I know in terms of $y$, the general forecasting equation is: $$ \hat{y}_t = \mu + \varphi_1 y_{t-1} + \dots + \varphi_p y_{t-p} - \theta_1e_{t-1} - \dots - \theta_qe_{t-q}. $$ I also know that ARIMA(1,0,0) = first-order autoregressive model: $$ \hat{y}_t = \mu + \varphi_1y_{t-1}. $$
Can someone tell me how I can write the equation for an ARIMA (1, 0, 1)?
I don't know what you mean by the general forecasting equation and I am not sure I understand your notation. Are you staying in the ARIMA realm?
The AR(1) model ARIMA(1,0,0) has the form: $Y_t = r Y_{t-1} + e_t$ where $r$ is the autoregressive parameter and $e_t$ is the pure error term at time $t$.
For ARIMA(1,0,1) it is simply $Y_t = r Y_{t-1} + e_t + a e_{t-1}$ where $a$ is the moving average parameter.
