Forecasting $X_{t+2}$ for causal AR(p)

Question

Let $X_t$ be a causal $AR(p)$ process. Compute a linear forecast $X_{t+2}$ based on $X_1, X_2, ..., X_t$ for $t \geq p+1$.

If $AR(p)$ is causal it means that it can be rewritten as a linear process: $X_t=\sum\limits_{i=0}^{\infty}\psi_iZ_{t-i}$ where $\sum\limits_{i=0}^{\infty}|\psi_i|<\infty$ and $Z_t$ is a white noise.

My prediction should be a linear combination of $X_1,...,X_t: \hat X_{t+2}=m+\sum\limits_{i=1}^t a_iX_{i}$ so it is a projection onto space spanned by $(X_1, ..., X_n)$. But how to proceed?

Answer 1

Consider $X_{t+2}$ term: \begin{equation} \begin{split} X_{t+2} & = \sum_{i=1}^p \psi_iX_{t+2-i} \,+Z_{t+2} = \psi_1X_{t+1} + \sum_{i=2}^p\psi_iX_{t+2-i} \,+Z_{t+2} \\ & = \psi_1\left(\sum_{i=1}^p \psi_iX_{t+1-i} + Z_{t+1}\right) + \sum_{i=1}^{p-1}\psi_{i+1}X_{t+1-i} + Z_{t+2} \\ & = \underbrace{\psi_1\sum_{i=1}^p \psi_iX_{t+1-i}}_{\in\;sp(X_1\ldots X_t)} + \underbrace{\sum_{i=1}^{p-1}\psi_{i+1}X_{t+1-i}}_{\in\;sp(X_1\ldots X_t)} + \underbrace{\psi_1Z_{t+1} + Z_{t+2}}_{\bot\;sp(X_1\ldots X_t)}. \end{split} \end{equation} We know that $\psi_1Z_{t+1} + Z_{t+2}\;\bot\;sp(X_1\ldots X_t)$ from causality.

Hence the forecast is: \begin{equation} \begin{split} P_{\text{sp}(X_1\ldots X_t)}X_{t+2} = \psi_1\sum_{i=1}^p \psi_iX_{t+1-i} + \sum_{i=1}^{p-1}\psi_{i+1}X_{t+1-i}. \end{split} \end{equation}

If you know Mr. John Mielniczuk (as I suppose he is an author of this task), give him my regards.

Answer 2

This looks like a self-study question, so I will give you a hint:

1. Start with $\hat X_{t+1}$ as a linear combination of $X_1,\dots,X_t$ using the definition of AR(p).
2. Then do $\hat X_{t+2}$ as a linear combination of $X_1,\dots,X_{t+1}$ in the same way.
3. Replace $X_{t+1}$ in the previous step with $\hat X_{t+1}$ from the first step.