Problems on the model with trend

Estimation ML/Bayesian estimation
2
  1. No, in the model everything is consistent. This part is about imposing cointegration in the model. If the mean in the data is different across the two variables, then shocks need to account for that.
  2. Your model needs to feature a BGP in the first place. In Iacoviello/Neri you need to adjust the utility function to get one.
  3. You could do that, but may have an issue identifying parameters related to the trend.
  4. Have a look at А model without balanced growth - #6 by jpfeifer