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Evolve according to the equations

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

u0= 0,

ut = ut-1 + ε1,t,     for      t ≥ 1,

et = ρet-1 + ε2,t,    for   t ≥ 1,

where |ρ| < 1 and { ε1,t} and { ε2,t} are white noise sequences, independent of each other (so, E[ε1,t ε2,s]=0  for all t, s.

Consider also processes {Xt} and {Yt} which obey the following equations

Xt + βYt = ut

Xt + αYt = et ,

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

(i) Show that

Xt =[ α/(α- β)] ut - [β/(α- β)] et

Yt =[-1/(α- β)] ut + [1/ (α- β)] et.

Reference no: EM131533584

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