# Forecast in $AR(1)$ process witn non-zero mean

For an $AR(1)$ model with $Y_t=12.2$ , $\phi=-0.5$ and $\mu=10.8$.

a) Find $\hat{Y}_t(1)$, $\hat{Y}_t(2)$, $\hat{Y}_t(10)$

I'm a litle lost in forecasting ARIMA models. What I think it is

$$Y_t-\mu=\phi(Y_{t-1}-\mu)+\epsilon_t$$ where $\epsilon_t$ is white noise.

$$\hat{Y}_t(1)=E[Y_{t+1}|Y_1,\dots,Y_t]=E[\mu+\phi(Y_t-\mu)+\epsilon_{t+1}|Y_1,\dots,Y_t]$$ $$=\mu+\phi(Y_t-\mu)$$

and the general formula is $$\hat{Y}_t(l)=\mu+\phi^l(Y_t-\mu)$$

So $$\hat{Y}_t(1)=10.8+(-0.5)(12.2-10.8)=10.1$$ $$\hat{Y}_t(2)=10.8+(-0.5)^2(12.2-10.8)=11.15$$ $$\hat{Y}_t(10)=10.8+(-0.5)^{10}(12.2-10.8)=10.801$$

So $$\hat{Y}_t(l)\rightarrow \mu$$

Is this results right?

For any ARIMA model that I want to do forecast I just take expectation in this way?

• yes. And AR(1) models are forecast to decay towards their unconditional mean for $\mid \phi \mid < 1$. – Matthew Gunn Dec 22 '16 at 19:17
• @MatthewGunn In any ARIMA/SARIMA model to find the forecasts at hand , I just need to take the expectation in the same way? – user72621 Dec 22 '16 at 19:27
• MA terms are trickier. AR(n) reduces to a vector AR(1) (i.e. VAR). – Matthew Gunn Dec 22 '16 at 20:36