Let the log dividend growth evolve according to $\Delta d_{t+1} = \epsilon_{d, t+1}$ where $\epsilon_{d, t+1}$ is just white noise. Let the log returns be $r_{t+1} = x_t + y_t + \epsilon_{r, t+1}$ where $x_t = b_x x_{t-1} + \delta_{x, t}$ and $y_t = b_y y_{t-1} + \delta_{y, t}$ and $\epsilon_{r, t+1}$, $\delta_{x, t}$, and $\delta_{y, t}$ are all white noise. Solve for the dividend price ratio, $d_t - p_t$ and show that it is an $ARMA(p, q)$ process, find $p$ and $q$.
What I did was to start off with the famous Campbell-Shiller decomposition: $d_t - p_t = E_t \sum_{j=1}^{\infty} \rho^{j-1}(r_{t+j} - \Delta d_{t+j})$ and one can easily show that $E_t(r_{t+j}) = b_x^{j-1} x_t + b_y^{j-1} y_t$ and $E_t(\Delta d_{t+j}) = 0$ and so we get
$$d_t-p_t = \frac{x_t}{1-\rho b_x} + \frac{y_t}{1-\rho b_y} $$
I'm now stuck with trying to show that this is an ARMA(p, q) process. What I've tried is substituting in $x_t$ and $y_t$ to get:
$$d_t - p_t = b_x \frac{x_{t-1}}{1-\rho b_x} + b_y \frac{y_{t-1}}{1-\rho b_y}+ \frac{\delta_{x, t}}{1-\rho b_x} + \frac{\delta_{y, t}}{1-\rho b_y}$$
But I am stuck on manipulating the RHS to form a $d_{t-1} - p_{t-1}$, specifically, I know that $d_{t-1}-p_{t-1} = \frac{x_{t-1}}{1-\rho b_x} + \frac{y_{t-1}}{1-\rho b_y}$ but the coefficients $b_x$ and $b_y$ on the RHS makes things difficult.
EDIT: I've done a bit more working out and here is what I've done:
\begin{align*} d_t - p_t & = \frac{x_t}{1-\rho b_x} + \frac{y_t}{1-\rho b_y} \\ & = \frac{b_x x_{t-1} + \delta_{x, t}}{1-\rho b_x} + \frac{b_y y_{t-1} + \delta_{y, t}}{1-\rho b_y} \\ & = b_x \frac{x_{t-1}}{1-\rho b_x} + b_y \frac{y_{t-1}}{1-\rho b_y} + \frac{\delta_{x, t}}{1-\rho b_x} + \frac{\delta_{y, t}}{1-\rho b_y} \\ & = b_x\left(d_{t-1} - p_{t-1} - \frac{y_{t-1}}{1-\rho b_y} \right) + b_y\left(d_{t-1} - p_{t-1} - \frac{x_{t-1}}{1-\rho b_x} \right)+ \frac{\delta_{x, t}}{1-\rho b_x} + \frac{\delta_{y, t}}{1-\rho b_y} \\ & = (b_x + b_y)(d_{t-1} - p_{t-1})-b_x\left(\frac{b_y y_{t-2} + \delta_{y, t-1}}{1-\rho b_y} \right)-b_y\left(\frac{b_x x_{t-2} + \delta_{x, t-1}}{1-\rho b_x} \right) + \frac{\delta_{x, t}}{1-\rho b_x} + \frac{\delta_{y, t}}{1-\rho b_y} \\ & = (b_x + b_y)(d_{t-1} - p_{t-1}) - b_xb_y\left(\frac{x_{t-2}}{1-\rho b_x} + \frac{y_{t-2}}{1-\rho b_y} \right) - b_x \frac{\delta_{y, t-1}}{1-\rho b_y}- b_y \frac{\delta_{x, t-1}}{1-\rho b_x}+ \frac{\delta_{x, t}}{1-\rho b_x} + \frac{\delta_{y, t}}{1-\rho b_y} \\ & = (b_x + b_y)(d_{t-1} - p_{t-1}) - b_xb_y(d_{t-2} - p_{t-2}) - b_x \frac{\delta_{y, t-1}}{1-\rho b_y}- b_y \frac{\delta_{x, t-1}}{1-\rho b_x}+ \frac{\delta_{x, t}}{1-\rho b_x} + \frac{\delta_{y, t}}{1-\rho b_y} \end{align*}
So it seems like there are two lags for $d_t-p_t$ but I am not sure how to manipulate the moving average terms...