Economerics 04,, 45-50; doi:0.3390/economerics03045 OPEN ACCESS economerics ISSN 5-46 www.mdpi.com/journal/economerics Aricle Asymmery and Leverage in Condiional Volailiy Models Micael McAleer,,3,4 Deparmen of Quaniaive Finance, Naional Tsing Hua Universiy, Hsincu 0, Taiwan; E-Mail: micael.mcaleer@gmail.com; Tel.: +886-3-56534; Fax: +886-3-5683 Economeric Insiue, Erasmus Scool of Economics, Erasmus Universiy Roerdam, Burgemeeser Oudlaan 50, 306 PA Roerdam, Te Neerlands 3 Tinbergen Insiue, Roerdam 3000 DR, Te Neerlands 4 Deparmen of Quaniaive Economics, Compluense Universiy of Madrid, Madrid, 8040, Spain Received: 8 Sepember 04; in revised form: 9 Sepember 04 / Acceped: 9 Sepember 04 / Publised: 4 Sepember 04 Absrac: Te ree mos popular univariae condiional volailiy models are e generalized auoregressive condiional eeroskedasiciy (GARCH) model of Engle (98) and Bollerslev (986), e GJR (or resold GARCH) model of Glosen, Jagannaan and Runkle (99), and e exponenial GARCH (or EGARCH) model of Nelson (990, 99). Te underlying socasic specificaion o obain GARCH was demonsraed by Tsay (987), and a of EGARCH was sown recenly in McAleer and Hafner (04). Tese models are imporan in esimaing and forecasing volailiy, as well as in capuring asymmery, wic is e differen effecs on condiional volailiy of posiive and negaive effecs of equal magniude, and purporedly in capuring leverage, wic is e negaive correlaion beween reurns socks and subsequen socks o volailiy. As ere seems o be some confusion in e lieraure beween asymmery and leverage, as well as wic asymmeric models are purpored o be able o capure leverage, e purpose of e paper is ree-fold, namely, () o derive e GJR model from a random coefficien auoregressive process, wi appropriae regulariy condiions; () o sow a leverage is no possible in e GJR and EGARCH models; and (3) o presen e inerpreaion of e parameers of e ree popular univariae condiional volailiy models in a unified manner. Keywords: condiional volailiy models; random coefficien auoregressive processes; random coefficien complex nonlinear moving average process; asymmery; leverage JEL classificaions: C; C5; C58; G3
Economerics 04, 46. Inroducion Te ree mos popular univariae condiional volailiy models are e generalized auoregressive condiional eeroskedasiciy (GARCH) model of Engle (98) [] and Bollerslev (986) [], e GJR (or resold GARCH) model of Glosen, Jagannaan and Runkle (99) [3], and e exponenial GARCH (or EGARCH) model of Nelson (990, 99) [4,5]. Te underlying socasic specificaion o obain GARCH was demonsraed by Tsay (987) [6], and a of EGARCH was sown recenly in McAleer and Hafner (04) [7]. Tese models are imporan in esimaing and forecasing volailiy and in capuring asymmery, wic is e differen effecs on condiional volailiy of posiive and negaive effecs of equal magniude; furermore, ey are purporedly imporan in capuring leverage, wic is e negaive correlaion beween reurns socks and subsequen socks o volailiy. Te purpose of e paper is ree-fold, namely, () o derive e GJR model from a random coefficien auoregressive process, wi appropriae regulariy condiions; () o sow a leverage is no possible in e GJR and EGARCH models; and (3) o presen e inerpreaion of e parameers of e ree popular univariae condiional volailiy models in a unified manner. Te derivaion of ree well known condiional volailiy models, namely GARCH, GJR and EGARCH, from eir respecive underlying socasic processes raises wo imporan issues: () e regulariy condiions for eac condiional volailiy model can be derived in a sraigforward manner; and () e GJR and EGARCH models can be sown o capure asymmery, bu ey can also be sown o be unable o capure leverage. Te paper is organized as follows. In Secion, e GARCH, GJR and EGARCH models are derived from differen socasic processes, e firs wo from random coefficien auoregressive processes and e ird from a random coefficien complex nonlinear moving average process. I is sown a asymmery is possible for GJR and EGARCH, bu a leverage is no possible. Issues relaed o esimaion are also discussed, wi a view o guaraneeing posiive esimaes of condiional volailiy. Some concluding commens are given in Secion 3.. Socasic Processes for Condiional Volailiy Models.. Random Coefficien Auoregressive Process and GARCH Consider e condiional mean of financial reurns as in e following: were e reurns, y = se a ime, and y E ) ( y I () P ), log P represens e log-difference in sock prices ( I is e informaion is condiionally eeroskedasic. In order o derive condiional volailiy specificaions, i is necessary o specify e socasic processes underlying e reurns socks,.
Economerics 04, 47 Consider e following random coefficien auoregressive process of order one: () were ~ iid ( 0, ). Tsay (987) [6] sowed a e ARCH() model of Engle (98) [] could be derived from Equaion () as: were is condiional volailiy, and E( I ) (3) I is e informaion se a ime. Te use of an infinie lag leng for e random coefficien auoregressive process in Equaion (), wi appropriae resricions on e random coefficiens, can be sown o lead o e GARCH model of Bollerslev (986) []. As e ARCH and GARCH models are symmeric, in a posiive and negaive socks of equal magniude ave idenical effecs on condiional volailiy, ere is no asymmery, and ence also no leverage, wereby negaive socks increase condiional volailiy and posiive socks decrease condiional volailiy (see Black (976) [8]). I is wor noing a a leas one of or mus be posiive for condiional volailiy o be posiive, wi > 0 and > 0 regarded as sufficien condiions for posiiviy of condiional volailiy. From e specificaion of Equaion (), i is clear a bo and sould be posiive as ey are e variances of wo differen socasic processes. From a pracical perspecive, a failure o impose e posiiviy resricions on e parameers can increase e probabiliy of obaining negaive esimaes of condiional volailiy. For example, in e curren version R04a of MATLAB, no posiiviy is imposed in esimaing e parameers of GARCH. Similar commens apply o oer sandard economeric, financial economeric and saisical sofware packages, and o e GJR and EGARCH models a are discussed below... Random Coefficien Auoregressive Process and GJR Te GJR model of Glosen, Jagannaan and Runkle (99) [3] can be derived as a simple exension of e random coefficien auoregressive process in Equaion (), wi an indicaor variable I ) a disinguises beween e differen effecs of posiive and negaive reurns socks on condiional volailiy, namely: I( ) ( (4) were I( ) = wen < 0, I( ) = 0 wen 0.
Economerics 04, 48 Te condiional expecaion of e squared reurns socks in (3), wic is ypically referred o as e GJR (or resold GARCH) model, can be sown o be an exension of Equaion (3), as follows: E( I ) I( ) (5) Te use of an infinie lag leng for e random coefficien auoregressive process in Equaion (4), wi appropriae resricions on e random coefficiens, can be sown o lead o e sandard GJR model wi lagged condiional volailiy. As GARCH is nesed wiin GJR, e inerpreaion of e coefficiens in e wo models is essenially e same, apar from e parameer associaed wi asymmery. I is wor noing a a leas one of (,, ) mus be posiive for condiional volailiy o be posiive, wi > 0, > 0 and > 0 regarded as sufficien condiions for posiiviy of condiional volailiy. From e specificaion of Equaion (4), i is clear a all ree parameers sould be posiive as ey are e variances of ree differen socasic processes. Te GJR model is asymmeric, in a posiive and negaive socks of equal magniude ave differen effecs on condiional volailiy. Terefore, asymmery exiss for GJR if: asymmery for GJR: 0. A special case of asymmery is leverage, wic is e negaive correlaion beween reurns socks and subsequen socks o volailiy (see Black (976) [8]). Te condiions for leverage in e GJR model in Equaion (5) are: leverage for GJR: 0 and 0. I is clear a leverage is no possible for GJR as bo and, wic are e variances of wo socasic processes, mus be posiive. As in e case of GARCH, e posiiviy resricions on e parameers of GJR are ypically no imposed in esimaion using sandard economeric, financial economeric and saisical sofware packages..3. Random Coefficien Complex Nonlinear Moving Average Process and EGARCH Anoer condiional volailiy model a can accommodae asymmery is e EGARCH model of Nelson (990, 99) [4,5]. McAleer and Hafner (04) [7] sowed a EGARCH could be derived from a random coefficien complex nonlinear moving average (RCCNMA) process, as follows: (6) were is a complex-valued funcion of.
Economerics 04, 49 Te condiional variance of e squared reurns socks in Equaion (6) is given as: I is wor noing a e ransformaion of approximaion given by: can be used o replace E (7) log ( I ) log( ( in Equaion (7) wi + in Equaion (7) is no logarimic, bu e log )). Te use of an infinie lag for e RCCNMA process in Equaion (6) would yield, afer suiable logarimic approximaion, e sandard EGARCH model wi lagged condiional volailiy. EGARCH differs from GARCH and GJR in a, given e logarimic ransformaion, no sign resricions on (,, ) are necessary for condiional volailiy o be posiive. However, i is clear from e RCCNMA process in Equaion (6) a all ree parameers sould be posiive as ey are e variances of ree differen socasic processes. Terefore, asymmery exiss for EGARCH if: asymmery for EGARCH: 0. Te condiions for leverage in e EGARCH model in Equaion (7) are: leverage for EGARCH: 0 and. As acknowledged in McAleer and Hafner (04) [7], leverage is no possible as bo and, wic are e variances of wo socasic processes, mus be posiive. As EGARCH is non-nesed wi bo GARCH and GJR, e inerpreaion of e coefficiens in EGARCH is no e same as in e oer wo condiional volailiy models, aloug e definiions of asymmery and leverage are idenical. Te derivaions in Secion are inended o keep e number of iid processes o a minimum for ease of presenaion. As in e case of GARCH and GJR, e posiiviy resricions on e parameers of GJR are ypically no imposed in esimaion using sandard economeric, financial economeric and saisical sofware packages. 3. Conclusions Te paper was concerned wi e ree mos widely-used univariae condiional volailiy models, namely e GARCH, GJR (or resold GARCH) and EGARCH models. Tese models are imporan in esimaing and forecasing volailiy, as well as in capuring asymmery, wic is e differen effecs on condiional volailiy of posiive and negaive effecs of equal magniude, and purporedly in capuring leverage, wic is e negaive correlaion beween reurns socks and subsequen socks o volailiy. As discussed in Secion, a failure o impose e posiiviy resricions on e parameers of e condiional volailiy models can increase e probabiliy of obaining negaive esimaes of condiional volailiy. In sandard economeric, financial economeric and saisical sofware packages, i is ypically e case a no posiiviy is imposed in esimaing e parameers of e ree mos popular condiional volailiy models.
Economerics 04, 50 As ere seems o be some confusion in e lieraure beween asymmery and leverage, as well as wic asymmeric models are purpored o be able o capure leverage, e purpose of e paper was ree-fold, namely, () o derive e GJR model from a random coefficien auoregressive process, wi appropriae regulariy condiions; () o sow a leverage is no possible in e GJR and EGARCH models; and (3) o presen e inerpreaion of e parameers of e ree popular univariae condiional volailiy models in a unified manner. Acknowledgemens Te auor wises o ank Massimiliano Caporin, Marc Paolella, Pawel Polak, e Edior-in-Cief, Kerry Paerson, and a reviewer for elpful commens and suggesions. For financial suppor, e auor wises o acknowledge e Ausralian Researc Council and e Naional Science Council, Taiwan. Conflics of Ineres Te auor as no conflics of ineres. References. Engle, R.F. Auoregressive condiional eeroscedasiciy wi esimaes of e variance of Unied Kingdom inflaion. Economerica 98, 50, 987 007.. Bollerslev, T. Generalised auoregressive condiional eeroscedasiciy. J. Econom. 986, 3, 307 37. 3. Glosen, L.R.; Jagannaan, R.; Runkle, D.E. On e relaion beween e expeced value and volailiy of nominal excess reurn on socks. J. Financ.99, 46, 779 80. 4. Nelson, D.B. ARCH models as diffusion approximaions. J. Econom. 990, 45, 7 38. 5. Nelson, D.B. Condiional eeroskedasiciy in asse reurns: A new approac. Economerica 99, 59, 347 370. 6. Tsay, R.S. Condiional eeroscedasic ime series models. J. Am. Sa. Assoc. 987, 8, 590 604. 7. McAleer, M.; Hafner, C. A one line derivaion of EGARCH. Economerics 04,, 9 97. 8. Black, F. Sudies of Sock Marke Volailiy Canges. In Proceedings of e American Saisical Associaion, Business and Economic Saisics Secion, Wasingon, DC, USA, 976; pp. 77 8. 04 by e auors; licensee MDPI, Basel, Swizerland. Tis aricle is an open access aricle disribued under e erms and condiions of e Creaive Commons Aribuion license (p://creaivecommons.org/licenses/by/4.0/).