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2.5 Variance decomposition and innovation
accounting
Consider the VAR(p) model
Φ(L)yt = εt,(75)
where
Φ(L) = I − Φ1L − Φ2L2 − · · · − ΦpLp(76)
is the (matric) lag polynomial.
Provided that the stationary conditions hold
we have analogously to the univariate case
the vector MA representation of yt as
yt = Φ−1(L)εt = εt +∞∑
i=1
Ψiεt−i,(77)
where Ψi is an m × m coefficient matrix.
53
The error terms εt represent shocks in the
system.
Suppose we have a unit shock in εt then its
effect in y, s periods ahead is
∂yt+s
∂εt= Ψs.(78)
Accordingly the interpretation of the Ψ ma-
trices is that they represent marginal effects,
or the model’s response to a unit shock (or
innovation) at time point t in each of the
variables.
Economists call such parameters are as dy-
namic multipliers.
54
For example, if we were told that the first
element in εt changes by δ1, that the sec-
ond element changes by δ2, . . . , and the mth
element changes by δm, then the combined
effect of these changes on the value of the
vector yt+s would be given by
Δyt+s =∂yt+s
∂ε1tδ1 + · · · + ∂yt+s
∂εmtδm = Ψsδ,
(79)
where δ′ = (δ1, . . . , δm).
55
The response of yi to a unit shock in yj is
given the sequence, known as the impulse
multiplier function,
ψij,1, ψij,2, ψij,3, . . .,(80)
where ψij,k is the ijth element of the matrix
Ψk (i, j = 1, . . . , m).
Generally an impulse response function traces
the effect of a one-time shock to one of the
innovations on current and future values of
the endogenous variables.
56
What about exogeneity (or Granger-causality)?
Suppose we have a bivariate VAR system
such that xt does not Granger cause y. Then
we can write⎛⎜⎝ yt
xt
⎞⎟⎠ =
⎛⎝ φ
(1)11 0
φ(1)21 φ
(1)22
⎞⎠ (
yt−1xt−1
)+ · · ·
+
⎛⎝ φ
(p)11 0
φ(p)21 φ
(p)22
⎞⎠ (
yt−pxt−p
)+
(ε1tε2t
).
(81)
Then under the stationarity condition
(I − Φ(L))−1 = I +∞∑
i=1
ΨiLi,(82)
where
Ψi =
⎛⎝ ψ
(i)11 0
ψ(i)21 ψ
(i)22
⎞⎠.(83)
Hence, we see that variable y does not react
to a shock of x.
57
Generally, if a variable, or a block of variables,
are strictly exogenous, then the implied zero
restrictions ensure that these variables do not
react to a shock to any of the endogenous
variables.
Nevertheless it is advised to be careful when
interpreting the possible (Granger) causali-
ties in the philosophical sense of causality
between variables.
Remark 2.6: See also the critique of impulse response
analysis in the end of this section.
58
Orthogonalized impulse response function
Noting that E(εtε′t) = Σε, we observe that
the components of εt are contemporaneously
correlated, meaning that they have overlap-
ping information to some extend.
59
Example 2.9. For example, in the equity-bond data
the contemporaneous VAR(2)-residual correlations are
=================================FTA DIV R20 TBILL
---------------------------------FTA 1DIV 0.123 1R20 -0.247 -0.013 1TBILL -0.133 0.081 0.456 1=================================
Many times, however, it is of interest to know
how ”new” information on yjt makes us re-
vise our forecasts on yt+s.
In order to single out the individual effects
the residuals must be first orthogonalized,
such that they become contemporaneously
uncorrelated (they are already serially uncor-
related).
Remark 2.7: If the error terms (εt) are already con-
temporaneously uncorrelated, naturally no orthogo-
nalization is needed.
60
Unfortunately orthogonalization, however, is
not unique in the sense that changing the
order of variables in y chances the results.
Nevertheless, there usually exist some guide-
lines (based on the economic theory) how the
ordering could be done in a specific situation.
Whatever the case, if we define a lower tri-
angular matrix S such that SS′ = Σε, and
νt = S−1εt,(84)
then I = S−1ΣεS′−1, implying
E(νtν′t) = S−1E(εtε′t)S′−1 = S−1ΣεS′−1 = I.
Consequently the new residuals are both un-
correlated over time as well as across equa-
tions. Furthermore, they have unit variances.
61
The new vector MA representation becomes
yt =∞∑
i=0
Ψ∗i νt−i,(85)
where Ψ∗i = ΨiS (m × m matrices) so that
Ψ∗0 = S. The impulse response function of yi
to a unit shock in yj is then given by
ψ∗ij,0, ψ∗
ij,1, ψ∗ij,2, . . .(86)
62
Variance decomposition
The uncorrelatedness of νts allow the error
variance of the s step-ahead forecast of yit to
be decomposed into components accounted
for by these shocks, or innovations (this is
why this technique is usually called innova-
tion accounting).
Because the innovations have unit variances
(besides the uncorrelatedness), the compo-
nents of this error variance accounted for by
innovations to yj is given by
s∑k=0
ψ∗2ij,k.(87)
Comparing this to the sum of innovation re-
sponses we get a relative measure how im-
portant variable js innovations are in the ex-
plaining the variation in variable i at different
step-ahead forecasts, i.e.,
63
R2ij,s = 100
∑sk=0 ψ∗2
ij,k∑mh=1
∑sk=0 ψ∗2
ih,k
.(88)
Thus, while impulse response functions trace
the effects of a shock to one endogenous
variable on to the other variables in the VAR,
variance decomposition separates the varia-
tion in an endogenous variable into the com-
ponent shocks to the VAR.
Letting s increase to infinity one gets the por-
tion of the total variance of yj that is due to
the disturbance term εj of yj.
64
On the ordering of variables
As was mentioned earlier, when there is con-
temporaneous correlation between the resid-
uals, i.e., cov(εt) = Σε �= I the orthogo-
nalized impulse response coefficients are not
unique. There are no statistical methods to
define the ordering. It must be done by the
analyst!
65
It is recommended that various orderings should
be tried to see whether the resulting interpre-
tations are consistent.
The principle is that the first variable should
be selected such that it is the only one with
potential immediate impact on all other vari-
ables.
The second variable may have an immediate
impact on the last m − 2 components of yt,
but not on y1t, the first component, and so
on. Of course this is usually a difficult task
in practice.
66
Selection of the S matrix
Selection of the S matrix, where SS′ = Σε,
actually defines also the ordering of variables.
Selecting it as a lower triangular matrix im-
plies that the first variable is the one affecting
(potentially) the all others, the second to the
m − 2 rest (besides itself) and so on.
67
One generally used method in choosing S
is to use Cholesky decomposition which re-
sults to a lower triangular matrix with posi-
tive main diagonal elements.∗
∗For example, if the covariance matrix is
Σ =
(σ2
1 σ12
σ21 σ22
)then if Σ = SS′, where
S =
(s11 0s21 s22
)We get
Σ =
(σ2
1 σ12
σ21 σ22
)=
(s11 0s21 s22
) (s11 s21
0 s22
)
=
(s211 s11s21
s21s11 s221 + s2
22
).
Thus s211 = σ2
1, s21s11 = σ21 = σ12, ands221 + s2
22 = σ22. Solving these yields s11 = σ1,
s21 = σ21/σ1, and s22 =√
σ22 − σ2
21/σ21. That is,
S =
(σ1 0
σ21/σ1 (σ22 − σ2
21/σ21)
1
2
).
68
Example 2.10: Let us choose in our example two or-derings. One with stock market series first followedby bond market series
[(I: FTA, DIV, R20, TBILL)],
and an ordering with interest rate variables first fol-lowed by stock markets series
[(II: TBILL, R20, DIV, FTA)].
In EViews the order is simply defined in the Cholesky
ordering option. Below are results in graphs with I:
FTA, DIV, R20, TBILL; II: R20, TBILL DIV, FTA,
and III: General impulse response function.
69
Impulse responses:
-2
-1
0
1
2
3
4
5
6
7
1 2 3 4 5 6 7 8 9 10
Response of DFTA to DFTA
-2
-1
0
1
2
3
4
5
6
7
1 2 3 4 5 6 7 8 9 10
Response of DFTA to DDIV
-2
-1
0
1
2
3
4
5
6
7
1 2 3 4 5 6 7 8 9 10
Response of DFTA to DR20
-2
-1
0
1
2
3
4
5
6
7
1 2 3 4 5 6 7 8 9 10
Response of DFTA to DTBILL
-0.4
0.0
0.4
0.8
1.2
1.6
1 2 3 4 5 6 7 8 9 10
Response of DDIV to DFTA
-0.4
0.0
0.4
0.8
1.2
1.6
1 2 3 4 5 6 7 8 9 10
Response of DDIV to DDIV
-0.4
0.0
0.4
0.8
1.2
1.6
1 2 3 4 5 6 7 8 9 10
Response of DDIV to DR20
-0.4
0.0
0.4
0.8
1.2
1.6
1 2 3 4 5 6 7 8 9 10
Response of DDIV to DTBILL
-2
-1
0
1
2
3
4
1 2 3 4 5 6 7 8 9 10
Response of DR20 to DFTA
-2
-1
0
1
2
3
4
1 2 3 4 5 6 7 8 9 10
Response of DR20 to DDIV
-2
-1
0
1
2
3
4
1 2 3 4 5 6 7 8 9 10
Response of DR20 to DR20
-2
-1
0
1
2
3
4
1 2 3 4 5 6 7 8 9 10
Response of DR20 to DTBILL
-2
-1
0
1
2
3
4
5
6
1 2 3 4 5 6 7 8 9 10
Response of DTBILL to DFTA
-2
-1
0
1
2
3
4
5
6
1 2 3 4 5 6 7 8 9 10
Response of DTBILL to DDIV
-2
-1
0
1
2
3
4
5
6
1 2 3 4 5 6 7 8 9 10
Response of DTBILL to DR20
-2
-1
0
1
2
3
4
5
6
1 2 3 4 5 6 7 8 9 10
Response of DTBILL to DTBILL
Response to Cholesky One S.D. Innovations ± 2 S.E.
Order {FTA, DIV, R20, TBILL}
-2
-1
0
1
2
3
4
5
6
7
1 2 3 4 5 6 7 8 9 10
Response of DFTA to DFTA
-2
-1
0
1
2
3
4
5
6
7
1 2 3 4 5 6 7 8 9 10
Response of DFTA to DDIV
-2
-1
0
1
2
3
4
5
6
7
1 2 3 4 5 6 7 8 9 10
Response of DFTA to DR20
-2
-1
0
1
2
3
4
5
6
7
1 2 3 4 5 6 7 8 9 10
Response of DFTA to DTBILL
-0.4
0.0
0.4
0.8
1.2
1.6
1 2 3 4 5 6 7 8 9 10
Response of DDIV to DFTA
-0.4
0.0
0.4
0.8
1.2
1.6
1 2 3 4 5 6 7 8 9 10
Response of DDIV to DDIV
-0.4
0.0
0.4
0.8
1.2
1.6
1 2 3 4 5 6 7 8 9 10
Response of DDIV to DR20
-0.4
0.0
0.4
0.8
1.2
1.6
1 2 3 4 5 6 7 8 9 10
Response of DDIV to DTBILL
-2
-1
0
1
2
3
4
1 2 3 4 5 6 7 8 9 10
Response of DR20 to DFTA
-2
-1
0
1
2
3
4
1 2 3 4 5 6 7 8 9 10
Response of DR20 to DDIV
-2
-1
0
1
2
3
4
1 2 3 4 5 6 7 8 9 10
Response of DR20 to DR20
-2
-1
0
1
2
3
4
1 2 3 4 5 6 7 8 9 10
Response of DR20 to DTBILL
-2
-1
0
1
2
3
4
5
6
7
1 2 3 4 5 6 7 8 9 10
Response of DTBILL to DFTA
-2
-1
0
1
2
3
4
5
6
7
1 2 3 4 5 6 7 8 9 10
Response of DTBILL to DDIV
-2
-1
0
1
2
3
4
5
6
7
1 2 3 4 5 6 7 8 9 10
Response of DTBILL to DR20
-2
-1
0
1
2
3
4
5
6
7
1 2 3 4 5 6 7 8 9 10
Response of DTBILL to DTBILL
Response to Cholesky One S.D. Innovations ± 2 S.E.
Order {TBILL, R20, DIV, FTA}
70
Impulse responses continue:
-4
-2
0
2
4
6
8
1 2 3 4 5 6 7 8 9 10
Response of DFTA to DFTA
-4
-2
0
2
4
6
8
1 2 3 4 5 6 7 8 9 10
Response of DFTA to DDIV
-4
-2
0
2
4
6
8
1 2 3 4 5 6 7 8 9 10
Response of DFTA to DR20
-4
-2
0
2
4
6
8
1 2 3 4 5 6 7 8 9 10
Response of DFTA to DTBILL
-0.4
0.0
0.4
0.8
1.2
1.6
1 2 3 4 5 6 7 8 9 10
Response of DDIV to DFTA
-0.4
0.0
0.4
0.8
1.2
1.6
1 2 3 4 5 6 7 8 9 10
Response of DDIV to DDIV
-0.4
0.0
0.4
0.8
1.2
1.6
1 2 3 4 5 6 7 8 9 10
Response of DDIV to DR20
-0.4
0.0
0.4
0.8
1.2
1.6
1 2 3 4 5 6 7 8 9 10
Response of DDIV to DTBILL
-2
-1
0
1
2
3
4
1 2 3 4 5 6 7 8 9 10
Response of DR20 to DFTA
-2
-1
0
1
2
3
4
1 2 3 4 5 6 7 8 9 10
Response of DR20 to DDIV
-2
-1
0
1
2
3
4
1 2 3 4 5 6 7 8 9 10
Response of DR20 to DR20
-2
-1
0
1
2
3
4
1 2 3 4 5 6 7 8 9 10
Response of DR20 to DTBILL
-2
-1
0
1
2
3
4
5
6
7
1 2 3 4 5 6 7 8 9 10
Response of DTBILL to DFTA
-2
-1
0
1
2
3
4
5
6
7
1 2 3 4 5 6 7 8 9 10
Response of DTBILL to DDIV
-2
-1
0
1
2
3
4
5
6
7
1 2 3 4 5 6 7 8 9 10
Response of DTBILL to DR20
-2
-1
0
1
2
3
4
5
6
7
1 2 3 4 5 6 7 8 9 10
Response of DTBILL to DTBILL
Response to Generalized One S.D. Innovations ± 2 S.E.
71
General Impulse Response Function
The general impulse response function aredefined as‡‡
GI(j, δi,Ft−1) = E[yt+j|εit = δi,Ft−1] − E[yt+j|Ft−1].
(89)
That is difference of conditional expectation
given an one time shock occurs in series j.
These coincide with the orthogonalized im-
pulse responses if the residual covariance ma-
trix Σ is diagonal.
‡‡Pesaran, M. Hashem and Yongcheol Shin (1998).Impulse Response Analysis in Linear MultivariateModels, Economics Letters, 58, 17-29.
72
Variance decomposition graphs of the equity-bond data
-20
0
20
40
60
80
100
120
1 2 3 4 5 6 7 8 9 10
Percent DFTA variance due to DFTA
-20
0
20
40
60
80
100
120
1 2 3 4 5 6 7 8 9 10
Percent DFTA variance due to DDIV
-20
0
20
40
60
80
100
120
1 2 3 4 5 6 7 8 9 10
Percent DFTA variance due to DR20
-20
0
20
40
60
80
100
120
1 2 3 4 5 6 7 8 9 10
Percent DFTA variance due to DTBILL
-20
0
20
40
60
80
100
120
1 2 3 4 5 6 7 8 9 10
Percent DDIV variance due to DFTA
-20
0
20
40
60
80
100
120
1 2 3 4 5 6 7 8 9 10
Percent DDIV variance due to DDIV
-20
0
20
40
60
80
100
120
1 2 3 4 5 6 7 8 9 10
Percent DDIV variance due to DR20
-20
0
20
40
60
80
100
120
1 2 3 4 5 6 7 8 9 10
Percent DDIV variance due to DTBILL
-20
0
20
40
60
80
100
1 2 3 4 5 6 7 8 9 10
Percent DR20 variance due to DFTA
-20
0
20
40
60
80
100
1 2 3 4 5 6 7 8 9 10
Percent DR20 variance due to DDIV
-20
0
20
40
60
80
100
1 2 3 4 5 6 7 8 9 10
Percent DR20 variance due to DR20
-20
0
20
40
60
80
100
1 2 3 4 5 6 7 8 9 10
Percent DR20 variance due to DTBILL
-20
0
20
40
60
80
100
1 2 3 4 5 6 7 8 9 10
Percent DTBILL variance due to DFTA
-20
0
20
40
60
80
100
1 2 3 4 5 6 7 8 9 10
Percent DTBILL variance due to DDIV
-20
0
20
40
60
80
100
1 2 3 4 5 6 7 8 9 10
Percent DTBILL variance due to DR20
-20
0
20
40
60
80
100
1 2 3 4 5 6 7 8 9 10
Percent DTBILL variance due to DTBILL
Variance Decomposition ± 2 S.E.
73
Accumulated Responses
Accumulated responses for s periods ahead
of a unit shock in variable i on variable j
may be determined by summing up the cor-
responding response coefficients. That is,
ψ(s)ij =
s∑k=0
ψij,k.(90)
The total accumulated effect is obtained by
ψ(∞)ij =
∞∑k=0
ψij,k.(91)
In economics this is called the total multi-
plier.
Particularly these may be of interest if the
variables are first differences, like the stock
returns.
For the stock returns the impulse responses
indicate the return effects while the accumu-
lated responses indicate the price effects.
74
All accumulated responses are obtained by
summing the MA-matrices
Ψ(s) =∞∑
k=0
Ψk,(92)
with Ψ(∞) = Ψ(1), where
Ψ(L) = 1 + Ψ1L + Ψ2L2 + · · ·.(93)
is the lag polynomial of the MA-representation
yt = Ψ(L)εt.(94)
75
On estimation of the impulse
response coefficients
Consider the VAR(p) model
yt = Φ1yt−1 + · · · + Φpyt−p + εt(95)
or
Φ(L)yt = εt.(96)
76
Then under stationarity the vector MA rep-
resentation is
yt = εt + Ψ1εt−1 + Ψ2εt−2 + · · ·(97)
When we have estimates of the AR-matrices
Φi denoted by Φ̂i, i = 1, . . . , p the next prob-
lem is to construct estimates for MA matrices
Ψj. It can be shown that
Ψj =j∑
i=1
Ψj−iΦi(98)
with Ψ0 = I, and Φj = 0 when i > p. Conse-
quently, the estimates can be obtained by re-
placing Φi’s by the corresponding estimates.
77
Next we have to obtain the orthogonalized
impulse response coefficients. This can be
done easily, for letting S be the Cholesky de-
composition of Σε such that
Σε = SS′,(99)
we can write
yt =∑∞
i=0 Ψiεt−i
=∑∞
i=0 ΨiSS−1εt−i
=∑∞
i=0 Ψ∗i νt−i,
(100)
where
Ψ∗i = ΨiS(101)
and νt = S−1εt. Then
Cov(νt) = S−1ΣεS′−1
= I.(102)
The estimates for Ψ∗i are obtained by replac-
ing Ψt with its estimates and using Cholesky
decomposition of Σ̂ε.
78
Critique of Impulse Response Analysis
Ordering of variables is one problem.
Interpretations related to Granger-causality
from the ordered impulse response analysis
may not be valid.
If important variables are omitted from the
system, their effects go to the residuals and
hence may lead to major distortions in the
impulse responses and the structural inter-
pretations of the results.
79
