The long run

equilibrium relationship between spot VIX and VIX future prices is given from

the following equation:

(1)

Where:

Ø

: the VIX

future prices

Ø

: the spot

VIX price

It is well known

that the above equation cannot be tested by ordinary least squares if at least one

of the variables is not stationary. So, the first step in time series testes is

to test for stationarity. The null hypothesis of a unit root test supports the

nonstationary. The models used for the unit root tests are the following:

(without trend)

(with trend)

Where

stand for the spot VIX and the nine nearest to

maturity VIX Future contracts prices respectively. Having run Augmented Dicey-Fuller unit root tests on spot VIX and

ten VIX futures prices indexes. All t-statistics are below 1% critical values,

as a result the null hypothesis cannot be rejected for any of the ten indexes.

Since the null hypothesis cannot be rejected all the time series of VIX and

futures prices are not stationary.

A nonstationary

time series which has stationary first difference, is said to be integrated to order

1, it is denoted as I(1). Having run Augmented Dicey-Fuller unit root tests on

the first differences of VIX and Fi. The null hypothesis is rejected for all

the indexes at the 1% significance level, so there is no unit root problem. In

conclusion all the ten indexes are I(1) processes.

In 1987 Enger

and Granger proved that if we have two I(1) processes and their liner combination

is I(0) (stationary), the two time series are cointegrated. From the economical

perspective, two time series are said to be cointegrated if they have a long-term

or else equilibrium relationship between them. One way to test if two time

series are cointegrated is to construct test statistics from the residuals of

their regression. Let

denote the estimated residuals from equation

(1), a test for no cointegration is given from a test for unit root of those

residuals. The ADF regression equation is:

Test

statistics is a t-ratio test for a=0 (t-test). The critical values are -3.34 for

5% confidence interval and -3.04 for 1% confidence interval. Significant

negative test statistics suggest cointegration (rejection of the unit root hypothesis).

Table IV and V presents the Enger-Granger cointegration results for different

pairs of time series. From the tables belowDT1

DT1Ti sx?li? na kanw edw?