Consider the second$-$order stochastic difference equation: $y_t=1\mathrm{.}5y_{t-1}-0\mathrm{.}5y_{t-2}+lo{n}_t.$

a$.$ Find the characteristic roots of the homogeneous equation.

b$.$ Re$-$express the second$-$order stochastic difference equation in terms of the lag$-$polynomial as $(L)y_t=lo{n}_t.$ Please write out the specific equation for $(L)$ in terms of the lag operator $L.$

c$.$ Demonstrate that the roots of $(L)=0$ are just the reciprocals of your answer in part a

d$.$ Find $E(y_t),E(y_{t+1}),$Var$(y_t),$Var$(y_{t+1}),$ and Cov$(y_{t+1},y_t).$