VAR or VECM for a system of I(0) and I(1) variables? [duplicate]

I have 4 variables (time series). All the variables depend on each other. Two of them are non - stationary and the others are stationary . If I apply log transformation in the non-stationary variables, one of them will stationary after this, the other not. So in this case i would have 3 stationary variables (one of them in logs) and the other variable non - stationary. The goal is forecast all variables.

1. I selected the optimal lag length based on AIC.
2. I test for Cointegration with all original variables using Johansen's test, the output is "Trace test indicates 1 cointegration eqn(s) at the 0.05 level". How i should interpret it?
3. Should I use VAR or VECM if the goal is forecast all variables? And how can i build these models?
4. Can you please give any good research?
• Excellent! It's just what I need. @Richard said that we should build a model with the characteristics that he mentioned, however, Is it your own consideration or are you refering to a book/paper? If the second is the case, can you please provide the source? – CarV Nov 16 '17 at 19:37
• This is my own consideration based on textbooks, papers, lecture notes, and plenty of hands-on experience with time series. The principles I lay out in my answer are rather simple and I doubt anyone would dispute them. However, giving references for each of them would be difficult as this is a combination of common knowledge and basic facts. – Richard Hardy Nov 16 '17 at 19:57
• Thank you very much Professor @Richard. And excuse me if I am a bit annoying, about your answers in b and c, should I consider them as a VEC model, or just a system of simultaneous equations? My regards. – CarV Nov 16 '17 at 20:10
• If you only take the equations for the cointegrated variables, then it is a VECM with extra regressors. But since equations for these regressors also belong in the model, I do not think you can call this a VECM anymore. So it is some hybrid model with no proper name. – Richard Hardy Nov 16 '17 at 20:21
• Got it. I'll try with your suggestions. – CarV Nov 16 '17 at 20:34