Best linear prediction from a finite past
= Best linear prediction from a finite past
{title2=$\widehat X_{t\mid T}=\operatorname{proj}_{\operatorname{span}(X_1,\ldots,X_T)}X_t$}
The best mean-square <linear predictor> is <orthogonal projection> onto the span of the available centered observations, plus the known mean. Orthogonal innovations give a convenient basis of that span. Prediction at a future horizon must use only actually observed variables; it need not be a one-step predictor.