Dara, my own working model is that external forcing is acting as a running boundary condition which guides the solution.
The shortest scale forcing that I see is the 28 month average QBO period. The characteristic response period is 51 months.
These are oscillatory forcings and responses, so unless you have lots of coefficients in the expansion of the extrapolated function, the prediction will likely diverge past a few years.
The trend is from the memory-less Markov to the more deterministic quasi-periodic.
What I would recommend is to create a trial function that you can express analytically and see if your method can track it.
A symbolic regression machine learning tool such as Eureqa is very difficult to fake out with trial solutions because it relentlessly tries out all possible sinusoidal combinations until it finds the one you picked.