o publish all kinds of bad news on he weekends. The consequence on he marke will be a lower reurn and higher volailiy on Monday. This phenomenon someimes ascribed o a leverage effec is compleely ignored in he GARCH processes, he sign of reurns playing any role on he volailiy. We verify ha he reurns on he index are no symmeric as indicae he negaive values of he cross correlogram beween he squared residuals and he residuals of he model. An addiional sage is o es ha according o he days of he week, he poenial asymmeric responses of volailiy can be differen. For ha, we use regressions defined by :

6 ss 1 1Ds, c w e, e iid normal (0, e ) () S when 0 and 0 where S oherwise and D s, represen he 1 days of he week. s = 1,,...,5. Applying eqn.(), we observe ha he Monday effec has an asymmeric impac on he volailiy since w is negaive, even if his asymmery feaure is only significan a 10%: S 1 1D (3) 1, (3.6) ( 1.90) To model boh he seasonaliy and leverage effec on he volailiy, we propose an asymmeric seasonal GARCH (1,1) model. The condiional volailiy of he index CAC 40 is se as : D w S D (4) 0 1 1, , 1 In comparison wih he GJR GARCH (1,1) model, we add seasonal erms in he variance equaion. The poenial seasonaliy on Tuesday is represened by he coefficiens while w 1 esimae he asymmeric seasonal impac on he condiional variance. The effec of a posiive shock is represened by he coefficien 1 and of a negaive shock by ( 1 +w 1 ). So, in his model he impac of shocks depends on he Monday effec. In able 6 we repor esimaes of he model. Table 6: coefficiens Esimaes for he seasonal asymmeric GARCH model saisic* E w E *significance a 5%

7 Looking a he able 6, we observe ha he coefficiens in he mean equaion are widely significan (a 10% for ). In he variance equaion, he seasonal heeroscedasiciy is significan on Monday and Tuesday. The resuls indicae ha he sign of he innovaion has an influence on he volailiy of reurns. A posiive shock a 1% increases he volailiy a 0.09% while a negaive shock a 1% increase he volailiy a 0.17%. Then he degree of asymmery is equal of The sudy of he sandardized residuals sample saisics of he seasonal asymmeric GARCH model, show significan decrease of kurosis from o , he skewness from o and Jarque Bera [16] from o The Ljung Box [18] es wih sandardized residuals and squared sandardized residuals are employed o verify ha here is no auocorrelaion and no ARCH effecs. As he able 7 shows, our model has aken care of he non-linear dependence and here is no significan auocorrelaion. Table 7: Tess on he sandardized residuals Average Sandard errors skewness kurosis Jarque bera LB(30)* LB (30)* *are compared wih (1) = 3.67 We can confirm hese resuls by able 8, applying on he sandardized residuals, non-linear ess suggesed by Ashley and Paerson [1]. Tess Table 8: McLeod-Li ( L=4) Non-lineariy ess on sandardized residuals Engle ( P=5 ) Tsay ( K=5 ) BDS (m=, = 1) Boosrap Asympoic * Significance a 5% level To evaluae he TR propery of model-sandardized residuals, he CCK es is no direcly applicable. So we use a modified version of he CCK es proposed by Chen [8]. Neverheless, in able 9, we show ha he modified CCK es sill deecs some non-linear dependence no capured by he BDS es.

8 Table 9: The Modified CCK es on he sandardized residuals TR es (C exp, k) k = 0.5 = * -3.85* 3 -.8* -3.3* * * * significance a 5% level For some k, he modified CCK es rejecs he model. However, here is a difference beween able 3 and able 9. The saisics C exp,k derived of he modified CCK es are all smaller han hose for he reurns. So, he model has capured some (bu no a all) ime irreversibiliy in he reurn series. 4 Conclusion The goal of his paper has been o characerize a volailiy model by is abiliy o capure he seasonaliy in boh he condiional mean and he condiional variance equaion. We have shown ha he Monday effec and seasonaliy on Tuesday appear in hese wo equaions. Neverheless, while he seasonaliies are inroduced in an addiive manner in he condiional mean equaion, he Monday effec has an asymmeric impac in he condiional volailiy. To ake ino accoun hese feaures, we propose a seasonal asymmeric GARCH model. This model appears o capure a large par of non- lineariies presen in he variance, even if i seems o neglec oher asymmeries sources. For furher research, i would be ineresing o es he predicion of he model for forecasing he volailiy ou of sample. Furhermore, similar applicaions o larger markes such as hose in Europe will be anoher exension. References [1] Ashley R.A, Paerson D.M (000) : A nonlinear Time Series Workshop, a oolki for deecing and idenifying nonlinear serial dependance, kluwer academic publishers [] Bessembinder H, Herzel MG (1993): Reurn auocorrelaions around nonrading days, review of Financial sudies, 6, pp

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