### ch13

```An Introduction to Macroeconometrics: VEC
and VAR Models
Prepared by Vera Tabakova, East Carolina University

13.1 VEC and VAR Models

13.2 Estimating a Vector Error Correction model

13.3 Estimating a VAR Model

13.4 Impulse Responses and Variance Decompositions
Principles of Econometrics, 3rd Edition
Slide 13-2
yt  10 11xt  ety , ety ~ N (0, 2y )
(13.1a)
xt  20  21 yt  etx , etx ~ N (0, 2x )
(13.1b)
Principles of Econometrics, 3rd Edition
Slide 13-3
yt  10  11 yt 1  12 xt 1  vty
xt  20  21 yt 1  22 xt 1  vtx
(13.2)
yt  11yt 1  12 xt 1  vty
xt  21yt 1  22 xt 1  vtx
Principles of Econometrics, 3rd Edition
(13.3)
Slide 13-4
yt  0  1 xt  et
(13.4)
yt  10  11 ( yt 1  0  1 xt 1 )  vty
(13.5a)
xt   20   21 ( yt 1  0  1 xt 1 )  vtx
Principles of Econometrics, 3rd Edition
Slide 13-5
yt  10  (11  1) yt 1  110  111 xt 1  vty
(13.5b)
xt   20   21 yt 1   210  ( 211  1) xt 1  vtx
yt  (10  110 )  (11  1) yt 1  111 xt 1  vty
(13.5c)
xt  ( 20   210 )   21 yt 1  ( 211  1) xt 1  vtx
Principles of Econometrics, 3rd Edition
Slide 13-6
yt  10  11eˆt 1  vty
(13.6a)
xt  20  21eˆt 1  vtx
(13.6b)
Principles of Econometrics, 3rd Edition
Slide 13-7
Figure 13.1 Real Gross Domestic Products (GDP)
Principles of Econometrics, 3rd Edition
Slide 12-8
Aˆt  0.985Ut ,
R2  0.995
eˆt  .128eˆt 1
(tau ) (  2.889)
Principles of Econometrics, 3rd Edition
(13.7)
(13.8)
Slide 13-9
At  0.492  0.099eˆt 1
(t )
(2.077)
(13.9)
U t  0.510  0.030eˆt 1
(t )
Principles of Econometrics, 3rd Edition
(0.789)
Slide 13-10
Figure 13.2 Real GDP and the Consumer Price Index (CPI)
Principles of Econometrics, 3rd Edition
Slide 12-11
eˆt  Gt  1.631  0.623Pt
eˆt  0.009eˆt 1
(13.10)
(tau ) (  0.977)
Principles of Econometrics, 3rd Edition
Slide 13-12
Pt  0.001  0.827Pt 1  0.046Gt 1
(t ) (2.017) (18.494)
(1.165)
Gt  0.010  0.327Pt 1  0.228Gt 1
(t )
(7.845) (  4.153)
Principles of Econometrics, 3rd Edition
(13.11a)
(3.256)
(13.11b)
Slide 13-13

13.4.1 Impulse Response Functions
 13.4.1a The Univariate Case
yt  yt 1  vt
The series is subject it to a shock of size ν in period 1.
t  1, y1  y0  v1  v
t  2, y2  y1  v
t  3, y3  y2  (y1 )  2v
...
the shock is v, v, 2v,
Principles of Econometrics, 3rd Edition
Slide 13-14
Figure 13.3 Impulse Responses for an AR(1) model
(y = .9y(–1)+e) following a unit shock
Principles of Econometrics, 3rd Edition
Slide 13-15
yt  10  11 yt 1  12 xt 1  vty
(13.12)
xt   20   21 yt 1   22 xt 1  vtx
Principles of Econometrics, 3rd Edition
Slide 13-16
Let v1 y   y , vty  0 for t  1, vt x  0 for all t:
t 1
y1  v1y   y
x1  v1x  0
t2
y2  11 y1  12 x1  11 y  12 0  11 y
x2   21 y1   22 x1   21 y   22 0   21 y
t 3
y3  11 y2  12 x2  1111 y  12 21 y
x3   21 y2   22 x2   2111 y   22 21 y
...
impulse response to y on y:  y {1, 11 ,  1111  12 21  , }
impulse response to y on x:  y {0,  21 ,   2111   22 21  , }
Principles of Econometrics, 3rd Edition
Slide 13-17
Let v1x   x , vtx  0 for t  1, vty  0 for all t:
t 1
y1  v1y  0
x1  vtx   x
t2
y2  11 y1  12 x1  11 0  12 x  12 x
x2  21 y1  22 x1  21 0   22 x   22 x
...
impulse response to x on y:  x {0, 12 ,  1112  1222  , }
impulse response to x on x:  x {1, 22 ,  2112   2222  , }
Principles of Econometrics, 3rd Edition
Slide 13-18
Response of y to y
Response of y to x
1.0
.6
.5
0.8
.4
0.6
.3
0.4
.2
0.2
.1
0.0
.0
5
10
15
20
25
30
5
Response of x to y
10
15
20
25
30
25
30
Response of x to x
.5
2.4
2.0
.4
1.6
.3
1.2
.2
0.8
.1
0.4
.0
0.0
5
10
15
20
25
30
5
10
15
20
Figure 13.4 Impulse Responses to Standard Deviation Shock
Principles of Econometrics, 3rd Edition
Slide 13-19
 13.4.2a The Univariate Case
yt  yt 1  vt
ytF1  Et [yt  vt 1 ]
yt 1  Et [ yt 1 ]  yt 1  yt  vt 1
ytF2  Et [yt 1  vt 2 ]  Et [(yt  vt 1 )  vt 2 ]  2 yt
yt 2  Et [ yt 2 ]  yt 2  2 yt  vt 1  vt 2
Principles of Econometrics, 3rd Edition
Slide 13-20
 13.4.2b The Bivariate Case
ytF1  Et [11 yt  12 xt  vty1 ]  11 yt  12 xt
xtF1  Et [ 21 yt   22 xt  vtx1 ]   21 yt   22 xt
FE1y  yt 1  Et [ yt 1 ]  vty1;
var( FE1y )   2y
FE1x  xt 1  Et [ xt 1 ]  vtx1;
var( FE1x )   2x
Principles of Econometrics, 3rd Edition
Slide 13-21
 13.4.2b The Bivariate Case
ytF2  Et [11 yt 1  12 xt 1  vty2 ]
 Et [11  11 yt  12 xt  vty1   12  21 yt  22 xt  vtx1   vty2 ]
 11  11 yt  12 xt   12  21 yt  22 xt 
Principles of Econometrics, 3rd Edition
Slide 13-22
 13.4.2b The Bivariate Case
xtF 2  Et [21 yt 1  22 xt 1  vtx2 ]
 Et [21 (11 yt  12 xt  vty1 )   22 ( 21 yt   22 xt  vtx1 )  vtx2 ]
 21 (11 yt  12 xt )   22 ( 21 yt   22 xt )
Principles of Econometrics, 3rd Edition
Slide 13-23
 13.4.2b The Bivariate Case
FE2y  yt  2  Et [ yt  2 ]  [11vty1  12vtx1  vty2 ]
2 2
2
var( FE2y )  11
 y  12
 2x   2y
FE2x  xt  2  Et [ xt  2 ]  [ 21vty1   22vtx1  vtx2 ]
2
2
var( FE2x )   21
 2y   22
 2x   2x
Principles of Econometrics, 3rd Edition
Slide 13-24
 13.4.2c The General Case
 The example above assumes that x and y are not
contemporaneously related and that the shocks are
uncorrelated. There is no identification problem and the
generation and interpretation of the impulse response
functions and decomposition of the forecast error variance are
straightforward. In general, this is unlikely to be the case.
Contemporaneous interactions and correlated errors
complicate the identification of the nature of shocks and
hence the interpretation of the impulses and decomposition of
the causes of the forecast error variance.
Principles of Econometrics, 3rd Edition
Slide 13-25







Dynamic relationships
Error Correction
Forecast Error Variance
Decomposition
Identification problem
Impulse Response Functions
VAR model
VEC Model
Principles of Econometrics, 3rd Edition
Slide 13-26
Principles of Econometrics, 3rd Edition
Slide 13-27
yt  1 xt  1 yt 1   2 xt 1  ety
xt  2 yt  3 yt 1   4 xt 1  etx
(13A.1)
 1 1   yt   1 2   yt 1  ety 
 1   x       x    x 
4   t 1 
 2
 t   3
et 
ety 
 1 2 
 1 1 
B
; A 
; E   x


2 1 
 3  4 
et 
Principles of Econometrics, 3rd Edition
Slide 13-28
yt  1 yt 1  2 xt 1  vty
xt  3 yt 1  4 xt 1  vtx
(13A.2)
vty 
 1 2 
C
; V   x

3 4 
 vt 
C  B1 A, V  B 1E
Principles of Econometrics, 3rd Edition
Slide 13-29
```