I asked a similar question before but i did not get any answers. Should i delete the old question?
$x_t = c + \theta_{1}x_{t-1} + \phi_1\varepsilon_{t-1}+ \phi_2\varepsilon_{t-2} + \varepsilon_t$
$\hat{x_t} = c + \theta_{1}x_{t-1} + \phi_1\varepsilon_{t-1}+ \phi_2\varepsilon_{t-2}$
t X
[1,] 1 44
[2,] 2 55
[3,] 3 66
[4,] 4 77
I want to predict x5. How should the equation be like? lets say c=5 and $\theta_{1} =2$ and $\phi_1=3$ and $\phi_2=4$
I am sure that $x_{t-1}$ is 77 but the part that confuses me is $\varepsilon_{t-1}$ and $\varepsilon_{t-2}$!
Should we predict the $x_{t-1}$ to calculated its error/residual first then put the residuals values in the equation to forecast? If that's the case then How we can calculate x1 (the first data point at time 1)?
I am talking about MA and ARMA. Even thinking about the estimation of $\phi$ is weird to me. in AR process is just like linear regression where we make another $x_{t-1}$ variable and regress it on $x_t$, but in the case of MA or ARMA models i can't get it.
PS. i know the $x_t$ is not stationary. It's just an example.
