Consider a stationary $AR(2)$ process given by $$X_{t} - X_{t-1} + 0.25X_{t-2} = 5 + a_{t}$$ where $a_{t} \sim WN(0,1)$ (white noise).
I am interested in obtaining the causal representation of $\{X_{t}\}$; that is, representing $X_{t} = \displaystyle\sum_{j=0}^{\infty} \psi_{j} a_{t-j}$ (i.e., $MA(\infty)$ process). This is guaranteed by the assumption of stationarity.
I want to determine $\psi_{j}$ for $j = 1,2,3,4,5$. I can do this pretty easily, but I'm being thrown off by the presence of the 5 on the right-hand side. Do I ignore it or do I account for it in some way?
As in your other question, the solution is to center the process $(X_t)$. Here, at stationarity, $E(X_t)=20$ for every $t$ hence one should consider $Y_t=X_t-20$ and note that $$Y_{t} - Y_{t-1} + \tfrac14 Y_{t-2} = a_t.$$ Now you should be back to known territory, the final result being $$X_t=20+\sum_{j=0}^\infty\frac{j+1}{2^j}a_{t-j}.$$