《金融投资学》教学资源（英文文献）Asymmetric Volatility and Risk in Equity Markets

Asymmetric Volatility and Risk in Equity Markets return are related to past conditional variances and covariances,past squared residuals and cross residuals,and past squared asymmetric shocks and cross-asymmetric shocks. Apart from its technical advantages that simplify estimation [see Engle and Kroner (1995)],the BEKK model is better suited for our purposes than alternative multivariate GARCH models.The (diagonal) VECH model of Bollerslev,Engle,and Wooldridge (1988)cannot capture volatility feedback effects at the firm level.The factor ARCH model [Engle,Ng,and Rothschild (1990)]assumes that the covariance matrix is driven by the conditional variance process of one portfolio (the market portfolio),making it impossible to test for firm-specific leverage effects.The constant correlation model of Bollerslev (1990)restricts the correlation between two asset returns to be constant over time.Braun, Nelson,and Sunier (1995)use univariate asymmetric GARCH models coupled with a specification for the conditional beta that accommodates asymmetry.As we suggest above,it is more natural to model asymmetry in covariances,as is possible in the BEKK framework. One drawback of the BEKK model is the large number of parame- ters that must be estimated.For a system of m equations,there are (9m2+m 2)/2 parameters.For example,a system of 4 equations has 75 parameters.To keep the size of the parameter space manageable,we impose additional constraints.We assume that lagged market-level shocks and variables enter all conditional variance and covariance equations,but that individual portfolio shocks and variables have ex- planatory power only for their own variances and covariances with the market. The parameter matrices B,C,and D now have the form,for ex- ample, DMM 0 0 B= bMn 0 This reduces the parameter space considerably while leaving flexibility in modeling the processes of all conditional variances and covariances with the market.For a system of 4 equations,there are 39 parameters instead of 75.We analyze the implied volatility dynamics in more detail in the next subsection. 6 Note that the asymmetric shock is defined using the negative shocks,as opposed to Glosten. Jagannathan,and Runkle (1993),who use positive shocks.This is consistent with the idea that the strong form of asymmetric volatility,discussed above,is most likely to arise from the direct leverage effect,see below.We also estimated a model where positive and negative shocks were simply allowed to have different coefficients.Since it yielded qualitatively similar results,we do not report it here. 11

Asymmetric Volatility and Risk in Equity Markets return are related to past conditional variances and covariances, past squared residuals and cross residuals, and past squared asymmetric shocks and cross-asymmetric shocks.6 Apart from its technical advantages that simplify estimation see
Engle and Kroner 1995 , the BEKK model is better suited for our Ž . purposes than alternative multivariate GARCH models. The diagonal Ž . VECH model of Bollerslev, Engle, and Wooldridge 1988 cannot Ž . capture volatility feedback effects at the firm level. The factor ARCH model Engle, Ng, and Rothschild 1990 assumes that the covariance
Ž . matrix is driven by the conditional variance process of one portfolio the Ž market portfolio , making it impossible to test for firm-specific leverage . effects. The constant correlation model of Bollerslev 1990 restricts the Ž . correlation between two asset returns to be constant over time. Braun, Nelson, and Sunier 1995 use univariate asymmetric GARCH models Ž . coupled with a specification for the conditional beta that accommodates asymmetry. As we suggest above, it is more natural to model asymmetry in covariances, as is possible in the BEKK framework. One drawback of the BEKK model is the large number of parame￾ters that must be estimated. For a system of m equations, there are Ž 2 9m  m  2.2 parameters. For example, a system of 4 equations has 75 parameters. To keep the size of the parameter space manageable, we impose additional constraints. We assume that lagged market-level shocks and variables enter all conditional variance and covariance equations, but that individual portfolio shocks and variables have ex￾planatory power only for their own variances and covariances with the market. The parameter matrices B, C, and D now have the form, for ex￾ample, bM M 0  0 b b M 1 11  0 B  . .. . . . . .. . . ..  0 bMn nn 0  b This reduces the parameter space considerably while leaving flexibility in modeling the processes of all conditional variances and covariances with the market. For a system of 4 equations, there are 39 parameters instead of 75. We analyze the implied volatility dynamics in more detail in the next subsection. 6 Note that the asymmetric shock is defined using the negative shocks, as opposed to Glosten, Jagannathan, and Runkle 1993 , who use positive shocks. This is consistent with the idea that Ž . the strong form of asymmetric volatility, discussed above, is most likely to arise from the direct leverage effect, see below. We also estimated a model where positive and negative shocks were simply allowed to have different coefficients. Since it yielded qualitatively similar results, we do not report it here. 11

The Review of Financial Studies/v 13 n 1 2000 Given this firm-level volatility model,leverage effects are now easily incorporated.Define 1+LRM. 0 0 0 1+LR1, 。。 0 0 0 1+LRn.t Then E =E(el1)=I1.Hence,if firm variances were to be constant,leverage affects conditional variances and covariances exactly as in the Christie (1982)model.If firm variances move around, their changes have a higher impact on stock return volatility when leverage is also higher.Note that the model remains econometrically attractive,guaranteeing symmetry,and hence positive definiteness,as in a standard BEKK model. 1.3 Empirical hypotheses 1.3.1 Asymmetry,volatility feedback,and leverage.If B =C=D =0 (no GARCH,no volatility feedback),the model reduces to the Christie (1982)leverage model under riskless debt.That is, =(1+LRi)2.for i=M,1.....n. (12) We provide tests of this hypothesis,but also separately test for the significance of GARCH effects (B=C=0)and asymmetries in the firm variance process (D =0).The latter model would constitute a GARCH model where all asymmetric effects are accounted for by leverage effects.That is,a simple likelihood ratio test can determine whether volatility feedback (which must enter through the parameters in D)is statistically significant.Furthermore,when the asymmetric effects at the firm level are purely caused by the volatility feedback effect,we would expect the diagonal elements on D(except d)to be zero-we test this hypothesis as well. To gain further insight into the relative importance of feedback effects versus leverage effects,let's analyze the volatility dynamics in more detail.Using the relation between firm and stock return shocks, we can write volatility at the stock return level as E,=1-1(nn)-1+B*-(B*)川 +C*15-1e-1(C*)'+D*1-1-1(D*)',(13) where,for example, B*1=l,-1Bl2 12

The Reiew of Financial Studies 13 n 1 2000 Given this firm-level volatility model, leverage effects are now easily incorporated. Define 1  LRM , t 0  0 0 1  LR1, t  0 l  . . .. . t . . .. . . ..  0 0 0  1  LRn, t   Then Ýt tt t  EŽ
I 1.  lt1Ýt t l 1. Hence, if firm variances were to be constant, leverage affects conditional variances and covariances exactly as in the Christie 1982 model. If firm variances move around, Ž . their changes have a higher impact on stock return volatility when leverage is also higher. Note that the model remains econometrically attractive, guaranteeing symmetry, and hence positive definiteness, as in a standard BEKK model. 1.3 Empirical hypotheses 1.3.1 Asymmetry, volatility feedback, and leverage. If B  C  D  0 Ž . no GARCH, no volatility feedback , the model reduces to the Christie Ž . 1982 leverage model under riskless debt. That is, 2 2 2 i, t1  Ž . Ž. 1  LRi, t i   for i  M,1,..., n. 12 We provide tests of this hypothesis, but also separately test for the significance of GARCH effects Ž . B  C  0 and asymmetries in the firm variance process Ž . D  0 . The latter model would constitute a GARCH model where all asymmetric effects are accounted for by leverage effects. That is, a simple likelihood ratio test can determine whether volatility feedback which must enter through the parameters Ž in D. is statistically significant. Furthermore, when the asymmetric effects at the firm level are purely caused by the volatility feedback effect, we would expect the diagonal elements on D Ž . except dM M to be zero
we test this hypothesis as well. To gain further insight into the relative importance of feedback effects versus leverage effects, let’s analyze the volatility dynamics in more detail. Using the relation between firm and stock return shocks, we can write volatility at the stock return level as Ý  l l   B Ý B Ž. Ž. t t1 t1 t1 t1 t1  C  C  D   D t1 t1 t1Ž . Ž . Ž. t1 t1 t1 t1 t1 , 13 where, for example, B  l Bl1 . t1 t1 t2 12

Asymmetric Volatility and Risk in Equity Markets The volatility at the market level consequently follows a univariate, leverage-adjusted,asymmetric GARCH model: 07,=(LM,-1)0wM lM.1- -2 (b品w质-1+cue-1+dwn-）, where is the first diagonal element of n'and i.-represents the relevant diagonal element in l,_1.Apart from the "Christie term," leverage enters in two ways in the conditional variance model.First,the historical leverage level of the market is embedded in and n,so that similar firm shocks generate larger volatility effects whenever leverage happens to be higher.Second,an increase in lever- age at time t-1 increases the normal GARCH effect with the ratio (/2)2.Volatility at the portfolio level is equally intuitive. Given the symmetry of the model,we only consider the terms contain- ing past variances: i+ 2 bci-1 it-2 The first term is the only term that would be present in a factor ARCH model.Since o.-reflects market leverage at time t-2,the model adjusts the factor ARCH effect upward only when the current portfolio leverage is higher than the past market leverage level.The second term (involving the past covariance)and the third term (involving the past idiosyncratic variance)are adjusted similarly.The second term reveals the importance of interaction terms,such as n and eeM,even in the variance equations.To present the volatility dynamics graphically,we therefore make use of news impact surfaces as in Kroner and Ng(1998). The news impact surface graphs the conditional variance as a func- tion of the shocks,keeping the other inputs to the conditional variance equation (conditional variances and covariances)constant at their un- conditional means.In all of our graphs,we will normalize the value when the shocks are zero to be zero.We call the effect of the e and n shocks the "direct effect."Of course,our variance equation also incor- porates leverage ratios.We augment the news impact curves with the effect of changes in leverage using a second-order Taylor approximation to the nonlinear relation between leverage ratios and shocks,evaluated at the sample mean.? Since we compute returns as the logarithm of gross returns,the level of leverage ratio as a function of the return shock is LR()=IR[1+e2-6]. where LR is evaluated at the sample mean of leverage ratios. 13

Asymmetric Volatility and Risk in Equity Markets The volatility at the market level consequently follows a univariate, leverage-adjusted, asymmetric GARCH model: 2 2 M , t M  Ž . l , t1  M M 2 lM , t1  2 2 22 22 Ž . bMM M  c  d  , , t1 MM M , t1 MM M , t1 ž / lM , t2 where  MM i is the first diagonal element of and l represents , t1 the relevant diagonal element in l . Apart from the ‘‘Christie term,’’ t1 leverage enters in two ways in the conditional variance model. First, the historical leverage level of the market is embedded in  2 , 2 M , , t1 M, t1 and 2 M , so that similar firm shocks generate larger volatility effects , t1 whenever leverage happens to be higher. Second, an increase in lever￾age at time t  1 increases the normal GARCH effect with the ratio Ž .2 lM l . Volatility at the portfolio level is equally intuitive. , t1 M, t2 Given the symmetry of the model, we only consider the terms contain￾ing past variances: 2 2 2 lll i, t1 i, t1 i, t1 2 2 22 bMi M 2 b b   b  . , t1 ii Mi iM , t1 ii ii, t1 ž / ž/ l ll l M , t2 i, t2 M , t2 i, t2 The first term is the only term that would be present in a factor ARCH model. Since  2 M reflects market leverage at time t  2, the model , t1 adjusts the factor ARCH effect upward only when the current portfolio leverage is higher than the past market leverage level. The second term Ž .Ž involving the past covariance and the third term involving the past idiosyncratic variance are adjusted similarly. The second term reveals . the importance of interaction terms, such as   and , even in the iM iM variance equations. To present the volatility dynamics graphically, we therefore make use of news impact surfaces as in Kroner and Ng 1998 . Ž . The news impact surface graphs the conditional variance as a func￾tion of the shocks, keeping the other inputs to the conditional variance equation conditional variances and covariances constant at their un- Ž . conditional means. In all of our graphs, we will normalize the value when the shocks are zero to be zero. We call the effect of the and  shocks the ‘‘direct effect.’’ Of course, our variance equation also incor￾porates leverage ratios. We augment the news impact curves with the effect of changes in leverage using a second-order Taylor approximation to the nonlinear relation between leverage ratios and shocks, evaluated at the sample mean.7 7 Since we compute returns as the logarithm of gross returns, the level of leverage ratio as a function of the return shock is 2 LRŽ .  LR
1   , t tt where LR is evaluated at the sample mean of leverage ratios. 13

The Review of Financial Studies/v 13 n 1 2000 1.3.2 Covariance and beta asymmetry.The covariance dynamics im- plied by the model can be written as OMi.t=li.1-1M.1-1iM +bM.-biM.-1 +b城M,-1b-10,-1+c嘴M,t-1ci,1-1,-l +cM,t-1c毫，-1eM,1-1e,4-1+dM,-1d,4-1ni-1 +dM,t-1d.-1M,-1.-1, where,for example, bMM.1-1= IM.-bMM IM.t-2 1i.t-LbMi b城.-1=1M.-2 These dynamics are quite general.There is a constant term that reflects leverage effects as in Christie (1982).The first variance term represents a"factor ARCH"term.When the conditional market variance was high last period,so will be the current market variance and all covariances between stock returns and the market return.The leverage adjustments correct for the fact that leverage may have changed since last period. Hence there is an indirect source of a leverage effect in the covariance equation:with a positive market shock,market leverage decreases and the "factor ARCH"effect is downweighted and vice versa.Further- more,since the ratio li.1/.-2 multiplies the market variance term, high leverage firms will tend to exhibit larger "factor ARCH"effects. The second term is a persistence term;shocks to the covariance persist over time and they are scaled up or down by changes in both market and firm leverage.Finally,the shock terms allow for different effects on the covariance depending on the particular combination of market and individual shocks.Generally we would like our estimate of the condi- tional covariance to be increased when these shocks are of the same sign and to be decreased otherwise.Ideally the model should accommo- date a different covariance response depending on whether the underly- ing shocks are positive or negative. To see how this generalized BEKK model accomplishes this,let ui=leil and consider the covariance response to all possible combina- tions of positive and negative market and individual shocks.We ignore the leverage corrections in the table. 14

The Reiew of Financial Studies 13 n 1 2000 1.3.2 Covariance and beta asymmetry. The covariance dynamics im￾plied by the model can be written as   l l   b b  2 M i, t i, t1 M , t1 iM MM , t1 M i, t1 M M , t1  b b  2  c  c  2 M M , t1 ii, t1 M i, t1 M M , t1 M i, t1 M , t1  c  c   d d 2 M M , t1 ii, t1 M , t1 i, t1 M M , t1 M i, t1 M , t1  d d M M   , , t1 ii, t1 M , t1 i, t1 where, for example,  lM , t1 bM M , t1  bM M lM , t2 l  i, t1 bM i  b . , t1 M i lM , t2 These dynamics are quite general. There is a constant term that reflects leverage effects as in Christie 1982 . The first variance term represents Ž . a ‘‘factor ARCH’’ term. When the conditional market variance was high last period, so will be the current market variance and all covariances between stock returns and the market return. The leverage adjustments correct for the fact that leverage may have changed since last period. Hence there is an indirect source of a leverage effect in the covariance equation: with a positive market shock, market leverage decreases and the ‘‘factor ARCH’’ effect is downweighted and vice versa. Further￾more, since the ratio l l multiplies the market variance term, i, t1 M, t2 high leverage firms will tend to exhibit larger ‘‘factor ARCH’’ effects. The second term is a persistence term; shocks to the covariance persist over time and they are scaled up or down by changes in both market and firm leverage. Finally, the shock terms allow for different effects on the covariance depending on the particular combination of market and individual shocks. Generally we would like our estimate of the condi￾tional covariance to be increased when these shocks are of the same sign and to be decreased otherwise. Ideally the model should accommo￾date a different covariance response depending on whether the underly￾ing shocks are positive or negative. To see how this generalized BEKK model accomplishes this, let u 

and consider the covariance response to all possible combina- i i tions of positive and negative market and individual shocks. We ignore the leverage corrections in the table. 14