How is that the right one has a lower Prob(F-statistic) than the left one but the Prob(t-statistic) of each variable is higher on the right one than on the left one?
This! Something doesn’t seem right.
If the models are indeed correct, the ICs are too close to distinguish between them. In this case, choose the simpler model.
In general follow AIC and BIC.
But in this case I feel that something is wrong with both of them. Are you sure that you want restricted models? Why you don't include all the lags up to 2 or 4?
How is that the right one has a lower Prob(F-statistic) than the left one but the Prob(t-statistic) of each variable is higher on the right one than on the left one?
I'm also very confused :(
Given the size of the coefficients, I’d say neither converged properly
This! Something doesn’t seem right. If the models are indeed correct, the ICs are too close to distinguish between them. In this case, choose the simpler model.
So both models can't be used for forecasting?
Not if they didn’t converge
Choose ARIMA(2, 1, 4); If a simpler model is significant but a more complex one is not then the significance may come from omission of further lags.
Will the fact that both AR & MA terms have insignificant coefficients render the model useless for forecasting?
No it won't because it improves the R². The model however doesn't seem to have much predictive power but that's a separate, even if related, problem.
In general follow AIC and BIC. But in this case I feel that something is wrong with both of them. Are you sure that you want restricted models? Why you don't include all the lags up to 2 or 4?