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The sales of a certain product are represented by the model $$Z_t=3+a_t+0.5a_{t-1}-0.25a_{t-2}$$ where $a_t\sim WN(0,4)$ (White Noise).

Given the data $Z_1=3.25,Z_2=4.75,Z_3=2.25$ and $Z_4=1.75$. Find $\hat{Z}_4(l)$ for $l=1,2,3,100$.

This is an $MA(2)$ model with nonzero mean. The MMSE forecast expression that I founded are $$\hat{Z}_t(1)=3+0.5a_t-0.25a_{t-1}$$ $$\hat{Z}_t(2)=3+0.5a_t$$ $$\hat{Z}_t(l)=3\quad l\geq 3$$

So the question that came to my mind is, how will I calculate this if I do not know the values of $a_t$?

I need to assume that the initial values of $ a_t $ are zero?

EDIT: I was reading something about back-forecasting to estimate the values of $a_t$ and then the forecasts, but I don't understood well how to apply it. Is this the way?

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