Question $6$

$[20$ Marks$]$

$8\mathrm{.}1.$ Give an example of a stationary and non$-$stationery process

$8\mathrm{.}2.$ Consider the process $x_t=-x_{t-1}-\frac{1}{5}{Z}_{t-1}+Z_t,$ where $Z_t$ is a white noise process.

$8\mathrm{.}2\mathrm{.}1.$ Is the process invertible? If yes, find the coefficients associated with the model. $(10)$

$8\mathrm{.}2\mathrm{.}2.$ Is the process casual? $(3)$

$($c$)$ Is the first difference of the process, $y_t=x_t-x_{t-1}$ stationary?

$(3)$