Why VAR Models Can Mix Variables with Different Integration Orders

Summary

The document considers whether every variable in a vector autoregression must have the same integration order. It explains that researchers may include growth rates, logarithmic levels, and levels together, as in the cited euro-area forecasting study. Matching integration orders is often motivated by the desire for a stable process, with stability linked to covariance stationarity under standard theory.

The response says differing orders are not automatically prohibited, though the underlying economic model may impose relevant restrictions. The principal consequence is interpretive: coefficients and impulse responses describe effects in different forms, such as growth-rate changes versus level effects. The note gives examples of published VAR work but does not discuss model specification, cointegration treatment, or diagnostics, so it should not be read as a general guarantee that mixed-order specifications are appropriate.

Key ideas

• VAR variables do not universally need to share the same integration order.
• Stability is a common reason to examine integration properties in VAR modeling.
• Economic theory may constrain which transformations or orders are appropriate.
• Mixed transformations affect how coefficients and impulse responses should be interpreted.

Tags

# Why is Banque de France using BVAR with different orders of integration?


# Why is Banque de France using BVAR with different orders of integration?












Don't all the variables used have to be of the same order of integration in VAR models ? In this paper Bayesian VAR Forecasts, Survey Information and Structural Change in the Euro Area Gergely Ganics and Florens Odendah are using $\Delta$loglevel, loglevel and level data as we can see on page 28 (31).

Thank you in advance for the clarification !

## Answer by Jur (score 1, accepted)

https://quant.stackexchange.com/a/57692

So I asked on reddit, and got this answer from Rasseren :

The integration order of the endogenous variables is most often used to ensure reasonable stability (all eigenvalues of the companion form <1 in modulus) of the process; remember stability implies covariance stationarity in theory.

There is nothing, except maybe the underlying theoretical model e.g. Taylor-rule, prohibiting differing integration orders. In fact, most high-profile VAR studies utilizes different integration orders, see e.g. Stock & Watson (JEP, 2001), Bernanke et al. (Quarterly Journal of Economics, 2005) even the seminal studies of Christopher Sims (1972,1980).

The difference lies only in the interpretation of coefficients and IRFs (growth rates vs. level effects).

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.