Principles behind time-series forecasting intervals

Question

So, this is truly a bit of a general question, but I am not aware of the guiding principles (if there are any) behind forecasting intervals. For whatever time-series model one might be using, whether it be an ARIMA or Markov-switching ARIMA-GARCH model, are there rules for how forecasting intervals are created for them, or is it all on a case-by-case basis?

I always intuitively believed forecasting errors were based on the error terms (most of the time standard-normal), which would provide a neat basis across all models, but I want to be certain if this is true.

Thank you in advance.

Answer 1

There are different ways of calculating prediction-intervals.

1. The parametric way. Fit a forecasting model to your data. Take in-sample residuals. Fit a distribution to these residuals (typically a normal one), assume that this distributional assumption also holds in the future. Forecast the moments of this distribution out, extract quantiles of the resulting predictive density.

   In forecasting the moments, the mean is usually the point forecast your forecasting model predicts, and higher moments depend on the autocorrelation structure you fitted, which will typically result in prediction intervals that widen over time.

2. The semi-parametric way. Fit a model and take residuals as above, but do not fit a distribution to the residuals. Instead, take (and interpolate as necessary) quantiles from the in-sample residuals and add/subtract these from your point forecast.

   This is semi-parametric because the point forecast comes from a model that is usually fitted using some distributional assumption.

3. Density forecasting, which is not always easy to cleanly separate from option 1 above. Do not fit a model with the aim of a point forecast. Instead, fit and forecast a full distribution, from which you can always extract prediction intervals. For instance, I have forecasted the densities of retail sales using Negbin regression and other approaches (Kolassa, 2016). I could have extracted PIs, but that was not the focus.

4. Direct quantile predictions, e.g., using quantile regression, where you fit separate models to predict (say) the 5% and the 95% quantiles. This is a pretty uncommon approach in time series forecasting.

Which approach performs best in a given situation will depend on your data. If you have little data or want extreme tail probabilities, then the parametric approach may be the only one you can take. If your distribution does not follow the assumptions in your model (e.g., you are using ARIMA for a conditionally lognormal distribution), then option 1 will of course be off. And so on.