Prediction for Causal AR(p)
Result
For a causal AR(p) process with $n \geq p$, the one-step-ahead best linear predictor is:
$$P_n X_{n+1} = \sum_{i=1}^p \phi_i X_{n+1-i} = \phi_1 X_n + \phi_2 X_{n-1} + \cdots + \phi_p X_{n+1-p}$$Why This Works
$X_{n+1} = \phi_1 X_n + \cdots + \phi_p X_{n+1-p} + Z_{n+1}$.
Apply the BLP operator $P(\cdot | X_n, \ldots, X_1)$:
- $P(X_j | X_n, \ldots, X_1) = X_j$ for $j \leq n$ (property (vii))
- $P(Z_{n+1} | X_n, \ldots, X_1) = E(Z_{n+1}) = 0$ (for causal process, $Z_{n+1}$ is uncorrelated with $X_n, \ldots, X_1$, so property (iii) applies)
Therefore: $P_n X_{n+1} = \phi_1 X_n + \cdots + \phi_p X_{n+1-p}$.
Implication
For causal AR(p), the BLP uses only the last $p$ observations. The prediction formula is exactly the AR recursion with the noise term set to its expected value (zero).