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

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) = σ

2 v

, and wt is Gaussian white noise and independent of vt

, with var(wt) = σ

2 w, and |φ| < 1 and Ex0 = 0. Prove that the spectrum of the observed series yt is fy(ω) = σ

2

|1 − θe

−2πiω |

2

|1 − φe

−2πiω |

2

, where

θ =

c ±

√

c 2 − 4 2

, σ2 =

σ

2 v φ

θ

, and c =

σ

2 w + σ

2 v

(1 + φ

2

)

σ

2 v φ

.