How do I state an ARIMA(0,1,4)x(0,1,1)12 in terms of Yt etc and not in terms of the backshift operator I would like to state an $\text{ARIMA}\left\{(0,1,4)\times(0,1,1)\right\}$ model (written in format $(p,d,q)\times(P,D,Q)$ model without using the backshift operator.
 A: I took your model form and entered it into AUTOBOX and asked for the model to be presented as an ADL (Auto-regressive Distributed Model ) or PDL (Polynomial Distributed Model).
 .  The actual coefficients depend of course on the data BUT you get the idea that Y is on the left-hand side and lags of Y and lags of the error terms are on the right-hand side. This was programmed so that the equation could be "sold" or "explained" to the customer , leading to acceptance of the results.
It is sometimes possible to express the equation as a pure right-hand side where there are mo error terms just lags of Y. The problem arises when the order of this pure lag model exceeds the number of observations. In Box-Jenkins language this is called the Pi weights.
I have shown these weights up to lag 50. In effect an ARIMA model is simply a weighted sum of the past much like an old-fashioned  k period weighted average. The second value shown .85284 is the weight of the first lag of Y , etc.. Also note that the weights are not decreasing in absolute value
lag 12  1.148
lag 24  1.1699
lag 36  1.1732
lag 48  1.1737

suggesting that this model when applied to the data set that I used here violates the invertability requirements as the past past is more important than the recent past.

