In econometrics and other applications of multivariate time series analysis , a variance decomposition or forecast error variance decomposition (FEVD ) is used to aid in the interpretation of a vector autoregression (VAR) model once it has been fitted.[1] variance decomposition indicates the amount of information each variable contributes to the other variables in the autoregression. It determines how much of the forecast error variance of each of the variables can be explained by exogenous shocks to the other variables.
Calculating the forecast error variance
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For the VAR (p) of form
y
t
=
ν
+
A
1
y
t
−
1
+
⋯
+
A
p
y
t
−
p
+
u
t
{\displaystyle y_{t}=\nu +A_{1}y_{t-1}+\dots +A_{p}y_{t-p}+u_{t}}
This can be changed to a VAR(1) structure by writing it in companion form (see general matrix notation of a VAR(p))
Y
t
=
V
+
A
Y
t
−
1
+
U
t
{\displaystyle Y_{t}=V+AY_{t-1}+U_{t}}
A
=
[
A
1
A
2
…
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_{2}&\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}}}
V
=
[
ν
0
⋮
0
]
{\displaystyle V={\begin{bmatrix}\nu \\0\\\vdots \\0\end{bmatrix}}}
U
t
=
[
u
t
0
⋮
0
]
{\displaystyle U_{t}={\begin{bmatrix}u_{t}\\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}
V
{\displaystyle V}
U
{\displaystyle U}
k
p
{\displaystyle kp}
The mean squared error of the h-step forecast of variable
j
{\displaystyle j}
M
S
E
[
y
j
,
t
(
h
)
]
=
∑
i
=
0
h
−
1
∑
l
=
1
k
(
e
j
′
Θ
i
e
l
)
2
=
(
∑
i
=
0
h
−
1
Θ
i
Θ
i
′
)
j
j
=
(
∑
i
=
0
h
−
1
Φ
i
Σ
u
Φ
i
′
)
j
j
,
{\displaystyle \mathbf {MSE} [y_{j,t}(h)]=\sum _{i=0}^{h-1}\sum _{l=1}^{k}(e_{j}'\Theta _{i}e_{l})^{2}={\bigg (}\sum _{i=0}^{h-1}\Theta _{i}\Theta _{i}'{\bigg )}_{jj}={\bigg (}\sum _{i=0}^{h-1}\Phi _{i}\Sigma _{u}\Phi _{i}'{\bigg )}_{jj},}
and where
e
j
{\displaystyle e_{j}}
th column of
I
k
{\displaystyle I_{k}}
j
j
{\displaystyle jj}
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'}
Σ
u
{\displaystyle \Sigma _{u}}
u
t
{\displaystyle u_{t}}
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}
The amount of forecast error variance of variable
j
{\displaystyle j}
l
{\displaystyle l}
ω
j
l
,
h
,
{\displaystyle \omega _{jl,h},}
ω
j
l
,
h
=
∑
i
=
0
h
−
1
(
e
j
′
Θ
i
e
l
)
2
/
M
S
E
[
y
j
,
t
(
h
)
]
.
{\displaystyle \omega _{jl,h}=\sum _{i=0}^{h-1}(e_{j}'\Theta _{i}e_{l})^{2}/MSE[y_{j,t}(h)].}
See also
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^ Lütkepohl, H. (2007) New Introduction to Multiple Time Series Analysis , Springer. p. 63.
![{\displaystyle \mathbf {MSE} [y_{j,t}(h)]=\sum _{i=0}^{h-1}\sum _{l=1}^{k}(e_{j}'\Theta _{i}e_{l})^{2}={\bigg (}\sum _{i=0}^{h-1}\Theta _{i}\Theta _{i}'{\bigg )}_{jj}={\bigg (}\sum _{i=0}^{h-1}\Phi _{i}\Sigma _{u}\Phi _{i}'{\bigg )}_{jj},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/83a54977393cc426642c00435eb061c072deeccb)
![{\displaystyle \omega _{jl,h}=\sum _{i=0}^{h-1}(e_{j}'\Theta _{i}e_{l})^{2}/MSE[y_{j,t}(h)].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/344f3c3885ba18e7782aaaa3068724d6db1143fe)