Prediction Error

The prediction error measures the quality of a forecast by comparing the predicted value to the actual value.

1. Definition

For a predictor $\hat{X}_{n+h}$ of $X_{n+h}$:

$$e_{n+h} = X_{n+h} - \hat{X}_{n+h}$$

The MSE is the expected squared prediction error:

$$\text{MSE} = \mathbb{E}[e_{n+h}^2] = \mathbb{E}[(X_{n+h} - \hat{X}_{n+h})^2]$$

The best linear predictor $P_n X_{n+h}$ minimizes MSE among all linear predictors.

For $X_t = \phi X_{t-1} + Z_t$ with $\{Z_t\} \sim \text{WN}(0, \sigma^2)$:

1-step ahead: $\text{MSE} = \sigma^2$
The prediction error $e_{n+1} = Z_{n+1}$, which is uncorrelated with all past observations

Only reporting point predictions without uncertainty. Always accompany forecasts with prediction intervals or MSE estimates.