Hwang (2001) proposes the FIFGARCH model to represent long memory asymmetric conditional variance. Although he claims that this model nests many previous models, we show that it does not and that the model is badly specified. We propose and alternative specification

Pérez, Ana

Ruiz, Esther

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Asymmetric long memory GARCH: a reply to Hwang's model

Ruiz Ortega, Esther

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Working Paper  01-62 (29)Statistics and Econometrics SeriesNovember 2001Departamento de Estadística y EconometríaUniversidad Carlos III de MadridCalle Madrid, 12628903 Getafe (Spain)Fax (34) 91 624-98-49 ASYMMETRIC LONG MEMORY GARCH: A REPLY TO HWANG’S MODEL Esther Ruiz and Ana Pérez*AbstractHwang (2001) proposes the FIFGARCH model to represent long memoryasymmetric conditional variance. Although he claims that this model nests many previousmodels, we show that it does not and that the model is badly specified. We propose andalternative specification.Keywords: EGARCH, FGARCH, FIGARCH, FIEGARCH*Ruiz, Departamento de Estadística y Econometría, Universidad Carlos III de Madrid. C/Madrid, 126 28903 Madrid. España. Tel: 34-91-624 98 51, Fax: 34-91-624 98 49, e-mail:ortega@est-econ.uc3m.es;  Pérez, Departamento de Economía Aplicada (Estadística yEconometría), Universidad de Valladolid.Asymmetric long memory GARCH: A reply toHwang's modelEsther Ruiz(a)and Ana Perez(b)(a) Universidad Carlos III de Madrid(b)Universidad de ValladolidNovember 2001AbstractHwang (2001) proposes the FIFGARCH model to represent longmemory asymmetric conditional variances. Although he claims thatthis model nests many previous models, we show that it does notand that the model is badly specied. We propose an alternativespecication.Keywords: EGARCH, FGARCH, FIGARCH, FIEGARCHJEL classication: C22Author for correspondence: Departamento de Estadistica y Econometria, UniversidadCarlos III de Madrid, C/ Madrid 126, 28903 Getafe, Madrid (Spain), Tel: 34 91 6249851, Fax: 34 91 624 9849, e-mail: ortega@est-econ.uc3m.es. Financial support fromproject PB98-0026 from the Spanish Government is gratefully acknowledged by the rstauthor. We are very grateful to Daniel Pe~na for helpful comments on the paper. Theusual disclaims apply.11 IntroductionThe GARCH model was originally proposed by Engle (1982) and Bollerslev(1986) to represent the dynamic evolution of conditional variances. One ofthe most useful applications of GARCH models is to represent high frequencynancial returns. The GARCH model has been extended in many directions.Two of the most popular generalizations deal with the asymmetric responseof the conditional variance, 2t; to positive and negative returns and with thepersistence often observed in the autocorrelations of squared returns.The rst asymmetric model proposed in the nancial econometrics liter-ature was the Exponential GARCH (EGARCH) model of Nelson (1991). Tosimplify the exposition, we will focus on the simplest EGARCH(1,1) modeland their equivalents. In the EGARCH(1,1) model, the series of returns isgiven by "t= zttwhere ztis an independent white noise process with unitvariance and symmetric distribution and the conditional variance of returnsis given by:log(2t) = ! + (1  L) 1(1 +  L)g(zt 1) (1)where g(zt) = zt+  [jztj   E(jztj)] and L is the lag operator such thatLjxt= xt j. The parameter  measures the asymmetric response of volatility.Later, Henstchel (1995) proposed the Family GARCH (FGARCH) modelthat nests many previous heteroscedastic models. The conditional varianceis given by:t  1= !0+ t 1f(zt 1) + Æt 1  1(2)f(zt) = jzt  bj   c(zt  b)The asymmetry is introduced by shifting and rotating the absolute valueof the shock through the parameters b and c respectively. The parameter represents the Box and Cox (1964) transformation for the conditional vari-ance so that the limit when  goes to zero is the logarithmic transformation.The parameter  serves to transform the function f(). Henstchel (1995)shows that model (2) encompasses, among many others, the GARCH model,the EGARCH model, the threshold GARCH (TGARCH) model of Zakoian(1994) and the Nonlinear GARCH (NGARCH) model of Higgins and Bera(1992).2With respect to the persistence of volatility, Baillie et al. (1996) proposethe Fractionally Integrated GARCH (FIGARCH) model where squared re-turns have long memory. The conditional variance of the FIGARCH(1,d,1)model is given by2t= !+h1  (1  L) 1(1  L) (1  L)di"2t(3)where d is the long memory parameter such that 0 d < 1: The FIGARCH(1,d,1)model in (3) has been derived by introducing a fractional root in the autore-gressive polynomial of the ARMA representation of squared observations ofa GARCH model. However, in the appendix we show how it is possibleto obtain the following alternative representation of the FIGARCH(1,d,1)model:2t= !00+ (1  L) 1(1  L) d(1  ÆL)2t 1(z2t 1  1) (4)At the moment, the only long memory asymmetric model of the GARCHfamily is the Fractionally Integrated EGARCH (FIEGARCH) model pro-posed by Bollerslev and Mikkelsen (1996). The conditional variance of theFIEGARCH(1,d,1) model is given bylog(2t) = ! + (1  L) 1(1  L) d(1 +  L)g(zt 1) (5)Hwang (2001) proposes a new asymmetric long memory model, the asym-metric FIFGARCH model, and claims that it nests the FIGARCH and FIE-GARCH models. In section 2, we will show that this model is badly speciedand does not encompass the previously mentioned models. We propose analternative specication, based on the asymmetric FIFGARCH model, thatseems to fulll the objectives stated by Hwang (2001).2 The asymmetric long memory GARCHmodel.Hwang (2001) proposes the asymmetric FIFGARCH(1,d,1) model to repre-sent both the long memory and asymmetry properties of conditional standarddeviations. The specication of the conditional variance is given by:t= +h1  (1  ÆL) 1(1  L)(1  L)dif(zt)t(6)Looking at equation (6), it seems that Hwang (2001) has tried to gen-eralize the FGARCH model in equation (2) to allow for long memory, by3introducing the fractional root, (1   L)d, in the volatility equation. Alter-natively, the FIFGARCH(1,d,1) model could be seen as a generalization ofthe FIGARCH model in (3) with the conditional variance depending on atransformation similar to the one proposed by Henstchel (1995), instead ofdepending on squared returns, so that the model could deal with asymme-try. In any case, Hwang (2001) does not refer to any of the previous relatedpapers in this area.Notice that model (6) is not able to represent adequately the simultaneouspresence of long memory and asymmetry in the conditional variance. First ofall, the long memory property is somehow dened in a strange way becausethe factor (1   L)ddoes not aect the corresponding transformation of theconditional standard deviation, but to the function f(). The way that thelong memory is introduced in equation (6) seems to be similar to the way thatthe GARCH model is generalized to the FIGARCH model in (3). However,this latter generalization is based on the ARMA representation of squaredreturns implied by the GARCH model; see, Baillie et al. (1996) and theappendix.Secondly, Hwang (2001) is not considering a Box-Cox transformation ofthe conditional standard deviation and, as a consequence, model (6) is badlymisspecied when  = 0. In this case, the model becomes:1 = +h1  (1  ÆL) 1(1  L)(1  L)dif(zt) (7)and the conditional standard deviation has disappeared. Therefore, the FIE-GARCH model cannot be obtained from model (6) when  = 0 as Hwang(2001) claims. But even if the asymmetric FIFGARCH(1,d,1) model weredened in such a way that, when  = 0, the logarithmic transformation wouldbe obtained and considering  = 1 and b = 0, the model would become:log(2t) = +h1  (1  ÆL) 1(1  L)(1  L)di(jztj   czt) log(2t) (8)which is not a FIEGARCH model.Finally, notice that the FIGARCH model in (3) can be obtained as aparticular case of model (6) when  =  = 2 and c = b = 0. However, thismodel is not able to represent asymmetries in the response of conditionalvolatilities to negative and positive returns. On the other hand, if d = 0 inequation (6) and the Box-Cox transformation is applied to tin the left handside of such equation, then the short memory Hentchel's FGARCH model4will be obtain. However, it is not obvious that the asymmetric FIFGARCHmodel is able to encompass adequately models with conditional variancesthat simultaneously are asymmetric and have long memory.Alternatively, we propose the following model based on a direct gen-eralization of the FGARCH model introducing a fractional unit root andsubtracting the unconditional mean of f(zt) from the function f(zt):t  1= !0+h(1  ÆL) 1(1  L) d(1  L)it 1ff(zt 1) E(f(zt 1))g(9)If  = 0;  = 1, b = 0 and  = 1, we obtain the FIEGARCH model.Also, if  =  = 2, b = c = 0 and  = 1, the FIGARCH model in (4) isobtained. Finally, if d =  =  = 0, the FGARCH model in (2) would beobtained. Remember that, as previously mentioned, the FGARCH modelnests the short memory GARCH and EGARCH models.However, it will be necessary to analyze in detail the statistical propertiesof this model as well as the properties of the estimators of the parametersbefore the model could be applied to the analysis of real time series.3 ConclusionsThe asymmetric FIFGARCH model has been proposed by Hwang (2001) torepresent adequately the asymmetry and persistence often observed in theconditional variances of high frequency series of returns. It seems that theFIFGARCH model is trying to generalize the model proposed by Henstchel(1995) in the same way the GARCH model has been generalized to the FI-GARCH model. However, contrary to the claims made by Hwang (2001), wehave shown that the asymmetric FIFGARCH is not able to encompass pre-vious models proposed in the literature with the exception of the FIGARCHmodel. We think that the new asymmetric FIFGARCH should be deeplyrevised before it can be safely applied to the analysis of real nancial timeseries. We propose an alternative specication that encompasses the mostpopular long memory and asymmetric models for conditional variances.4 AppendixThe GARCH(2,1) model is given by:5"t= ztt(10)2t= ! + 1"2t 1+ 2"2t 2+ 2t 1Alternatively, model (10) can be written as an ARMA(2,1) for squaredobservations as follows:"2t= ! + 1"2t 1+ 2"2t 2+ ("2t 1  t 1) + t(11)where t= "2t  2t: Reorganizing terms in (11), it is possible to obtain thefollowing expression:[1  (1+ )L  2L2]"2t= !+ (1  L)t(12)If there is a unit root, the autoregressive polynomial in (12) can be factor-ized as [1  (1+)L 2L2] = (1 L)(1 L) and the following expressionis obtained for "2t:(1  L)(1  L)"2t= !+ (1  L)t(13)However, if there is a fractional root instead of a unit root, the followingmodel for squared observations is obtained(1  L)(1  L)d"2t= !+ (1  L)t(14)Finally, substituting t= "2t  2tin (14), the FIGARCH(1,d,1) modelin (3) is directly obtained. However, it is possible to obtain an alternativeexpression of the FIGARCH(1,d,1) model by considering directly the expres-sion of the variance in (10):2t= ! + (1+ 2L)"2t 1+ 2t 1+ (1+ 2L)2t 1  (1+ 2L)2t 1=(15)= ! + (1+ 2L)2t 1(z2t 1  1) + ( + 1+ 2L)2t 1(16)Reorganizing terms in (15), the following expression is obtained:??2t= !+ [1  ( + 1)L  2L2] 1(1+ 2L)2t 1(z2t 1  1) (17)6If similarly to what we have done in expression (14), we factorize the au-toregressive polynomial allowing for a fractional root, then the FIGARCH(1,d,1)model in (4) is obtained:2t= !+ (1  L) 1(1  L) d(1  ÆL)2t 1(z2t 1  1) (18)where  = 1and Æ =  21:7References[1] Baillie, R.T., T. Bollerslev and H.O. Mikkelsen, 1996, Fractionally inte-grated autoregressive conditional heteroskedasticity, Journal of Econo-metrics 74, 3-30.[2] Bollerslev, T., 1986, Generalized autoregressive conditional het-eroskedasticity, Journal of Econometrics 31, 307-327.[3] Bollerslev, T. and H.O. Mikkelsen, 1996, Modeling and pricing longmemory in stock market volatility, Journal of Econometrics 73, 151-184.[4] Box, G.E.P. and D.R. Cox, 1964, An Analysis of transformations, Jour-nal of the Royal Statistical Society B 26, 211-243.[5] Engle, R. F., 1982, Autoregressive conditional heteroskedasticity withestimates of the variance of U.K. ination, Econometrica 50, 987-1007.[6] Hentschel, L., 1995, All in the family. Nesting symmetric and asymmet-ric GARCH models, Journal of Financial Economics 39, 71-104.[7] Higgins, M.L. and A.K. Bera, 1992, A class of nonlinear ARCH models,International Economic Review 33, 137-158.[8] Hwang, Y., 2001, Asymmetric long memory GARCH in exchange return,Economics Letters 71, 1-5.[9] Nelson, D.B., 1991, Conditional heteroskedasticity in asset returns: Anew approach, Econometrica 59, 347-370.[10] Zakoian, J.M., 1994, Threshold heteroskedastic models, Journal of Eco-nomic Dynamics and Control 18, 931-955.8

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