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Does anyone know what the consequences for the properties of the OLS estimator are (biased/consistency/...) if you run following regression inflation = beta0 + beta1 * u + beta2 * s + epsilon, with inflation being I(1) process and u and s I(0) processes?

thanks in advance

You should preferably run a regression with both inflation and the regressor being I(0) (stationary). Is there a problem with first differencing inflation to get it to be I(0)?

Worst case scenario is to have both inflation and your regressor as I(1). You'd be running "spurious" regressions.