# Reformulation of ARIMA for an ML context

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 $$$$(1 - \sum_{i=1}^{p}{\varphi_i B^i})y_t = (1 + \sum_{j=1}^{q}\vartheta_j B^j ) \varepsilon_{t}.$$$$ where $$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?