Forecasting with ARIMA Models

Before we get into fitting and predicting with actual ARIMA models in further lessons, it is worth exploring some general results. This will be very useful to understand what to expect from each type of model in different forecast horizons.

When working with ARIMA models and a quadratic loss (such as MSE), a very neat result arises. By minimizing the expected loss function, we get that the optimal prediction is the conditional expectation of $y_{T+h}$:

Note that we have denoted expectations with a subscript $T$. This represents all the information we have until the moment that we make the prediction. Also, the expression above does not require any particular functional form of $\hat y_{T+h}$, only that the loss is quadratic.

MA($q$) processes

To illustrate how forecasting works with an MA($q$) process, let’s take the case of an MA($1$ ...