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Variance Decomposition or Forecast error variance decomposition indicates the amount of information each variable contributes to the other variables in a Vector autoregression (VAR) models. [ 1]
Calculating the Forecast error variance
For the VAR (p) of form
y
t
=
ν
+
A
1
y
t
−
1
+
⋯
+
A
p
y
t
−
p
+
u
{\displaystyle y_{t}=\nu +A_{1}y_{t-1}+\dots +A_{p}y_{t-p}+u}
Change this to a VAR (1) by writing it in companion form (see General matrix notation of a VAR(p) )
Y
t
=
ν
+
A
Y
t
−
1
+
U
{\displaystyle Y_{t}=\mathbf {\nu } +AY_{t-1}+U}
A
=
[
A
1
A
1
…
A
p
−
1
A
p
I
k
0
…
0
0
0
I
k
0
0
⋮
⋱
⋮
⋮
0
0
…
I
k
0
]
{\displaystyle A={\begin{bmatrix}A_{1}&A_{1}&\dots &A_{p-1}&A_{p}\\\mathbf {I} _{k}&0&\dots &0&0\\0&\mathbf {I} _{k}&&0&0\\\vdots &&\ddots &\vdots &\vdots \\0&0&\dots &\mathbf {I} _{k}&0\\\end{bmatrix}}}
Y
=
[
y
1
⋮
y
p
]
{\displaystyle Y={\begin{bmatrix}y_{1}\\\vdots \\y_{p}\end{bmatrix}}}
ν
=
[
ν
0
⋮
0
]
{\displaystyle \mathbf {\nu } ={\begin{bmatrix}\nu \\0\\\vdots \\0\end{bmatrix}}}
ν
=
[
ν
0
⋮
0
]
{\displaystyle \mathbf {\nu } ={\begin{bmatrix}\nu \\0\\\vdots \\0\end{bmatrix}}}
U
=
[
u
0
⋮
0
]
{\displaystyle U={\begin{bmatrix}u\\0\\\vdots \\0\end{bmatrix}}}
where
y
t
{\displaystyle y_{t}}
ν
{\displaystyle \nu }
u
{\displaystyle u}
k
{\displaystyle k}
A
{\displaystyle A}
k
p
{\displaystyle kp}
k
p
{\displaystyle kp}
Y
{\displaystyle Y}
ν
{\displaystyle \mathbf {\nu } }
U
{\displaystyle U}
k
p
{\displaystyle kp}
Calculate the mean squared error of the variables,
M
S
E
[
y
j
,
t
(
h
)
]
{\displaystyle \mathbf {MSE} [y_{j,t}(h)]}
Σ
y
(
h
)
{\displaystyle \Sigma _{y}(h)}
M
S
E
[
y
j
,
t
(
h
)
]
=
diag
(
Σ
y
(
h
)
)
{\displaystyle \mathbf {MSE} [y_{j,t}(h)]=\operatorname {diag} (\Sigma _{y}(h))}
Σ
y
(
h
)
=
∑
i
=
0
h
−
1
Φ
i
Σ
u
Φ
i
′
{\displaystyle \Sigma _{y}(h)=\sum _{i=0}^{h-1}\Phi _{i}\Sigma _{u}\Phi _{i}'}
where
Φ
i
=
J
A
i
J
′
{\displaystyle \Phi _{i}=JA^{i}J'}
J
=
[
I
k
0
…
0
]
{\displaystyle J={\begin{bmatrix}\mathbf {I} _{k}&0&\dots &0\end{bmatrix}}}
J
{\displaystyle J}
k
{\displaystyle k}
k
p
{\displaystyle kp}
Σ
u
{\displaystyle \Sigma _{u}}
u
{\displaystyle u}
The amount of forecast error variance of variable
j
{\displaystyle j}
k
{\displaystyle k}
ω
j
k
,
h
{\displaystyle \omega _{jk,h}}
ω
j
k
,
h
=
∑
i
=
0
h
−
1
(
e
j
′
Θ
i
e
k
)
2
/
M
S
E
[
y
j
,
t
(
h
)
]
{\displaystyle \omega _{jk,h}=\sum _{i=0}^{h-1}(e_{j}'\Theta _{i}e_{k})^{2}/MSE[y_{j,t}(h)]}
where
e
j
{\displaystyle e_{j}}
j
th
{\displaystyle {\text{j}}^{\text{th}}}
I
k
{\displaystyle I_{k}}
Θ
i
=
Φ
i
P
{\displaystyle \Theta _{i}=\Phi _{i}P}
P
{\displaystyle P}
Cholesky decomposition of
Σ
u
{\displaystyle \Sigma _{u}}
Σ
u
=
P
P
′
{\displaystyle \Sigma _{u}=PP'}
See also
Notes
^ Lütkepohl, H, "New Introduction to Multiple Time Series Analysis", Springer, 2007, p. 63.
![{\displaystyle \mathbf {MSE} [y_{j,t}(h)]}](/media/api/rest_v1/media/math/render/svg/97f59090605adc63801cbac232816595e4c8d104)
![{\displaystyle \mathbf {MSE} [y_{j,t}(h)]=\operatorname {diag} (\Sigma _{y}(h))}](/media/api/rest_v1/media/math/render/svg/f0810812aa989650352afabbf22967ebd28c2742)
![{\displaystyle \omega _{jk,h}=\sum _{i=0}^{h-1}(e_{j}'\Theta _{i}e_{k})^{2}/MSE[y_{j,t}(h)]}](/media/api/rest_v1/media/math/render/svg/96dca873f5c4dec0ee383ea289ee9866f91888ea)
