the Hong Kong Stock Exchange to illustrate the nature of stock market volatility. Regression-based ... suggests that a negative shock to stock returns will generate more volatility than ...... den (Eds), North Holland, Amsterdam. Campbell, J. and ...
Applied Financial Economics, 1998, 8, 145—153
Modelling the asymmetry of stock market volatility OLA N HENRY Department of Economics, ºniversity of Melbourne, Parkville, »ictoria 3052, Australia
Recent studies suggest that a negative shock to stock prices will generate more volatility than a positive shock of equal magnitude. This paper uses daily data from the Hong Kong Stock Exchange to illustrate the nature of stock market volatility. Regression-based tests for integration in variance are applied, providing contrasting results to the usual test based on the Wald statistic. A partially non-parametric model of the relationship between news and volatility is estimated and used in conjunction with tests for the sensitivity to both the size and sign of a shock as a metric to judge various candidate characterizations of the underlying data generating process. I. INTRO DUCTI ON There appears to be widespread agreement that the volatility of asset returns is, to a degree, forecastable. Recently, a great deal of attention has been focused on this topic, see Bollerslev et al. (1992, 1994) for detailed surveys. Theory suggests that the price of an asset is a function of the volatility, or risk, of the asset. Consequently, an understanding of how volatility evolves over time is central to the decision making process. Moreover, optimal inference about the conditional mean of a variable requires that the conditional second moment be correctly specified. Misspecified models of stock volatility may lead to incorrect, or invalid, conclusions about stock return dynamics. Black (1976), Christie (1982), Nelson (1991), Pagan and Schwert (1990), Sentena (1992), Campbell and Hentschel (1992) and Engle and Ng (1993) all report evidence that suggests that a negative shock to stock returns will generate more volatility than a positive shock of equal magnitude. Black and Christie suggest that as stock prices fall, the weight attached to debt in the capital structure increases. This increase in leverage will lead equity holders, who bear the residual risk of the firm, to anticipate higher expected future returns volatility. Both authors find that their predictions are satisfied for data on individual stock returns. However, there remains no general agreement as to how the predictability of volatility should be modelled and, in particular, how to condition such models for the asymmetric nature of stock return volatility. In Section II of this paper various models of stock return volatility, both symmetric and asymmetric are outlined. Section III describes 0960—3017 ( 1998 Routledge
the data. Section IV discusses the estimation and testing procedure and presents initial empirical results. Section V presents the partially non-parametric model of stock return volatility, and estimates of the relationship between news and volatility for each of the candidate models. The final section provides a brief summary and conclusion.
II . MODEL LI NG AS Y MMET R Y IN T HE V O LA T IL IT Y O F EQU I TY R ET U R N S Let R be the continuously compounded rate of return on t a stock, or a portfolio of stocks, over a single period from time t!1 to t. Furthermore, denote the information available to investors at time t!1, when the investment decision is made, as I . The expected return and volatility of returns t~1 pertinent to such decisions are the conditional mean and variance of R given I , denoted as y "E(R D I ) and t t~1 t t t~1 h "Var(R D I ) respectively. Using these definitions the t t t~1 unexpected return at time t is R !y "e . This paper follows t t t Engle and Ng (1993) in treating e as a collective measure of t news. An unexpected increase in returns (a positive value of e ) t indicates the arrival of good news, while e (0 indicates bad t news. The magnitude of D e D implies important news in the t sense that it will reflect a significant change in price. Engle (1982) presents the ARCH model which specifies the conditional variance, h , as a distributed lag over past t squared innovations e2 as shown by t~i p h "u# + a e2 (1) t i t~i i/1 145
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146 where u'0, a , 2 , a *0 are constant parameters. The 1 p conditional variance under the ARCH(p) model reflects only information from time t!p to t!1 with more importance being placed on the most recent innovations, implying a (a for i'j. Given the degree of difficulty in i j selecting the optimal lag length p, and ensuring the nonnegativity of the coefficients of the conditional variance equation, Bollerslev (1986) presented the Generalized ARCH or GARCH model p q h "u# + a e2 # + b h (2) t i t~i j t~j i/1 j/1 where u'0, a , 2 , a *0, b , 2 , b *0 are constant 1 p 1 q parameters. The GARCH model corresponds to an infinite order ARCH model. A common parameterization for the GARCH model that has been adopted is the GARCH(1,1) specification under which the effect of a shock to volatility declines geometrically over time.1 The ARCH(p) and GARCH(p,q) models impose symmetry on the conditional variance structure which may not be appropriate for modelling and forecasting stock return volatility. Nelson (1991) proposes the exponential GARCH or EGARCH model as a way to deal with this problem. Under the EGARCH(1,1) the conditional variance is given by
CK
e
K
D
t~1 !J2/n Jh t~1 e #b log(h )#d t~1 (3) t~1 Jh t~1 where u, a, b and d are constant parameters. The EGARCH model has two distinct advantages over the GARCH model. First, the logarithmic construction of Equation 3 ensures that the estimated conditional variance is strictly positive, thus the non-negativity constraints used in the estimation of the ARCH and GARCH models are not necessary. Secondly, since the parameter d typically enters Equation 3 with a negative sign, bad news, e (0, generates more volatility t than good news. The generalized quadratic ARCH or GQARCH(1,1) model of Sentena (1992) takes the form log(h )"u#a t
h "u#a(e #d)2#bh (4) t t~1 t~1 where u'0, a*0, b*0 are constant parameters. The estimated value of the parameter d is usually negative, thus Equation 4 responds asymmetrically to positive and negative shocks of equal magnitude. Glosten et al. (1993), here-
after GJR, propose an alternative model h "u#ae2 #bh #dN e2 (5) t t~1 t~1 t~1 t~1 where N is a dummy variable that takes the value of t~1 unity if e (0 and zero otherwise. The GJR model is t~1 closely related to the threshold ARCH, or TARCH model of Rabemananjara and Zakoian (1993) and Zakoian (1994). Provided that d'0, the GJR model generates higher values for h given e (0, than for a positive shock of equal t t~1 magnitude. As with the ARCH and GARCH models the parameters of the conditional variance, Equation 5, are subject to non-negativity constraints. Suppose information is held constant at time t!2 and before, Engle and Ng (1993) describe the relationship between e and h as the news impact curve. It is the purpose t~1 t of this study to illustrate the difficulties in deciding upon the shape and location of the relationship between e and t~1 h . The news impact curves of the GARCH(1,1) and t GQARCH models are symmetric and centred at e "0 t~1 and e "!d, respectively. The news impact curves of the t~1 EGARCH(1,1) and GJR models are centred at e "0. t~1 The EGARCH(1,1) has a steeper slope for e (0, prot~1 vided that d(0 in Equation 3, while the GJR has different slopes for its positive and negative sides. Table 1 presents the relevant news impact curves, evaluating the lagged conditional variance h , at its unconditional level p2. t II I. DA TA DESCRI PTION The data consist of 1415 observations of the closing value of the Hang Seng Index, from the Hong Kong stock market, sampled daily from 01/01/90 to 12/6/95. The data are transformed to continuously compounded returns, calculated as R "log(P /P ), where P represents the value of the index t t t~1 t at time t. Pagan and Schwert (1990) suggest a method whereby a measure of the unpredictable element of stock returns may be obtained. To adjust for a possible ‘day-ofthe-week’ effect R is initially regressed on a constant, and t four day-of-the-week dummy variables and the residual, u , t is obtained. A possible moving average error due to nonsynchronous trading is approximated using a second stage autoregression. The residuals from the regression of u on t a constant and u , 2 , u , r , are the unpredictable t~1 t~5 t stock return data. Table 2 presents summary statistics for r .2 t The unconditional mean of r is zero, by construction. The t unconditional variance is 2.0097, but visual inspection of the time series plot of the data (Fig. 1) suggests that the volatility of r displays the clustering phenomenon associated t
1The parameter subscripts are not necessary for the ARCH(1) and GARCH(1,1) models and are suppressed for the remainder of the paper. 2Both r and u were considered in the preliminary work on this paper, with no qualitative effect on the results. For brevity only the results t t for r are reported. t
Modelling the asymmetry of stock market volatility
147
Table 1. News impact curves Model
News impact curve
GARCH(1,1)
h "A#ae2 t t~1 where A"u#bp2
EGARCH(1,1)
h "A . exp t h "A . exp t
C C
D D
(d#a) )e , for e '0 t~1 t~1 p (d!a) )e , for e (0 t~1 t~1 p
C
where A,p2b ) exp u!a
SD 2 n
GQARCH
h "A#a(e #d)2 t t~1 where A"u#bp2
GJR
h "A#ae2 , for e '0 t t~1 t~1 h "A#(a#d)e2 , for e (0 t t~1 t~1 where A"u#bp2
Fig. 1. ¹ime series plot of r t
with GARCH processes. Large shocks of either sign tend to be followed by large shocks, and small shocks of either sign tend to follow small shocks. The observation of volatility clustering is reinforced by the plot of the absolute value of r (Fig. 2). There is signifit cant evidence of ARCH in the data, as shown by the test for tenth order ARCH and the Ljung—Box Q statistic on the squared return data. There is, however, no evidence of serial correlation in the mean as shown by the Ljung—Box Q statistic for the prefiltered return data. Furthermore, the null hypothesis of no higher order non-linear dependence in r , t i.e. dependence between u and u2 was satisfied at the 5% t t level using Ramsey’s (1969) RESET test. The estimated unconditional density function for r is t clearly skewed to the left and markedly leptokurtic when compared with the standard normal distribution, as shown in Fig. 3. This is reinforced by the Bera—Jarque test for normality, which is significant at any reasonable level of confidence.
IV . E S TI M A TI O N A ND HY P OT HE SIS T E ST IN G The conditional mean return, which is assumed to be entirely unpredictable, is modelled as3 r "e t t
(6)
Stacking the parameters of the various models into a vector, (, the log likelihood for a sample of ¹ observations (conditional on initial values) is proportional to T ¸(()" + M!log D h D!(e2/h )N (7) t t t t/1 which assumes conditional normality of the forecast errors. The Broyden, Fletcher, Goldfarb and Shanno (BFGS) algorithm — see Press et al. (1988) for details — was used to obtain the parameter estimates and relevant standard errors. Preliminary results suggest that the assumption of normally distributed standardized innovations z "e /Jh may t t t be tenuous. Weiss (1986) and Bollerslev and Wooldridge (1992) argue that asymptotically valid inference regarding normal Quasi-Maximum Likelihood Estimates (QMLE) resulting from Equation 7, say (ª , may be based upon ‘robustified’ versions of the standard test statistics. Under fairly weak conditions, an asymptotic robust covariance matrix for the parameter estimates is consistently estimated by A((K )~1B((ª )A((ª )~1, where A((ª ) and B((ª ) denote the hessian and the outer product of the gradients, respectively, evaluated at (ª . This robust variance—covariance estimator may be used to calculate Wald statistics using ¼"c((ª )@[+ c((ª )A((ª )~1B((ª )A((ª )~1+ c((ª )@]~1c((ª ) ( ( (8) where, under the null hypothesis, c((ª )"0 and + c((ª ) 0 ( represents the gradient of the likelihood function with
3Earlier drafts of this paper consider GARCH-M conditional mean equations of the form r "k#jh #e t t t Wald and likelihood ratio tests uniformly satisfied the null hypothesis H : j"0, for any level of significance. Given that E(h )"e2 , this 0 t t reinforces the result of the RESET test. There is little evidence of linear or non-linear dependence in the mean of the prefiltered returns series. Furthermore, specifying the conditional mean equation as a GARCH-M as opposed to Equation 6 had no qualitative effects on the conclusions drawn here.
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148 Table 2. Summary statistics for r t rN
t
0.000
var(r ) t 2.01
Sk !0.357 [0.000]
Ku
B—J
Q(10)
Q2(10)
A(10)
R(2)
4.080 [0.000]
1010.7 [0.000]
3.063 [0.980]
195.82 [0.000]
116.05 [0.000]
3.02 [0.08]
Notes: Marginal significance levels displayed as [ . ]. Sk and Ku are tests for zero skewness and excess kurtosis. B—J is the Bera—Jarque test for normality, distributed as s2(2). Q(10) and Q2(10) are Ljung—Box tests for serial correlation in the returns and squared returns data respectively, distributed as s2(10). A(10) is Engle’s (1982) test for tenth order ARCH, distributed as s2(10). R(2) is Ramsey’s (1969) RESET test for non-linear dependence in the conditional mean of r distributed as s2(1). t
Fig. 2. ¹ime series plot of D r D t
Engle and Ng (1993) describe three tests which examine whether it is possible to predict the squared normalized residual z2"e2/h using some variables observed in the past t t t which are not included in the regression model. Define N t~1 as in Equation 5, and let P "1!N . In the sign t~1 t~1 bias test z2 is regressed on a constant and N . If the t t~1 coefficient on N is significant then positive and negative t~1 innovations affect future volatility differently to the prediction of the model. The negative size bias test examines whether the magnitude of negative innovations causes the bias to predicted volatility. The test examines the significance of N e in the regression of z2 on a constant t~1 t~1 t and N e . The positive size bias test examines the t~1 t~1 significance of P e in the regression of z2 on a cont~1 t~1 t stant and P e . Engle and Ng (1993) show that a t~1 t~1 joint test for size and sign bias, based on the Lagrange Multiplier principal, may be obtained by ¹.R2 from the regression z2"/ #/ N #/ N e #/ P e #g t 0 1 t~1 2 t~1 t~1 3 t~1 t~1 t
Fig. 3. Estimated unconditional distribution of r t
respect to the restrictions. Given correct specification of the conditional mean and variance equations ¼ has an asymptotic chi-square distribution under the null hypothesis, whether or not the conditional normality assumption holds.
The estimation and test results for the models defined in Equations 2—6, displayed in Table 3, suggest that all the models pass standard Ljung—Box specification tests on z t and z2 at the 5% level. The test for negative sign bias is t significant at the 5% level for the GARCH(1,1) model, implying that the assumption of symmetric volatility may not be congruent with the asymmetry in the data. The specification tests based on the Ljung—Box statistic do not appear to capture this bias. Likewise, the parameters b and d in the GJR conditional variance equation are, at best, marginally significant, indicating that the GJR model is a poor characterization of the underlying data generating process. If + p a #+ q b "1 in Equation 2 then, using the i/1 i j/1 j terminology of Engle and Bollerslev (1986), the model is said to be integrated in variance. The null hypothesis of variance non-stationarity in models (2)—(5) is tested using a robustified Wald test of the form of Equation 8. Lumsdaine (1991) and Lee and Hansen (1994) examine the distribution theory for the QML estimator of GARCH(1,1) models. Providing certain assumptions hold, both studies conclude that, even
Modelling the asymmetry of stock market volatility
149
in the case of IGARCH, QML estimation will be asymptotically normal. Deb (1995) provides evidence that the QML estimator of the EGARCH(1,1) has poor finite sample properties when the data generating process has conditional excess kurtosis. Pagan (1995) argues that the estimation problem for the EGARCH model may not be well defined under the null of integration in variance. Moreover an integrated EGARCH process is neither strictly stationary nor covariance stationary. Psaradakis and Tzavalis (1995) suggest a regression-based test for integration in variance for the exponential family of ARCH models. They argue that such tests have well defined limiting distributions under the null hypothesis, which may not be the case for the Wald test based on the (quasi) maximum likelihood estimator. Psaradakis and Tzavalis base their inference on the logarithmic GARCH(1,1) process, written as h "exp(u#a ln e2 #b ln h ) t~1 t~1 t
(9)
where b"o!a. Based upon the ARMA form of Equation 9 an autoregression such as ln e2"h#c¹#o ln e2 #u t~1 t t
(10)
provides an alternative test, since if o"1, the process in Equation 10 is integrated. Such a test may be based on the Phillips—Perron (1988) non-parametric unit root test, (PP), or indeed the augmented Dickey—Fuller unit root test (ADF), (Dickey and Fuller, 1981). Note that in Equation 10 h, c and o are constant parameters and ¹ is a time trend. Psaradakis and Tzavalis provide Monte Carlo evidence demonstrating that tests based on autoregressive approximations of the ARMA representation have minimal size distortion, and appeared most robust to model misspecification. Their procedure extends to the EGARCH(1,1) case since, following Pantula (1986), Equation 3 may be rearranged to yield ln e2"u#o ln e2 #dz #a[ D z D!E D z D] t~1 t~1 t~1 t~1 t #m !om t t~1
(11)
where m "ln(z2). If o"1 then the process in (11) is intet t grated. Since the noise function in Equation 11 can be shown to have a MA(1) representation, Psaradakis and Tzavalis argue that regression-based criteria provide appropriate statistics for testing the hypothesis of integration in conditional variance. Given that returns are the product of two stationary processes e "z Jh , the GARCH model represents a t t t strictly stationary process. However, under IGARCH the variance of returns does not exist, thus the process is not covariance stationary, see Nelson (1990) and Pagan (1995) for further details. Nelson (1990) demonstrates that even
under the null of integration in variance the series e2 is t stationary. Thus, using the ARMA form of Equations 2, 4 or 5, the null hypothesis H : o"1 may be tested using 0 standard distribution theory as suggested by Tzavalis and Wickens (1993). Glosten et al. (1993) (GJR) compare the degree of persistence across various models informally by regressing the estimate of h , hK on a constant t t and hK . As the slope coefficient from this regression, c, t~1 approaches one, the degree of persistence in variance approaches infinity. The Wald tests for infinite persistence in the GARCH(1,1) and GJR models appear to support the null hypothesis of integrated variance, while the opposite result is obtained for the EGARCH and GQARCH models. The regressionbased tests for infinite persistence fail to satisfy the null hypothesis for all models. The QML-based Wald test may be less robust to misspecification of the conditional variance equation than the regression-based tests, which in line with the conclusion of Psaradakis and Tzavalis (1995). Further investigation of this potential bias using Monte-Carlo methods is beyond the scope of this paper and is left for future research. Using a likelihood ratio criterion, where appropriate, the asymmetric volatility models appear to be more data consistent than the symmetric GARCH model. Moreover, the residuals from the estimation of the asymmetric models appear free from negative size bias. As a further diagnostic check for the adequacy of the various parameterizations of the conditional variance equations the moment type specification test suggested by Pagan and Sabau (1992) was computed from the regression eL 2"/ #/ hK #g t 0 1 t t
(12)
where eL 2 and hK are the squared innovations and the estit t mated conditional variances, respectively, from the models reported in Table 3. Under the null hypothesis of correct specification, the moment condition E(e2 D I )"h implies t t~1 t that / "0 and / "1. The results of the ordinary least 0 1 squares estimation of (12) presented in Table 4 suggest that / "0 and / "1 for all the models. Note that the appar0 1 ent misspecification of the GARCH(1,1) and GJR models is not highlighted by the test. However, the Pagan-Sabau statistic may have better power as a specification test for ARCH-M models.
V. E ST IM AT IN G N E WS IMP A CT C U R VE S Engle and Ng (1993) propose a partially non-parametric (PNP) model, shown as Equation 13, which uses linear splines with kinks at e equal to 0, $p, $2p, $3p t~1 and $4p to estimate the shape of the news impact curve. The relationship between news and volatility is treated non-parametrically, while the long memory component is
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150 Table 3. Estimates of the parametric volatility models GARCH u
0.1063 (12.4380) M5.7405N 0.1100 (10.0692) M5.7616N 0.8385 (162.6138) M74.6997N
a b d Log likelihood Sk Ku Q(10) Q2(10) IGARCH Wald H : o"1 0 c Size bias Negative sign bias Positive sign bias Joint test &s2(3)
!1087.6164 !0.6193 [0.0000] 5.8266 [0.0000] 9.4931 [0.4860] 3.8787 [0.9543] 1.1549 [0.2825] !16.4759 0.9209 M0.0104N 1.3887 !2.7245 !1.0765 5.2305 [0.1556]
EGARCH 0.0442 (4.1861) M1.6818N 0.2116 (6.9803) M4.4702N 0.9431 (69.8148) M30.5978N !0.3183 (!4.0377) M!2.7568N !1075.2466 !0.7203 [0.0000] 5.2405 [0.0000] 8.8215 [0.5491] 3.6905 [0.9491] 14.3277 [0.0002] !21.3370 0.9336 M0.0096N 0.7297 !1.2195 !0.2290 1.5840 [0.6630]
GJR 0.1248 (3.4356) M1.0780N 0.0591 (2.7377) M1.5678N 0.8272 (23.9300) M9.7845N 0.1040 (3.0238) M1.2074N !1079.5250 !0.5392 [0.0000] 5.4218 [0.0000] 9.3190 [0.5021] 3.2662 [0.9745] 1.5145 [0.2185] !16.2794 0.9156 M0.0107N 0.8221 !1.1642 !0.2619 1.5096 [0.6801]
GQARCH 0.0983 (2.7870) M1.2442N 0.1129 (4.9011) M2.6284N 0.8261 (22.0015) M9.4241N !0.4763 (!3.6042) M!2.2234N 1080.2552 !0.5058 [0.0000] 5.1728 [0.0000] 9.5780 [0.4783] 3.6425 [0.9620] 8.3710 [0.0038] !16.8200 0.9271 M0.0100N 0.5587 !1.2158 !0.2178 1.5369 [0.6738]
Notes: See notes to Table 2. Asymptotic t-ratios, based on the inverse of the Hessian A((ª )~1 are displayed as ( . ). Robust t-ratios displayed as M . N. Marginal significance levels displayed as [ . ]. IGARCH is a robust Wald test of the null H : a#b"1. 0
Table 4. Moment specification test for the estimated conditional variance models
/ 0 / 1
GARCH-M
GARCH
0.2062 M0.6611N 0.8890 M5.1888N
0.2062 M0.6611N 0.8890 M5.1889N
EGARCH !0.1767 M!0.5173N 1.0984 M5.7246N
GJR
GQARCH
0.2290 M0.7367N 0.8688 M5.1398N
0.1572 M0.4782N 0.9162 M5.1989N
Notes. Heteroscedasticity consistent t-ratios, calculated using the White (1980) estimator are reported as M . N
parametric. The PNP model is of the form m` h "u#bh # + h P (e !ip) t t~1 i it~1 t~1 i/0 m~ (e #ip) #+ dN i it~1 t~1 i/0
(13)
In this case the PNP model is estimated for m`"m~"4, with the kinks at e equal to 0, $0.5p, $p, $1.5p t~1 and $2p, yielding ten coefficients of the news impact curve. The results are displayed in Table 5. Comparison of the coefficients corresponding to P (e !ip) and N (e #ip) suggests that the it~1 t~1 it~1 t~1 news impact curve for the Hong Kong market is indeed
Modelling the asymmetry of stock market volatility asymmetric. The parameters h , 2 , h are, at best, margin1 4 ally significant, suggesting that the reaction to good news is reasonably independent of the magnitude of the shock. The parameters d , 2 , d are all strongly significant, indicating 1 4 that the magnitude of bad news has a non-negligible effect and suggesting that the news impact curve is likely to be steeper for e '0. This conclusion is subject to a caveat: t~1 while there may be a reasonable number of observations for
151 P (e !ip) and N (e #ip), for i"0, 0.5 and 1, it~1 t~1 it~1 t~1 the likelihood of there being a large number of extreme shocks is low (i"1.5, 2.0). However, d and d are reason3 4 ably significant using asymptotic or robust standard errors. The news impact curve for the PNP model may be calculated as m` m~ h "A# + h P (e !ip)# + d N (e #ip) t i it~1 t~1 i it~1 t~1 i/0 i/0 (14)
Table 5. Estimates of the PNP model of the news impact curve u b h 0 h 1 h 2 h 3 h 4
0.0557 (3.7532) M2.7915N 0.8463 (50.3924) M27.6643N 0.1174 (3.5535) M2.5410N !0.1070 (!0.9279) M!0.5107N 0.4144 (2.7929) M2.1383N !0.1633 (!1.5602) M!1.4385N !0.3343 (!1.0511) M!0.8439N
d 0 d 1 d 2 d 3 d 4
!0.1973 (!3.5864) M!2.5478N !0.3744 (!4.3020) M!2.5249N 0.8171 (1.9401) M1.3324N !2.7559 (!2.7817) M!1.7882N 2.1662 (2.3857) M1.5948N
Notes: Asymptotic t-ratios displayed as ( . ). Robust t-ratios displayed as M . N.
where A"u#bp2 with i"0, $0.5, $1, $1.5, and $2. Table 6 reports news impact curves calculated for the various models for a range of values for e , assuming that t~1 p2"h "2.00972. t Relative to the asymmetric models, the symmetric GARCH(1,1) model tends to overstate the variance for e '0 and to understate the variance for e (0. Howt~1 t~1 ever, for large negative values of e the EGARCH model t~1 returns unreasonably large estimates of h . For a value of t !10 for e the estimated conditional variance from the t~1 EGARCH news impact curve is 71.6421, almost 36 times the size of the unconditional variance. Additionally, for large positive values of e the estimated conditional variance t~1 declines for the EGARCH, which is unattractive. The news impact curve estimates suggest that the EGARCH, model is too extreme in the tails, and thus is an inadequate characterization of the conditional variance of the Hong Kong stock market. The GQARCH model appears to be the most adequate representation of the underlying data generating mechanism, given the insignificant parameter estimates in the conditional variance equation of the GJR model.
Table 6. Estimated news impact curves e t~1
GARCH
EGARCH
GJR
GQARCH
PNP
!10 !9 !8 !7 !6 !5 !4 !3 !2 !1 0 1 2 3 4 5 6 7 8 9 10
12.7918 10.7018 8.8317 7.1817 5.7516 4.5416 3.5516 2.7815 2.2315 1.9015 1.7915 1.9015 2.2315 2.7815 3.5516 4.5416 5.7516 7.1817 8.8317 10.7018 12.7918
71.6421 49.2976 33.9221 23.3421 16.0619 11.0523 7.6052 5.2332 3.6010 2.4779 1.7051 1.5814 1.4668 1.3605 1.2618 1.1704 1.0855 1.0068 0.9338 0.8661 0.8033
18.0984 14.9993 12.2264 9.7797 7.6592 5.8650 4.3970 3.2552 2.4397 1.9503 1.7872 1.8464 2.0237 2.3194 2.7333 3.2654 3.9158 4.6844 5.5713 6.5765 7.6999
14.1543 11.9008 9.8732 8.0175 6.4957 5.1457 4.0216 3.1234 2.4511 2.0047 1.7842 1.7895 2.0208 2.4779 3.1609 4.0698 5.2045 6.5652 8.1517 9.9641 12.0024
4.0243 3.6801 3.3359 2.9917 2.6475 2.3032 1.9590 1.6148 1.2706 0.9264 0.5821 0.6268 0.6265 2.3109 2.2346 2.1583 2.0819 2.0056 1.9293 1.8529 1.7766
152 VI . S UMM A RY AND C ONCLU SION S This paper applies the news impact curve of Engle and Ng (1993) as a metric for the specification of models of the conditional volatility of stock returns. The standard GARCH(1,1) model, which imposes symmetry on the conditional variance of stock returns, is shown to produce biased estimates of h when stock price movements are large t and negative (e (0). The estimated news impact curve t~1 for the GARCH(1,1) suggests that h is underestimated for t large negative shocks and overestimated for large positive shocks. This bias is not detected by the Ljung—Box Q(10) statistic on the standardized squared residuals, or the Pagan and Sabau (1992) moment specification test. Moreover, a robustified Wald test for integration in variance suggests that shocks to volatility are infinitely persistent, in the sense that the optimal k-step-ahead linear forecast of the conditional variance continues to depend on the initial conditions for all forecast horizons. Using the regression based methodology suggested by Psaradakis and Tzavalis (1995) the null of infinite persistence in variance is not satisfied. Asymmetric models were fitted to the pre-filtered daily returns data from the Hong Kong Stock market. Initially the EGARCH model appears the most valid, in terms of the usual diagnostic statistics. Further examination of the various news impact curves suggests that the EGARCH(1,1) model is overly sensitive to extremely large positive and negative shocks. The conditional variance equation of the GJR model contains two parameters b and d that are, at best, marginally significant. Consequently, the GQARCH model, which passes all the tests and appears relatively congruent with the asymmetry inherent in the data, appears to be the most adequate characterization of the underlying data generating process.
AC KN OWL ED GEM EN TS I am grateful to Charles Ward and Christopher Brooks of the University of Reading, seminar participants at the University of Melbourne and University of Adelaide, and participants at the 1996 Australasian Meeting of the Econometric Society for comments on earlier drafts of this paper. All estimation was performed using RATS version 4.2 on a 486 PC. The data and estimation routines are available on request from the author. The usual disclaimer applies for any remaining errors or omissions.
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