In some exam review I received the following question
- Assume that $\{X_{t}\}$ is a stationary time series, with mean $\mu$ and acvf $\gamma(h)$. Let $P_{n}X_{n+h}$ be the optimal (in terms of MSE) linear prediction of $X_{n+h}$ based on $X_{1},...,X_{n}$. Show explicitly how to write down a set of equations which define $P_{n}X_{n+h}$ in the case $n=2$ and $h=1$.
Now I know that the best linear predictor of $X_{n+h}$ given the set $X_{n}, \text{...} , X_{1}$ is \begin{equation*}Pred(X_{n+h}|X_{n}, \text{...}, X_{1}) = \mu + (a_{1}, \text{...}, a_{n})^{T} \begin{pmatrix} X_{n}-\mu \\ X_{n-1}-\mu \\ . \\ . \\ X_{1} - \mu \end{pmatrix} \end{equation*} where $\mathbf{a}$ satisfies the avcf equation. I also know the MSE is given by $$\gamma(0) - \mathbf{a}^{T}\gamma(n,h)$$.
What I'm unsure about is what exactly a set of equations defining my optimal predictor would look like. Do I simply plug in $n=2$ and $h=1$ to rewrite the definition I've given, or is there something more to it?