It looks like predict() maybe be called at each step of the iteration in ARIMA with all the dates making machinery too. Need a dates-free implementation for "fittedvalues" or something for the estimation to be quicker.
Correction. It's in the determination of start_params not in the actual fitting process.
Parking this here. The (out of sample) forecasting functions are a good candidate for Cython.
For the Kalman Filter, we should explore two speed-ups.
Closed by a738b4f. There's another open issue for start_params so I'll look into that when I get to it. See #1301 and #1366.