1. Definition
- A forecast error is the difference between the actual observed value and the forecasted (predicted) value.
$e_t = y_t – \hat{y}_t$
Where:
- $y_t$ = actual value at time $t$.
- $\hat{y}_t$ = forecasted value at time $t$.
- $e_t$ = forecast error (residual).
2. Interpretation
- If $e_t > 0$: forecast underestimated the actual value.
- If $e_t < 0$: forecast overestimated the actual value.
- If $e_t = 0$: perfect forecast.
3. Properties of Good Forecast Errors
- Should be unbiased: mean of errors ≈ 0.
- Should be uncorrelated over time (no systematic patterns left).
- Should have constant variance (homoscedastic).
- If errors show trends, autocorrelation, or bias → model is misspecified.
4. Example
Suppose actual vs predicted sales (units):
| Time | Actual ($y_t$t) | Forecast ($\hat{y}_t$) | Error ($e_t$) |
|---|---|---|---|
| 1 | 100 | 95 | +5 (underestimated) |
| 2 | 120 | 130 | -10 (overestimated) |
| 3 | 90 | 88 | +2 |
Errors = [5, -10, 2].
5. Forecast Error Metrics
Since raw errors can be positive/negative, we summarize with metrics:
- Mean Error (ME): average of errors (bias measure).
- $ME = \frac{1}{n}\sum e_t$
- Mean Absolute Error (MAE): average absolute errors.
- $MAE = \frac{1}{n}\sum |e_t|$
- Mean Squared Error (MSE): average squared errors (penalizes large errors).
- $MSE = \frac{1}{n}\sum e_t^2$
- Root Mean Squared Error (RMSE): square root of MSE.
- Mean Absolute Percentage Error (MAPE):
- $MAPE = \frac{100}{n} \sum \left| \frac{e_t}{y_t} \right|$
- Mean Absolute Scaled Error (MASE): scale-free, good for comparing models.
6. In Forecasting Competitions
Forecast error is central. For example:
- M-series competitions use MASE, sMAPE.
- Low forecast error = better accuracy.
Summary:
Forecast error = actual − predicted. It shows how far off forecasts are from reality. Good models produce errors that are small, unbiased, and random. Errors are summarized using metrics like MAE, RMSE, MAPE, or MASE to evaluate and compare forecasting models.