  In this paper, we consider a framework adapting the notion of cointegration
when two asset prices are generated by a driftless It\^{o}-semimartingale
featuring jumps with infinite activity, observed regularly and synchronously at
high frequency. We develop a regression based estimation of the cointegrated
relations method and show the related consistency and central limit theory when
there is cointegration within that framework. We also provide a Dickey-Fuller
type residual based test for the null of no cointegration against the
alternative of cointegration, along with its limit theory. Under no
cointegration, the asymptotic limit is the same as that of the original
Dickey-Fuller residual based test, so that critical values can be easily
tabulated in the same way. Finite sample indicates adequate size and good power
properties in a variety of realistic configurations, outperforming original
Dickey-Fuller and Phillips-Perron type residual based tests, whose sizes are
distorted by non ergodic time-varying variance and power is altered by price
jumps. Two empirical examples consolidate the Monte-Carlo evidence that the
adapted tests can be rejected while the original tests are not, and vice versa.
