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Principles of Econometrics, 4th Edition Page 1Chapter 13: Vector Error Correction and Vector
Autoregressive Models
Chapter 13Vector Error Correction and
Vector Autoregressive Models
Walter R. Paczkowski Rutgers University
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Principles of Econometrics, 4th Edition Page 2Chapter 13: Vector Error Correction and Vector
Autoregressive Models
13.1 VEC and VAR Models13.2 Estimating a Vector Error Correction Model13.3 Estimating a VAR Model13.4 Impulse Responses and Variance
Chapter Contents
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Principles of Econometrics, 4th Edition Page 3Chapter 13: Vector Error Correction and Vector
Autoregressive Models
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:
210 11 , ~ (0, )y y
t t t t yy x e e N 2
20 21 , ~ (0, )x xt t t t xx y e e N
Eq. 13.1a
Eq. 13.1b
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Principles of Econometrics, 4th Edition Page 4Chapter 13: Vector Error Correction and Vector
Autoregressive Models
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
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Principles of Econometrics, 4th Edition Page 5Chapter 13: Vector Error Correction and Vector
Autoregressive Models
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
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Principles of Econometrics, 4th Edition Page 6Chapter 13: Vector Error Correction and Vector
Autoregressive Models
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
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Principles of Econometrics, 4th Edition Page 7Chapter 13: Vector Error Correction and Vector
Autoregressive Models
13.1 VEC and VAR Models
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Principles of Econometrics, 4th Edition Page 8Chapter 13: Vector Error Correction and Vector
Autoregressive Models
Consider the system of equations:
– 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)
13.1VEC and VAR
Models
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
Eq. 13.2
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Principles of Econometrics, 4th Edition Page 9Chapter 13: Vector Error Correction and Vector
Autoregressive Models
If y and x are nonstationary I(1) and not cointegrated, then we work with the first differences:
13.1VEC and VAR
Models
11 1 12 1
21 1 22 1
yt t t t
xt t t t
y y x v
x y x v
Eq. 13.3
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Principles of Econometrics, 4th Edition Page 10Chapter 13: Vector Error Correction and Vector
Autoregressive Models
Consider two nonstationary variables yt and xt that are integrated of order 1 so that:
13.1VEC and VAR
Models
Eq. 13.4 0 1t t ty x e
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Principles of Econometrics, 4th Edition Page 11Chapter 13: Vector Error Correction and Vector
Autoregressive Models
The VEC model is:
which we can expand as:
The coefficients α11, α21 are known as error correction coefficients
13.1VEC and VAR
Models
Eq. 13.5a10 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
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
Eq. 13.5b
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Principles of Econometrics, 4th Edition Page 12Chapter 13: Vector Error Correction and Vector
Autoregressive Models
Let’s consider the role of the intercept terms– Collect all the intercept terms and rewrite
Eq. 13.5b as:
13.1VEC and VAR
Models
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
Eq. 13.5c
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Principles of Econometrics, 4th Edition Page 13Chapter 13: Vector Error Correction and Vector
Autoregressive Models
13.2 Estimating a Vector Error Correction
Model
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Principles of Econometrics, 4th Edition Page 14Chapter 13: Vector Error Correction and Vector
Autoregressive Models
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:
13.2Estimating a Vector Error Correction
Model
Eq. 13.6a 10 11 1ˆ yt t ty e v
20 21 1ˆ xt t tx e v Eq. 13.6b
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Principles of Econometrics, 4th Edition Page 15Chapter 13: Vector Error Correction and Vector
Autoregressive Models
13.2Estimating a Vector Error Correction
Model
13.2.1Example
FIGURE 13.1 Real gross domestic products (GDP = 100 in 2000)
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Principles of Econometrics, 4th Edition Page 16Chapter 13: Vector Error Correction and Vector
Autoregressive Models
To check for cointegration we obtain the fitted equation (the intercept term is omitted because it has no economic meaning):
A formal unit root test is performed and the estimated unit root test equation is:
13.2Estimating a Vector Error Correction
Model
Eq. 13.7
Eq. 13.8
ˆ 0.985t tA U
1ˆ ˆ.128
( ) ( 2.889)t te e
tau
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Principles of Econometrics, 4th Edition Page 17Chapter 13: Vector Error Correction and Vector
Autoregressive Models
13.2Estimating a Vector Error Correction
ModelFIGURE 13.2 Residuals derived from the cointegrating relationship
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Principles of Econometrics, 4th Edition Page 18Chapter 13: Vector Error Correction and Vector
Autoregressive Models
The estimated VEC model for {At, Ut} is:
13.2Estimating a Vector Error Correction
Model
Eq. 13.9
1
1
ˆ0.492 0.099( ) (2.077)
ˆ0.510 0.030( ) (0.789)
t t
t t
A et
U et
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Principles of Econometrics, 4th Edition Page 19Chapter 13: Vector Error Correction and Vector
Autoregressive Models
13.3 Estimating a VAR Model
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Principles of Econometrics, 4th Edition Page 20Chapter 13: Vector Error Correction and Vector
Autoregressive Models
The VEC is a multivariate dynamic model that incorporates a cointegrating equationWe now ask: what should we do if we are interested in the interdependencies between y and x, but they are not cointegrated?
13.3Estimating a VAR
Model
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Principles of Econometrics, 4th Edition Page 21Chapter 13: Vector Error Correction and Vector
Autoregressive Models
13.3Estimating a VAR
ModelFIGURE 13.3 Real personal disposable income and real personal consumption expenditure (in logarithms)
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Principles of Econometrics, 4th Edition Page 22Chapter 13: Vector Error Correction and Vector
Autoregressive Models
The test for cointegration for the case normalized on C is:
– This is a Case 2 since the cointegrating relationship contains an intercept term
13.3Estimating a VAR
Model
Eq. 13.10
1 1
ˆ 0.404 1.035
ˆ ˆ ˆ0.088 0.299Δ( ) ( 2.873)
t t t
t t t
e C Y
e e etau
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Principles of Econometrics, 4th Edition Page 23Chapter 13: Vector Error Correction and Vector
Autoregressive Models
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:
13.3Estimating a VAR
Model
Eq. 13.11a
Eq. 13.11b
1 1
1 1
ˆΔ 0.005 0.215Δ 0.149Δ
6.969 2.884 2.587ˆΔ 0.006 0.475Δ 0.217Δ
6.122 4.885 2.889
t t t
t t t
C C Y
t
Y C Y
t
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Principles of Econometrics, 4th Edition Page 24Chapter 13: Vector Error Correction and Vector
Autoregressive Models
13.4 Impulse Responses and Variance
Decompositions
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Principles of Econometrics, 4th Edition Page 25Chapter 13: Vector Error Correction and Vector
Autoregressive Models
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
13.4Impulse Responses
and Variance Decompositions
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Principles of Econometrics, 4th Edition Page 26Chapter 13: Vector Error Correction and Vector
Autoregressive Models
Impulse response functions show the effects of shocks on the adjustment path of the variables
13.4Impulse Responses
and Variance Decompositions
13.4.1Impulse Response
Functions
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Principles of Econometrics, 4th Edition Page 27Chapter 13: Vector Error Correction and Vector
Autoregressive Models
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:
13.4Impulse Responses
and Variance Decompositions
13.4.1aThe Univariate Case
1 0 1
2 1
23 2 1
2
1, 2,
3, ( )...
the shock is , , ,
t y y v vt y y v
t y y y v
v v v
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Principles of Econometrics, 4th Edition Page 28Chapter 13: Vector Error Correction and Vector
Autoregressive Models
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
13.4Impulse Responses
and Variance Decompositions
13.4.1aThe Univariate Case
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Principles of Econometrics, 4th Edition Page 29Chapter 13: Vector Error Correction and Vector
Autoregressive Models
13.4Impulse Responses
and Variance Decompositions
13.4.1aThe Univariate Case
FIGURE 13.4 Impulse responses for an AR(1) model yt = 0.9yt-1 + et following a unit shock
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Principles of Econometrics, 4th Edition Page 30Chapter 13: Vector Error Correction and Vector
Autoregressive Models
Consider an impulse response function analysis with two time series based on a bivariate VAR system of stationary variables:
13.4Impulse Responses
and Variance Decompositions
13.4.1bThe Bivariate Case
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
Eq. 13.12
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Principles of Econometrics, 4th Edition Page 31Chapter 13: Vector Error Correction and Vector
Autoregressive Models
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
13.4Impulse Responses
and Variance Decompositions
13.4.1bThe Bivariate Case
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Principles of Econometrics, 4th Edition Page 32Chapter 13: Vector Error Correction and Vector
Autoregressive Models
Consider the case when there is a one–standard deviation shock (alternatively called an innovation) to y:
13.4Impulse Responses
and Variance Decompositions
13.4.1bThe Bivariate Case
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Principles of Econometrics, 4th Edition Page 33Chapter 13: Vector Error Correction and Vector
Autoregressive Models
13.4Impulse Responses
and Variance Decompositions
13.4.1bThe Bivariate Case
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
02 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 vt 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
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Principles of Econometrics, 4th Edition Page 34Chapter 13: Vector Error Correction and Vector
Autoregressive Models
13.4Impulse Responses
and Variance Decompositions
13.4.1bThe Bivariate Case
FIGURE 13.5 Impulse responses to standard deviation shock
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Principles of Econometrics, 4th Edition Page 35Chapter 13: Vector Error Correction and Vector
Autoregressive Models
13.4Impulse Responses
and Variance Decompositions
13.4.1bThe Bivariate Case
1
1 1
1
2 11 1 12 1 11 12 12
2 21 1 22 1 21 22 22
Now 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 vt y y x
x y x
x y
12 11 12 12 22
22 21 12 22 22
, , }
impulse response to on : {1, , , }xx x
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Principles of Econometrics, 4th Edition Page 36Chapter 13: Vector Error Correction and Vector
Autoregressive Models
13.4Impulse Responses
and Variance Decompositions
13.4.2Forecast 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
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Principles of Econometrics, 4th Edition Page 37Chapter 13: Vector Error Correction and Vector
Autoregressive Models
13.4Impulse Responses
and Variance Decompositions
13.4.2aUnivariate Analysis Consider a univariate series: yt = ρyt-1 + vt
– The best one-step-ahead forecast (alternatively the forecast one period ahead) is:
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
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Principles of Econometrics, 4th Edition Page 38Chapter 13: Vector Error Correction and Vector
Autoregressive Models
13.4Impulse Responses
and Variance Decompositions
13.4.2aUnivariate 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
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Principles of Econometrics, 4th Edition Page 39Chapter 13: Vector Error Correction and Vector
Autoregressive Models
13.4Impulse Responses
and Variance Decompositions
13.4.2bBivariate 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:
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
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Principles of Econometrics, 4th Edition Page 40Chapter 13: Vector Error Correction and Vector
Autoregressive Models
13.4Impulse Responses
and Variance Decompositions
13.4.2bBivariate Analysis
The two–step ahead forecast for y is:
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
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Principles of Econometrics, 4th Edition Page 41Chapter 13: Vector Error Correction and Vector
Autoregressive Models
13.4Impulse Responses
and Variance Decompositions
13.4.2bBivariate Analysis
The two–step ahead forecast for x is:
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
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Principles of Econometrics, 4th Edition Page 42Chapter 13: Vector Error Correction and Vector
Autoregressive Models
13.4Impulse Responses
and Variance Decompositions
13.4.2bBivariate Analysis
The corresponding two-step-ahead forecast errors and variances are:
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
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Principles of Econometrics, 4th Edition Page 43Chapter 13: Vector Error Correction and Vector
Autoregressive Models
13.4Impulse Responses
and Variance Decompositions
13.4.2bBivariate 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:
– The proportion of the two-step forecast error variance of y explained by the ‘‘other’’ shock is:
2 2 2 2 2 2 2 211 11 12y y y x x
2 2 2 2 2 2 212 11 12x y x x
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Principles of Econometrics, 4th Edition Page 44Chapter 13: Vector Error Correction and Vector
Autoregressive Models
13.4Impulse Responses
and Variance Decompositions
13.4.2bBivariate Analysis
Similarly, the proportion of the two-step forecast error variance of x explained by its own shock is:
The proportion of the forecast error of x explained by the other shock is:
2 2 2 2 2 2 2 222 21 22x x y x x
2 2 2 2 2 2 221 21 22y y x x
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Principles of Econometrics, 4th Edition Page 45Chapter 13: Vector Error Correction and Vector
Autoregressive Models
13.4Impulse Responses
and Variance Decompositions
13.4.2cThe 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
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Principles of Econometrics, 4th Edition Page 46Chapter 13: Vector Error Correction and Vector
Autoregressive Models
Key Words
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Principles of Econometrics, 4th Edition Page 47Chapter 13: Vector Error Correction and Vector
Autoregressive Models
Keywords
dynamic relationshipserror correctionforecast error variance decompositionidentification problem
impulse response functionsVAR modelVEC model
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Principles of Econometrics, 4th Edition Page 48Chapter 13: Vector Error Correction and Vector
Autoregressive Models
Appendix
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Principles of Econometrics, 4th Edition Page 49Chapter 13: Vector Error Correction and Vector
Autoregressive Models
A bivariate dynamic system with contemporaneous interactions (also known as a structural model) is written as:
– In matrix terms:
13AThe
Identification Problem
Eq. 13A.11 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
11
yt t t
xt t t
y y ex x e
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Principles of Econometrics, 4th Edition Page 50Chapter 13: Vector Error Correction and Vector
Autoregressive Models
More compactly, BYt = AYt-1 + Et where:
13AThe
Identification Problem
1 21
3 42
1
1
yt t
xt t
y eY B A E
x e
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Principles of Econometrics, 4th Edition Page 51Chapter 13: Vector Error Correction and Vector
Autoregressive Models
A VAR representation (also known as reduced-form model) is written as:
13AThe
Identification Problem
1 1 2 1
3 1 4 1
yt t t t
xt t t t
y y x v
x y x v
Eq. 13A.2
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Principles of Econometrics, 4th Edition Page 52Chapter 13: Vector Error Correction and Vector
Autoregressive Models
In matrix form, this is Yt = Cyt-1 + Vt where:
13AThe
Identification Problem
1 2
3 4
ytxt
vC V
v
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Principles of Econometrics, 4th Edition Page 53Chapter 13: Vector Error Correction and Vector
Autoregressive Models
There is a relationship between Eq. 13A.1 and Eq. 13A.2 with:
13AThe
Identification Problem
1 1, t tC B A V B E