L-8 VECM Formulation, Hypothesis Testing, and Forecasting - KH.pptx

L8 – Vector Error Correction
Models: Formulation,
Hypothesis Testing, and
Forecasting
Presenter
Kaddour Hadri
Course on Macroeconomic Forecasting
This training material is the property of the International Monetary Fund (IMF) and is
intended for use in IMF Institute for Capacity Development courses. Any reuse requires the
permission of the IMF Institute for Capacity Development.

Introduction
In L4 we considered single equation approach to testing
cointegration, and estimating the cointegrating vector.
In L7 we considered VAR approach to testing cointegration
(Johansen) and estimating the vector error correction model.
In this lecture we will:
• compare the two methodologies described above
• consider a third way, the autoregressive distributed lag
representation
2
This training material is the property of the International Monetary Fund (IMF) and is intended
for use in IMF Institute courses. Any reuse requires the permission of the IMF Institute.”

Outline of the lecture
1. Single equation approach: Engle-Granger methodology
2. The VAR/VECM approach: Johansen methodology
3. The Error-Correction / Autoregressive Distributed Lag
approach
JV12.10 - MF 3

Part I. Single Equation Approach:
the Engle-Granger
Methodology
JV12.10 - MF 4

Example: short and long interest rates
Consider interest rate (annualized) on:
• 3 month US Treasury bills (short)
• 10 year US Treasury securities (long)
JV12.10 - MF 5
0
4
8
12
16
20
55 60 65 70 75 80 85 90 95 00 05 10
SHORT LONG
Each rate seem to move in
some unpredictable way:
no tendency to converge to
a constant, persistence of
movements
However, the two rates seem
to be moving together

Example: short and long interest rates
For both variables we cannot reject the that they are I(1)
JV12.10 - MF 6

-4
-2
0
2
4
6
0
4
8
12
16
20
65 70 75 80 85 90 95 00 05 10
Residual Actual Fitted
Example: interest rates
JV12.10 - MF 7
The residuals from the OLS regression of short on long are I(0), and
the estimated long-run relationship is (standard errors within
parenthesis –579 obs):
(0.152) (0.020)
1.398 0.99
t t
short long
  

Normalization of the cointegrating vector
If x1, x2, … xn are cointegrated, then for the cointegrating vector
β and any λ  0, the following are stationary variables
JV12.10 - MF 8
1 1 2 2 ... n n
z x x x
  
   
1 1 2 2 ... n n
e x x x z
   
    
Hence, the following are sensible representations of the long-
run relationship
1 2 2 ... n n
x x x
 
   
1
i
i



 
2 1 1 ... n n
x x x
 
   
2
i
i



 
1 1 2 2 1 1
...
n n n
x x x x
    
   
i
i
n



 

JV12.10 - MF 9
Using the result of the previous slide, consider regressing long
on short (standard errors within parenthesis):
(0.103) (0.017)
2.513 0.806
t t
long short
 
Because OLS minimizes the sum of squared errors in the direction
of the dependent variable, so the slope here is different from the
slope of the other regression. However, 1
0.806
0.99

Also, the no-cointegration
test based on the residuals
can be rejected at 5
percent significance level
(the critical level is -3.35)

Broad issues with single equation approach
Let us consider estimating the error correction model:
JV12.10 - MF 10
3 3
1 1
( ) ( ) ( ) ( )
t t t i t i i t i
i i
d short c short long d short d long
    
 
 
     
 
The coefficient α has the correct sign and the constant c is
(correctly so) not significantly different from zero.

If instead we use the other specification:
JV12.10 - MF 11
3 3
1 1
( ) ( ) ( ) ( )
t t t i t i i t i
i i
d long c long short d short d long
    
 
 
     
 
The coefficient α has the correct sign and the constant c is (correctly
so) not significantly different from zero
The t-Statistics on α is also a test for cointegration: if α = 0 in both
equations, neither short nor long respond to the cointegrating
vector. Together, these tests reject the absence of cointegration.

The Engle-Granger approach:
1.Conduct a unit root test on the residual of an OLS of y on x
2.If the test is rejected (y and x are cointegrated), estimate the
long-run relationship (with OLS, FM, DOLS)
In general suffers from some shortfalls:
• The results are sensible to the specification and may be
contradictory
• Bias in small sample lowers the power of the test
• It does not specify/assume any direction of causality
• It assumes a proportional adjustment of y to y - βx
• It does not help understand the dynamic of y and x
• If there are n variables and there are k cointegrating vector
one has to choose one among them
JV12.10 - MF 12

Part II. The VAR/VECM Approach:
Johansen Methodology
JV12.10 - MF 13

VAR specification: summary
Consider the following VAR for n variables:
JV12.10 - MF 14
t 0 1 t-1 2 t-2 t
x = A + A x + A x +ε
Consider adding and subtracting lags:
1 1 1 1
 
   
t t 0 1 t-1 t 2 t- 2 t- 2 t-2 t
x x = A + A x x + A x A x A x +ε
2 1 1
( ) 
   
t 0 1 n t 2 t- t
x = A + A A -I x A x +ε
 
t 0 t-1 t-1 t
x = A + Πx + Γ x +ε
• if rank() = 0 ( = 0 matrix) the n variables are not cointegrated
• if rank() = n ( is invertible) the n variables are stationary
• if rank() = r there are r cointegrating vectors and
 
t 0 t-1 t-1 t
x = A +αβ'x + Γ x +ε

VAR specification: summary
If the variables in the VAR are cointegrated the two
representations are equivalent:
JV12.10 - MF 15
t 0 1 t-1 2 t-2 t
x = A + A x + A x +ε
 
t 0 t-1 t-1 t
x = A + Πx + Γ x +ε
Notice that
• cointegration implies restriction on the parameters of the VAR
(A1 + A2 - In cannot be invertible)
• If variables are cointegrated, it would NOT be appropriate to
estimate the VAR in differences only: this would correspond to
omitting the error correction part of the dynamic
• If variables are not cointegrated, estimating the VAR in differences
is more appropriate (as  = 0 there is one less matrix to estimate)
1 2 n
Π = A + A -I 2 2
Γ = -A

Consider again the short-run and long-run interest rate, and the
3-order VAR representation and the corresponding VECM (lag
length chosen based on Schwartz Information criterion)
JV12.10 - MF 16
The Johansen test
indicates that the
variables are
cointegrated
1 2 2
  
t 0 t-1 t-1 t- t
x = A +αβ'x + Γ x + Γ x +ε
t
t
short
long
 
 
 
t
x =
t 0 1 t-1 2 t-2 3 t-3 t
x = A + A x + A x + A x +ε

The estimates of the cointegrating vector and factor loadings are:
JV12.10 - MF 17
Notice that the factor loadings indicate that, in response to a
positive deviation from equilibrium, short decreases and long
increases. Also, all residuals appear to be stationary.
-4
-3
-2
-1
0
1
2
3
4
5
55 60 65 70 75 80 85 90 95 00 05 10
COINTEQ01 RESID01 RESID02

Notice that if we change the normalization of the cointegrating
vector, the estimates are not contradictory (1.002 = 1/0.998)
JV12.10 - MF 18
Notice also that we can conduct standard tests on coefficients; for
example, we cannot reject that the β coefficient on short is -1.

Weak exogeneity
Consider the VECM for two cointegrated variables
JV12.10 - MF 19
1 1
1 1
1 1
1 1
1 1
1 1
( )
( )
t
t
p p
t y y t t i t i i t i y
i i
p p
t x x t t i t i i t i x
i i
y c y x y x
x c y x y x
    
    
 
   
 
 
   
 
        
        
 
 
Suppose that αx = 0: y does all the adjustment necessary to
restore equilibrium. We say that x is weakly exogenous.
If x is weakly exogenous, we can estimate a model for the
behavior of y without the need to model and estimate the
behavior of x.

The short run interest rate appears to be weakly exogenous: the
factor loading for short is not significantly different from zero:
• short term interest rate is weakly exogenous
JV12.10 - MF 20

Granger causality
Consider the two variable VECM, and consider the equation for x
only:
To be able to say that y does not Granger cause x, two conditions
must be satisfied:
• all the coefficients λi = 0 (as seen in L6)
• x must be weakly exogenous: αx = 0, or x should not respond
to deviation of y from equilibrium, otherwise the past level of
y, relative to that of x, would determine changes in x
JV12.10 - MF 21
1 1
1 1
1 1
( ) t
p p
t x x t t i t i i t i x
i i
x c y x y x
    
 
   
 
        
 

A Granger Causality test indicates that, in the equation for the
difference of short, the coefficients λi of the lagged differences
of long n the variable short are jointly not significantly different
from 0.
JV12.10 - MF 22
This and the fact that short is
weakly exogenous indicate
that long does not Granger
cause short.
Of course, short does
Granger causes long (if not,
short and long would not
be cointegrated.

Part III. The error correction /
autoregressive distributed
lag approach
JV12.10 - MF 23

Reducing the number of equations
Consider the VECM for two cointegrated variables
JV12.10 - MF 24
1 1 1 1
1 1 1 1
( )
( )
t
t
t y y t t t t y
t x x t t t t x
y c y x y x
x c y x y x
    
    
   
   
        
        
Suppose that the errors are correlated, in particular
( , ) 0
t
x t
Cov v
 
It is possible to find a way to orthogonalize the two errors so that
they are not correlated (see Appendix and L6 for details)
11 12
21 22
t t
t t
y y
x x
u
c c
c c u


   
 

   
 
   
 
   
( , ) 0
t t
y x
Corr u u 
t t
y x t
v
 
 

Reducing the number of equations
Let us substitute into
JV12.10 - MF 25
1 1 1 1
1 1 1 1
( )
( )
t
t
t y t t t t y
t x t t t t x
y y x y x
x y x y x
    
    
   
   
       
       
to obtain
t t
y x t
v
 
 
1 1 1 1
( ) t
t y t t t t x t
y y x y x
     
   
        
 
1 1 1 1
1 1 1 1
( )
( )
t y t t t t
t x t t t t t
y y x y x
x y x y x
   
     
   
   
      
        
      
1 1 1 1
t y x t t t t t t
y y x y x x
        
   
            
Using the equation for x the equation for y becomes

The autoregressive distributed lag model
Notice that
JV12.10 - MF 26
      
1 1 1 1
t y x t t t t t t
y y x y x x
        
   
            
is a restricted version of the autoregressive distributed lag model
0 1 1 2 1
1 0
p q
t t t j t j j t j t
j j
y y x y x
     
   
 
        
 

Problems in estimating one equation
Consider for estimating
JV12.10 - MF 27
In general, you are likely to encounter:
1. Identification problem: you cannot identify the factor
loadings for y and x (β1 = αy - αx)
2. Simultaneity problem: x and y are likely to be jointly
determined, as xt is likely correlated with t
0 1 1 2 1
1 0
p q
t t t j t j j t j t
j j
y y x y x
     
   
 
        
 

The role of weak exogeneity
Consider the ADL model
JV12.10 - MF 28
Suppose that:
• x is weakly exogenous, that is αx = 0
• x is predetermined, that is (in slide 26, c21 = 0,
so that the innovation in x does not depend on the
innovation in y)
Then:
• β1 = αy and the long run relationship is given by
(the identification problem is partly solved)
• xt is not related to vt (the simultaneity problem is solved)
22
t t
x x
c u
 
2
1




0 1 1 2 1
1 0
p q
t t t j t j j t j t
j j
y y x y x
     
   
 
        
 

The benefits of using weak exogeneity
Compare the error term of
JV12.10 - MF 29
with those of
• because is composed of xt, xt-1, and vt, by controlling
for them we obtain more precise estimates of the coefficients
• the long run coefficient β is estimated within the equation:
there is no common factor restriction; that is, the dynamic of y
is not dictated by the estimated equilibrium error zt = yt - βxt
1 1 1 1
1 1 1 1
( )
( )
t
t
t y y t t t t y
t x x t t t t x
y c y x y x
x c y x y x
    
    
   
   
        
        
t
y

0 1 1 2 1
1 0
p q
t t t j t j j t j t
j j
y y x y x
     
   
 
        
 

Common factor restriction
Consider estimating the long run relationship and α from
JV12.10 - MF 30
1 0 1 1
( )
t t t t
long c long x
   
 
     
0 1
t t t
long short
  
  
0 1 1 1
t t t t
long c long x
  
 
    
The estimate of α is very similar to that of β0.
However, estimates of the cointegrating
relationship are different:
0.021
0.806
0.014


Estimating cointegration
From
JV12.10 - MF 31
it is still possible to test no-cointegration by testing
H0: β1 = 0
and using specific tables for critical values.
If β1 = 0 there is no error adjustment; also, because all y and
x are stationary, the coefficient on xt-1 will not be significantly
different from zero.
The dynamic of x can be estimated separately for forecasting
purposes.
0 1 1 2 1
1 0
p q
t t t j t j j t j t
j j
y y x y x
     
   
 
        
 

Using the fact that short appears to be weakly exogenous, we can
write the model in ADL form
JV12.10 - MF 32
0 1 1 2 1 2 1 2 2
1 2 1 2 2
t t t t t
t t t t
long long short long long
short short short
    
   
   
 
       
      

Collecting the estimated coefficient one obtains
JV12.10 - MF 33
 
1 1
0.032 1.904 0.925 ...
t t t
long long short
 
     
• The t-Statistic for β1 is -3.866 and the appropriate critical value
(5%) is -3.221: we can reject that β1 = 0
• The t-Statistic for β2 is -3.911:
we can reject that β2 = 0
• To test whether rates move
1:1, we can use an F-test for
the restriction β1 - β2 = 0; this
hypothesis is rejected

Part IV. Comparison of the three
approaches and
conclusions
JV12.10 - MF 34

Let us compare the results of the three approaches by topic
JV12.10 - MF 35
Single equation:
Engle-Granger
VAR/VECM
approach:
Johansen
ADL approach
Cointegration: are
variables cointegrated
Yes Yes Yes
Adjustment to
equilibrium: which
variable adjust?
Not specified Both Long-run rates
Speed of adjustment
-0.028 (short)
-0.030 (long)
0.027 (short)
-0.019 (long)
-0.032 (long)
Long-run coefficient β
0.990 (short)
0.806 (long)
basically, 1 0.925

Conclusion
JV12.10 - MF 36
Single equation:
Engle-Granger
VAR/VEC
M
approach:
Johansen
ADL approach
Is cointegration test
allowed?
Yes Yes Yes
Are all cointegrating vector
captured?
No: need to
normalize each
variable and estimate
long-run relationship
Yes
Not a sensible
question…
Does it describe which
variable adjust?
No Yes It is assumed
How good are estimates?
Need to correct for
bias in small samples
Fairly good
Lower standard
errors
Assumptions about
causality of variables
None
None, but
testable
Assumed

Orthogonalization of the errors
From
JV12.10 - MF 38
compute
11 12
21 22
t t
t t
x x
y y
u
c c
c c u


   
 

   
 
   
 
   
( , ) 0
t t
y x
Cov u u 
E[ ' ]
x xy
t t
xy y
 
 
 
   
 
ε ε Ω
11 12
21 22
2 2 2
2 2 2
11 21 12 22
( ) ( ) ( )
( ) ( ) ( )
( , ) ( ) ( )
t t t
t t t
t t t t
x x x y
y y x y
xy x x x y
Var c Var u c Var u
Var c Var u c Var u
Cov c c Var u c c Var u
 
 
  
  
  
  

Orthogonalization of the errors
Recall from L6 that given there exist a
JV12.10 - MF 39
11
21 22
0
t t
t t
x x
y y
u
a
a a u


   
 

   
 
   
 
   
( , ) 0
t t
y x
Cov u u 
triangular matrix A and a diagonal matrix D such that  = ADA’
and
E[ ' ]
x xy
t t
xy y
 
 
 
   
 
ε ε Ω
11
21 22
t t
t t t
x x
y x y
a u
a u a u



 
11
21
22
11
t t
t t t
x x
y x y
a u
a
a u
a

 

 
( , ) 0
t t
y x
Cov u  
so that
t t
y x t
v
 
 

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