ARIMA model interpretation I have a question about ARIMA models. Let's say I have a time series $Y_t$ that I would like to forecast and an $\text{ARIMA}(2,2)$ model seems like a good way to conduct the forecasting exercise. 
$$
\Delta Y_t = \alpha_1  \Delta Y_{t-1} + \alpha_2 \Delta Y_{t-2} + \nu_{t} + \theta_1 \nu_{t-1} + \theta_2 \nu_{t-2}
$$
Now the lagged $Y$'s imply that my series today is influenced by prior events. This makes sense. But what is the interpretation of the errors? My prior residual (how off I was in my calculation) is influencing the value of my series today? How are the lagged residuals calculated in this regression as it is the product / remainder of the regression? 
 A: I totally agree with the sentiment of the previous commentators. I would like to add that all ARIMA model can also be represented as a pure AR model. These weights are referred to as the Pi weights as compared to the pure MA form (Psi weights) . In this way you can view (interpret) an ARIMA model as an optimized weighted average of the past values. In other words rather than assume a pre-specified length and values for a weighted average , an ARIMA model delivers both the length ($n$) of the weights and the actual weights ($c_1,c_2,...,c_n$). 
$$Y(t) =c_1 Y(t−1) + c_2 Y(t-2) + c_3 Y(t-3)+ ... + c_n Y(t-n) + a(t)$$
In this way an ARIMA model can be explained as the answer to the question


*

*How many historical values should I use to compute a weighted sum of the past?

*Precisely what are those values?

A: Note that due to Wold's decomposition theorem you can rewrite any stationary ARMA model as a $MA(\infty)$ model, i.e. :
$$\Delta Y_t=\sum_{j=0}^{\infty} \psi_j\nu_{t-j}$$
In this form there are no lagged variables, so any interpretation involving notion of a lagged variable is not very convincing. However looking at the $MA(1)$  and the $AR(1)$ models separately:
$$Y_t=\nu_t+\theta_{1}\nu_{t-1}$$
$$Y_t=\rho Y_{t-1}+\nu_{t}=\nu_t+\rho \nu_{t-1}+ \rho^2 \nu_{t-1}+...$$
you can say that error terms in ARMA models explain "short-term" influence of the past, and lagged terms explain "long-term" influence. Having said that I do not think that this helps a lot and usually nobody bothers with the precise interpretation of ARMA coefficients. The goal usually is to get an adequate model and use it for forecasting. 
