Information Technology Reference
In-Depth Information
18.2 Integrated and Cointegrated VAR
Let
p
be a positive integer, and let
y
t
denote the
K
-variate time series (i.e., real-
izations of
K
-dimensional process
Y
). A
vector autoregressive
model of order
p
,
denoted VAR(
p
), is formally defined as follows:
(
t
)
y
t
=
ν
+
A
1
y
t
−
1
+
...
+
A
p
y
t
−
p
+
ε
t
,
t
=
0
,±
1
,±
2
,...,
(18.1)
y
Kt
)
is a
ν
=(
ν
1
,...,
ν
K
)
is a fixed
where
y
t
=(
y
1
t
,...,
(
K
×
1
)
random vector,
(
K
×
1
)
vector representing a nonzero mean
EY
(
t
)
,the
A
i
,
i
=
1
,...,
p
are fixed
ε
t
=(
ε
1
t
,...,
ε
Kt
)
is a
K
-dimensional
(
K
×
K
)
-dimensional coefficient matrices, and
[
ε
s
ε
t
]=
[
ε
s
ε
t
]=
Σ
ε
).
white noise
process (i.e.,
E
[
ε
t
]=
0,
E
0, for
s
=
t
, and
E
Σ
ε
is nonsingular. In addition, the fol-
lowing three important conditions are imposed on the time series in the VAR model:
It is assumed that the covariance matrix
•
Y
(
t
)
is a stable process;
•
Y
(
t
)
is stationary;
•
the underlying white noise process
ε
t
is
Gaussian
.
However, in practice, many time series data are fit better by unstable non-
stationary processes. For instance, integrated and cointegrated processes are found
especially useful in econometric studies, and for such processes the stability and
stationarity conditions are violated.
Note that the VAR(
p
) process (18.1) satisfies the
stability condition
when its
re-
verse characteristic polynomial
det
A
p
z
p
has no roots on and inside
a complex unit circle. If an unstable process has a single unit root and all the other
roots outside of the complex unit circle, then such process exhibits a behavior simi-
lar to that of a random walk. In other words, the variance of such process increases
linearly to infinity, and the correlation between the variables
Y
(
I
K
−
A
1
z
−...
)
(
t
)
and
Y
(
t
±
h
)
tends
to1as
t
. On the other hand, when the root of reverse characteristic polynomial
lies inside the unit circle, the process becomes explosive, i.e., its variance increases
exponentially. In real-life applications, the former case is of the most practical
interest.
This renders the following definition of an integrated process.
A 1D process with
d
roots on the unit circle is said to be
integrated of order d
(denoted as
I
→
∞
).
It can be shown [17] that the integrated
I
(
d
)
of order
d
with all
roots of its reverse characteristic polynomial being equal to 1 can be made stable by
differencing the original process
d
times. For example, the integrated
I
(
d
)
process
Y
(
t
)
(
1
)
process
Y
(
t
)
becomes stable after taking the first differences
(
1
−
L
)
Y
(
t
)=
Y
(
t
)
−
Y
(
t
−
1
)
,
where
L
represents the lag operator. More generally, for the
I
(
d
)
process
Y
(
t
)
,
d
Y
(
−
)
(
)
(
)
its transformation
pro-
cess in the univariate case is an
autoregressive integrated moving average
process
ARIMA(
p
,
d
,
q
).
It is noteworthy to point out that taking differences may distort the relationship
among the variables (i.e., 1D components) in some VAR(
p
) models. In particular,
1
L
t
is stable. An example of an integrated
I
d
