Dynamic systems: The VAR representation, and
solution.
Ragnar Nymoen
Department of Economics, UiO
8 September 2009
ECON 3410/4410: Lecture 4
These notes complete the introduction to the solution of
dynamic systems.
Main reference is IDM, Ch 2.8.5,
but we will refer to these concepts and principles many times
when we work through IAM.
ECON 3410/4410: Lecture 4
The vector autoregressive model, an example
Consider again the dynamic simultaneous equation model from
Lecture 3, with 0 = 0 to save space.
Ct
=
INCt
1 INCt
+ Ct
1
+ "t ;
(1)
= Ct + Jt ,
(2)
In matrix notation:
1
1
1
1
INCt
Ct
0 0
0
=
INCt 1
Ct 1
1
0
+
0
1
Jt +
"t
The reduced form is then
INCt
Ct
1
=
1
+
1
1
1
1
1
1
1
0 0
0
1
1
0
INCt 1
Ct 1
Jt +
1
1
(3)
1
1
Which is called a vector autoregressive model, VAR.
ECON 3410/4410: Lecture 4
1
0
1
"t
Reduced form equation and …nal equation
VAR models play a large role in modern macroeconomics.
It is a general result that the reduced form of dynamic
simultaneous equation model is a VAR.
The rows of the VAR contain the reduced form equations for
each endogenous variable.
The de…nition of a reduced form equation is that it contains
only predetermined and exogenous variables on the right hand
side.
In general, a reduced form equation is not the same as a …nal
equation, which represent the solution for that endogenous
variable.
A …nal equation contain only lags of the endogenous variable
that we want to solve and exogenus variables, on the right
hand side, and no other pre-determined variables.
ECON 3410/4410: Lecture 4
Solving the example VAR I
1
1
"
=
1
1
1
1
1
1
1
1
1
"
1
1
1
1
1
1
#
1
1
1
1
1
1
1
1
0 0
0
=
"
1
1
1
1
#
0
0
1
1
1
1
#
The VAR representation of the simple Keynesian model is seen to
be:
"
#
"
#
1
1
0
INCt
INC
t 1
1 1
1 1
1 1
=
+
Jt
1
1
Ct
0
C
t
1
1 1
1 1
1 1
"
#
1
1
+
1
1
1
1
1
1
1
1
1
"t
1
ECON 3410/4410: Lecture 4
Solving the example VAR II
We see that in this example, the …rst row gives the reduced
form equation for INCt .
The second row is both the reduced form equation for Ct , and
the …nal equation for Ct — the same that we found by
substitution in Lecture 3.
In this example we therefore don’t need the …nal equation for
INCt :
We use the …nal equation for consumption to solve for C1 , C2 ,
C3 ; ::: (assuming that t = 0 is the initial period), and then
substitute that solution sequence into the …rst row of the VAR,
lagged one period.
In general however, we need to use the explicit …nal equations
for all the endogenous variable that we want to …nd the
solution for.
ECON 3410/4410: Lecture 4
A general method for solving the VAR I
IDM Ch 2.8.5, and abstracting from the exogenous variable:
yt
=
11 yt 1
+
12 xt 1
+ "yt;
(4)
xt
=
21 yt 1
+
22 xt 1
+ "xt :
(5)
Assume that there are no shocks in the solution period. (4) and
(5) also hold for period t 1.
yt
1
=
11 yt 2
+
12 xt 2
xt
1
=
21 yt 2
+
21 xt 2 :
From the …rst equation obtain
1
(
xt 2 =
11 yt 2
yt
1 );
12
and substitute this for xt
xt
1
=
2
21 yt 2
in the second equation to obtain.
22 11
12
Substitution for xt
1
yt
2
+
22
yt
1:
12
in (4) gives the …nal equation for yt .
ECON 3410/4410: Lecture 4
A general method for solving the VAR II
Final equation for yt in the VAR given by (4) and (5).
yt = (
11
+
22 )yt 1
+(
12 21
11 22 )yt 2 ,
(6)
and a similar procedure gives the …nal equation for xt .
Equation (6) has 2nd order dynamics!
2nd order dynamics is implies by combing two equations with
1st order dynamics
This is general: A VAR consisting of three equations with …rst
order dynamics, implies a dynamic solution which generally
follows a …nal equation with 3rd order dynamics.
Since macro models often contain many equations, this means
that the dynamic solutions om models will have complex
dynamics (even if the building blocks have simple dynamics)
ECON 3410/4410: Lecture 4
A general method for solving the VAR III
Returning to the …nal equation for yt in (4) and (5)
yt = (
11
+
22 )yt 1
+(
12 21
11 22 )yt 2 :
Since y0 and y 1 are given by history, this equation gives the
unique solution for y1 , y2 ,.......
Finding the solution is not di¢ cult: use forward recursion!
Proving whether the solution is stable, unstable or explosive is
more di¢ cult than in the case with …rst order dynamics.
We note withut proof that the condition for stability is that
the two roots of the so called characteristic polynomial are
both less than zero in magnitude.
Alternatively: the conditions can be expressed as:
1
1
(
12 21
11 22 )
> 0
(
11
+
22 )
+(
12 21
11 22 )
> 0
1+(
11
+
22 )
+(
12 21
11 22 )
> 0:
ECON 3410/4410: Lecture 4
The multipliers from a VAR I
We now re-introduce the exogenous variable in the VAR
yt
=
11 yt 1
+
12 xt 1
+
10 wt
+ "yt;
(7)
xt
=
21 yt 1
+
22 xt 1
+
20 wt
+ "xt :
(8)
The two impact multipliers for w are given directly form the
speci…cation: they are 10 and 20 .
Remember that you may …rst have to …nd the reduced form of
a simultaneous equation model to obtain (7) and (8).
The long-run multipliers are obtained by assuming dynamic
stability, and writing the long-run version of the model as
y
=
11 y
+
12 x
+
10 w ;
(9)
x
=
21 y
+
22 x
+
20 w .
(10)
where y , x and w denote the two steady-state values. (In
IDM w = mw ).
ECON 3410/4410: Lecture 4
The multipliers from a VAR II
By solving the two simultaneous equations (9) and (10) we
…nd the long-run solution as
y
x
=
=
(
11 +
11 +
22
1)
12 20
10
22
11
22 +
21 20
+(
11
11
22 +
22
12 21
1)
1
10
12 21
1
w;
(11)
w:
(12)
and therefore also the long-run multipliers. What are they?
Note how the distinction between the short-run and the
long-run version of the model is uses to de…ne the two classes
of multipliers.
What about the cumulated interim multipliers. How can they
be obtained?
ECON 3410/4410: Lecture 4