The long run
equilibrium relationship between spot VIX and VIX future prices is given from
the following equation:
: the VIX
: the spot
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:
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.
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:
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?