2. Structural VAR representation
Suppose we have the following bivariate VAR(1) model of the
following form (Note: we suppress the intercept term for
simplicity):
In the framework, y and x are specified to be related
contemporaneously as well as with lags and u1t and u2t are
structural shocks of yt and xt respectively and
OLS estimation method may not be used to estimate the above
since the error terms are correlated with the right-hand-side
contemporaneous variables.
ttttt ubxyxay 11111211112 +++−= −− αα
ttttt ubxyyax 22212212121 +++−= −− αα
10
01
,
0
0
~
2
1
N
u
u
t
t
3. SVAR and Reduced-Form VAR
• Expressing (1) and (2) in matrix notation, we have:
• Rearranging terms:
+
=
−
−
t
t
t
t
t
t
u
u
b
b
x
y
x
y
a
a
2
1
22
11
1
1
2221
1211
21
12
0
0
1
1
αα
αα
+
=
−
−
−
−
t
t
t
t
t
t
u
u
b
b
a
a
x
y
a
a
x
y
2
2
22
11
1
21
12
1
1
2221
1211
1
21
12
0
0
1
1
1
1
αα
αα
+
=
−
−
t
t
t
t
t
t
x
y
x
y
2
1
1
1
2221
1211
ε
ε
ββ
ββ Reduced form
VAR/Tradition
al VAR
SVAR or
primitive
VAR
4. Objectives of VAR Analysis
To uncover dynamic interactions among the variables
examined:
Two VAR approaches:
- minimal theoretical restrictions: traditional VAR
- theoretical identifying restrictions: Structural VAR
(Theoretic-dependent)
Similar analytical principle – analyzing the effects of
structural shocks on the variables under study.
5. Main issue in VAR/SVAR
The basic issue in VAR modeling is structural shock
identification.
Note that, the reduced form errors make up of structural
errors. Accordingly, a shock to a reduced form “error”
cannot be taken as a structural shock to a particular
variable:
−
−
−
=
t
t
t
t
u
u
b
b
a
a
aa 2
1
22
11
21
12
21122
1
0
0
1
1
1
1
ε
ε
6. Structural shock identification
Note that:
This is AB model proposed by Amisano and Giannini (1997)).
Let B be an identity matrix. To just identify the structural shocks, the
number of restrictions on the off-diagonal elements of matrix A is n(n-
1)/2 since the estimated variance-covariance matrix of reduced form
residuals has n(n+1)/2 unique elements.
In general, the number of restrictions for exact identification is 2n2
-
n(n+1)/2 on A and B matrices.
=
−
t
t
t
t
u
u
b
b
a
a
2
1
22
11
1
21
12
2
1
0
0
1
1
ε
ε
=
t
t
t
t
u
u
b
b
a
a
2
1
22
11
2
1
21
12
0
0
1
1
ε
ε
tt BuA =ε
7. Case 1: a12 = a21 = 0
• In this case, the contemporaneous value of x will not
appear in the y equation and, likewise, the
contemporaneous value of y will not be in the x equation.
Hence, the reduced form residuals are identical to
structural residuals (or shocks).
+−
−
−
=
tt
tt
t
t
uua
uau
aa 2121
2121
21122
1
1
1
ε
ε
=
t
t
t
t
u
u
2
1
2
1
ε
ε
8. Case 2: a12 = 0
This is a recursive identification as suggested by
Sims(1980), the so-called Cholesky factorization.
In this case, x is affected contemporaneously by structural
shock in y but not the reverse (y is also affected by
structural shock in x with lag).
=
t
t
t
t
u
u
a 2
1
2
1
21 1
01
ε
ε
ttt
tt
uua
u
21212
11
+−=
=
ε
ε
9. Theoretical Restrictions
In general, the model contains more than two variables.
The essence of SVAR is to use theoretical restrictions to
identify the shocks.
Since these restrictions are theoretical-dependent,
illustration need to be given from the theoretical model
adopted to assess the relationship among the variables.
To be fruitful, the models involving more than two variables
will be used.
10. Example I: Kozluk and Mehrotra (2009)
Kozluk, T., and Mehrotra, A., (2009), The impact of monetary policy
shocks on East and South-east Asia. Economics of Transition, 17(1),
121-145.
11. Step I: Estimate the VAR
• The model contains 6 variable.
• The construction of the AB model and its restrictions
should be based on theories (see the paper).
• Estimate the 6-variable VAR model containing the six
variables (i.e. the reduced form VAR)
• It is suggested that the information criterion is used to the
determine the VAR lag order.
• This step is standard in any VAR analysis, i.e. an
estimation of the reduced-form VAR.
• From the reduced form VAR, a structural identification is
made.
12. Step II: Construct the A and B Matrices
• There are various ways to construct the A and B
matrices.
• Take note the matrices are square matrices with
the size equals to the number of endogenous
variables. In our case, it is 6.
• There are various ways to construct the matrices.
The simple way is to use matrix object.
• Click object/new object and choose Matrix-
Vector-Coef.
• Eviews will bring up the following (next page).
15. Fill in the restrictions
as specified in the
model with NA for the
value to be estimated
16. Close the Matrix Object.
MATA (i.e. Matrix A) will
appear in the workfile.
Construct Matrix B in the
same manner.
17. Step III: Estimate the structural
factorization
After the estimation of the VAR in Step I and
construction of the A and B matrices in step II,
proceed to estimate the AB model by estimating
structural factorization.
Do not close your VAR result window as the
structural factorization needs to be estimated
from there.
Click Proc/Estimate Structural Factorization.
Eviews will bring to…(next page).
19. Step IV: Generate IRF and VDC
• The previous step gives you the estimated of
contemporaneous relations as specified in the AB format.
• In order the generate the IRF and VDC based on your
identification scheme (the SVAR), make sure that the
impulse definition is STRUCTURAL DECOMPOSITION.
• Shock will be named as Shock 1, Shock 2 and so on
corresponding to the order of the variables entered when
estimating the VAR in the first step.
