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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)?

Richard Hardy
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Kam
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  • See [this](http://stats.stackexchange.com/questions/69407/how-do-i-write-a-mathematical-equation-for-arima-2-1-0-x-0-2-2-period-12) and [this](http://stats.stackexchange.com/questions/61510/how-do-i-write-a-mathematical-equation-for-arima-0-2-1-x-0-0-1-period-12?rq=1) which are possibly duplicates. See also the related questions to the above ones in the right panel. – Richard Hardy Dec 14 '16 at 20:01

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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.

Michael R. Chernick
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