330 LARGE AND SMALL CAP STOCKS IN EUROPESince the CAPM does not restrict the time variation in second moments,
we employ a multivariate GARCH model. The three most widely used mod-
els are: (1) the VECH model proposed by Bollerslevet al.(1988), (2) the
constant correlation model, CCORR, proposed by Bollerslev (1990), and (3)
the BEKK model of Engle and Kroner (1995). Each model imposes different
restrictions on the conditional covariance and gives very different variance
and covariance estimates. More recently, Kroner and Ng (1998) have derived
another multivariate GARCH model, the Asymmetric Dynamic Covariance
Matrix model, ADC. This model encompasses the above models in the
sense that, under certain restrictions, any particular model can be obtained.
These authors introduce asymmetries following the Glosten, Jagannathan
and Runkel (1993) approach.
The bivariate ADC can be written as:
[
σ1,^2 t σ12,t
· σ2,^2 t
]
=[√
θ11,t 0
0√
θ22,t][
1 ρ 12
ρ 12 1][√
θ11,t 0
0√
θ22,t]+[
0 φ 12
φ 12 0]
◦[
θ11,t θ12,t
· θ22,t]=[
θ11,t φ 12 θ12,t+ρ 12√
θ11,t√
θ22,t
· θ22,t](17.2)where
[
θ11,t θ12,t
· θ22,t]
=[
ω 11 ω 12
· ω 22]
+[
b 11 b 12
b 21 b 22]′[
σ^2 1,t− 1 σ12,t− 1
· σ2,^2 t− 1][
b 11 b 12
b 21 b 22]+[
a 11 a 12
a 21 a 22]′[
ε^2 1,t− 1 ε1,t− 1 ε2,t− 1
· ε^2 2,t− 1][
a 11 a 12
a 21 a 22]+[
g 11 g 12
g 21 g 22]′[
η^2 1,t− 1 η1,t− 1 η2,t− 1
· η^2 2,t− 1][
g 11 g 12
g 21 g 22]where◦is the Hadamard product operator (element-by-element matrix
multiplication),ωij,bij,aijandgijfori,j=1, 2 are parameters,ε1,tandε2,t
are the unexpected shocks series, η1,t=max[0, −ε1,t] and η2,t=max[0,
−ε2,t] are the Glosten, Jagannathan and Runkel (1993) dummy series col-
lecting a negative asymmetry from the shocks and, finally, σij,t for all
i,j=1, 2 are the conditional second moment series. The specification test
proposed by Kroner and Ng (1998) is as follow: (1) ifρ 12 =b 12 =b 21 =
a 12 =a 21 =g 12 =g 21 =0, a restricted asymmetric VECH is obtained, (2) if
φ 12 =b 12 =b 21 =a 12 =a 21 =g 12 =g 21 =0, the asymmetric CCORR model is
derived; (3) ifφ 12 =1 andρ 12 =0 the asymmetric BEKK model is obtained.