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To fit an ARMA model to a time series, the time series should be stationary to start with. If we obtain a reasonable model fit by looking at mean and variance, ACF plot, Ljung-Box test, and normality test, the residuals should mimic a white noise series in all these aspects.

If our residuals is broadly a white noise, does this guarantee the stationarity of the fitted model (in a practical sense, not in a sense of mathematical rigour)? Or is it irrelevant (e.g. it only suggests a reasonable model to account for all the information in the time series, but if this is the case, and given the time series is stationary, doesn't the white noise suggest stationarity already?)

I have a readily runnable example here, which show a model fit that has white noise residuals however the characteristic roots seemed suggesting otherwise (looking at m1).

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