an introduction to macroeconometrics: vec and var models prepared by vera tabakova, east carolina...

Chapter 13

An Introduction to Macroeconometrics: VEC and VAR Models

Prepared by Vera Tabakova, East Carolina University

Chapter 13: An Introduction to Macroeconometrics: VEC and VAR Models

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

Slide 13-2Principles of Econometrics, 3rd Edition

Chapter 13: An Introduction to Macroeconometrics: VEC and VAR Models


(13.1a)

(13.1b)

210 11 , ~ (0, )y y

t t t t yy x e e N

220 21 , ~ (0, )x x

t t t t xx y e e N



(13.2)

(13.3)

10 11 1 12 1

20 21 1 22 1

yt t t t

xt t t t

y y x v

x y x v

11 1 12 1

21 1 22 1

yt t t t

xt t t t

y y x v

x y x v



(13.4)

(13.5a)

0 1t t ty x e

10 11 1 0 1 1

20 21 1 0 1 1

( )

( )

yt t t t

xt t t t

y y x v

x y x v



(13.5b)

(13.5c)

10 11 1 11 0 11 1 1

20 21 1 21 0 21 1 1

( 1)

( 1)

yt t t t

xt t t t

y y x v

x y x v

10 11 0 11 1 11 1 1

20 21 0 21 1 21 1 1

( ) ( 1)

( ) ( 1)

yt t t t

xt t t t

y y x v

x y x v

13.2 Estimating a Vector Error Correction Model


(13.6a)

(13.6b)

10 11 1ˆ yt t ty e v

20 21 1ˆ xt t tx e v

13.2.1 Example

Figure 13.1 Real Gross Domestic Products (GDP)


13.2.1 Example


(13.7)

(13.8)

2ˆ 0.985 , 0.995t tA U R

1ˆ ˆ.128

( ) ( 2.889)t te e

tau

13.2.1 Example


(13.9)

1

1

ˆ0.492 0.099

( ) (2.077)

ˆ0.510 0.030

( ) (0.789)

t t

t t

A e

t

U e

t


Figure 13.2 Real GDP and the Consumer Price Index (CPI)




(13.10)1

ˆ 1.631 0.623

ˆ ˆ0.009

( ) ( 0.977)

t t t

t t

e G P

e e

tau



(13.11a)

1 10.001 0.827 0.046

( ) (2.017) (18.494) (1.165) t t tP P G

t

(13.11b)

1 10.010 0.327 0.228

( ) (7.845) ( 4.153) (3.256) t t tG P G

t

13.4 Impulse Responses and Variance Decompositions 13.4.1 Impulse Response Functions

13.4.1a The Univariate Case

The series is subject it to a shock of size ν in period 1.


1t t ty y v

1 0 1

2 1

23 2 1

2

1,

2,

3, ( )

...

the shock is , , ,

t y y v v

t y y v

t y y y v

v v v


Figure 13.3 Impulse Responses for an AR(1) model (y = .9y(–1)+e) following a unit shock


13.4.1b The Bivariate Case


(13.12)

10 11 1 12 1

20 21 1 22 1

yt t t t

xt t t t

y y x v

x y x v



1

1 1

1 1

2 11 1 12 1 11 12 11

2 21 1 22 1 21 22 21

3 11 2 12 2 11 11 12 21

Let , 0 for 1, 0 for all :

1

0

2 0

0

3

y y xy t t

yy

x

y y

y y

y y

v v t v t

t y v

x v

t y y x

x y x

t y y x

3 21 2 22 2 21 11 22 21

11 11 11 12 21

21 21 11 22 21

...

impulse response to on : {1, , , }

impulse response to on : {0, , , }

y y

y

y

x y x

y y

y x



1

1 1

1

2 11 1 12 1 11 12 12

2 21 1 22 1 21 22 22

12

Let , 0 for 1, 0 for all :

1 0

2 0

0

...

impulse response to on : {0, ,

x x yx t t

y

xt x

x x

x x

x

v v t v t

t y v

x v

t y y x

x y x

x y

11 12 12 22

22 21 12 22 22

, }

impulse response to on : {1, , , }xx x


Figure 13.4 Impulse Responses to Standard Deviation Shock


0.0

0.2

0.4

0.6

0.8

1.0

5 10 15 20 25 30

Response of y to y

.0

.1

.2

.3

.4

.5

.6

5 10 15 20 25 30

Response of y to x

.0

.1

.2

.3

.4

.5

5 10 15 20 25 30

Response of x to y

0.0

0.4

0.8

1.2

1.6

2.0

2.4

5 10 15 20 25 30

Response of x to x

13.4.2 Forecast Error Variance Decompositions



1

1 1

1 1 1 1

22 1 2 1 2

22 2 2 1 2

[ ]

[ ]

[ ] [ ( ) ]

[ ]

t t t

Ft t t t

t t t t t t

Ft t t t t t t t t

t t t t t t t

y y v

y E y v

y E y y y v

y E y v E y v v y

y E y y y v v




1 11 12 1 11 12

1 21 22 1 21 22

21 1 1 1 1

21 1 1 1 1

[ ]

[ ]

[ ] ; var( )

[ ] ; var( )

F yt t t t t t t

F xt t t t t t t

y y yt t t t y

x x xt t t t x

y E y x v y x

x E y x v y x

FE y E y v FE

FE x E x v FE




2 11 1 12 1 2

11 11 12 1 12 21 22 1 2

11 11 12 12 21 22

[ ]

[ ]

F yt t t t t

y x yt t t t t t t t

t t t t

y E y x v

E y x v y x v v

y x y x




2 21 1 22 1 2

21 11 12 1 22 21 22 1 2

21 11 12 22 21 22

[ ]

[ ( ) ( ) ]

( ) ( )

F xt t t t t

y x xt t t t t t t t

t t t t

x E y x v

E y x v y x v v

y x y x




2 2 2 11 1 12 1 2

2 2 2 2 22 11 12

2 2 2 21 1 22 1 2

2 2 2 2 22 21 22

[ ] [ ]

var( )

[ ] [ ]

var( )

y y x yt t t t t t

yy x y

x y x xt t t t t t

xy x x

FE y E y v v v

FE

FE x E x v v v

FE


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.


Keywords


Dynamic relationships Error Correction Forecast Error Variance

Decomposition Identification problem Impulse Response Functions VAR model VEC Model

Chapter 13 Appendix


Appendix 13A The Identification Problem


Principles of Econometrics, 3rd Edition Slide 13-28

(13A.1)1 1 1 2 1

2 3 1 4 1

yt t t t t

xt t t t t

y x y x e

x y y x e

1 2 11

3 4 12

1

1

yt t t

xt t t

y y e

x x e

1 21

3 42

1; ;

1

ytxt

eB A E

e


Principles of Econometrics, 3rd Edition Slide 13-29

(13A.2)1 1 2 1

3 1 4 1

yt t t t

xt t t t

y y x v

x y x v

1 2

3 4

; ytxt

vC V

v

1 1, C B A V B E

an introduction to macroeconometrics: vec and var models prepared by vera tabakova, east carolina...

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univariate case slide

bivariate case slide

unit shock slide

standard deviation shock

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consumer price index

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