Consider the same model as in the preceding problem.

(a) Prove the optimal smoothed estimator of the form xbt = X∞

s=−∞

asyt−s has as = σ2 w

σ2

θ|s|

1 − θ2 .

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

2} = σ2 vσ2 w

σ2(1 − θ2)

.

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

(b) with that of the optimal finite estimator of the form xbt = a1yt−1 + a2yt−2 when σ2 v = .053, σ2 w = .172, and φ1 = .9.

Section 4.12