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In this technical question I asked how to build a VAR model after differencing.

I have a conceptual question regarding differencing and VAR models.

When I apply the 1st order differencing, my ACF plots look like this

and this

My tests tell me that the process is stationary.

I decided to apply the 2nd order and now my first variable looks better, but the second looks worse on ACF plots.

and this

My question:

Is it possible/appropriate to use different levels of differencing for different variables in VAR model?

Appreciate any tips!

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Yes, it is possible. There is no statistical argument regarding which transformation of a variable should be treated as the original variable. For example, given a multivariate time series, you can take any single series and integrate it or difference it before showing the whole system to me. Then you can ask me your original question. Not knowing what the original variables were, I would likely have to difference some variables but not other until they all are on the same level of integration. If you transformed the variables in a different way before showing them to me, my transformations before fitting a VAR model would have to be different. Also, consider that the variables you have obtained from somewhere might have been integrated or differenced by another person who assembled the data. We cannot always backtrack the origins of each variable. What matters is the level of integration of the variables that enter the model after appropriate transformations, not before.

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