Nonlinear cointegration in financial time series 269cointegration, as presented in the previous section, we adapted the two-stage procedure
suggested by Engle and Granger to the nonlinear framework. In the first stage, the
local linear model was initially estimated amongst the series under investigation; then
the stationarity was tested using the residuals of the estimated linear local model. If
the null hypothesis of nonstationarity was discarded, the second stage of the procedure
was conducted: it consisted in verifying the cointegration hypothesis by performing
a second regression as in (3). The results of the two-stage procedure are shown in 3.
They highlight that the use of local linear models has enabled the identification of
nonlinear cointegration relationships among 40 binary time series combinations. This
confirms the initial assumption, i.e., that the time series lacking linear cointegration
in fact present a nonlinear relationship.
For the time series that presented nonlinear cointegration, an unrestricted local
error correction model was also estimated to obtain both the long-run dynamic re-
lationship and the function of the speed of adjustment. Below, for brevity, only one
case is presented. The considered period went from 01/07/2000to 31/12/2002. has to
be interpreted as acceptation of the nullhypothesis of the absence of co-integration
with ap-value≥ 0 .15.
Fig. 1.Time series of stocks priceFig. 2.Speed of adjustment function