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If someone wants to run a regression he has to check
if the auxiliary variables of the regression are I(0)1,
if not he has to use the first differences of those variables. It is of grate
importance for the auxiliary variables to be stationary. In case of non-stationarity,
any deviation from equilibrium will
not be temporary. This of course is the safe way which has been used in many
regressions of time series ever since Granger &Newbold published their
paper about the problem of spurious regression. Apparently, that technic cannot
be considered flowless. The need of using the levels and not the first
differences of the variables “created” the meaning of cointegration. 



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In most of cases, the linear combination of two variables which are I(1)
is also I(1). In general, variables with different orders of integration are combined,
their combination order of integration equals the largest. So, if


,  we have

 variables each integrated of order

 , so that

The integration
order of


. Solving the
above requisition with respect to

, we have:


  The equation can be considered as a new
regression where

 is a disturbance term. This disturbance term
has two unwanted properties: first it is not stationary in most of the cases
and secondly is autocorrelated since all the


Let’s consider
an example:

The sample
regression function of the above equation will be written as follows:

If we solve
the above requisition with respect to

 we have:

We have
expressed the residuals like a linear combination of our auxiliary variables.
In most of the regressions the combination of

 variables will itself be

 but this is not very convenient. The ideal
case would be the residuals to be

 This is the case when the variables are cointegrated.

Back in 1987,
Engle and Granger proposed the following definition about cointegration.


 be a

vector of
variables integrated of order




is at least one vector of coefficients

 such that

In reality most
of the financial variables have one unit root so they are 

.  Having this in mind, a set of variables is considered
cointegrated if their linear combination is stationary.

It has been observed that many time series may not be stationary, but
they maybe related  in the long run. A cointegration
relationship may also be considered as a long term or else equilibrium phenomenon
because it is possible that cointegrated variables may seem unrelated in the short
run but this is not the case for the long run.

In this point it is important to distinct the meaning of spurious relations
with cointegrated ones. The spurious
regression problem is appeared when totally unrelated time series
may appear to be related using conventional testing procedures. And from
the other hand, we face genuine relationships which arise when the time series
are cointegrated.


and Granger test


Engle and Granger in 1987, recommended that «If a set of variables are cointegrated,
then there exists a valid error correction representation of the data, and
viceversa». To put it different, if two variables are cointegrated there must
be some force that will make the equilibrium error to go back to zero.

Engle and Granger in 1987, also suggested a two-step model for cointegration
analysis. For example, let’s say that we have an independent variable

 and a dependent one


First of all,
it should be estimated the long-run equilibrium equation:

We run a OLS
regression and we have:

We solve the
above equation with respect to 

  and we have:

In practice a
cointegration test is a test which examines if the residuals have not got unit root.
To examine this, we run a ADF test on the residuals, but we use the MacKinnon
(1991) critical values. If the hypothesis of the existence of cointegration cannot
be rejected, the OLS estimator, is said to be super-consistent. This means that
for a very big sample it is not necessary to include

 variables in our model.

The only important thing from the above test is the stationarity of the
residuals, if they are stationary (no unit root) we can move to the second
step. So, we save the residuals from the OLS and we prosed to the second step.

The second
step we use the unit root process for the stationarity of the residuals to the
next equation:

The above
equation does not have constant term because of the fact that, the residuals
have been calculated with the method of ordinary lest squares, so they have zero
mean. The test suggested from the Engle Granger is a little bit different from
those of the one of Dickey-Fuller. The hypothesis of this test is:



 (no cointegration)



The null hypothesis
can be rejected only when

 (? is the critical value of Engle-Granger table).

The Engle-
Granger Test can be also used for more than two variables. The process is the
same with the one we have described.

In conclusion the cointegration process is a way to estimate the long
run relation between two or even more variables. Engle and Granger in 1987 proved
that if two variables are cointegrated, then they have a long run relation
equilibrium, while in short run this may not be true. To check if their is a short
run disequilibrium we can use an Error Correction Mechanism (ECM). The
equilibrium error can be used to combine the long run with the short run with
the help of ECM.

The equation of this model is:



: is the
equilibrium error


: is the short
run coefficient which has to be between 0 and -1.



: are the
first differences of


 which are not stationary

We now can now
use ordinary least squares since all the variables are


It is important
to point out that long run equilibrium is tested trough the p-value of coincidence

. If

 is significant then


 in the long run. In addition, the coefficient

 measures the speed of adjustment to the long
run equilibrium. The higher this coefficient the faster the return to the equilibrium.

1 Integration is when in a
univariate context,


 if its (d-1)th difference is

 That is






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