Forecasting Volatility in Tehran Stock Market with GARCH Models

Transcription

1 J. Basic. Appl. Sci. Res., (1) , 01 01, TexRoad Publicaion ISSN Journal of Basic and Applied Scienific Researc Forecasing Volailiy in Teran Sock Marke wi GARCH Models Medi Parvares 1,*,Moreza Bavagar 1, Deparmen of Accouning, Roodan Branc, Islamic Azad Universiy, Roodan, Iran ABSTRACT Te aim of is paper is Forecasing Volailiy in Teran Sock Marke wi GARCH Models. Te daa consis of 1884 daily observaions of e closing value of e Teran sock marke from 9/8/1997 o 8/0/010. Te sample is divided in wo pars. Te firs 800 observaions (from 9/8/1997 o 8/0/009) are used as e in-sample for esimaion purposes, wile e remaining 315 observaions (from 8/1/009 o 8/0/010) are aken as e ou-of-sample for forecas evaluaion purposes. Te esimaion and es resuls for models sugges a e leverage effec erm,, is significan in EGARCH model(even wi a one-sided es). So ere does appear o be an asymmeric effec in Teran sock marke. In addiion Evaluaion forecasing wi MSE crieria indicae a GARCH models in is paper ave a same forecasing power, bu wen Log- Likeliood is evaluaion crieria, CGARCH as e bes forecasing power. KEY WORDS: Forecasing, Volailiy, Teran Sock Marke, GARCH Models. 1. INTRODUCTION Te abiliy o forecas financial marke volailiy is imporan for porfolio selecion, valuaion of socks, asse managemen, predicabiliy of risk premiums and designing opimal dynamic edging sraegies for opions and Fuures. Wile mos researcers agree a volailiy is predicable in many asse markes (see for example e survey by Bollerslev e al.,199, 1994), ey differ on ow is volailiy predicabiliy sould be modeled. Over e pas several decades e evidence for predicabiliy as led o variey of approaces. Te mos ineresing of ese approaces are e asymmeric or leverage volailiy models, in wic good news and bad news ave differen predicabiliy for fuure volailiy (see,for example, Black, 1976, Crisle, 197, Nelson, 1991, Pagan and Scwer, 1990, Senena, 199, Campbell and Herscel, 199, Engle, 1993, Henry, 1998, and Friedmann, Sanddorf-Köle, 00). In mos ese sudies researcers ave documened srong evidence a volailiy is asymmeric in euiy markes: negaive reurns are generally associaed wi upward revisions of e condiional volailiy wile posiive reurns are associaed wi smaller upward or even downward revisions of e condiional volailiy (see, for example, Cox and Ross, 1976, Engle and Ng, 1993, Henry, 1998,). Researcers (see Black, 1976 Crisie, 198, and Scwer, 1989) believe a e asymmery could be due o canges in leverage in response o canges in e value of euiy. Oers ave argued a e asymmery could arise from e feedback from volailiy o sock price wen canges in volailiy induce canges in risk premiums (see Pindyck, 1984, Frenc e al., 1987, Campbell and Henscel, 199, and Wu, 001). Te presence of asymmeric volailiy is mos apparen during a marke crisis wen large declines in sock prices are associaed wi a significan increase in marke volailiy. Asymmeric volailiy can poenially explain e negaive skewness in sock reurn daa, as discussed in Harvey and Siddiue (1999). Tere is no general agreemen as o ow e predicabiliy sould be modeled and, in paricular ow o condiion suc models for asymmeric naure of e sock reurn volailiy. In is paper we compare e performance of GARCH, TARCH, EGARCH and componen ARCH (CARCH) fied o daily Teran Sock Marke (TSM) reurns and es weer asymmery is presen. Tere are no any sudies wic focus explicily on modeling e volailiy in e TSM. Tis paper is organized as follows. In secion II of is paper various models of sock reurn volailiy, bo symmeric and asymmeric are oulined. Secion III describes e daa. Secion IV presens empirical resuls and esimaes of e relaionsip beween news and volailiy for e candidae models. Te final secion provides a brief summary and conclusion.. Modeling volailiy R Le be e rae of reurn of a sock, or a porfolio of socks from ime 1 o and 1 be e pas Informaion se conaining e Realized value of all relevan variables up o ime 1. So e condiional mean and *Corresponding Auor: Medi Parvares, Posal address: Parvares alley, Daroopaks sree, Bandarabbas, Iran, Posal code: , Telepone: , Fax: parvares009@gmail.com, 150

2 Parvares and Bavagar, 01 y E( R ), var( R ) variance are respecively. Given is definiion, e unexpeced reurn a ime is R y. Tis paper follows Engle and Ng (1993) in reaing as a collecive measure of bad news (unexpeced decrease in reurns) if < 0 and good news (unexpeced increase in price) if 0. Furer, a large value of implies a e news is significan or big in e sense a i produce a large unexpeced cange in reurns. on reurns we presen ARCH models. ARCH models were Inroduced by In order o model e effec of Engle (198) and generalized as GARCH models by Bollerslev (1986). In developing GARCH (p, ) we will ave o provide mean and variance Euaion R, ε = η, η ~N(0,1) (1) p i i j j i1 j1 (), i, j, x were are consan parameers and conains exogenous and predeermined regressors. As is variance i mus be nonnegaive wic impose e following condiions: 0 1,... 0, p and 1,..., 0. Te condiional variance under ARCH (p) model reflecs only informaion from ime p o 1 ai a j wi more imporance being placed on e mos recen innovaion implying for i j. To avoid long lag lengs on in ARCH (p) and difficuly in selecing e opional leng p, and ensuring e non-negaiviy of coefficiens of condiional variance euaion (), Bollerslev (1986) presen GARCH(P, ). A common parameerizaion for e GARCH model a as been adoped in mos applied sudies is e GARCH (1, 1) specificaion under wic e effec of a sock o volailiy declines geomerically over ime. One problem wi ARCH (p) and GARCH (p, ) is a good news and bad news wi some absolue size ave e same effec on.tis fac is symmeric effec. However, e marke may reac differenly o good and bad news. I is imporan, o be able o es for and allow asymmery in e ARCH ype specificaion. Nelson (1991) proposes e exponenial GARCH (EGARCH) model as a way o deal wi is problem. Under e EGARCH (1, 1) e is given as: 1 1 log( 1 ) log (3) Te EGARCH news Impac differs from e GARCH new Impac in four ways: (1) i is no symmeric. () Big news can ave a muc greaer impac an in e GARCH model. (3) Log consrucion of Euaion 3 ensures a e esimaed is sricly posiive, us non-negaiviy consrains used in e esimaion of e ARCH and GARCH are no necessary. (4) Since e parameer of ypically eners euaion 3 wi a negaive sign, bad news generaes more volailiy an good news. Glosen, jagannaan and Runkle (1993 ), ereafer GJR, Defined GJR Asymmeric Volailiy model as follow: 1 1 s 1 1 (4) s 1 Were 0 if, and 0 oerwise. Te GJR model is closely relaed o e resold ARCH or TARCH model of Rabemananjara and Zakoian (1993) and Zakoian (1994). Provideda 0, e GJR model generaes iger values for given -1 < 0 an for a > 0 of eual magniudes. 151

3 J. Basic. Appl. Sci. Res., (1) , 01 Te Componen GARCH (CGARCH) model by Engle and Lee (1993) decomposes reurns uncerainy ino a sor-run and a long-run componen by permiing ransiory deviaions of e condiional volailiy around a imevarying rend, a, modeled as: ( 1 1) ( 1 1) 1 ( 1 ) ( 1 1 Here is sill e volailiy, wile Te firs euaion describes e ransiory componen, ) (5) (6) akes e place of and is e ime varying long run volailiy. wic converges o zero wi powers of ( ). Te second euaion describes e long run componen, wic converges o wi powers of. Typically approaces very slowly. We can combine e ransiory and permanen is beween 0.99 and 1 so a euaions and wrie ( 1 )(1 ) ( ) 1 ( ( ) ) 1 ) 1 ( ( ) ) (7) Wic sows a e componen model is a (nonlinear) resriced GARCH (, ) model.in addiion, GARCH(1, 1) is a special case of e CARCH in wic 0. We can include exogenous variables in e condiional variance euaion of componen models, eier in e permanen or ransiory euaion (or bo). Te variables in e ransiory euaion will ave an impac on e sor run movemens in volailiy, wile e variables in e permanen euaion will affec e long run levels of volailiy. Te asymmeric componen combines e componen model wi e asymmeric TARCH model. Tis specificaion inroduces asymmeric effecs in e ransiory euaion and esimaes models of e form: R x (8) ( 1 ) ( 1 1 ) 1z1 (9) ( ) ( ) d ( ) ( ) z (10) Were z are e exogenous variables and d is e dummy variable indicaing negaive socks. 0 indicaes e presence of ransiory leverage effecs in e condiional variance. Suppose informaion is eld consan a ime and before, Engle and Ng (1993) describe e relaionsip beween 1 and as e news impac curve. Te news impac curves of GARCH and CGARCH models are symmeric and cenered a differen slopes Te news impac curves of EGARCH and TARCH are asymmeric wi 3. Daa Descripion Te daa consis of 1884 daily observaions of e closing value of e Teran sock marke from 9/8/1997 o 8/0/010. Te sample is divided in wo pars. Te firs 800 observaions (from 9/8/1997 o 8/0/009) are used as e in-sample for esimaion purposes, wile e remaining 315 observaions (from 8/1/009 o 8/0/010) are aken as e ou-of-sample for forecas evaluaion purposes. Te reurn is calculaed as p r 100 [log( )] p 1 P is e value of index a ime Table 1 sows some descripive saisics of e TSM rae of reurn. were Te mean is uie small and e sandard deviaion is around 0.6. Te kurosis is significanly iger an e normal value of 3 indicaing a fa-ailed disribuion are necessary o correcly describe condiional disribuion of r. e skewness is significan, small and posiive, sowing a e upper ail of empirical disribuion of e reurn is longer an e lower ail, a is posiive reurns are more likely o be far below e mean an eir counerpars. 15

4 Parvares and Bavagar, 01 Mean Sandard Deviaion r Table 1.Descripive Saisics Min Max Sk Ku B J Q (1) LM(1) p-value: [0.00] [0.00] [0.00] Noe:Sk and Ku are skewness and excess kurosis. B-J is e Bera-Jarue es for normaliy disribued as (). Te Q (1) saisic is e Ljung-Box es on e suared residuals of e condiional mean regression up o (1). e welf order. for serial correlaion in e suared reurn daa, disribued as LM(1) saisic is e ARCH LM es up o welf lag and under e null ypoesis of no ARCH effecs i as a ( ) disribuion, were is e number of lags. LM (1) is e Lagrange Muliplier es for ARCH effecs in e OLS residuals from e regression of e reurns on a consan, wile Q (1) is e corresponding Ljung-Box saisic on e suared sandardized residuals. Bo ese saisic are igly significan suggesion e presence of ARCH effecs in e TSM reurns up o e welf order. 4. Empirical Resuls We ave used uasi-maximum likeliood (QML) covariance and sandard errors using e meods described by Bollerslev and Woldridy (199) for esimaion GARCH models. Te esimaion and es resuls for models sugges a e leverage effec erm,, is significan in EGARCH model(even wi a one-sided es). So ere does appear o be an asymmeric effec. In EGARCH model e esimaed coefficien on e asymmery erm ˆ 1 ˆ 1 is , wic is significan a convenional levels. Table : Maximum Likeliood Esimaes of sandard GARCH Models wi Normal condiional disribuion. GARCH TARCH EGARCH CGARCH PARCH δ p-value p-value p-value p-value p-value p-value p-value ν p-value Log likeliood MSE Evaluaion forecasing wi MSE crieria indicae a GARCH models in is paper ave a same forecasing power, bu wi Log-Likeliood crieria, CGARCH as e bes forecasing power. 153

5 J. Basic. Appl. Sci. Res., (1) , conclusion We ave used uasi-maximum likeliood (QML) covariance and sandard errors using e meods described by Bollerslev and Woldridy (199) for esimaion GARCH models. Te esimaion and es resuls for models sugges a e leverage effec erm,, is significan in EGARCH model(even wi a one-sided es). So ere does appear o be an asymmeric effec in Teran sock marke. In addiion Evaluaion forecasing wi MSE crieria indicae a GARCH models in is paper ave a same forecasing power, bu wen Log-Likeliood is evaluaion crieria, CGARCH as e bes forecasing power. REFERENCES [1]. Black, F., Sudies of sock price volailiy canges. Proceedings of e 1976 Meeings of e AmericanSaisical Associaion, Business and Economical Saisics Secion, pp []. Bollerslev, T., Generalized auoregressive condiional eeroskedasiciy. Journal of Economerics 31, [3]. Bollerslev, T.,Wooldridge, J., 199. Quasi-maximum likeliood esimaion and inference in dynamic models wi ime-varying covariances. Economeric Reviews 11, [4]. Bollerslev, T., Cou, R.Y., Kroner, K.F., 199. ARCH modeling in finance. Journal of Economerics 5, [5]. Box, G., Pierce, D., Disribuion of residual auocorrelaions in auoregressive-inegraed moving average ime series models. Journal of e American Saisical Associaion 65, [6]. Breusc, T.S., Pagan, A.R., Te Lagrange muliplier es and is applicaions o model specificaion in economerics. Review of Economic Sudies 47, [7]. Campbell, J.Y., Henscel, L., 199. No news is good news: an asymmeric model of canging volailiy in sock reurns. Journal of Financial Economics 31, [8]. Campbell, J.Y., Lo, A.W., MacKinlay, A.C., Te Economerics of Financial Markes. Princeon Univ. Press, Princeon, NJ. [9]. Crisie, A.A., 198. Te socasic beavior of common sock variances value, leverage and ineres rae effecs. Journal of Financial Economics 10, [10]. Cox, J.C., Ross, S.A., Te valuaion of opions for alernaive socasic processes. Journal of Financial Economics 3, [11]. Engle, R., 198. Auoregression condiional eeroskedasiciy wi esimaes of e variance of e UK inflaion. Economerica 50, [1]. Engle, R.F., Kroner, K., Mulivariae simulaneous generalized ARCH. Economeric Teory 11, [13]. Engle, R.F., Ng, V.K., Measuring and esing e impac of news on volailiy. Journal of Finance 48, [14]. Frenc, K.R., Scwer, G.W., Sambaug, R., Expeced sock reurns and volailiy. Journal of Financial Economics 19, 3 9. [15]. Glosen, L.R., Jagannaan, R., Runkle, D.E., On e relaion beween e expeced value and e volailiy of e nominal excess reurn on socks. Journal of Finance 48, [16]. Friedmann, R. and Sanddorf-Köle, W.G. (00) Volailiy clusering and nonrading days in Cinese sock markes, Journal of Economics and Business, 54, [17]. Harvey, C., Siddiue, A., Auoregressive condiional skewness. Journal of Financial and Quaniaive [18]. Analysis 34, [19]. Henry, O., Modelling e asymmery of sock marke volailiy. Applied Financial Economics 8, [0]. Henscel, L., All in e family: nesing symmeric and asymmeric GARCH models. Journal of Financial Economics 39, [1]. Ljung, G., Box, G., On a measure of lack of fi in ime series models. Biomerika 66, []. Nelson, D.B., Condiional eeroskedasiciy in asse reurns: a new approac. Economerica 59, [3]. Pagan, A.R., Scwer, G.W., Alernaive models for condiional sock volailiy. Journal of Economerics 45, [4]. Pindyck, R.S., Risk, inflaion, and e sock marke. American Economic Review 74,

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