Rob J. Hyndman once wrote in "Why I don't like statistical tests" (emphasis is mine):
In forecasting, the only place in which I find testing useful is in determining the order of integration of a time series; i.e., choosing d in an ARIMA(p,d,q) model. If I could come up with some way of doing this effectively without using a unit-root test, I would gladly do so. But so far, I have not found a reliable alternative.
I know that AIC-based comparisons are tricky when models for different transformations of the data are considered; one needs to account for that using Jacobians or something like that -- I have never got my hands dirty with this, so I do not know precisely.
But are they prohibitively tricky? Or what is the reason for the lack of "reliable alternatives" above? What about the following AIC-based comparison to replace unit-root testing for model selection in forecasting: for a given time series $\{x_t\}_{t=1}^T$,
- Fit an ARIMA(1,0,0) model and obtain its likelihood for observations $t=2,\dots,T$
- Fit an ARIMA(0,1,0) model and obtain its likelihood for observations $t=2,\dots,T$ (not for $\Delta x_t$ but for $x_t$, of course)
- Construct AIC values for the two (which should be somehow tricky for case 2. -- or should it not?) and compare them.
(This is not supposed to be an alternative for testing the hypothesis of presence of a unit root. But I hope it could be an alternative in model selection.)
