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Interpret the forecasts as best linear prediction from a finite past. For a causal autoregressive process, the innovations are orthogonal to previous observations. If they also have zero conditional mean given the past, the same forecasts are conditional means; white noise by itself guarantees only the linear-prediction interpretation.

Let $m$ be the process mean and write its centered autoregression as $X_t-m={\sum }_{j=1}^p{\varphi }_j(X_{t-j}-m)+{ϵ}_t$. Set ${\hat{X}}_{T,h}=X_{T+h}$ for $h\le 0$, and recursively putThe forecast is a linear combination of the observed last $p$ values. Subtracting the recursion from the future autoregression shows that the prediction error is a linear combination of future innovations, hence orthogonal to every observed value. This proves the optimal projection property, rather than merely suggesting that future noise should be replaced by zero. In particular,The empty sum for $p=1$ is zero. Taking $m=0$ gives the centered convention. This is the recursive forecasts of an autoregressive process construction.

For the stable order-one equation, use its stationary causal solutionThe series converges in mean square because $\sum {\varphi }^{2j}<\infty$ and the white noise innovations have a common finite variance ${\sigma }^2$. Orthogonality of their distinct time indices givesAdding the deterministic mean therefore yields weak stationarity for $Z$, withThe autocorrelation function assumes ${\sigma }^2>0$; a zero-variance process has no normalized correlation. A recursion from an arbitrary nonstationary initial value would have transient moments, so the stationary-solution interpretation is important here.

Iteration gives $X_{T+k}={\varphi }^k{X}_T+{\sum }_{j=1}^k{\varphi }^{k-j}{ϵ}_{T+j}$. The innovation sum is orthogonal to the observed past, givingFor the integrated process, the backshift operator subtracts consecutive levels, so $(I-B)W_t=\mu +(I-B)Y_t=\mu +X_t=Z_t$. Consequently $W_{T+k}=W_T+{\sum }_{j=1}^k{Z}_{T+j}$. If the last increment is observed, substitute its forecast and sum the geometric progression:This is the forecasts of an integrated AR(1) process formula. Its final term is bounded as $k\to \infty$, even for negative $\varphi$, hence the forecast's long-run slope is the drift:With an initial level orthogonal to future innovations, the level formula is also the best linear forecast from the observed levels. Without that initial-level condition, it is the increment-based recursive forecast, and the levels could contain additional predictive information. The level formula uses $T\ge 2$, or a known $W_0$ when $T=1$. With just $W_1$ and no model for the initial level, the last increment cannot be recovered from the supplied observations; a one-observation level forecast would need that additional information.