Advances in Risk Management

310 VOLATILITY TRANSMISSION PATTERNS BETWEEN THE USA AND SPAIN

In the bivariate case, the BEKK model is written as follows:

[

h 11 t h 12 t

· h 22 t

]

=

[

c 11 c 12

0 c 22

]′[

c 11 c 12

0 c 22

]

+

[

b 11 b 12

b 21 b 22

]′[

h11,t− 1 h12,t− 1

· h22,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

]

(16.4)

whereci,j,bi,j,ai,jandgi,jfor alli,j=1, 2 are parameters,ε1,tandε2,tare the
unexpected shock series coming from equation (16.2),η1,t=max[0,−ε1,t]
andη2,t=max[0,−ε2,t] are the Glosten, Jagannathan and Runkle (1993)
dummy series collecting a negative asymmetry from the shocks and, finally,
hij,tfor alli,j=1, 2 are the conditional second-moment series.
Equation (16.4) allows for both own-market and cross-market influences
in the conditional variance, therefore allowing the analysis of volatility
spillovers between both markets. Moreover, the BEKK model guarantees by
construction that the variance-covariance matrix will be positive definite.
In equation (16.4), parametersci,j,bi,j,ai,jand gi,jfor alli,j=1, 2 can not be
interpreted individually. Instead, we have to interpret the non-linear func-
tions of the parameters which form the intercept terms and the coefficients
of the lagged variances, covariances and error terms. We follow Kearney
and Patton (2000) and calculate the expected value and the standard error
of those non-linear functions. The expected value of a non-linear function of
random variables is calculated as the function of the expected value of the
variables, if the estimated variables are unbiased. In order to calculate the
standard errors of the function, a first-order Taylor approximation is used.
This linearizes the function by using the variance-covariance matrix of the
parameters as well as the mean and standard error vectors.
The parameters of the bivariate BEKK system are estimated by maximiz-
ing the conditional log-likelihood function:

L(θ)=−

TN

2

ln( 2 π)−

1

2

∑T

t= 1

(

ln|Ht(θ)|+ε′tHt−^1 (θ)εt

)

(16.5)

whereTis the number of observations,Nis the number of variables in
the system andθdenotes the vector of all the parameters to be estimated.
Numerical maximization techniques were used to maximize this non-linear
log likelihood function based on the BFGS algorithm.