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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». In other words, this mean that if two variables are cointegrated
there must be some force that will return the equilibrium error 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

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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 testes if the residuals are stationary. To examine
this, we run a ADF test on the residuals, with the MacKinnon (1991) critical
values adjusted for the specific number of variables of our model. 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 T there is no need
to include

variables in the model

The only thing that matters from the above test is the stationarity of
the residuals, if they are stationary we move to the second step. So, we save
the residuals 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 include constant term because 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:

Ø

:

(no cointegration)

Ø

(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 followed for two variables.

In conclusion
the cointegration process is a way to estimate the long run relation re else equilibrium
between two or even more variables. Engle and Granger in 1987 proved that if two
variables are cointegrated, then those variables have a long run relation
equilibrium, while in short run this may not be the  case. The short run disequilibrium relation
can be proved with 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:

Where:

Ø

: is the
equilibrium error

Ø

: is the short
run coefficient which has to be less than zero and more than -1.

Ø

and

: are the
first differences of

and

which are

and

is

We now can now
use the OLS because all the variables are

.

It is important
to point out that long run equilibrium is tested trough the significance of coincidence

. If

is significant then

causes

in the long run. In addition, the coefficient

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