Biology Reference
In-Depth Information
In the above,
Y
(
t
)
and
υ
(
t
)
are (
kp
×
1) vectors,
A
is a (
kp
×
kp
) companion ma-
trix, and
I
is a (
p
p
) identity matrix. The VAR(
p
) process is said to be stationary
(covariance-stationary) if the absolute values of the eigenvalues of the companion
matrix
A
are lesser than 1.
×
3.1.2.2 Lag Order of a VAR Process
The literature contains numerous discussions on how to select a suitable lag or-
der for a covariance stationary VAR process. Among classic procedures, there are
information criterion such as AIC and BIC, also known as the
Schwarz criterion
(
Lutkepohl 2005
).
Information criteria are statistics that measure the distance between observations
and model classes. If the value of the information criterion is small, the distance is
small and the model class contains a model that fits the data well. Typical criteria,
such as AIC or BIC, consist of two additive parts. The first is a naıve goodness-of-fit
measure, and the second is a penalty term that increases with the model's complex-
ity. The most popular information criteria is the AIC due to Akaike:
2
n
m
ˆ
(
)=
|
Σ
(
)
|
+
,
AIC
m
log
m
(3.8)
where
m
denotes the number of free parameters in the model and
ˆ
denotes
the maximum likelihood estimate of the error covariance matrix. In principle, a
VA R (
p
)has
m
Σ
(
m
)
k
2
p
2 free parameters. Since we are only interested
in finding the optimal
p
, we can assume
k
(the number of variables at each time
point) is constant, discard the intercept, and not impose any constraint on the error
covariance matrix. Therefore, for a VAR(
p
) process, AIC is defined as
=
+
k
+
k
(
k
+
1
)
/
2
pk
2
n
ˆ
(
)=
|
Σ
(
)
|
+
.
AIC
p
log
p
(3.9)
An important property of AIC is its ability to select models with strong predictive
power. Some authors also suggest that AIC can select good models even for small
samples, possibly through the use of a second-order correction called AICc.
BIC is also commonly used because it is consistent, that is, the selected
p
will
be the true
p
with probability one as
n
tends toward infinity. For a VAR(
p
) process,
BIC is defined as
pk
2
log
(
n
)
ˆ
BIC
(
p
)=
log
|
Σ
(
p
)
|
+
.
(3.10)
n
3.1.2.3 Tests for Multivariate Normality in VARs
When using a statistical model on real-world data, it is important to check that the
assumptions of the model are satisfied. In the case of VAR processes, one of those
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