HELENA CHULIÁ ET AL. 311In order to estimate the model in equations (16.1), (16.2) and (16.3), it is
assumed that the vector of innovations is conditionally normal and a quasi-
maximum likelihood method is applied. Bollerslev and Wooldridge (1992)
show that the standard errors calculated using this method are robust even
when the normality assumption is violated.
16.3.2 Asymmetric volatility impulse response functions (AVIRF)
The Volatility Impulse-Response Function (VIRF), proposed by Lin (1997),
is a useful methodology for obtaining information on the second-moment
interaction between related markets. The VIRF, and its asymmetric ver-
sion, measure the impact of an unexpected shock on the predicted volatility.
This is:
Rs,3=∂vechE[Ht+s|ψt]
∂dg(εtε′t)(16.6)whereRs,3isa3×2 matrix,s=1, 2,...is the lead indicator for the condi-
tioning expectation operator,Htis the 2×2 conditional covariance matrix,
∂dg(εtε′t)=(ε^2 1,t, ε^2 2,t)′andψtis the set of conditioning information. The
vechoperator transforms a symmetricNxNmatrix into a vector by stack-
ing each column of the matrix underneath the other and eliminating all
supradiagonal elements.
In volatility symmetric structures, it is not necessary to distinguish
between positive and negative shocks, but with asymmetric structures the
VIRF can change with the sign of the shock. The asymmetric VIRF (AVIRF)
for the asymmetric BEKK model is taken from Meneu and Torró (2003) by
applying (16.6) to (16.3):
R+s,3={
as= 1
(a+b+ 1 / 2 g)R+s−1,3 s> 1
(16.7)R−s,3={
a+gs= 1
(a+b+ 1 / 2 g)R−s−1,3 s> 1
(16.8)where R+s,3(R−s,3) represents the VIRF for positive (negative) initial shocks
anda,bandgare 3×3 parameter matrices. Moreover,a=D+N(A′⊗A′)DN,
b=D+N(B′⊗B′)DN,g=D+N(G′⊗G′)DN, whereDNis a duplication matrix,
D+N is its Moore-Penrose inverse and⊗denotes the Kronecker product
between matrices, that is:
DN=
100
010
010
001
D+
N=
1000
0½00
00½0
0001
(16.9)