The long runequilibrium relationship between spot VIX and VIX future prices is given fromthe following equation: (1)Where:Ø : the VIXfuture pricesØ : the spotVIX priceIt is well knownthat the above equation cannot be tested by ordinary least squares if at least oneof the variables is not stationary. So, the first step in time series testes isto test for stationarity. The null hypothesis of a unit root test is that the variableis not stationary. The models used for the unit root tests are the following: (without trend) (with trend) Where stand for the spot VIX and the ten nearest tomaturity VIX Future contracts prices respectively. Having run Augmented Dicey-Fuller unit root tests on spot VIX andten 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 andfutures prices are not stationary. A nonstationarytime series which has stationary first difference, is said to be integrated to order1, it is denoted as I(1). Having run Augmented Dicey-Fuller unit root tests onthe first differences of VIX and Fi. The null hypothesis is rejected for allthe indexes at the 1% significance level, so there is no unit root problem. Inconclusion all the ten indexes are I(1) processes.
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In 1987 Engerand Granger proved that if we have two I(1) processes and their liner combinationis I(0) (stationary), the two time series are cointegrated. From the economicalperspective, two time series are said to be cointegrated if they have a long-termor else equilibrium relationship between them. One way to test if two timeseries are cointegrated is to conduct test statistics from the residuals oftheir regression.
Let denote the estimated residuals from equation(1), a test for no cointegration is given from a test for unit root of thoseresiduals. The ADF regression equation is: Teststatistics is a t-ratio test for a=0 (t-test). The critical values are -3.
34 for5% confidence interval and -3.04 for 1% confidence interval. Significantnegative test statistics suggest cointegration (rejection of the unit root hypothesis).Table IV and V presents the Enger-Granger cointegration results for differentpairs of time series. From the tables belowDT1 DT1Ti sx?li? na kanw edw?