# Question: consider the following ma2 process zt ut 1ut1...

###### Question details

Consider the following MA(2) process: ${z}_{t}={u}_{t}+{\alpha}_{1}{u}_{t-1}+{\alpha}_{2}{u}_{t-2}$, where ut is a zero-mean white noise process with variance ${\sigma}^{2}$.

(a) Calculate the conditional and unconditional means of ${z}_{t}$, that is, ${E}_{t}\left[{z}_{t+1}\right]$ and E[${z}_{t}$].

(b) Calculate the conditional and unconditional variances of ${z}_{t}$, that is, $Va{r}_{t}\left[{z}_{t+1}\right]$ and Var[${z}_{t}$].

(c) Derive the autocorrelation function of this process for all lags as functions of the parameters ${\alpha}_{1}$ and ${\alpha}_{2}$.