Applied Financial Economics, 1998, 8, 145 Ð 153

Modelling the asymmetry of stock marketvolatility

OLAN HENRY

Department of Economics, University of Melbourne, Parkville, V ictoria 3052, Australia

Recent studies suggest that a negative shock to stock prices will generate morevolatility than a positive shock of equal magnitude. This paper uses daily data fromthe Hong Kong Stock Exchange to illustrate the nature of stock market volatility.Regression-based tests for integration in variance are applied, providing contrastingresults to the usual test based on the Wald statistic. A partially non-parametric modelof the relationship between news and volatility is estimated and used in conjunctionwith tests for the sensitivity to both the size and sign of a shock as a metric to judgevarious candidate characterizations of the underlying data generating process.

I . INTRODUCTION

There appears to be widespread agreement that the vola-tility of asset returns is, to a degree, forecastable. Recently,a great deal of attention has been focused on this topic, seeBollerslev et al. (1992, 1994) for detailed surveys. Theorysuggests that the price of an asset is a function of thevolatility, or risk, of the asset. Consequently, an understand-ing of how volatility evolves over time is central to thedecision making process. Moreover, optimal inferenceabout the conditional mean of a variable requires that theconditional second moment be correctly speci® ed. Misspeci-® ed models of stock volatility may lead to incorrect, orinvalid, conclusions about stock return dynamics.

Black (1976), Christie (1982), Nelson (1991), Pagan andSchwert (1990), Sentena (1992), Campbell and Hentschel(1992) and Engle and Ng (1993) all report evidence thatsuggests that a negative shock to stock returns will generatemore volatility than a positive shock of equal magnitude.Black and Christie suggest that as stock prices fall, theweight attached to debt in the capital structure increases.This increase in leverage will lead equity holders, who bearthe residual risk of the ® rm, to anticipate higher expectedfuture returns volatility. Both authors ® nd that their predic-tions are satis® ed for data on individual stock returns.

However, there remains no general agreement as to howthe predictability of volatility should be modelled and, inparticular, how to condition such models for the asymme-tric nature of stock return volatility. In Section II of thispaper various models of stock return volatility, both sym-metric and asymmetric are outlined. Section III describes

the data. Section IV discusses the estimation and testingprocedure and presents initial empirical results. SectionV presents the partially non-parametric model of stockreturn volatility, and estimates of the relationship betweennews and volatility for each of the candidate models. The® nal section provides a brief summary and conclusion.

II . MODELLING ASYMMETRY IN THEVOLATILITY OF EQUITY RETURNS

Let Rt be the continuously compounded rate of return ona stock, or a portfolio of stocks, over a single period fromtime t - 1 to t. Furthermore, denote the information avail-able to investors at time t - 1, when the investment decisionis made, as It ± 1 . The expected return and volatility of returnspertinent to such decisions are the conditional mean andvariance of Rt given It ± 1 , denoted as yt = E (Rt | It ± 1 ) andht = Var(Rt | It ± 1 ) respectively. Using these de® nitions theunexpected return at time t is Rt - yt = e t . This paper followsEngle and Ng (1993) in treating e t as a collective measure ofnews. An unexpected increase in returns (a positive value of e t)indicates the arrival of good news, while e t < 0 indicates badnews. The magnitude of | e t | implies important news in thesense that it will re¯ ect a signi® cant change in price.

Engle (1982) presents the ARCH model which speci® esthe conditional variance, ht , as a distributed lag over pastsquared innovations e 2

t ± i as shown by

ht = v +p

+i = 1

a i e2t ± i (1)

0960 Ð 3017 Ó 1998 Routledge 145

1 The parameter subscripts are not necessary for the ARCH(1) and GARCH(1,1) models and are suppressed for the remainder of the paper.2 Both rt and ut were considered in the preliminary work on this paper, with no qualitative e� ect on the results. For brevity only the resultsfor rt are reported.

where v > 0, a 1 , ¼ , a p > 0 are constant parameters. Theconditional variance under the ARCH(p) model re¯ ectsonly information from time t - p to t - 1 with more im-portance being placed on the most recent innovations,implying a i < a j for i > j. Given the degree of di� culty inselecting the optimal lag length p, and ensuring the non-negativity of the coe� cients of the conditional varianceequation, Bollerslev (1986) presented the GeneralizedARCH or GARCH model

ht = v +p

+i = 1

a i e2t ± i +

q

+j = 1

b jht ± j (2)

where v > 0, a 1 , ¼ , a p > 0, b 1 , ¼ , b q > 0 are constantparameters. The GARCH model corresponds to an in® niteorder ARCH model. A common parameterization for theGARCH model that has been adopted is the GARCH(1,1)speci® cation under which the e� ect of a shock to volatilitydeclines geometrically over time.1

The ARCH(p) and GARCH(p,q) models impose sym-metry on the conditional variance structure which may notbe appropriate for modelling and forecasting stock returnvolatility. Nelson (1991) proposes the exponential GARCHor EGARCH model as a way to deal with this problem.Under the EGARCH(1,1) the conditional variance is givenby

log(ht) = v + a 3 ) e t ± 1

Ï ht ± 1) - Ï 2/p 4

+ b log (ht ± 1 ) + de t ± 1

Ï ht ± 1

(3)

where v , a , b and d are constant parameters. The EGARCHmodel has two distinct advantages over the GARCH model.First, the logarithmic construction of Equation 3 ensuresthat the estimated conditional variance is strictly positive,thus the non-negativity constraints used in the estimation ofthe ARCH and GARCH models are not necessary. Second-ly, since the parameter d typically enters Equation 3 witha negative sign, bad news, e t < 0, generates more volatilitythan good news.

The generalized quadratic ARCH or GQARCH(1,1)model of Sentena (1992) takes the form

ht = v + a (e t ± 1 + d )2 + b ht ± 1 (4)

where v > 0, a > 0, b > 0 are constant parameters. Theestimated value of the parameter d is usually negative, thusEquation 4 responds asymmetrically to positive and nega-tive shocks of equal magnitude. Glosten et al. (1993), here-

after GJR, propose an alternative model

ht = v + a e 2t ± 1 + b ht ± 1 + d Nt ± 1 e 2

t ± 1 (5)

where Nt ± 1 is a dummy variable that takes the value ofunity if e t ± 1 < 0 and zero otherwise. The GJR model isclosely related to the threshold ARCH, or TARCH model ofRabemananjara and Zakoian (1993) and Zakoian (1994).Provided that d > 0, the GJR model generates higher valuesfor ht given e t ± 1 < 0, than for a positive shock of equalmagnitude. As with the ARCH and GARCH models theparameters of the conditional variance, Equation 5, aresubject to non-negativity constraints.

Suppose information is held constant at time t - 2 andbefore, Engle and Ng (1993) describe the relationship be-tween e t ± 1 and ht as the news impact curve. It is the purposeof this study to illustrate the di� culties in deciding upon theshape and location of the relationship between e t ± 1 andht . The news impact curves of the GARCH(1,1) andGQARCH models are symmetric and centred at e t ± 1 = 0and e t ± 1 = - d , respectively. The news impact curves of theEGARCH(1,1) and GJR models are centred at e t ± 1 = 0.The EGARCH(1,1) has a steeper slope for e t ± 1 < 0, pro-vided that d < 0 in Equation 3, while the GJR has di� erentslopes for its positive and negative sides. Table 1 presentsthe relevant news impact curves, evaluating the lagged con-ditional variance ht , at its unconditional level s 2 .

III . DATA DESCRIPTION

The data consist of 1415 observations of the closing value ofthe Hang Seng Index, from the Hong Kong stock market,sampled daily from 01/01/90 to 12/6/95. The data are trans-formed to continuously compounded returns, calculated asRt = log (Pt/Pt ± 1 ), where Pt represents the value of the indexat time t. Pagan and Schwert (1990) suggest a methodwhereby a measure of the unpredictable element of stockreturns may be obtained. To adjust for a possible d̀ay-of-the-week’ e� ect Rt is initially regressed on a constant, andfour day-of-the-week dummy variables and the residual, ut ,is obtained. A possible moving average error due to non-synchronous trading is approximated using a second stageautoregression. The residuals from the regression of ut ona constant and ut ± 1 , ¼ , ut ± 5 , rt , are the unpredictablestock return data. Table 2 presents summary statisticsfor rt.

2

The unconditional mean of rt is zero, by construction. Theunconditional variance is 2.0097, but visual inspection of thetime series plot of the data (Fig. 1) suggests that the vola-tility of rt displays the clustering phenomenon associated

146 O. Henry

Table 1. News impact curves

Model News impact curve

GARCH(1,1) ht = A + a e 2t ± 1

where A = v + b s 2

EGARCH(1,1) ht = A . exp 3 (d + a )s

´ e t ± 1 4 , for e t ± 1 > 0

ht = A . exp 3 (d - a )s

´ e t ± 1 4 , for e t ± 1 < 0

where A º s 2 b ´exp 3 v - a ! 2p 4

GQARCH ht = A + a (e t ± 1 + d )2

where A = v + b s 2

GJR ht = A + a e 2t ± 1 , for e t ± 1 > 0

ht = A + (a + d ) e 2t ± 1 , for e t ± 1 < 0

where A = v + b s 2

Fig. 1. T ime series plot of rt

3 Earlier drafts of this paper consider GARCH-M conditional mean equations of the form

rt = m + l ht + e t

Wald and likelihood ratio tests uniformly satis® ed the null hypothesis H0 : l = 0, for any level of signi® cance. Given that E (ht) = e 2t , this

reinforces the result of the RESET test. There is little evidence of linear or non-linear dependence in the mean of the pre® ltered returnsseries. Furthermore, specifying the conditional mean equation as a GARCH-M as opposed to Equation 6 had no qualitative e� ects on theconclusions drawn here.

with GARCH processes. Large shocks of either sign tend tobe followed by large shocks, and small shocks of either signtend to follow small shocks.

The observation of volatility clustering is reinforced bythe plot of the absolute value of rt (Fig. 2). There is signi® -cant evidence of ARCH in the data, as shown by the test fortenth order ARCH and the Ljung Ð Box Q statistic on thesquared return data. There is, however, no evidence of serialcorrelation in the mean as shown by the Ljung Ð Box Q stat-istic for the pre® ltered return data. Furthermore, the nullhypothesis of no higher order non-linear dependence in rt ,i.e. dependence between ut and u2

t was satis® ed at the 5%level using Ramsey’s (1969) RESET test.

The estimated unconditional density function for rt isclearly skewed to the left and markedly leptokurtic whencompared with the standard normal distribution, as shownin Fig. 3. This is reinforced by the Bera Ð Jarque test fornormality, which is signi® cant at any reasonable level ofcon® dence.

IV . ESTIMATION AND HYPOTHESISTESTING

The conditional mean return, which is assumed to be entire-ly unpredictable, is modelled as3

rt = e t (6)

Stacking the parameters of the various models into a vector,C , the log likelihood for a sample of T observations (condi-tional on initial values) is proportional to

L (C ) =T

+t= 1

{ - log | ht | - (e 2t /ht)} (7)

which assumes conditional normality of the forecast errors.The Broyden, Fletcher, Goldfarb and Shanno (BFGS) algo-rithm Ð see Press et al. (1988) for details Ð was used to obtainthe parameter estimates and relevant standard errors.

Preliminary results suggest that the assumption of nor-mally distributed standardized innovations zt = e t/ Ï ht maybe tenuous. Weiss (1986) and Bollerslev and Wooldridge(1992) argue that asymptotically valid inference regardingnormal Quasi-Maximum Likelihood Estimates (QMLE)resulting from Equation 7, say C Ã , may be based uponr̀obusti® ed’ versions of the standard test statistics. Underfairly weak conditions, an asymptotic robust covariancematrix for the parameter estimates is consistently estimatedby A ( C = ) ± 1 B (C Ã )A ( C Ã ) ± 1 , where A ( C Ã ) and B (C Ã ) denote thehessian and the outer product of the gradients, respectively,evaluated at C Ã . This robust variance Ð covariance estimatormay be used to calculate Wald statistics using

W = c (C Ã )9 [=w c(C Ã )A ( C Ã ) ± 1 B (C Ã )A (C Ã ) ± 1 =

w c (C Ã )9 ] ± 1 c (C Ã )

(8)

where, under the null hypothesis, c( C Ã 0 ) = 0 and =w c( C Ã )

represents the gradient of the likelihood function with

Modelling the asymmetry of stock market volatility 147

Table 2. Summary statistics for rt

rkt var(rt) Sk Ku B Ð J Q (10) Q2 (10) A(10) R(2)

0.000 2.01 - 0.357 4.080 1010.7 3.063 195.82 116.05 3.02[0.000] [0.000] [0.000] [0.980] [0.000] [0.000] [0.08]

Notes: Marginal signi® cance levels displayed as [ . ]. Sk and Ku are tests for zero skewness and excess kurtosis. B Ð Jis the Bera Ð Jarque test for normality, distributed as x 2 (2). Q (10) and Q2 (10) are Ljung Ð Box tests for serialcorrelation in the returns and squared returns data respectively, distributed as x 2 (10). A(10) is Engle’s (1982) testfor tenth order ARCH, distributed as x 2 (10). R(2) is Ramsey’s (1969) RESET test for non-linear dependence in theconditional mean of rt distributed as x 2 (1).

Fig. 2. T ime series plot of | rt |

Fig. 3. Estimated unconditional distribution of rt

respect to the restrictions. Given correct speci® cation ofthe conditional mean and variance equations W has anasymptotic chi-square distribution under the null hypothe-sis, whether or not the conditional normality assumptionholds.

Engle and Ng (1993) describe three tests which examinewhether it is possible to predict the squared normalizedresidual z2

t = e 2t /ht using some variables observed in the past

which are not included in the regression model. De® ne Nt ± 1

as in Equation 5, and let Pt ± 1 = 1 - Nt ± 1 . In the signbias test z2

t is regressed on a constant and Nt ± 1 . If thecoe� cient on Nt ± 1 is signi® cant then positive and negativeinnovations a� ect future volatility di� erently to the predic-tion of the model. The negative size bias test examineswhether the magnitude of negative innovations causes thebias to predicted volatility. The test examines the signi® -cance of Nt ± 1 e t ± 1 in the regression of z2

t on a constantand Nt ± 1 e t ± 1 . The positive size bias test examines thesigni® cance of Pt ± 1 e t ± 1 in the regression of z2

t on a con-stant and Pt ± 1 e t ± 1 . Engle and Ng (1993) show that ajoint test for size and sign bias, based on the LagrangeMultiplier principal, may be obtained by T .R2 from theregression

z2t = / 0 + / 1 Nt ± 1 + / 2 Nt ± 1 e t ± 1 + / 3 Pt ± 1 e t ± 1 + h t

The estimation and test results for the models de® ned inEquations 2 Ð 6, displayed in Table 3, suggest that all themodels pass standard Ljung Ð Box speci® cation tests on ztand z2

t at the 5% level. The test for negative sign bias issigni® cant at the 5% level for the GARCH(1,1) model,implying that the assumption of symmetric volatility maynot be congruent with the asymmetry in the data. Thespeci® cation tests based on the Ljung Ð Box statistic do notappear to capture this bias. Likewise, the parameters b andd in the GJR conditional variance equation are, at best,marginally signi® cant, indicating that the GJR model isa poor characterization of the underlying data generatingprocess.

If +pi = 1 a i + +

qj = 1 b j = 1 in Equation 2 then, using the

terminology of Engle and Bollerslev (1986), the model is saidto be integrated in variance. The null hypothesis of variancenon-stationarity in models (2) Ð (5) is tested using a robusti-® ed Wald test of the form of Equation 8. Lumsdaine (1991)and Lee and Hansen (1994) examine the distribution theoryfor the QML estimator of GARCH(1,1) models. Providingcertain assumptions hold, both studies conclude that, even

148 O. Henry

in the case of IGARCH, QML estimation will be asymp-totically normal. Deb (1995) provides evidence that theQML estimator of the EGARCH(1,1) has poor ® nitesample properties when the data generating process hasconditional excess kurtosis. Pagan (1995) argues that theestimation problem for the EGARCH model may not bewell de® ned under the null of integration in variance. More-over an integrated EGARCH process is neither strictlystationary nor covariance stationary. Psaradakis andTzavalis (1995) suggest a regression-based test for integra-tion in variance for the exponential family of ARCHmodels. They argue that such tests have well de® ned limit-ing distributions under the null hypothesis, which may notbe the case for the Wald test based on the (quasi) maximumlikelihood estimator. Psaradakis and Tzavalis base theirinference on the logarithmic GARCH(1,1) process, writtenas

ht = exp(v + a ln e 2t ± 1 + b ln ht ± 1 ) (9)

where b = r - a . Based upon the ARMA form of Equation9 an autoregression such as

ln e 2t = u + g T + r ln e 2

t ± 1 + ut (10)

provides an alternative test, since if r = 1, the process inEquation 10 is integrated. Such a test may be based on thePhillips Ð Perron (1988) non-parametric unit root test, (PP),or indeed the augmented Dickey Ð Fuller unit root test(ADF), (Dickey and Fuller, 1981). Note that in Equation 10u , g and r are constant parameters and T is a time trend.Psaradakis and Tzavalis provide Monte Carlo evidencedemonstrating that tests based on autoregressive approxi-mations of the ARMA representation have minimal sizedistortion, and appeared most robust to model misspeci® ca-tion. Their procedure extends to the EGARCH(1,1) casesince, following Pantula (1986), Equation 3 may be rear-ranged to yield

ln e 2t = v + r ln e 2

t ± 1 + d zt ± 1 + a [ | zt ± 1 | - E | zt ± 1 |]

+ j t - r j t ± 1 (11)

where j t = ln(z2t ). If r = 1 then the process in (11) is inte-

grated. Since the noise function in Equation 11 can beshown to have a MA(1) representation, Psaradakis andTzavalis argue that regression-based criteria provide appro-priate statistics for testing the hypothesis of integration inconditional variance.

Given that returns are the product of two stationaryprocesses e t = zt Ï ht , the GARCH model represents astrictly stationary process. However, under IGARCH thevariance of returns does not exist, thus the process is notcovariance stationary, see Nelson (1990) and Pagan (1995)for further details. Nelson (1990) demonstrates that even

under the null of integration in variance the series e 2t is

stationary. Thus, using the ARMA form of Equations 2, 4or 5, the null hypothesis H0 : r = 1 may be tested usingstandard distribution theory as suggested by Tzavalisand Wickens (1993). Glosten et al. (1993) (GJR) comparethe degree of persistence across various models infor-mally by regressing the estimate of ht , h= t on a constantand h= t ± 1 . As the slope coe� cient from this regression, g ,approaches one, the degree of persistence in varianceapproaches in® nity.

The Wald tests for in® nite persistence in the GARCH(1,1)and GJR models appear to support the null hypothesis ofintegrated variance, while the opposite result is obtained forthe EGARCH and GQARCH models. The regression-based tests for in® nite persistence fail to satisfy the nullhypothesis for all models. The QML-based Wald test maybe less robust to misspeci® cation of the conditional varianceequation than the regression-based tests, which in line withthe conclusion of Psaradakis and Tzavalis (1995). Furtherinvestigation of this potential bias using Monte-Carlomethods is beyond the scope of this paper and is left forfuture research.

Using a likelihood ratio criterion, where appropriate, theasymmetric volatility models appear to be more data consis-tent than the symmetric GARCH model. Moreover, theresiduals from the estimation of the asymmetric modelsappear free from negative size bias. As a further diagnosticcheck for the adequacy of the various parameterizations ofthe conditional variance equations the moment type speci-® cation test suggested by Pagan and Sabau (1992) wascomputed from the regression

e W 2t = / 0 + / 1 h= t + h t (12)

where e W2t and h= t are the squared innovations and the esti-

mated conditional variances, respectively, from the modelsreported in Table 3. Under the null hypothesis of correctspeci® cation, the moment condition E (e 2

t | It ± 1 ) = ht impliesthat / 0 = 0 and / 1 = 1. The results of the ordinary leastsquares estimation of (12) presented in Table 4 suggest that/ 0 = 0 and / 1 = 1 for all the models. Note that the appar-ent misspeci® cation of the GARCH(1,1) and GJR models isnot highlighted by the test. However, the Pagan-Sabaustatistic may have better power as a speci® cation test forARCH-M models.

V. ESTIMATING NEWS IMPACT CURVES

Engle and Ng (1993) propose a partially non-parametric(PNP) model, shown as Equation 13, which uses linearsplines with kinks at e t ± 1 equal to 0, 6 s , 6 2 s , 6 3sand 6 4 s to estimate the shape of the news impact curve.The relationship between news and volatility is treatednon-parametrically , while the long memory component is

Modelling the asymmetry of stock market volatility 149

Table 3. Estimates of the parametric volatility models

GARCH EGARCH GJR GQARCH

v 0.1063 0.0442 0.1248 0.0983(12.4380) (4.1861) (3.4356) (2.7870){5.7405} {1.6818} {1.0780} {1.2442}

a 0.1100 0.2116 0.0591 0.1129(10.0692) (6.9803) (2.7377) (4.9011){5.7616} {4.4702} {1.5678} {2.6284}

b 0.8385 0.9431 0.8272 0.8261(162.6138) (69.8148) (23.9300) (22.0015){74.6997} {30.5978} {9.7845} {9.4241}

d - 0.3183 0.1040 - 0.4763( - 4.0377) (3.0238) ( - 3.6042){ - 2.7568} {1.2074} { - 2.2234}

Log likelihood - 1087.6164 - 1075.2466 - 1079.5250 1080.2552Sk - 0.6193 - 0.7203 - 0.5392 - 0.5058

[0.0000] [0.0000] [0.0000] [0.0000]Ku 5.8266 5.2405 5.4218 5.1728

[0.0000] [0.0000] [0.0000] [0.0000]Q (10) 9.4931 8.8215 9.3190 9.5780

[0.4860] [0.5491] [0.5021] [0.4783]Q2 (10) 3.8787 3.6905 3.2662 3.6425

[0.9543] [0.9491] [0.9745] [0.9620]IGARCH 1.1549 14.3277 1.5145 8.3710Wald [0.2825] [0.0002] [0.2185] [0.0038]H0 : r = 1 - 16.4759 - 21.3370 - 16.2794 - 16.8200g 0.9209 0.9336 0.9156 0.9271

{0.0104} {0.0096} {0.0107} {0.0100}Size bias 1.3887 0.7297 0.8221 0.5587Negative sign bias - 2.7245 - 1.2195 - 1.1642 - 1.2158Positive sign bias - 1.0765 - 0.2290 - 0.2619 - 0.2178Joint test 5.2305 1.5840 1.5096 1.5369~ x 2 (3) [0.1556] [0.6630] [0.6801] [0.6738]

Notes: See notes to Table 2. Asymptotic t-ratios, based on the inverse of the Hessian A (C Ã ) ± 1 are displayed as( . ). Robust t-ratios displayed as { .}. Marginal signi® cance levels displayed as [ . ]. IGARCH is a robust Waldtest of the null H0 : a + b = 1.

Table 4. Moment speci® cation test for the estimated conditional variance models

GARCH-M GARCH EGARCH GJR GQARCH

/ 0 0.2062 0.2062 - 0.1767 0.2290 0.1572{0.6611} {0.6611} { - 0.5173} {0.7367} {0.4782}

/ 1 0.8890 0.8890 1.0984 0.8688 0.9162{5.1888} {5.1889} {5.7246} {5.1398} {5.1989}

Notes. Heteroscedasticity consistent t-ratios, calculated using the White (1980) estimator are reported as { .}

parametric. The PNP model is of the form

ht = v + b ht ± 1 +m+

+i = 0

u iPi t ± 1 (e t ± 1 - is )

+m ±

+i = 0

d iNi t ± 1 (e t ± 1 + i s ) (13)

In this case the PNP model is estimated for m+ = m ± = 4,with the kinks at e t ± 1 equal to 0, 6 0.5 s , 6 s , 6 1.5 sand 6 2 s , yielding ten coe� cients of the news impact curve.The results are displayed in Table 5.

Comparison of the coe� cients corresponding toPit ± 1 (e t ± 1 - i s ) and Ni t ± 1 (e t ± 1 + is ) suggests that thenews impact curve for the Hong Kong market is indeed

150 O. Henry

Table 6. Estimated news impact curves

e t ± 1 GARCH EGARCH GJR GQARCH PNP

- 10 12.7918 71.6421 18.0984 14.1543 4.0243- 9 10.7018 49.2976 14.9993 11.9008 3.6801- 8 8.8317 33.9221 12.2264 9.8732 3.3359- 7 7.1817 23.3421 9.7797 8.0175 2.9917- 6 5.7516 16.0619 7.6592 6.4957 2.6475- 5 4.5416 11.0523 5.8650 5.1457 2.3032- 4 3.5516 7.6052 4.3970 4.0216 1.9590- 3 2.7815 5.2332 3.2552 3.1234 1.6148- 2 2.2315 3.6010 2.4397 2.4511 1.2706- 1 1.9015 2.4779 1.9503 2.0047 0.9264

0 1.7915 1.7051 1.7872 1.7842 0.58211 1.9015 1.5814 1.8464 1.7895 0.62682 2.2315 1.4668 2.0237 2.0208 0.62653 2.7815 1.3605 2.3194 2.4779 2.31094 3.5516 1.2618 2.7333 3.1609 2.23465 4.5416 1.1704 3.2654 4.0698 2.15836 5.7516 1.0855 3.9158 5.2045 2.08197 7.1817 1.0068 4.6844 6.5652 2.00568 8.8317 0.9338 5.5713 8.1517 1.92939 10.7018 0.8661 6.5765 9.9641 1.8529

10 12.7918 0.8033 7.6999 12.0024 1.7766

Table 5. Estimates of the PNP model of the news impact curve

v 0.0557(3.7532){2.7915}

b 0.8463(50.3924){27.6643}

u 0 0.1174 d 0 - 0.1973(3.5535) ( - 3.5864){2.5410} { - 2.5478}

u 1 - 0.1070 d 1 - 0.3744( - 0.9279) ( - 4.3020){ - 0.5107} { - 2.5249}

u 2 0.4144 d 2 0.8171(2.7929) (1.9401){2.1383} {1.3324}

u 3 - 0.1633 d 3 - 2.7559( - 1.5602) ( - 2.7817){ - 1.4385} { - 1.7882}

u 4 - 0.3343 d 4 2.1662( - 1.0511) (2.3857){ - 0.8439} {1.5948}

Notes: Asymptotic t-ratios displayed as ( . ). Robust t-ratios dis-played as { .}.

asymmetric. The parameters u 1 , ¼ , u 4 are, at best, margin-ally signi® cant, suggesting that the reaction to good news isreasonably independent of the magnitude of the shock. Theparameters d 1 , ¼ , d 4 are all strongly signi® cant, indicatingthat the magnitude of bad news has a non-negligible e� ectand suggesting that the news impact curve is likely to besteeper for e t ± 1 > 0. This conclusion is subject to a caveat:while there may be a reasonable number of observations for

Pit ± 1 (e t ± 1 - is ) and Nit ± 1 (e t ± 1 + is ), for i = 0, 0.5 and 1,the likelihood of there being a large number of extremeshocks is low (i = 1.5, 2.0). However, d 3 and d 4 are reason-ably signi® cant using asymptotic or robust standard errors.

The news impact curve for the PNP model may be cal-culated as

ht = A +m+

+i = 0

u iPit ± 1 (e t ± 1 - is ) +m ±

+i = 0

d iNit ± 1 (e t ± 1 + is )

(14)

where A = v + b s 2 with i = 0, 6 0.5, 6 1, 6 1.5, and6 2. Table 6 reports news impact curves calculated for thevarious models for a range of values for e t ± 1 , assuming thats 2 = ht = 2.00972.

Relative to the asymmetric models, the symmetricGARCH(1,1) model tends to overstate the variance fore t ± 1 > 0 and to understate the variance for e t ± 1 < 0. How-ever, for large negative values of e t ± 1 the EGARCH modelreturns unreasonably large estimates of ht . For a value of- 10 for e t ± 1 the estimated conditional variance from the

EGARCH news impact curve is 71.6421, almost 36 times thesize of the unconditional variance. Additionally, for largepositive values of e t ± 1 the estimated conditional variancedeclines for the EGARCH, which is unattractive. The newsimpact curve estimates suggest that the EGARCH, model istoo extreme in the tails, and thus is an inadequate character-ization of the conditional variance of the Hong Kong stockmarket. The GQARCH model appears to be the mostadequate representation of the underlying data generatingmechanism, given the insigni® cant parameter estimates inthe conditional variance equation of the GJR model.

Modelling the asymmetry of stock market volatility 151

VI . SUMMARY AND CONCLUSIONS

This paper applies the news impact curve of Engle and Ng(1993) as a metric for the speci® cation of models of theconditional volatility of stock returns. The standardGARCH(1,1) model, which imposes symmetry on the con-ditional variance of stock returns, is shown to producebiased estimates of ht when stock price movements are largeand negative (e t ± 1 < 0). The estimated news impact curvefor the GARCH(1,1) suggests that ht is underestimated forlarge negative shocks and overestimated for large positiveshocks. This bias is not detected by the Ljung Ð Box Q (10)statistic on the standardized squared residuals, or the Paganand Sabau (1992) moment speci® cation test. Moreover,a robusti® ed Wald test for integration in variance suggeststhat shocks to volatility are in® nitely persistent, in thesense that the optimal k-step-ahead linear forecast of theconditional variance continues to depend on the initialconditions for all forecast horizons. Using the regressionbased methodology suggested by Psaradakis and Tzavalis(1995) the null of in® nite persistence in variance is notsatis® ed.

Asymmetric models were ® tted to the pre-® ltered dailyreturns data from the Hong Kong Stock market. Initiallythe EGARCH model appears the most valid, in terms of theusual diagnostic statistics. Further examination of the vari-ous news impact curves suggests that the EGARCH(1,1)model is overly sensitive to extremely large positive andnegative shocks. The conditional variance equation of theGJR model contains two parameters b and d that are, atbest, marginally signi® cant. Consequently, the GQARCHmodel, which passes all the tests and appears relativelycongruent with the asymmetry inherent in the data, appearsto be the most adequate characterization of the underlyingdata generating process.

ACKNOWLEDGEMENTS

I am grateful to Charles Ward and Christopher Brooks ofthe University of Reading, seminar participants at the Uni-versity of Melbourne and University of Adelaide, andparticipants at the 1996 Australasian Meeting of the Econo-metric Society for comments on earlier drafts of this paper.All estimation was performed using RATS version 4.2 ona 486 PC. The data and estimation routines are available onrequest from the author. The usual disclaimer applies forany remaining errors or omissions.

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