Is the following approach to reformulate the ARIMA model for $\varepsilon_{t}$ correct?
I want to minimize $\sum{\varepsilon^{2}_{t}}$ but I only have a time series $y_t$ and the two are coupled such that \begin{equation} (1 - \sum_{i=1}^{p}{\varphi_i B^i})y_t = (1 + \sum_{j=1}^{q}\vartheta_j B^j ) \varepsilon_{t}. \end{equation} where $B$ is the backwords operator $B^i y_t = y_{t-i}$. It is implicitly assumed that the feature $y_t$ already displays a stationary behavior and zero mean. Now in Einstein notation for readability, \begin{eqnarray} \varepsilon_{t} &=& y_t - \varphi_i y_{t-i} - \vartheta_r \varepsilon_{t-r}\\ \varepsilon_{t-r} &=& y_{t-r} - \varphi_i y_{t-r-i} - \vartheta_j \varepsilon_{t-r-j}\\ \end{eqnarray} We can now expand $\varepsilon_t$ in orders of $\vartheta_j^k$ by recursively substituting for all $\varepsilon_{t-j}$. Given that we are looking to find the optimal $\varphi_p$ and $\vartheta_q$ \begin{eqnarray} \varepsilon_{t} &=& \underbrace{y_t - \varphi_p y_{t-p}}_{X_t} - \vartheta_{q_1}\left(X_{t-q_1 } - \vartheta_{q_2}\left(X_{t-q_1-q_2}- \vartheta_{q_1-q_2-q_3}(\dots)\right)\right)\\ \varepsilon_{t} &=& X_t - \vartheta_{q_1}\left(X_{t-q_1} - \vartheta_{q_2}\left(X_{t-q_1-q_2}- \vartheta_{q_3}(\dots)\right)\right)\\ \varepsilon_{t} &=&X_t - \sum_{k=1}^{\infty} \left(\prod_{j=1}^{k}\left(-\vartheta_{q_j}\right)\right)X_{t-\sum_{j=1}^{k}q_j} \end{eqnarray} This way the error is practically expressed as a convolution over the feature series. Since all $\vartheta_q < 1$, could we say an approximation for $\varepsilon_t$ can be expressed with a max error R when evaluted up to the k$^{th}$-order? How close are the values of the optimal $\varphi_p$ and $\vartheta_q$ to the approximated?
