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In monetary economics, I've seen the following 4-equation VAR with

log industrial production, log consumer price index, federal funds rate, excess bond premium (a la Gilchrist and Zakrajšek)

what troubles me is that the first two variables are clearly trending and non stationary, the third is borderline stationary, the fourth is stationary. Authors don't seem at all to comment on this mixture of stationarity aspects. and so I wonder whether such a VAR makes sense in econometric terms.

Examples of papers doing this include:

Gertler, M. and P. Karadi (2015). Monetary Policy Surprises, Credit Costs, and Economic Activity. American Economic Journal: Macroeconomics, 7 (1), 44–76.

Bauer, M. D. and E. T. Swanson (2022). A Reassessment of Monetary Policy Surprises and High-Frequency Identification, NBER Chapters, in: NBER Macroeconomics Annual 2022, 37, National Bureau of Economic Research, Inc

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