Finding the best linear predictor

Question

How do I find the best linear predictor of $X_{n+1}$ in terms of $X_{n-1}, X_n$, if $X_t$ is the MA(1) model $X_t = Z_t + \theta Z_{t−1}$.

Answer 1

Your question is about pure MA modelling, but I don't know if your true intentions are to do ARMA modelling or just pure MA modelling.

The answer to your question is as follows: We need to find coefficients $a_0,a_1,a_2$ in $$\hat{x}_{n+1} = a_0x_{n} + a_1 x_{n-1} = a_0x_{n} + (a_0 \theta + a_1) x_{n-1} + a_2\theta x_{n-2}$$ so as to minimize the mean-square error $\left|x_{n+1} - {x_n}\right|^2$

We write the desired values $x$ as a vector and the observations as another vector $z$. Let's denote $a_0 , a_0 \theta + a_1$ and $a_2\theta$ as $b_0,b_1,b_2$. We get

$$\widehat{\mathbf{x}}_{n+1} = b_0 \mathbf{z}_n + b_1 \mathbf{z}_{n-1} + b_2 \mathbf{z}_{n-2} , $$

where the notation $\mathbf{x}_n$ denotes all the values of $x$ from $0$ to $n$.

Aggregating the column vectors $\mathbf{z}_n,\mathbf{z}_{n-1},\mathbf{z}_{n-1}$ into a matrix $\mathbf{Z}$ and $\mathbf{b} = [b_0,b_1,b_2]^T$ gives the equation

$$\widehat{\mathbf{x}}_{n+1} = \mathbf{Z} \mathbf{b}.$$ We need to find $\mathbf{b}_{opt}$ where

$$\mathbf{b}_{opt} = \arg \min_\mathbf{b} ||\widehat{\mathbf{x}}_{n+1} - \mathbf{x}_{n+1}||^2 .$$

This is the classic least squares problem, that has a nice closed form solution, which you should be able to derive on your own, or find from google.