Consider the model yt = xt + vt, where xt = xt1 + wt, such that vt is Gaussian white noise and independent of xt

Consider the model yt = xt + vt, where xt = φxt−1 + wt, such that vt is Gaussian white noise and independent of xt with var(vt) = σ2 v, and wt is Gaussian white noise and independent of vt, with var(wt) = σ2 w, and |φ| < 1 and Ex0 = 0. Prove that the spectrum of the observed series yt is fy(ω) = σ2|1 − θe−2πiω|

2

|1 − φe−2πiω|

2 , where

θ = c ± √c2 − 4 2 , σ2 = σ2 vφ

θ , and c = σ2 w + σ2 v(1 + φ2)

σ2 vφ