Hi there stuck on the following:
Consider the model: $$y_{t}=(1+a)y_{t-1}-(a)y_{t-2}+\epsilon_{t}$$
where $\epsilon_{t}$ is a white noise problem:
1) Transform $y_t$ into some other series $w_t$ such that $w_t$ is stationary:
For this I considered creating a difference stationary process such that, $w_t=(1+a)y_{t-1}-(1+2a)y_{t-2}+ay_{t-3}+\epsilon_{t}- \epsilon_{t-1}$
2) Calculate the forecast function for $w_t$ and the corresponding variance of the forecast error.
Im fine with the forecast function just struggling with the variance, my forecast function is:
$$E_{t}(w_{t+j})= (1+a)y_{t}-(1+2a)y_{t-1}+ay_{t-2}-\epsilon_{t}$$ for j=1
$$E_{t}(w_{t+j})=-(1+2a)y_{t}+ay_{t-1}$$ for j=2
$$E_{t}(w_{t+j})=ay_{t}$$ for j=3
$$E_{t}(w_{t+j})=0$$ for j>3.
But then it asks for the variance of the forecast error, not sure where to go with this. It then goes on to ask to form the forecast function for $y_t$, which I have done, and calculate the corresponding variance of the forecast for lead times j=1,2,3?
Any help would be most appreciated, many thanks.