# Normalization choice in cointegration models

Im a student in econometrics and Im currently working on vector error correction models (VECM).

Im facing a problem with correction error coefficients signs. My teacher stated that they must be negative and significant. But I found out that their signs depend on the choice of the normalized coefficient.

For example, if the Johansen test tells me that there is one relation of cointegration between 3 variables (all I(1)), the sign of each error correction term will depends on the way I'm estimating. If I'm estimating a VECM between X Y Z, the first cointegrated coefficient (X) will be used for the normalization. If I'm running it in this order (Y Z X), the cointegrated coefficient of Y will be used.

Changing the order, usually change signs (global estimation results are the same but coefficients are different), so I'd like to know if there is a trick to know if my correction error terms are actually negative even if I change the order ?

As all variables are supposed endogenous, the order shouldn't be a problem.

Thanks

• This answer is closely related, but I am not sure if it adds much extra intuition to what you have already said yourself. Depending on normalization, a positive loading might make perfect sense. So perhaps your understanding is fine, and it is actually your professor who might need to rethink this?.. – Richard Hardy Jan 17 '18 at 18:37
• Yes I saw this post already, my professor told me whatever the order is, all coefficients must always be negative. Im pretty sure this isn't even possible mathematically speaking. I'm quite stuck since I tryed to find an article discussing about this, didn't find yet. – CDVC23 Jan 17 '18 at 19:27
• Sorry to hear the professor's opinion. I think you got it right. Unfortunately, I cannot remember any references to recommend. Some things are just too simple to be discussed in papers. – Richard Hardy Jan 17 '18 at 19:51