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I have property return variables and economic variables in natural log form, which are non-stationary in level and stationary in first differences, but are not cointegrated. To my understanding, this rules out the use of the VECM, and so should I estimate the VAR in levels, first differences or something else? Thanks!

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Your choice should be VAR in first differences.

If you do VAR in levels, you will have left-hand-side variables diverging from right-hand-side variables, which is bad (coefficient estimates will be ill-behaved and not really meaningful). You cannot do VECM since there is no stationary linear combination of variables in levels to put in the model. VAR in first differences is just like VECM but without the stationary combination of variables in levels. Since differenced variables are stationary, you can put them in a VAR and expect well-behaved coefficient estimators.

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  • $\begingroup$ Thanks Richard, what about if the variables are in fact non-stationary in levels, stationary in first differences but also cointegrated? After adding more variables, my data is now cointegrated. I believe this would prompt the use of the VECM, however would it still be appropriate to use the VAR in first differences? VAR in first differences gives me better results in the Impulse Response Functions $\endgroup$
    – James
    Sep 11 '19 at 1:40
  • $\begingroup$ @James, If they are I(1) and cointegrated, you would use VECM. It would not be appropriate to use VAR in first differences because it would be misspecified, namely, affected by omitted variable bias (missing the stationary combination of variables in levels, i.e. the error-correction term). This has been discussed in some previous posts on cointegration, VECM and VAR. $\endgroup$ Sep 11 '19 at 5:10

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