Suppose we have ARIMA(2,3,2) in a study. How to write the final formula for this model? The parameters are: AR1 and AR2 for auto-regressive part, MA1 and MA2 for the moving average part.
Most of similar posts in this platform, are hard to understand by beginner. For example, I have been addressed to this link, but it is somehow complicated, too. So I decided to create this post to keep it simple as possible for future references.
Maybe your confusion comes from the fact that in the ARIMA(2,3,2) one considers three times differencing of the original series. My approach is as follows: Say your original time series is $Y_t$, the first differencing yields say another time series say $X_t$ and so on.We define them clearly as such:
$X_t=\overbrace{Y_t-Y_{t-1}}^\text{first differencing}$, $Z_t=\overbrace{X_t-X_{t-1}}^\text{second differencing}$, lastly $V_t=\overbrace{Z_t-Z_{t-1}}^\text{Third differencing}$, so now imagine we fit an ARMA(2,2) in the time series denoted as $V_t$. I use the names you propose for the coefficients and we get:
$V_t=\underbrace{AR_1V_{t-1} + AR_2V_{t-2}}_\text{Autoregressive part} + \underbrace{MA_1\epsilon_{t-1} + MA_2\epsilon_{t-2}}_\text{Moving Average Part} + \epsilon_t$, where $\epsilon_t \sim WN(0,\sigma^2)$ i.e $\epsilon_t$ is white noise.
