2.5 Variance decomposition and innovation
accounting
Consider the VAR(p) model
(75)
Φ(L)yt = t,
where
(76)Φ(L) = I − Φ1L − Φ2L2 − · · · − ΦpLp
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
(77) yt =
Φ−1(L)
t
= t +
∞
i=1
Ψit−i,
where Ψi is an m × m coefficient matrix.
53
The error terms t represent shocks in the
system.
Suppose we have a unit shock in t then its
effect in y, s periods ahead is
∂ yt+s
= Ψs .
∂ t
Accordingly the interpretation of the Ψ matrices 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.
(78)
Economists call such parameters are as dynamic multipliers.
54
For example, if we were told that the first
element in t changes by δ1, that the second 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
∂ yt+s
δ1 + · · · +
δm = Ψsδ ,
∂1t
∂mt
(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,
(80)
ψij,1, ψij,2, ψij,3, . . .,
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
⎞
⎛
⎟
⎠ =
⎞
(1)
yt−1
⎝ φ11 0
⎠
+ ···
(1)
(1)
x
t−1
φ21 φ22
⎛
⎞
(p)
φ
0
yt−p
1t
⎠
+
.
+ ⎝ 11
(p)
(p)
x
t−p
2t
φ21 φ22
(81)
Then under the stationarity condition
(82)
(I − Φ(L))−1
where
(83)
=I+
∞
i=1
⎛
Ψi =
(i)
ψ
⎝ 11
(i)
ψ21
ΨiLi,
⎞
0
(i)
ψ22
⎠.
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) causalities 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(tt) = Σ, we observe that
the components of t are contemporaneously
correlated, meaning that they have overlapping 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
1
DIV
0.123 1
R20
-0.247 -0.013 1
TBILL -0.133 0.081 0.456 1
=================================
Many times, however, it is of interest to know
how ”new” information on yjt makes us revise 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 uncorrelated).
Remark 2.7: If the error terms (t) are already contemporaneously uncorrelated, naturally no orthogonalization 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 guidelines (based on the economic theory) how the
ordering could be done in a specific situation.
Whatever the case, if we define a lower triangular matrix S such that SS = Σ, and
(84)
ν t = S−1t,
then I = S−1ΣS−1, implying
E(ν tν t) = S−1E(tt)S−1 = S−1ΣS−1 = I.
Consequently the new residuals are both uncorrelated over time as well as across equations. Furthermore, they have unit variances.
61
The new vector MA representation becomes
(85)
yt =
∞
i=0
Ψ∗i ν t−i,
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
(86)
∗ , ψ∗ , ψ∗ , . . .
ψij,0
ij,1 ij,2
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 innovation accounting).
Because the innovations have unit variances
(besides the uncorrelatedness), the components of this error variance accounted for by
innovations to yj is given by
(87)
s
k=0
2
∗
ψij,k .
Comparing this to the sum of innovation responses we get a relative measure how important variable js innovations are in the explaining the variation in variable i at different
step-ahead forecasts, i.e.,
63
(88)
s
∗2
ψ
k=0 ij,k
2 = 100
Rij,s
m s
2 .
∗
h=1 k=0 ψih,k
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 variation in an endogenous variable into the component shocks to the VAR.
Letting s increase to infinity one gets the portion of the total variance of yj that is due to
the disturbance term j of yj .
64
On the ordering of variables
As was mentioned earlier, when there is contemporaneous correlation between the residuals, i.e., cov(t) = Σ = I the orthogonalized 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 interpretations 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 variables.
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 = Σ,
actually defines also the ordering of variables.
Selecting it as a lower triangular matrix implies 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 results to a lower triangular matrix with positive main diagonal elements.∗
∗ For
example, if the covariance matrix is
2
σ1 σ12
Σ=
σ21 σ22
then if Σ = SS , where
s11 0
S=
s21 s22
We get
2
σ1 σ12
s11 0
s11 s21
Σ=
=
s21 s22
0
s22
σ21 σ22
2
s11
s11 s21
=
.
s21s11 s221 + s222
Thus
s211 = σ12 ,
s21 s11 = σ21 = σ12,
and
Solving
s221 + s222 = σ22.
these yields s11 = σ1,
2 /σ 2 . That is,
s21 = σ21 /σ1 , and s22 = σ22 − σ21
1
σ1
0
S=
.
2
2 /σ 2 ) 12
σ21 /σ1 (σ2 − σ21
1
68
Example 2.10: Let us choose in our example two orderings. One with stock market series first followed
by bond market series
[(I: FTA, DIV, R20, TBILL)],
and an ordering with interest rate variables first followed 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:
Order {FTA, DIV, R20, TBILL}
Response to Cholesky One S.D. Innovations ± 2 S.E.
Response of DFTA to DFTA
Response of DFTA to DDIV
Response of DFTA to DR20
Response of DFTA to DTBILL
7
7
7
7
6
6
6
6
5
5
5
5
4
4
4
4
3
3
3
3
2
2
2
1
1
1
0
0
0
0
-1
-1
-1
-1
-2
-2
-2
-2
1
2
3
4
5
6
7
8
9
10
1
2
3
Response of DDIV to DFTA
4
5
6
7
8
9
10
2
1
1
2
Response of DDIV to DDIV
3
4
5
6
7
8
9
10
1
Response of DDIV to DR20
1.6
1.6
1.6
1.2
1.2
1.2
1.2
0.8
0.8
0.8
0.8
0.4
0.4
0.4
0.4
0.0
0.0
0.0
-0.4
1
2
3
4
5
6
7
8
9
10
2
Response of DR20 to DFTA
3
4
5
6
7
8
9
10
Response of DR20 to DDIV
2
3
4
5
6
7
8
9
10
1
Response of DR20 to DR20
4
4
4
3
3
3
2
2
2
2
1
1
1
0
0
0
-1
-1
-1
-2
-2
-2
-2
3
4
5
6
7
8
9
10
1
2
3
4
5
6
7
8
9
10
1
Response of DTBILL to DDIV
2
3
4
5
6
7
8
9
10
1
Response of DTBILL to DR20
6
6
6
5
5
5
5
4
4
4
4
3
3
3
3
2
2
2
2
1
1
1
0
0
0
0
-1
-1
-1
-1
-2
-2
-2
-2
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6
7
8
9
10
8
9
10
2
3
4
5
6
7
8
9
10
2
3
4
5
6
7
8
9
10
Response of DTBILL to DTBILL
6
1
7
1
0
-1
2
6
Response of DR20 to DTBILL
3
Response of DTBILL to DFTA
5
-0.4
1
4
1
4
0.0
-0.4
1
3
Response of DDIV to DTBILL
1.6
-0.4
2
1
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6
7
8
9
10
Order {TBILL, R20, DIV, FTA}
Response to Cholesky One S.D. Innovations ± 2 S.E.
Response of DFTA to DFTA
Response of DFTA to DDIV
Response of DFTA to DR20
Response of DFTA to DTBILL
7
7
7
7
6
6
6
6
5
5
5
5
4
4
4
4
3
3
3
3
2
2
2
2
1
1
1
0
0
0
0
-1
-1
-1
-1
-2
-2
1
2
3
4
5
6
7
8
9
10
1
-2
1
2
Response of DDIV to DFTA
3
4
5
6
7
8
9
10
-2
1
2
Response of DDIV to DDIV
3
4
5
6
7
8
9
10
1
Response of DDIV to DR20
1.6
1.6
1.6
1.6
1.2
1.2
1.2
1.2
0.8
0.8
0.8
0.8
0.4
0.4
0.4
0.4
0.0
0.0
0.0
-0.4
-0.4
1
2
3
4
5
6
7
8
9
10
2
Response of DR20 to DFTA
3
4
5
6
7
8
9
10
2
3
4
5
6
7
8
9
10
1
Response of DR20 to DR20
4
4
4
3
3
3
3
2
2
2
2
1
1
1
0
0
0
0
-1
-1
-1
-1
-2
2
3
4
5
6
7
8
9
10
2
3
4
5
6
7
8
9
10
2
3
4
5
6
7
8
9
10
1
Response of DTBILL to DR20
7
7
7
6
6
6
5
5
5
5
4
4
4
4
3
3
3
3
2
2
2
1
1
1
0
0
0
0
-1
-1
-1
-1
-2
-2
-2
-2
3
4
5
6
7
8
9
10
1
2
3
4
5
6
7
8
9
10
8
9
10
2
3
4
5
6
7
8
9
10
2
3
4
5
6
7
8
9
10
Response of DTBILL to DTBILL
6
2
7
-2
1
Response of DTBILL to DDIV
7
1
6
1
-2
1
Response of DTBILL to DFTA
5
Response of DR20 to DTBILL
4
1
4
-0.4
1
Response of DR20 to DDIV
-2
3
0.0
-0.4
1
2
Response of DDIV to DTBILL
2
1
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6
7
70
8
9
10
Impulse responses continue:
Response to Generalized One S.D. Innovations ± 2 S.E.
Response of DFTA to DFTA
Response of DFTA to DDIV
Response of DFTA to DR20
Response of DFTA to DTBILL
8
8
8
8
6
6
6
6
4
4
4
4
2
2
2
0
0
0
0
-2
-2
-2
-2
-4
-4
1
2
3
4
5
6
7
8
9
10
2
-4
1
2
Response of DDIV to DFTA
3
4
5
6
7
8
9
10
-4
1
2
Response of DDIV to DDIV
3
4
5
6
7
8
9
10
1
Response of DDIV to DR20
1.6
1.6
1.6
1.6
1.2
1.2
1.2
1.2
0.8
0.8
0.8
0.8
0.4
0.4
0.4
0.4
0.0
0.0
0.0
-0.4
-0.4
1
2
3
4
5
6
7
8
9
10
2
Response of DR20 to DFTA
3
4
5
6
7
8
9
10
2
3
4
5
6
7
8
9
10
1
Response of DR20 to DR20
4
4
4
3
3
3
3
2
2
2
2
1
1
1
1
0
0
0
0
-1
-1
-1
-1
-2
2
3
4
5
6
7
8
9
10
-2
1
Response of DTBILL to DFTA
2
3
4
5
6
7
8
9
10
2
3
4
5
6
7
8
9
10
1
Response of DTBILL to DR20
7
7
7
6
6
6
5
5
5
5
4
4
4
4
3
3
3
3
2
2
2
2
1
1
1
0
0
0
-1
-1
-1
-1
-2
-2
-2
-2
3
4
5
6
7
8
9
10
1
2
3
4
5
6
7
8
9
10
7
8
9
10
2
3
4
5
6
7
8
9
10
2
3
4
5
6
7
8
9
10
Response of DTBILL to DTBILL
6
2
6
-2
1
Response of DTBILL to DDIV
7
1
5
Response of DR20 to DTBILL
4
1
4
-0.4
1
Response of DR20 to DDIV
-2
3
0.0
-0.4
1
2
Response of DDIV to DTBILL
1
0
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6
71
7
8
9
10
General Impulse Response Function
The general impulse response function are
defined as‡‡
GI(j, δi, Ft−1) = E[yt+j |it = 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 impulse responses if the residual covariance matrix Σ is diagonal.
‡‡ Pesaran,
M. Hashem and Yongcheol Shin (1998).
Impulse Response Analysis in Linear Multivariate
Models, Economics Letters, 58, 17-29.
72
Variance decomposition graphs of the equity-bond data
Variance Decomposition ± 2 S.E.
Percent DFTA variance due to DFTA
Percent DFTA variance due to DDIV
Percent DFTA variance due to DR20
Percent DFTA variance due to DTBILL
120
120
120
120
100
100
100
100
80
80
80
80
60
60
60
60
40
40
40
40
20
20
20
0
0
0
0
-20
-20
-20
-20
1
2
3
4
5
6
7
8
9
10
1
Percent DDIV variance due to DFTA
2
3
4
5
6
7
8
9
10
20
1
Percent DDIV variance due to DDIV
2
3
4
5
6
7
8
9
10
1
Percent DDIV variance due to DR20
120
120
120
120
100
100
100
100
80
80
80
80
60
60
60
60
40
40
40
40
20
20
20
0
0
0
0
-20
-20
-20
-20
1
2
3
4
5
6
7
8
9
10
1
Percent DR20 variance due to DFTA
2
3
4
5
6
7
8
9
10
2
3
4
5
6
7
8
9
10
1
Percent DR20 variance due to DR20
100
100
100
80
80
80
80
60
60
60
60
40
40
40
40
20
20
20
20
0
0
0
-20
1
2
3
4
5
6
7
8
9
10
Percent DTBILL variance due to DFTA
2
3
4
5
6
7
8
9
10
Percent DTBILL variance due to DDIV
2
3
4
5
6
7
8
9
10
Percent DTBILL variance due to DR20
1
100
100
100
80
80
80
60
60
60
60
40
40
40
40
20
20
20
0
0
0
0
-20
-20
-20
-20
3
4
5
6
7
8
9
10
1
2
3
4
5
6
7
8
9
10
7
8
9
10
2
3
4
5
6
7
8
9
10
2
3
4
5
6
7
8
9
10
Percent DTBILL variance due to DTBILL
80
2
6
-20
1
100
1
5
0
-20
1
4
Percent DR20 variance due to DTBILL
100
-20
3
20
1
Percent DR20 variance due to DDIV
2
Percent DDIV variance due to DTBILL
20
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6
73
7
8
9
10
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 corresponding response coefficients. That is,
(90)
(s)
ψij
=
s
k=0
ψij,k .
The total accumulated effect is obtained by
(91)
(∞)
ψij
=
∞
k=0
ψij,k .
In economics this is called the total multiplier.
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 accumulated responses indicate the price effects.
74
All accumulated responses are obtained by
summing the MA-matrices
(92)
Ψ(s)
=
∞
k=0
Ψk ,
with Ψ(∞) = Ψ(1), where
(93)
Ψ(L) = 1 + Ψ1L + Ψ2L2 + · · ·.
is the lag polynomial of the MA-representation
(94)
yt = Ψ(L)t.
75
On estimation of the impulse
response coefficients
Consider the VAR(p) model
(95)
yt = Φ1yt−1 + · · · + Φpyt−p + t
or
(96)
Φ(L)yt = t.
76
Then under stationarity the vector MA representation is
(97)
yt = t + Ψ1t−1 + Ψ2t−2 + · · ·
When we have estimates of the AR-matrices
Φi denoted by Φ̂i, i = 1, . . . , p the next problem is to construct estimates for MA matrices
Ψj . It can be shown that
(98)
Ψj =
j
i=1
Ψj−iΦi
with Ψ0 = I, and Φj = 0 when i > p. Consequently, the estimates can be obtained by replacing 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 decomposition of Σ such that
Σ = SS,
(99)
we can write
∞
i=0 Ψi t−i
∞
−1
=
i=0 Ψi SS t−i
∞
∗ν
=
Ψ
i=0 i t−i,
yt =
(100)
where
(101)
Ψ∗i = ΨiS
and νt = S−1t. Then
(102)
Cov(νt) = S−1ΣS
−1
= I.
The estimates for Ψ∗i are obtained by replacing Ψt with its estimates and using Cholesky
decomposition of Σ̂.
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 interpretations of the results.
79