DEPARTMENT OF ECONOMICS AND FINANCE COLLEGE OF BUSINESS AND ECONOMICS UNIVERSITY OF CANTERBURY CHRISTCHURCH, NEW ZEALAND

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1 DEPARTMENT OF ECONOMICS AND FINANCE COLLEGE OF BUSINESS AND ECONOMICS UNIVERSITY OF CANTERBURY CHRISTCHURCH, NEW ZEALAND Asymmery and Leverage in Condiional Volailiy Models Michael McAleer WORKING PAPER No. 4/014 Deparmen of Economics and Finance College of Business and Economics Universiy of Canerbury Privae Bag 4800, Chrischurch New Zealand

2 WORKING PAPER No. 4/014 Asymmery and Leverage in Condiional Volailiy Models* Michael McAleer 1 Sepember 5, 014 Absrac: The hree mos popular univariae condiional volailiy models are he generalized auoregressive condiional heeroskedasiciy (GARCH) model of Engle (198) and Bollerslev (1986), he GJR (or hreshold GARCH) model of Glosen, Jagannahan and Runkle (199), and he exponenial GARCH (or EGARCH) model of Nelson (1990, 1991). The underlying sochasic specificaion o obain GARCH was demonsraed by Tsay (1987), and ha of EGARCH was shown recenly in McAleer and Hafner (014). These models are imporan in esimaing and forecasing volailiy, as well as capuring asymmery, which is he differen effecs on condiional volailiy of posiive and negaive effecs of equal magniude, and leverage, which is he negaive correlaion beween reurns shocks and subsequen shocks o volailiy. As here seems o be some confusion in he lieraure beween asymmery and leverage, as well as which asymmeric models are purpored o be able o capure leverage, he purpose of he paper is wo-fold, namely: (1) o derive he GJR model from a random coefficien auoregressive process, wih appropriae regulariy condiions; and () o show ha leverage is no possible in hese univariae condiional volailiy models.. Keywords: Condiional volailiy models, random coefficien auoregressive processes, random coefficien complex nonlinear moving average process, asymmery, leverage. JEL Classificaions: C, C5, C58, G3. Acknowledgemens: For financial suppor, he auhor wishes o acknowledge he Ausralian Research Council and he Naional Science Council, Taiwan. 1 Deparmen of Quaniaive Finance, Naional Tsing Hua Universiy, Taiwan; Economeric Insiue. Erasmus School of Economics, Erasmus Universiy Roerdam and Tinbergen Insiue, The Neherlands; and Deparmen of Quaniaive Economics, Compluense Universiy of Madrid *Corresponding Auhor: michael.mcaleer@gmail.com

3 1. Inroducion The hree mos popular univariae condiional volailiy models are he generalized auoregressive condiional heeroskedasiciy (GARCH) model of Engle (198) and Bollerslev (1986), he GJR (or hreshold GARCH) model of Glosen, Jagannahan and Runkle (199), and he exponenial GARCH (or EGARCH) model of Nelson (1990, 1991). The underlying sochasic specificaion o obain GARCH was demonsraed by Tsay (1987), and ha of EGARCH was shown recenly in McAleer and Hafner (014). These models are imporan in esimaing and forecasing volailiy, in capuring asymmery, which is he differen effecs on condiional volailiy of posiive and negaive effecs of equal magniude, and (possibly) in capuring leverage, which is he negaive correlaion beween reurns shocks and subsequen shocks o volailiy. The purpose of he paper is wo-fold, namely: (1) o derive he GJR model from a random coefficien auoregressive process, wih appropriae regulariy condiions; and () o show ha leverage is no possible in hese univariae condiional volailiy models. The derivaion of hree well known condiional volailiy models, namely GARCH, GJR and EGARCH, from heir respecive underlying sochasic processes raises wo imporan issues: (1) he regulariy condiions for each condiional volailiy model can be derived in a sraighforward manner; and () he GJR and EGARCH models can be shown o capure asymmery, bu hey can also be shown o be unable o capure leverage. The paper organized is as follows. In Secion, he GARCH, GJR and EGARCH models are derived from differen sochasic processes, he firs wo from random coefficien auoregressive processes and he hird from a random coefficien complex nonlinear moving average process. I is shown ha asymmery is possible for GJR and EGARCH, bu ha leverage is no possible. Some concluding commens are given in Secion 3. 1

4 . Sochasic Processes for Condiional Volailiy Models.1 Random Coefficien Auoregressive Process and GARCH Consider he condiional mean of financial reurns as in he following: y = E( y I 1 ) + ε (1) where he reurns, y = log P, represen he log-difference in sock prices ( P ), I 1 is he informaion se a ime -1, and ε is condiionally heeroskedasic. In order o derive condiional volailiy specificaions, i is necessary o specify he sochasic processes underlying he reurns shocks, ε. Consider he following random coefficien auoregressive process of order one: ε = φ ε 1 + η () where φ ~ iid ( 0, α ), η ~ iid ( 0, ω ). Tsay (1987) showed ha he ARCH(1) model of Engle (198) could be derived from equaion () as: h = E( I 1) = ω + αε 1 ε. (3) where h is condiional volailiy, and 1 I is he informaion se a ime -1. The use of an infinie lag lengh for he random coefficien auoregressive process in equaion (), wih

5 appropriae resricions on he random coefficiens, can be shown o lead o he GARCH model of Bollerslev (1986). As he ARCH and GARCH models are symmeric, in ha posiive and negaive shocks of equal magniude have idenical effecs on condiional volailiy, here is no asymmery, and hence also no leverage, whereby negaive shocks increase condiional volailiy and posiive shocks decrease condiional volailiy (see Black (1976)). I is worh noing ha a leas one of ω or α mus be posiive for condiional volailiy o be posiive. From he specificaion of equaion (), i is clear ha boh ω and α should be posiive as hey are he variances of wo differen sochasic processes.. Random Coefficien Auoregressive Process and GJR The GJR model of Glosen, Jagannahan and Runkle (199) can be derived as a simple exension of he random coefficien auoregressive process in equaion (), wih an indicaor variable I( ε ) 1 ha disinguishes beween he differen effecs of posiive and negaive reurns shocks on condiional volailiy, namely: ε ( + (4) = φ ε 1 + ψ I ε 1 ) ε 1 η where φ ~ iid ( 0, α ), ψ ~ iid ( 0, γ ), η ~ iid ( 0, ω ), I( ε ) 1 = 1 when ε 1 < 0, I( ε ) 1 = 0 when ε

6 The condiional expecaion of he squared reurns shocks in (3), which is ypically referred o as he GJR (or hreshold GARCH), can be shown o be an exension of equaion (3), as follows: h = E( I 1) = ω + α ε 1 + γ I( ε 1 ) ε 1 ε. (5) The use of an infinie lag lengh for he random coefficien auoregressive process in equaion (4), wih appropriae resricions on he random coefficiens, can be shown o lead o he sandard GJR model wih lagged condiional volailiy. I is worh noing ha a leas one of ( ω, α, γ ) mus be posiive for condiional volailiy o be posiive. From he specificaion of equaion (4), i is clear ha all hree parameers should be posiive as hey are he variances of hree differen sochasic processes. The GJR model is asymmeric, in ha posiive and negaive shocks of equal magniude have differen effecs on condiional volailiy. Therefore, asymmery exiss for GJR if: Asymmery for GJR: γ > 0. A special case of asymmery is leverage, which is he negaive correlaion beween reurns shocks and subsequen shocks o volailiy. The condiions for leverage in he GJR model in equaion (5) are: Leverage for GJR: α < 0 and α +γ > 0. I is clear ha leverage is no possible for GJR as boh α and γ, which are he variances of wo sochasic processes, mus be posiive..3 Random Coefficien Complex Nonlinear Moving Average Process and EGARCH 4

7 Anoher condiional volailiy model ha can accommodae asymmery is he EGARCH model of Nelson (1990, 1991). McAleer and Hafner (014) showed ha EGARCH could be derived from a random coefficien complex nonlinear moving average (RCCNMA) process, as follows: ε φ η ψ η + η (6) = where φ ~ iid ( 0, α ), ψ ~ iid ( 0, γ ), η ~ iid ( 0, ω ), η 1 is a complex-valued funcion of 1 η. The condiional variance of he squared reurns shocks in equaion (6) is given as: h ε. (7) = E( I 1) = ω + α η 1 + γ η 1 I is worh noing ha he ransformaion of h in equaion (7) is no logarihmic, bu he approximaion given by: log h = log(1 + ( h 1)) h 1 can be used o replace h in equaion (7) wih 1 + log h. The use of an infinie lag for he RCCNMA process in equaion (6) would yield he sandard EGARCH model wih lagged condiional volailiy. EGARCH differs from GARCH and GJR in ha, given he logarihmic ransformaion, no sign resricions on ( ω, α, γ ) are necessary for condiional volailiy o be posiive. However, i is 5

8 clear from he RCCNMA process in equaion (6) ha all hree parameers should be posiive as hey are he variances of hree differen sochasic processes. Therefore, asymmery exiss for EGARCH if: Asymmery for EGARCH: γ > 0. The condiions for leverage in he EGARCH model in equaion (7) are: Leverage for EGARCH: γ < 0 and γ < α < γ. As acknowledged in McAleer and Hafner (014), leverage is no possible as boh α and γ, which are he variances of wo sochasic processes, mus be posiive. 3. Concluding Remarks The paper was concerned wih he hree mos widely-used univariae condiional volailiy models, namely he GARCH, GJR (or hreshold GARCH) and EGARCH models. These models are imporan in esimaing and forecasing volailiy, as well as in capuring asymmery, which is he differen effecs on condiional volailiy of posiive and negaive effecs of equal magniude, and in capuring leverage, which is he negaive correlaion beween reurns shocks and subsequen shocks o volailiy. As here seems o be some confusion in he lieraure beween asymmery and leverage, as well as which asymmeric models are purpored o be able o capure leverage, he purpose of he paper was wo-fold, namely: (1) o derive he GJR model from a random coefficien auoregressive process, wih appropriae regulariy condiions; and () o show he GJR and EGARCH models are able o capure asymmery, bu are unable o capure leverage. 6

9 References Black, F. (1976), Sudies of sock marke volailiy changes, 1976 Proceedings of he American Saisical Associaion, Business and Economic Saisics Secion, pp Bollerslev, T. (1986), Generalised auoregressive condiional heeroscedasiciy, Journal of Economerics, 31, Engle, R.F. (198), Auoregressive condiional heeroscedasiciy wih esimaes of he variance of Unied Kingdom inflaion, Economerica, 50, Glosen, L., R. Jagannahan and D. Runkle (199), On he relaion beween he expeced value and volailiy of nominal excess reurn on socks, Journal of Finance, 46, McAleer, M. and C. Hafner (014), A one line derivaion of EGARCH, Economerics, (), Nelson, D.B. (1990), ARCH models as diffusion approximaions, Journal of Economerics, 45, Nelson, D.B. (1991), Condiional heeroskedasiciy in asse reurns: A new approach, Economerica, 59, Tsay, R.S. (1987), Condiional heeroscedasic ime series models, Journal of he American Saisical Associaion, 8,