There's no automated Stata version as far as I know.
The Hyndman-Khandakar algorithm that auto.arima() uses to pick p,d, and q is described here. The basic steps are:
- The number of differences d is determined using repeated KPSS tests.
- The values of p and q are then chosen by minimizing the AICc after
differencing the data d times. Rather than considering every
possible combination of p and q, the algorithm uses a stepwise
search to traverse the model space.
(a) The best model (with smallest AICc) is selected from the following
If d=0 then the constant c is included; if d≥1 then the constant c is
set to zero. This is called the "current model".
(b) Variations on the current model are considered:
- vary p and/or q from the current model by ±1
- include/exclude c from the current model.
The best model considered so far (either the
current model, or one of these variations) becomes the new current
(c) Repeat Step 2(b) until no lower AICc can be found.
All of the tests/statistics involved can be calculated with Stata, so you could achieve something similar by hand, and automate it with a bit more effort. For KPSS, use Chris Baum's
kpss from SSC. You can get the corrected AIC by using
estat ic and this formula.