Let {wt ; t = 0, 1, . . . } be a white noise process with variance 2 w and let || <

Let {wt

; t = 0, 1, . . . } be a white noise process with variance σ

2 w and let |φ| < 1 be a constant. Consider the process x0 = w0, and xt = φxt−1 + wt

, t = 1, 2, . . . .

We might use this method to simulate an AR(1) process from simulated white noise.

(a) Show that xt =

Ít j=0

φ

jwt−j for any t = 0, 1, . . . .

(b) Find the E(xt).

(c) Show that, for t = 0, 1, . . ., var(xt) =

σ

2 w

1 − φ

2

(1 − φ

2(t+1)

)

(d) Show that, for h ≥ 0, cov(xt+h, xt) = φ

h var(xt)

(e) Is xt stationary?

(f) Argue that, as t → ∞, the process becomes stationary, so in a sense, xt is “asymptotically stationary."
(g) Comment on how you could use these results to simulate n observations of a stationary Gaussian AR(1) model from simulated iid N(0,1) values.
(h) Now suppose x0 = w0/
p 1 − φ
2 . Is this process stationary? Hint: Show var(xt) is constant.