Let $\{Y_t\}$ be the first order autoregressive $(AR(1))$ process

$Y_t-=(Y_{t-1}-)+{}_t,||<mn\; id="EditorElement\_152623">1</mn>$

${}_{t}iid(0,{}_{}^2)$

and define $x_t$:$=Y_t-.$

$($a$)$ Find MA$($ representation of $x_t.$

$($b$)$ Using the $MA()$ representation in $3.($a$),$ compute the impulse response function

$IR{F}_j=\frac{del{x}_t}{\text{d}\text{e}\text{l}{}_{t-j}}.$

$($c$)$ Let $x_t$ be the time series of US GDP$,$ and ${}_{t-j}$ be some unexpected news $($say$,$ a sudden

drop of the federal fund rate$)$ from time $t-j.$ What will be the economic interpretation of

the $IR{F}_j$ above? What happens to this $IR{F}_j$ if $j$ gets larger and larger $(j)?$

$($d$)$ Compute Var$(x_t),$Cov$(x_t,x_{t-1})$ and Corr$(x_t,x_{t-1}).$