rolling window forecast

choose the best forecast of an AR(p) model, by comparing all AR(p) forecast with realized values

This function split the time series into rolling windows. Then, for each of these rolling windows, the algorithm analyzes some AR(p) processes. Then it produces a forecast for each of these processes and for each of the rolling windows. These forecast are compared with the realized values of the time series, and then the function looks for the "best" p-order. This "best" p is chosen by minimizing the quadratic loss function, that is, the squared difference between the forecast and the realized values. the model with smallest quadratic loss function is the best selected, and the it is performed a direct forecast in order to produce the final forecast.
Minimizing the quadratic loss function means that the best model is the one with less distance from the observed values.