Noncontemporaneous cointegration and the importance of timing

It is well known that in standard I(1) cointegration, the timing of variables in the cointegrating relation does not interfere with the cointegration property, whereas in an I(2) framework, it may. We show that, in a general (fractional) CI(d,b) model, it is the reduction in integration orders, b, implied by cointegration that determines whether timing matters, and not the integration order of the observed variables. We also derive implications of noncontemporaneous cointegration in both the time and frequency domains.