If I have a multi-step forecast of some timeseries, where the model is some auto-regressive function
y[t] = f(y[t-1],x1[t],x2[t])
is it a valide approach to use sections of historical data to train the model and than evaluate it on future section recursively to obtain the multi-step forecast. Doing this multiple times with different sections of my historical data I can obtain the forecast error
e[t] = y_hat[t] - y[t]
over the forecast horizon T. Can I use this error to estimate the prediction interval?