Consider the same model as in the preceding problem. (a) Prove the optimal smoothed estimator of the form xt = s= as yts has

Consider the same model as in the preceding problem.

(a) Prove the optimal smoothed estimator of the form xˆt =

Õ∞

s=−∞

as yt−s has as =

σ

2 w

σ2

θ

|s |

1 − θ

2

.

(b) Show the mean square error is given by E{(xt − xˆt)

2

} =

σ

2 vσ

2 w

σ2

(1 − θ

2

)

.

(c) Compare mean square error of the estimator in part

(b) with that of the optimal finite estimator of the form xˆt = a1 yt−1 + a2 yt−2 when σ

2 v = .053, σ2 w = .172, and φ1 = .9.