Page 144 - Special Topic Session (STS) - Volume 2

STS474 Takaki S. et al.
We repeat steps 2-3 until the minimum t-value is greater than a critical
value, for example 1.96.

3. Estimation
′
We shall propose estimation of the parameters (, , , , ) in SARMA-

GARCH models. Parameters are estimated by a two step procedure. First step
is the estimation of and and second step is that , , .

Now, let us derive quasi likelihood function by regarding ′s as Gaussian
,
2
variables with mean zero and variance . Then, the likelihood function of
,
SARMA-GARCH models is
1 2
2
log = log| − | + | − | + ∑ ∑ (− log 2 − , 2 ),
,
2
=1 =1 2 ,
2
where is the i-th element of ( − )( − ) . Here, the number of

,
parameters are 3 + 2 and optimization of all parameters simultaneously is a
difficult task, so we adopt a two step procedure to reduce the number of
parameters.
Parameters and are estimated in first step. The parameters are
2
estimated by the quasi-likelihood estimation method. Here, we regard as
,
constant heteroskedastic variances because GARCH processes are stationary
processes, namely variances in the model are different according to assets but
don't change over time. Gaussian likelihood function for first step estimation
is derived by regarding as independent Gaussian variables with mean zero
,
2
and variance . Then the log likelihood function is

2 (5)
2
log = log| − | + | − | + − ∑ log 2 ∑ ∑ ( , )

2 =1 =1 =1 2 2
̂
The QML estimator and ̂ maximizes the log likelihood function (5).
We move to estimation of GARCH parameters. We have already obtained
estimate of spatial parameters, λ and ρ. The residuals are obtained by
̂
̂ = ( − ̂)( − ) ,

̂
where and ̂ are estimates of spatial parameters in first step. we apply the
GARCH (1, 1) model to the residuals. Let in (3) be Gaussian white noise with
,
unit variance. Then is an GARCH (1, 1) process if
,
2
=
, ,
,
2
= + 2 + 2 .

,−1
,
,−1
Here, we use residuals ̂ in stead of . Then, the conditional log likelihood
,
,
function of GARCH models is given by
1 ̂ 2
2
log = ∑ {− log(2 − 2 , 2 } ,
,
2
=2 ,
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