First puzzle: I am taught that the lag order of VECM does not affect the cointegration rank because the lag order is for the differenced regressors. But, I see the contrary: I experimented with sample data of 4 variables, using lag orders between 1 and 12. At low lag orders the cointegration rank is low, but as I increase the lag length, cointegration rank goes up. Does anyone know of a paper or source discussing this issue? Any suggestions as to what to do in this situation?
Second puzzle: If I go by what information criteria indicate as an optimal lag length, establish the coinegration rank and then set up a VECM model, the residuals of the model are heteroscedastic and serially correlated. The info criteria based lag order is 2. To circumvent the issue, I have two options:
Go back, increase the lag order and set up a new VECM and do diagnostic tests
Just keep the cointegration rank as it is and increase the lag order until the residuals are homoscedastic and free of autocorrelation
What would be the optimal course of action?