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. 14The 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