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.

Cointegration

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

for

, we have

variables each integrated of order

, so that

The integration

order of

is

. Solving the

above requisition with respect to

, we have:

Where

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

are

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.

Let

be a

vector of

variables integrated of order

:

Ø

All

are

Ø

There

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.

Engle

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:

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 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:

Where:

Ø

: is the

equilibrium error

Ø

: is the short

run coefficient which has to be between 0 and -1.

Ø

and

: are the

first differences of

and

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

causes

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,

is

if its (d-1)th difference is

That is

is

stationary?

is

if

is

.