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Bivariate First-Order Vector Autoregression Model with Correlated Random Shocks

​
auto-correlation coefficient corr​
X
t
​,
X
t-1
​​
0.8
cross-correlation coefficient corr​
X
t
​,
Y
t-1
​​
0.2
shock correlation coefficient corr​
ϵ
x,t
​,
ϵ
y,t
​​
0.2
time
60
random seed
40
This Demonstration generates and visualizes a bivariate first-order vector autoregression (VAR) model with a symmetric coefficient matrix and correlated random shocks.
The model is
X
t
Y
t
=
ρ
ν
ν
ρ
X
t-1
Y
t-1
+
ϵ
X,t
ϵ
Y,t
, where
E[
ϵ
X,t
]=E[
ϵ
Y,t
]=0
,
var(
ϵ
X,t
)=var(
ϵ
Y,t
)=1
, and
-1<corr(
ϵ
X,t
,
ϵ
Y,t
)<1
. This simplified case with symmetric coefficient matrix is chosen to emphasize each coefficient's effect. The case where
corr(
ϵ
X,t
,
ϵ
Y,t
)=0
is the standard VAR(1) model with symmetric coefficient matrix. Positive and negative correlations between random shocks are included since they are widely used in economic and financial analysis as spillovers. The blue curve shows
X
t
and the purple curve shows
Y
t
.
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