Forecast Confidence

Estimating the next point in a time series is only half of the forecasting job. The confidence around that point is just as important, if not more. Let’s think in terms of the weather. Forecasting that tomorrow’s temperature will be 68˚F is useless if the forecast is ±30˚F. However, if the interval is ±2˚F, we won’t really mind if the temperature ends up being 66˚F instead of 68˚F.

As this simple example illustrates, we can’t provide a point forecast without a confidence interval. A confidence interval will tell us the window where the realization of $y_{T+1}$ is likely to fall. In this lesson, we show how to produce such intervals.

Definition

Let’s define our forecast error, $e_{T+1}$, as the difference between the realization of $y_{T+1}$ and our forecasted value $\hat y_{T+1}$ ...