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From my experience, when a time series $T$ is integrated/has a unit-root, one cannot model it with an ARMA model can usually need to turn to the first difference of $T$ (or some research I read uses $ln(T)$).

However, I just realized recently that in R's auto.arima function (under the "forecast" package), it is possible to run an ARIMA regression with covariates (by specifying the parameter xreg) even if the time series is integrated (e.g. an ARIMA(2,1,2) process).

My question is:

  1. Does this model still yield valid statistical results about the covariates' coefficients?

  2. If valid, how does auto.arima deal with the integrated process? I think it is probably not taking the first difference, but am not sure what is it.

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Note that auto.arima() does not fit an ARIMAX model, but a regression on the covariates with ARIMA errors. See Rob Hyndman's "The ARIMAX model muddle" blog post for the difference.

So you should be somewhat careful about inference for covariate coefficients, since standard $t$ tests assume iid errors, and the errors in your auto.arima() model are ARIMA. Note that auto.arima() reports estimated coefficients and their standard errors, but no $t$ statistics or $p$ values.

As to how auto.arima() deals with potential integration, a look at ?auto.arima is helpful:

   d: Order of first-differencing. If missing, will choose a value
      based on KPSS test.

The KPSS test is of course standard, since it is applied to the residuals of the regression on your covariates.

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