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Co integration: An overview
Co integration and Error correction: Representation, estimation and testing
X is a vector of economic variables, they may be said to be in equilibrium when;
 ' Xt  0
(1)
Occurs
In most time period X t will not be in equilibrium. Then
Zt   ' X t
(2) is the equilibrium error.
The error correction models allow long run components of X to obey equilibrium
constraints while short run components have a flexible dynamitic specification. A
condition for this to be true, called co integration was introduced by Granger
(1981), Granger and Weiss ( 1983). The phenomenon that non stationary process
has linear combinations that are stationary was called co integration by Granger
who used it for modeling long run economic relations.
1. Integration, co integration and Error Correction
If X t ; Yt
I (d ) , then their linear combinations Zt
However, it is possible that Zt
I (d  b), b
In particular, let us consider X t , Yt
1 X t   2Yt
I (d )
0
I (1) and Zt  X t  aYt
I (0), d  b  1.
The constant a is such that the bulk of the long run components of X t and Yt
cancel out.
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Definition: The components of the vector X t are said to be co integrated of order
d, b, denoted X t CI (d , b) if:
a) All components of X t are I(d)
b) There exists a vector   0 so that Zt   ' X t
I (d  b); b
0
The vector  is called the co integration vector.
If d=1=b, co integration would mean that if X t
I (1) , then the equilibrium error
would be I (0).
If X has N components, then there are
at maximum r  N 1 co integration
vectors. By construction the rank of  will be r and will be called the co
integrating rank of X t .
For a two variable system a typical error correction model would relate
Equilibrium Correction or Error Correction models
Let us consider two variables X t and Yt which I(1). The model that one may
consider is to use their differenced variables which are I(0):
Yt  X t   t
(1)
One definition of the long run is employed in econometrics implies that the
variables have converged upon long term values and are no longer changing.
Thus, yt  yt 1; xt  xt 1 . This implies that y  x  0 so that every thing in (10
cancels in such way that there is no long run solution in 91) and does not say
nothing about whether x and y have an equilibrium relationship.
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There is a class of models that can overcome this problem by using
combinations of first differenced and lagged levels of cointegrated variables.
For example, consider the following equation:
yt  1xt  2 ( yt 1   xt 1 )   t
(2)
This model is known as an Error correction model (ECM) or an equilibrium
correction model. yt 1   xt 1 is known as the error correction term. . Considering
that y and x are cointegrated with cointegrating coefficient  , then yt 1   xt 1 is
stationary. OLS method can be used to estimate the coefficients of (2).
It is possible to have an intercept in either the cointegrating term ( yt 1     xt 1 )
or in the model ( yt   0  1xt   2 ( yt 1   xt 1 )   t or both.
Multivariate case
Let us consider X t ( X1 ,..., X m )
And X t    1 X t 1  2 X t 2  ...   t
With  t
(3)
NID(0, )
The equation (3) can be re write into a Vector Error Correction Model (VECM) as
follow:
X t    1 X t 1  2 X t 2  ...   t
 X t  X t 1    X t 1  1 X t 1  2 X t 1  2 X t 2  2 X t 2   t
 X t    ( I  1  2 ) X t 1  2 ( X t 1  X t 2 )   t
 X t     X t 1  X t 1   t
The VECM is
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X t     X t 1  X t 1   t
(4)
With   ( I  1  2 );   2
Because X t
I (1) , the left hand of (4) is stationary. If   0 and the model holds,
then  X t 1 must be stationary. In general case we can decompose    ' where
 and  are mxr matrices, r  m
Is the rank of  .
 X t 1   ' X t 1
 contains the long run relationships between the components of X t and 
contains the short run adjusting parameters towards the long run steady
relationship  ' X t .
1. If r=m,  is full rank and X t
I (0)
2. If r=0, then   0 , X t are not cointegrated
3. If 0 r m 1,  has reduced rank, and there are r cointegrating vectors.
In testing cointegration, the first step is to find the cointegration order, r using
Johansson’s (1988) procedure, where also estimates of  and  are obtained.