Ch13

Report
Chapter 13
Vector Error Correction and Vector
Autoregressive Models
Walter R. Paczkowski
Rutgers University
Principles of Econometrics, 4th Edition
Chapter 13: Vector Error Correction and Vector
Autoregressive Models
Page 1
Chapter Contents




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
Principles of Econometrics, 4th Edition
Chapter 13: Vector Error Correction and Vector
Autoregressive Models
Page 2
A priori, unless we have good reasons not to, we
could just as easily have assumed that yt is the
independent variable and xt is the dependent
variable
– Our models could be:
Eq. 13.1a
yt  10 11xt  ety , ety ~ N (0, 2y )
Eq. 13.1b
xt  20  21 yt  etx , etx ~ N (0, 2x )
Principles of Econometrics, 4th Edition
Chapter 13: Vector Error Correction and Vector
Autoregressive Models
Page 3
For Eq. 13.1a, we say that we have normalized on
y whereas for Eq. 13.1b we say that we have
normalized on x
Principles of Econometrics, 4th Edition
Chapter 13: Vector Error Correction and Vector
Autoregressive Models
Page 4
We want to explore the causal relationship
between pairs of time-series variables
– We will discuss the vector error correction
(VEC) and vector autoregressive (VAR)
models
Principles of Econometrics, 4th Edition
Chapter 13: Vector Error Correction and Vector
Autoregressive Models
Page 5
Terminology
– Univariate analysis examines a single data
series
– Bivariate analysis examines a pair of series
– The term vector indicates that we are
considering a number of series: two, three, or
more
• The term ‘‘vector’’ is a generalization of the
univariate and bivariate cases
Principles of Econometrics, 4th Edition
Chapter 13: Vector Error Correction and Vector
Autoregressive Models
Page 6
13.1
VEC and VAR Models
Principles of Econometrics, 4th Edition
Chapter 13: Vector Error Correction and Vector
Autoregressive Models
Page 7
13.1
VEC and VAR
Models
Consider the system of equations:
yt  10  11 yt 1  12 xt 1  vty
Eq. 13.2
xt  20  21 yt 1  22 xt 1  vtx
– Together the equations constitute a system
known as a vector autoregression (VAR)
• In this example, since the maximum lag is of
order 1, we have a VAR(1)
Principles of Econometrics, 4th Edition
Chapter 13: Vector Error Correction and Vector
Autoregressive Models
Page 8
13.1
VEC and VAR
Models
If y and x are nonstationary I(1) and not
cointegrated, then we work with the first
differences:
yt  11yt 1  12 xt 1  vty
Eq. 13.3
Principles of Econometrics, 4th Edition
xt  21yt 1  22 xt 1  vtx
Chapter 13: Vector Error Correction and Vector
Autoregressive Models
Page 9
13.1
VEC and VAR
Models
Consider two nonstationary variables yt and xt that
are integrated of order 1 so that:
Eq. 13.4
Principles of Econometrics, 4th Edition
yt  0  1 xt  et
Chapter 13: Vector Error Correction and Vector
Autoregressive Models
Page 10
13.1
VEC and VAR
Models
The VEC model is:
yt  10  11 ( yt 1  0  1 xt 1 )  vty
Eq. 13.5a
xt   20   21 ( yt 1  0  1 xt 1 )  vtx
which we can expand as:
yt  10  (11  1) yt 1  110  111 xt 1  vty
Eq. 13.5b
xt   20   21 yt 1   210  ( 211  1) xt 1  vtx
The coefficients α11, α21 are known as error
correction coefficients
Principles of Econometrics, 4th Edition
Chapter 13: Vector Error Correction and Vector
Autoregressive Models
Page 11
13.1
VEC and VAR
Models
Let’s consider the role of the intercept terms
– Collect all the intercept terms and rewrite
Eq. 13.5b as:
yt  (10  110 )  (11  1) yt 1  111 xt 1  vty
Eq. 13.5c
xt  ( 20   210 )   21 yt 1  ( 211  1) xt 1  vtx
Principles of Econometrics, 4th Edition
Chapter 13: Vector Error Correction and Vector
Autoregressive Models
Page 12
13.2
Estimating a Vector Error Correction
Model
Principles of Econometrics, 4th Edition
Chapter 13: Vector Error Correction and Vector
Autoregressive Models
Page 13
13.2
Estimating a Vector
Error Correction
Model
There are many econometric methods to estimate
the error correction model
– A two step least squares procedure is:
• Use least squares to estimate the
cointegrating relationship and generate the
lagged residuals
• Use least squares to estimate the equations:
Eq. 13.6a
yt  10  11eˆt 1  vty
Eq. 13.6b
xt  20  21eˆt 1  vtx
Principles of Econometrics, 4th Edition
Chapter 13: Vector Error Correction and Vector
Autoregressive Models
Page 14
13.2
Estimating a Vector
Error Correction
Model
FIGURE 13.1 Real gross domestic products (GDP = 100 in 2000)
13.2.1
Example
Principles of Econometrics, 4th Edition
Chapter 13: Vector Error Correction and Vector
Autoregressive Models
Page 15
13.2
Estimating a Vector
Error Correction
Model
Eq. 13.7
To check for cointegration we obtain the fitted
equation (the intercept term is omitted because it
has no economic meaning):
Aˆ  0.985U
t
t
A formal unit root test is performed and the
estimated unit root test equation is:
Eq. 13.8
Principles of Econometrics, 4th Edition
eˆt  .128eˆt 1
(tau ) (  2.889)
Chapter 13: Vector Error Correction and Vector
Autoregressive Models
Page 16
13.2
Estimating a Vector
Error Correction
Model
FIGURE 13.2 Residuals derived from the cointegrating relationship
Principles of Econometrics, 4th Edition
Chapter 13: Vector Error Correction and Vector
Autoregressive Models
Page 17
13.2
Estimating a Vector
Error Correction
Model
The estimated VEC model for {At, Ut} is:
At  0.492  0.099eˆt 1
(t )
(2.077)
Eq. 13.9
U t  0.510  0.030eˆt 1
(t )
Principles of Econometrics, 4th Edition
(0.789)
Chapter 13: Vector Error Correction and Vector
Autoregressive Models
Page 18
13.3
Estimating a VAR Model
Principles of Econometrics, 4th Edition
Chapter 13: Vector Error Correction and Vector
Autoregressive Models
Page 19
13.3
Estimating a VAR
Model
The VEC is a multivariate dynamic model that
incorporates a cointegrating equation
We now ask: what should we do if we are
interested in the interdependencies between y and
x, but they are not cointegrated?
Principles of Econometrics, 4th Edition
Chapter 13: Vector Error Correction and Vector
Autoregressive Models
Page 20
13.3
Estimating a VAR
Model
FIGURE 13.3 Real personal disposable income and real personal
consumption expenditure (in logarithms)
Principles of Econometrics, 4th Edition
Chapter 13: Vector Error Correction and Vector
Autoregressive Models
Page 21
13.3
Estimating a VAR
Model
The test for cointegration for the case normalized
on C is:
eˆt  Ct  0.404  1.035Yt
Eq. 13.10
eˆt  0.088eˆt 1  0.299Δeˆt 1
(tau ) (  2.873)
– This is a Case 2 since the cointegrating
relationship contains an intercept term
Principles of Econometrics, 4th Edition
Chapter 13: Vector Error Correction and Vector
Autoregressive Models
Page 22
13.3
Estimating a VAR
Model
For illustrative purposes, the order of lag in this
example has been restricted to one
– In general, we should test for the significance
of lag terms greater than one
– The results are:
Eq. 13.11a
Eq. 13.11b
Principles of Econometrics, 4th Edition
ΔCˆ t  0.005  0.215ΔCt 1  0.149ΔYt 1
 t   6.969   2.884 
 2.587 
ΔYˆt  0.006  0.475ΔCt 1  0.217ΔYt 1
 t   6.122   4.885 
Chapter 13: Vector Error Correction and Vector
Autoregressive Models
 2.889 
Page 23
13.4
Impulse Responses and Variance
Decompositions
Principles of Econometrics, 4th Edition
Chapter 13: Vector Error Correction and Vector
Autoregressive Models
Page 24
13.4
Impulse Responses
and Variance
Decompositions
Impulse response functions and variance
decompositions are techniques that are used by
macroeconometricians to analyze problems such
as the effect of an oil price shock on inflation and
GDP growth, and the effect of a change in
monetary policy on the economy
Principles of Econometrics, 4th Edition
Chapter 13: Vector Error Correction and Vector
Autoregressive Models
Page 25
13.4
Impulse Responses
and Variance
Decompositions
13.4.1
Impulse Response
Functions
Impulse response functions show the effects of
shocks on the adjustment path of the variables
Principles of Econometrics, 4th Edition
Chapter 13: Vector Error Correction and Vector
Autoregressive Models
Page 26
13.4
Impulse Responses
and Variance
Decompositions
13.4.1a
The Univariate Case
Consider a univariate series: yt = ρyt-1 + vt
– The series is subject to a shock of size v in
period 1
– At time t = 1 following the shock, the value of y
in period 1 and subsequent periods will be:
t  1, y1  y0  v1  v
t  2, y2  y1  v
t  3, y3  y2  (y1 )  2 v
...
the shock is v, v, 2 v,
Principles of Econometrics, 4th Edition
Chapter 13: Vector Error Correction and Vector
Autoregressive Models
Page 27
13.4
Impulse Responses
and Variance
Decompositions
13.4.1a
The Univariate Case
The values of the coefficients {1, ρ, ρ2, ...} are
known as multipliers and the time-path of y
following the shock is known as the impulse
response function
Principles of Econometrics, 4th Edition
Chapter 13: Vector Error Correction and Vector
Autoregressive Models
Page 28
13.4
Impulse Responses
and Variance
Decompositions
FIGURE 13.4 Impulse responses for an AR(1) model yt = 0.9yt-1 + et
following a unit shock
13.4.1a
The Univariate Case
Principles of Econometrics, 4th Edition
Chapter 13: Vector Error Correction and Vector
Autoregressive Models
Page 29
13.4
Impulse Responses
and Variance
Decompositions
13.4.1b
The Bivariate Case
Consider an impulse response function analysis
with two time series based on a bivariate VAR
system of stationary variables:
yt  10  11 yt 1  12 xt 1  vty
Eq. 13.12
xt   20   21 yt 1   22 xt 1  vtx
Principles of Econometrics, 4th Edition
Chapter 13: Vector Error Correction and Vector
Autoregressive Models
Page 30
13.4
Impulse Responses
and Variance
Decompositions
13.4.1b
The Bivariate Case
The mechanics of generating impulse responses in
a system is complicated by the facts that:
1. one has to allow for interdependent dynamics
(the multivariate analog of generating the
multipliers)
2. one has to identify the correct shock from
unobservable data
Together, these two complications lead to what is
known as the identification problem
Principles of Econometrics, 4th Edition
Chapter 13: Vector Error Correction and Vector
Autoregressive Models
Page 31
13.4
Impulse Responses
and Variance
Decompositions
13.4.1b
The Bivariate Case
Consider the case when there is a one–standard
deviation shock (alternatively called an
innovation) to y:
Principles of Econometrics, 4th Edition
Chapter 13: Vector Error Correction and Vector
Autoregressive Models
Page 32
13.4
Impulse Responses
and Variance
Decompositions
13.4.1b
The Bivariate Case
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, 4th Edition
Chapter 13: Vector Error Correction and Vector
Autoregressive Models
Page 33
13.4
Impulse Responses
and Variance
Decompositions
FIGURE 13.5 Impulse responses to standard deviation shock
13.4.1b
The Bivariate Case
Principles of Econometrics, 4th Edition
Chapter 13: Vector Error Correction and Vector
Autoregressive Models
Page 34
13.4
Impulse Responses
and Variance
Decompositions
13.4.1b
The Bivariate Case
Now 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, 4th Edition
Chapter 13: Vector Error Correction and Vector
Autoregressive Models
Page 35
13.4
Impulse Responses
and Variance
Decompositions
13.4.2
Forecast Error
Variance
Decompositions
Another way to disentangle the effects of various
shocks is to consider the contribution of each type
of shock to the forecast error variance
Principles of Econometrics, 4th Edition
Chapter 13: Vector Error Correction and Vector
Autoregressive Models
Page 36
13.4
Impulse Responses
and Variance
Decompositions
13.4.2a
Univariate Analysis
Consider a univariate series: yt = ρyt-1 + vt
– The best one-step-ahead forecast (alternatively
the forecast one period ahead) is:
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, 4th Edition
Chapter 13: Vector Error Correction and Vector
Autoregressive Models
Page 37
13.4
Impulse Responses
and Variance
Decompositions
13.4.2a
Univariate Analysis
In this univariate example, there is only one shock
that leads to a forecast error
– The forecast error variance is 100% due to its
own shock
– The exercise of attributing the source of the
variation in the forecast error is known as
variance decomposition
Principles of Econometrics, 4th Edition
Chapter 13: Vector Error Correction and Vector
Autoregressive Models
Page 38
13.4
Impulse Responses
and Variance
Decompositions
13.4.2b
Bivariate Analysis
We can perform a variance decomposition for our
special bivariate example where there is no
identification problem
– Ignoring the intercepts (since they are
constants), the one–step ahead forecasts are:
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, 4th Edition
Chapter 13: Vector Error Correction and Vector
Autoregressive Models
Page 39
13.4
Impulse Responses
and Variance
Decompositions
13.4.2b
Bivariate Analysis
The two–step ahead forecast for y is:
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, 4th Edition
Chapter 13: Vector Error Correction and Vector
Autoregressive Models
Page 40
13.4
Impulse Responses
and Variance
Decompositions
13.4.2b
Bivariate Analysis
The two–step ahead forecast for x is:
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, 4th Edition
Chapter 13: Vector Error Correction and Vector
Autoregressive Models
Page 41
13.4
Impulse Responses
and Variance
Decompositions
13.4.2b
Bivariate Analysis
The corresponding two-step-ahead forecast errors
and variances are:
FE2y  yt  2  Et [ yt  2 ]  [11vty1  12 vtx1  vty 2 ]
2 2
2
var( FE2y )  11
 y  12
 2x   2y
FE2x  xt  2  Et [ xt  2 ]  [21vty1  22 vtx1  vtx 2 ]
var( FE2x )  2212y  222 2x   2x
Principles of Econometrics, 4th Edition
Chapter 13: Vector Error Correction and Vector
Autoregressive Models
Page 42
13.4
Impulse Responses
and Variance
Decompositions
13.4.2b
Bivariate Analysis
This decomposition is often expressed in
proportional terms
– The proportion of the two step forecast error
variance of y explained by its ‘‘own’’ shock is:
2 2
2
2 2
2 2
2











 11 y y   11 y 12 x x 
– The proportion of the two-step forecast error
variance of y explained by the ‘‘other’’ shock
is:
2
2
2 2
2
2
2









 12 x   11 y 12 x x 
Principles of Econometrics, 4th Edition
Chapter 13: Vector Error Correction and Vector
Autoregressive Models
Page 43
13.4
Impulse Responses
and Variance
Decompositions
13.4.2b
Bivariate Analysis
Similarly, the proportion of the two-step forecast
error variance of x explained by its own shock is:
2
2
2
2
2
2
2
2











 22 x x   21 y 22 x x 
The proportion of the forecast error of x explained
by the other shock is:
2
2
2
2
2
2
2









 21 y   21 y 22 x x 
Principles of Econometrics, 4th Edition
Chapter 13: Vector Error Correction and Vector
Autoregressive Models
Page 44
13.4
Impulse Responses
and Variance
Decompositions
13.4.2c
The General 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, 4th Edition
Chapter 13: Vector Error Correction and Vector
Autoregressive Models
Page 45
Key Words
Principles of Econometrics, 4th Edition
Chapter 13: Vector Error Correction and Vector
Autoregressive Models
Page 46
Keywords
dynamic
relationships
error correction
forecast error
variance
decomposition
identification
problem
Principles of Econometrics, 4th Edition
impulse response
functions
VAR model
VEC model
Chapter 13: Vector Error Correction and Vector
Autoregressive Models
Page 47
Appendix
Principles of Econometrics, 4th Edition
Chapter 13: Vector Error Correction and Vector
Autoregressive Models
Page 48
13A
The Identification
Problem
A bivariate dynamic system with
contemporaneous interactions (also known as a
structural model) is written as:
yt  1 xt  1 yt 1   2 xt 1  ety
Eq. 13A.1
xt  2 yt  3 yt 1   4 xt 1  etx
– In matrix terms:
 1 1   yt   1 2   yt 1  ety 
 1   x       x    x 
4   t 1 
 2
 t   3
et 
Principles of Econometrics, 4th Edition
Chapter 13: Vector Error Correction and Vector
Autoregressive Models
Page 49
13A
The Identification
Problem
More compactly, BYt = AYt-1 + Et where:
 yt 
 1
 1 1 
Y   B
A

2 1 
 xt 
 3
Principles of Econometrics, 4th Edition
Chapter 13: Vector Error Correction and Vector
Autoregressive Models
2 
ety 
E  x

4 
et 
Page 50
13A
The Identification
Problem
A VAR representation (also known as reducedform model) is written as:
Eq. 13A.2
Principles of Econometrics, 4th Edition
yt  1 yt 1  2 xt 1  vty
xt  3 yt 1  4 xt 1  vtx
Chapter 13: Vector Error Correction and Vector
Autoregressive Models
Page 51
13A
The Identification
Problem
In matrix form, this is Yt = Cyt-1 + Vt where:
 1
C
3
Principles of Econometrics, 4th Edition
2 
vty 
V  x

4 
 vt 
Chapter 13: Vector Error Correction and Vector
Autoregressive Models
Page 52
13A
The Identification
Problem
There is a relationship between Eq. 13A.1 and
Eq. 13A.2 with:
C  B1 A, Vt  B1Et
Principles of Econometrics, 4th Edition
Chapter 13: Vector Error Correction and Vector
Autoregressive Models
Page 53

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