Best linear prediction from a finite past (${\hat{X}}_{t\mid T}={\mathrm{proj}}_{\mathrm{span}(X_1,\dots ,X_T)}{X}_t$)

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.

For a causal autoregressive process with mean $m$ and innovations orthogonal to the observed past, define ${\hat{X}}_{T,h}=X_{T+h}$ for $h\le 0$ and replace future innovations by zero. The displayed recursion is its best linear prediction from a finite past when at least the last $p$ observations are available. Future innovation sums are orthogonal to those observations, which proves the projection property. Mere white noise does not guarantee that a linear forecast equals a conditional mean: that stronger interpretation requires martingale-difference or independent innovations.

If the increments of $W$ have mean $\mu$ and centered increments satisfy a causal autoregressive process of order one with $|\varphi |<1$, sum their recursive forecasts to obtain the displayed expression. For fixed forecast origin, ${\hat{W}}_{T,k}/k\to \mu$. One must observe the last increment, requiring $T\ge 2$ or a known initial level $W_0$; a single observed level without an initial-level model does not identify that increment.

Starting observations at a finite time, subtract the best linear predictor based on the available earlier observations. The resulting residual is orthogonal to their linear span. The unit-triangular relation between observations and residuals makes the residuals an orthogonal basis of the same finite observation space. These differ from infinite-past innovations until the initialization effect disappears.

In a centered MA(1) process, $X_{T+2}$ is uncorrelated with every observation through time $T$. Its best linear predictor from that information is therefore zero, with error variance ${\sigma }^2(1+{\theta }^2)$. One-step innovation formulas using $X_{T+1}$ cannot be substituted when that observation is unavailable.

Starting at time one, the orthogonal residuals satisfy $U_t=X_t-{\lambda }_t{U}_{t-1}$, with ${\lambda }_t={\gamma }_1/{\phi }_{t-1}$ and ${\phi }_t={\gamma }_0-{\gamma }_1{\lambda }_t$. Earlier innovation directions have zero covariance with the new observation. Explicitly, with $S_m={\sum }_{j=0}^m{\theta }^{2j}$, ${\phi }_t={\sigma }^2{S}_t/S_{t-1}$ and ${\lambda }_t=\theta S_{t-2}/S_{t-1}$ for $t\ge 2$. These formulas follow by induction from $S_t=(1+{\theta }^2)S_{t-1}-{\theta }^2{S}_{t-2}$, with $S_0=1$ and $S_1=1+{\theta }^2$. This includes $\theta =\pm 1$, where ${\phi }_t={\sigma }^2(t+1)/t$.

For an MA(1) process, the recursion ${\lambda }_t=\theta /(1+{\theta }^2-\theta {\lambda }_{t-1})$ has candidate limits $\theta$ and $1/\theta$. The limit in $[-1,1]$ is $\theta$ when $|\theta |\le 1$ and $1/\theta$ otherwise. The limiting innovation variance is ${\sigma }^2\max (1,{\theta }^2)$, expressing the same covariance law through an invertible reciprocal representation.