6.1 Consider a system process given by xt = −.9xt−2 + wt t = 1, . . . , n where x0 ∼ N(0, σ2 0), x−1 ∼ N(0, σ2 1), and wt is Gaussian white noise with variance σ2w

. The system process is observed with noise, say, yt = xt + vt, where vt is Gaussian white noise with variance σ2 v. Further, suppose x0, x−1, {wt} and {vt} are independent.

(a) Write the system and observation equations in the form of a state space model.

(b) Find the values of σ2 0 and σ2 1 that make the observations, yt, stationary.

(c) Generate n = 100 observations with σw = 1, σv = 1 and using the values of σ2 0 and σ2 1 found in (b). Do a time plot of xt and of yt and compare the two processes. Also, compare the sample ACF and PACF of xt and of yt.

(d) Repeat (c), but with σv = 10.