One-step and multistep forecasts for a causal AR(1)

Reindexing the convergent causal series gives $X_t-\mu =\varphi (X_{t-1}-\mu )+{ϵ}_t$; repeating this identity gives $X_{n+h}-\mu ={\varphi }^h(X_n-\mu )+R_h$, where $R_h={\sum }_{r=0}^{h-1}{\varphi }^r{ϵ}_{n+h-r}$. Every innovation in $R_h$ has index greater than $n$, while the causal series for each $X_j-\mu$, $1\le j\le n$, uses only indices at most $j$.

For each $j\le n$, every finite partial sum for $X_j-\mu$ uses innovation indices at most $j$, so its covariance with each future term of $R_h$ is zero by the uncorrelated-noise assumption; [F4] passes this covariance identity to the $L^2$ limit. Thus $\mathrm{Cov}(R_h,X_j)=0$ and $\mathbb{E}[R_h]=0$, making the residual orthogonal to the constant and every history variable.

Let $H=\mu +{\varphi }^h(X_n-\mu )$ and let any competing affine forecast $Z$ be in the history span; then $D:=H-Z$ is also in that span. Step 2.1 gives $\mathbb{E}[R_h]=0$ and $\mathrm{Cov}(R_h,X_j)=0$ for every $j$; bilinearity [F3] gives $\mathrm{Cov}(R_h,D)=0$, hence $\mathbb{E}[R_{h}D]=0$. Since $X_{n+h}-Z=R_h+D$, expansion yields $\mathbb{E}[(X_{n+h}-Z{)}^2]=\mathbb{E}[R_h^2]+\mathbb{E}[D^2]\ge \mathbb{E}[R_h^2]$, with equality exactly when $D=0$ as an $L^2$ class. Thus $H$ is the unique best class by [F2]. The innovations in $R_h$ are pairwise uncorrelated, so [F3] gives $\mathrm{Var}(R_h)={\sigma }^2{\sum }_{r=0}^{h-1}{\varphi }^{2r}$. This proves both formulas without Gaussian or independent innovations.