Prediction Intervals

The prediction error is approximately Gaussian:

$$X_{N+h} - P_N X_{N+h} \sim \mathcal{N}(0, \text{MSE})$$

Standardize: $\frac{X_{N+h} - P_N X_{N+h}}{\sqrt{\text{MSE}}} \sim \mathcal{N}(0, 1)$

$$P\!\left(-z_{\alpha/2} < \frac{X_{N+h} - P_N X_{N+h}}{\sqrt{\text{MSE}}} < z_{\alpha/2}\right) = 1 - \alpha$$

Rearrange:

$$P\!\left(P_N X_{N+h} - z_{\alpha/2}\sqrt{\text{MSE}} < X_{N+h} < P_N X_{N+h} + z_{\alpha/2}\sqrt{\text{MSE}}\right) = 1 - \alpha$$

$$P_N X_{N+h} \;\pm\; z_{\alpha/2}\sqrt{\text{MSE}}$$

$\alpha = 0.05$, $z_{0.025} = \Phi^{-1}(0.975) = 1.96$:

$$P_N X_{N+h} \;\pm\; 1.96\sqrt{\text{MSE}}$$

In practice, assumptions (Gaussian, known ACVF) are approximate
ACVF is estimated from data → sample ACVF fed into prediction formula
The Gaussian assumption can be checked via QQ plots or normality tests on residuals

Prediction Intervals#