A lot of economic variable responses to unexpected shocks display hump shapes. The simplest model to explain such behavior is an $AR(2)$ process:

$x_t=0\mathrm{.}3x_{t-1}+0\mathrm{.}5x_{t-2}+{}_t$

where ${}_t$ is white noise, i$.$i$.$d$.$ from $N(0,{}^2).$

a$)$ What is the impulse$-$response function of $x_{t+j}$ to a shock in ${}_t,j=1,2,$dots $?$ Plot the impulse response for $20$ periods.

b$)$ What is the limiting response to a shock when $j$ approaches $+?$ What can you say about correlation between $x_t$ and $x_{t+j}$ when $j$ approaches $+?$

c$)$ For a general $AR(2)$ process:

$x_t=b_1{x}_{t-1}+b_2{x}_{t-2}+{}_t$

What are the restrictions on $b_1$ and $b_2$ such that $x_t$ is stationary?