Let $\{x_t\}$ be generated by a moving average process as given in the following equation:

$x_t=+{}_t+{}_1{}_{t-1}+{}_2{}_{t-2}.$

where $\{{}_t\}$ are independently identically distributed random variables with $E({}_t)=0,$ and

var$({}_t)={}^2.$

a$)[5\%]$ Calculate the expected value and the variance of $\{x_t\}.$

b$)[5\%]$ What is it meant by saying that a time series process like $\{x_t\}$ is invertible? What

condition would assure that $\{x_t\}$ is invertible? Assume that ${}_1=0\mathrm{.}75$ and ${}_2=0\mathrm{.}25,$ does the

process $\{x_t\},$ satisfies this condition?

c$)[10\%]$ Carefully explain the difference between the autocorrelation function $($ACF$)$ and the

partial autocorrelation function $($PACF$)$ for a process like $\{x_t\}.$ What shape do you expect both

ACF and PACF to take for $\{x_t\}?$ Derive the first $4$ autocorrelations for this process $({}_1$ up to

${}_4).$

d$)[5\%]$ Obtain expressions for the $1,2,3$ and $4$ steps ahead forecast for $\{x_t\}.$