# ARMA one step ahead forecast and forecast error

#### Clarkson

##### New Member
For a ARMA(1,1) process with constant $$\theta$$ is
$$X_t=\alpha X_{t-1}+\theta +Z_t+\beta Z_{t-1}$$ where $$Z_T$$ is white noise with mean 0 and variance$$\sigma ^2$$.
1)Find the one step ahead forecast
2)Find the expected value and variance of one step ahead forecast error

Here's what I did:
1)$$\hat X_{t+1}=\hat \alpha X_t+\theta+\hat \beta Z_t$$.
I would like to know if this is the correct one step ahead forecast.
Also does $$\theta$$ becomes $$\hat \theta$$?

I think not because it is a constant.
For second part:
error=$$X_{t+1} -$$$$\hat X_{t+1}=X_t(\alpha-\hat \alpha)+Z_t(\beta-\hat \beta)+Z_{t+1}$$.

Is this forecast error correct?

Then E(error)=0
and V(error)=$$(\alpha-\hat \alpha)^2 V(X_t)+(\beta-\hat \beta)^2\sigma^2+\sigma^2$$