From Cointegration to estimation

[TomDoan]

Hello Tom,

I just completed first part with 5 variables, out of 2 were stationary, 3 that were non-stationary were cointegrated and had 1 cointegrating relationship.

Now i am in the second part. I add a variable that is non-stationary. with 2 stationary variables and 4 non-stationary variables, I am getting 2 cointegrating vectors among these 4 variables.

Where should i start my reading? The Katarina Textbook is very hard and CATs manual also does not have a specific section handling r=2 case in CATs. Enders book helped me a lot with understanding of Johansen test, applying Engle-Granger method to my first case, however there is not much explanation or any program file to handle r=2 case there either.

So I need guidance, where should i start my reading to understand the cointegrating space and how to put and test restrictions on the vectors?

your prompt response will be highly appreciated.

Regards,

If you have six variables, two of which are I(0) and four of which are I(1), then the "cointegrating" rank has to be at least 2, since each stationary variable is (in effect) cointegrated with itself. So your four I(1) variables aren't (or at least don't appear to be) cointegrated.

[ateeb]

So it means even if 2 variables are stationary in levels, they will still be tested for cointegration?

I was only testing cointegration among 4 variables that are non stationary in levels.

Thanks. I await your response.

[TomDoan]

If you do @JOHMLE with six variables, you can get rank 2 if two of the variables are I(0) (which means that there is no cointegration among the others). If you only use the four I(1) variables, then of course you would not pick up the behavior of the (omitted) I(0) variables. The point of including I(0) variables in a cointegration analysis is to pick up a more systematic part of the (presumed) I(0) residual---you just have to take account of how they affect the inference on the rank.

[ateeb]

So how to go about disentangling this problem?

For the model where i had 2 stationary and 3 non stationary variables i had 1 cointegrating vector among 3 non stationary variables. So that led to a single ect term but now its about playing inside the cointegrating space.

Where can i get help understanding how to start and go on with analysis?

Regards

[TomDoan]

You then have three "error correction" terms in the system---one for each of the stationary variables (with just themselves) and one for cointegrating vector involving the I(1) variables. The latter can include the I(0) variables if that's how you estimated it.

[ateeb]

Actually the problem lies here.

1. When we have all 6 non stationary variables and there is a single cointegrating vector among them, its easy, we can use the ect term and thats all good because there is a single long run relationship among all variables.

2. Now suppose we have 6 variables, 2 stationary and only 4 non stationary but cointegrated variables with 1 cointegrating vector, in this case what i did was to run var kn 1st differences and then modify equations with levels for stationary variables with ect only for 4 variables that were cointegrated. Thats fine.

3. Now situation is that out of 6, 2 are stationary, 4 are non stationary but have 2 cointegrating vectors meaning 2possible long run relationships. So if i add both ects shouldn't that state that the given 4 economic variables have 2 long run relationships and it would matter that which one should be considered as correct right? So in such a case what to do? Like what is the econometrics here onwards for VAR?

Regards

[TomDoan]

If you analyze six and get rank two cointegration and two of the six are stationary, then there is, in fact, no cointegration because each stationary variable adds 1 to the "cointegrating" rank. If this is different than what you get by doing the four I(1) variables separately, then that's an indication that the cointegration found in the latter case is because there is too much noise in the residuals without the added explanatory power of the stationary variables.

[ateeb]

Understood. But now how to proceed with analysis in such a case?

I want to read or do anything to do this ...

[TomDoan]

Isn't that a result? If apparent cointegration disappears once you add some stationary variables to the model, that's similar to an effect in labor economics disappearing once you control for covariates.

[ateeb]

So if 4 variables taken without 2 stationary variables show 2 cointegrating vectors that would need 2 ect terms to be added to VAR?

Ok but how to determine which one we believe in? Whichnone is infact the LR relationship between the variables?

[TomDoan]

You seem to be unwilling to accept the fact that "there is no long-run relationship" is a possibility, but it sounds like that's what the data shows. Again, that *is* a result.

[ateeb]

I am not giving all variables to the test for cointegration. Only 4 non stationary, so you mean that missing the two stationary variables is causing this?

[TomDoan]

Yes. Read the earlier response.

[ateeb]

Dear Tom,

Now i am trying to run the cointegration tests as described by you:

Consider this and if you can help me get through this i will be thankful.

When i am considering all 5 variables (3 non-stationary and 2 stationary) in my cointegration test i.e. in CATs:

If i use deterministic term as none:
it gives me 2 cointegrating vectors, which means there is no cointegration? right? because there are two stationary variables in there.

If i use deterministic term as cimean:
it gives me 4 cointegrating relationships, which means there are 2 cointegrating vectors? right?

Similarly, when i use only 3 non-stationary variables and use none or cimean as deterministic term, i get a single cointegrating vector. Which you think is not the right way to do it as there is noise due to omission of stationary variables.

I am not a bad student, neither i have problem of econometrics, i am quite clear on that but this coding part has become headache for me. So please help me. So now my question is that should i consider the model with 5 variables and no cointegration as in the first case? or should i include the constant and consider 2 cointegrating vectors? once i get the answer i will try to write code and share and understand whats going on ...

Waiting.

Regards,

Ateeb

[TomDoan]

Dear Tom,

Now i am trying to run the cointegration tests as described by you:

Consider this and if you can help me get through this i will be thankful.

When i am considering all 5 variables (3 non-stationary and 2 stationary) in my cointegration test i.e. in CATs:

If i use deterministic term as none:
it gives me 2 cointegrating vectors, which means there is no cointegration? right? because there are two stationary variables in there.

That would be the correct conclusion, though DET=NONE is almost never the correct choice.

If i use deterministic term as cimean:
it gives me 4 cointegrating relationships, which means there are 2 cointegrating vectors? right?

That would be the interpretation.

Similarly, when i use only 3 non-stationary variables and use none or cimean as deterministic term, i get a single cointegrating vector. Which you think is not the right way to do it as there is noise due to omission of stationary variables.

Again correct (though again DET=NONE is probably not the best choice).

It's important to understand that inference regarding cointegration is (very) asymptotic. If you have a series which is the sum of a small variance random walk and a large variance stationary process, eventually (as T gets large) the random walk will dominate. In any fixed sample, it might not. Adding stationary variables to the analysis might eliminate quite a bit of the stationary variance in the residuals which would make it easier to see if what remains has a unit root or not.