Assume we have $k$ I(1) variables, cointegrated of rank $r = 1$.
By cointegration, I know that the error-correction term (ECT) is itself a I(0) univariate process. Assume now I am interested in the time-series properties of the ECT term, namely how many lags, and its persistence. I estimate the lags with AIC, and assess persistence looking at the univariate Impulse Response Function (IRF), i.e. I look at how quickly shocks on the ECT go to zero.
How can I relate the univariate properties of the ECT term (lag, IRF), with those of the VECM (lag, IRF)?
My understanding from Hansen (2005) is that the number $p$ of lags on the AR(p) for the ECT should be the same (eventually -1 depending on notation) as for the $p$ lags of the VECM (compare Theorem 1 with Corrollary 1 in Hansen).
But what about the IRF? What if I learn that the ECT has a high persistence (slow univariate IRF)? What does this imply in terms of the IRF of the VECM?
- Hansen (2005) Granger's representation theorem: A closed‐form expression for I(1) processes, Econometrics Journal 2005, 8:1