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I am estimating the following VAR model:

\begin{equation*} x_t = k + A_1 x_{t-1} + A_2 x_{t-2} + \dots + A_p x_{t-p} + \epsilon_t, \end{equation*} where $x_t$ is a vector of variables and notation is standard. I have three variables in $x_t$: Two $I(1)$ processes and one $I(0)$ process. The Johansen cointegration test yields rank 1, such that there is one cointegrating relationship.

I am aware that if I rewrite the model to a vector error correction model (VECM), then inference is valid on the parameters using t-values and standard normal distributions.

My question, however, is whether (standard) inference is available directly on the parameters of the VAR model, that is $A_1,A_2,\dots,A_p$, given cointegration?

Thank you!

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If you have some variables in the system that are I(0), and some that are I(1), then you cannot use VAR, unless you ensure all the variables are I(0).

The Johansen test can find cointegration relations only if all the variables are non-stationary and integrated at the same order, so your data seems to be some borderline case. To be certain, I'd suggest differencing the I(1) variables and then using VAR.

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    $\begingroup$ Are you sure? Consider this note by Helmut Lütkepohl (core.ac.uk/download/pdf/41100689.pdf), which says: "A set of I(1) variables is called cointegrated if a linear combination exists which is I(0). If a system consists of both I(0) and I(1) variables, any linear combination which is I(0) is called a cointegration relation." He then proceeds to present a VECM model where "it is assumed that all variables are at most I(1)" and goes on to discuss the Johansen test. $\endgroup$ – Greased Dog Apr 6 at 12:30
  • $\begingroup$ A system can contain both I(0) and I(1) variables and one can build a VECM with them, but a I(0) variable cannot be cointegrated with some other variable, either stationary or non-stationary one. Please see this answer and the discussion after it: stats.stackexchange.com/a/301491/101145 $\endgroup$ – vpekar Apr 6 at 22:06
  • $\begingroup$ Whether an I(0) variable can be cointegrated with some other variable clearly depends on the definition of cointegration - it says so directly in the quote from Lütkepohl (obviously, if an I(0) variable is to be cointegrated with a variable, the other variable also has to be I(0) then). Your link offers no value in this discussion, as it uses the other definition of cointegration where I(0) variables cannot be cointegrated. Note, however, that in my case, it doesn't really matter, since I'm not saying that the I(0) variable is cointegrated, I merely say that the two I(1) variables cointegrate. $\endgroup$ – Greased Dog Apr 7 at 10:14
  • $\begingroup$ Lütkepohl's (less conventional) definition you cited allows for two I(0) series to be called co-integrated, but that does not change the fact that I(0) cannot cointegrate with I(1) - the link I included discusses this. So my point is: treat with caution the results you describe in your question, e.g., use different stationarity and cointegration tests to confirm the results. If stationarity cannot be reliably established, you cannot use a VAR, if co-integration cannot be established, then you cannot use a VECM, and so stationarize the series, build a VAR, and interpret it in the usual way. $\endgroup$ – vpekar Apr 7 at 11:22

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