long run relationship with stationary variable

[abi]

Dear Tom,

I have 5 variables, the dependent (Y) is I(1), one independent (X1) is I(0), and the others (X2,X3,X4) are I(1). The result of CATS shows 2 cointegration vectors. Since each stationary variable is cointegrated with itself, there are some questions that i would be greatly appreciated if you answer.

1) Is it possible to estimate a long run relationship with a stationary variable?

2) If yes, how can i include the I(0) variable into "error correction" terms?

Thanks,

[TomDoan]

If Y and X are cointegrated, then there is some b such that Y-Xb is I(0). There are I(0) processes which are quite persistent---not as persistent as an I(1) process, but they can be close. An error correction model for Y will take the form

delta Y = a (Y{1}-bX{1}),lags of delta Y, lags of delta X+error

where the error is supposed to be white noise. Everything in this, in the form in which it appears, is I(0). For any stationary variable Z, and any coefficient g,

Y-bX-gZ

is also stationary. (Note that the b is theoretically the same regardless of g). If gZ explains much of the Y-Xb "error", then including Z in the model will improve the ability to do inference on b (and, as a result a). If Z has nothing to offer, then, asymptotically it costs nothing, but in small samples, it could hurt.

[abi]

Dear Tom,
Thank you very much for reply,
Well, it seems the answer is yes, that's a good news for me .
As you know, it is important that we allow the components of a vector process to be integrated of different orders and we should be able to analyse for instance stationary as well as non-stationary variables. However i couldn't find any article that is using I(0) and I(1) component. Is there any problem with this issue?

[TomDoan]

It's fairly rare. VAR (at opposed to VECM) analysis doesn't usually involve any attempt to differentiate between I(1) and I(0) variables since it doesn't really matter (asymptotically). Most VECM analysis is employed with data which are pretty much expected to be I(1). This typically comes up when someone is trying to run a static Y on X regression (rather than a full VAR or VECM) without dealing with the dynamics. See Spurious Regressions.

[abi]

Thank's Tom,

I'm being a little confused!. Juselius believes if we using an I(0) variable, then the statistical inference will produce logically inconsistent results. However, Johansen argue that we can allow the components of a
vector process to be integrated of different orders and hence we should be able to analyse stationary and non-stationary variables in the same model. What's your opinion Tom?