Consider a random walk ut and a stationary ar1 time series

Question

Consider a random walk {u_t} and a stationary AR(1) time series {e_t}, which evolve according to the following equations:

u₀= 0,

u_t = u_t-1 + ε_1,t,     for      t ≥ 1,

e_t = ρe_t-1 + ε_2,t,    for   t ≥ 1,

where |ρ| < 1 and { ε_1,t} and { ε_2,t} are white noise sequences, independent of each other (so, E[ε_1,t ε_2,s]=0  for all t, s.

Consider also processes {X_t} and {Y_t} which obey the following equations

X_t + βY_t = u_t

X_t + αY_t = e_t ,

where α and β are some constants, αβ , and {u_t} and {e_t} are the processes defined above.

(i) Show that

X_t =[ α/(α- β)] u_t - [β/(α- β)] e_t

Y_t =[-1/(α- β)] u_t + [1/ (α- β)] e_t.