Serial correlation in cointegrating relationship

[Bekah500]

HI,
I am estimating a cointegrating relationship but I really need to include a lag of the dependent variable in the long run relationship. I was wondering if this was 'allowed'? When I miss it out the resulting residuals are stationary even though they are serially correlated so I can model these dynamics in the ECM part but it feels like they should be accounted for in stage 1. Ill probably figure it out but any help would be appreciated!

[Bekah500]

But it doesn't make much sense to add it because of it being a unit root, and it also makes the sign of one of the other variables wrong.
But I just had a thought-I don't have to use the resulting residuals for an ECM? Cant I just create an ARMA model of the residuals and combine it with the LR part?

[Bekah500]

Im very new to ECMs so Im still getting my head around it but I guess the purpose of adding lagged changes of the dependent variable to the ECM is a way of capturing these dynamics? (Ill find out when I look at the final residuals anyway)