For the AR(1) model yt = yt1 + t t = 1, 2, . . . , T; with || <1 and t IIN(0,

For the AR(1) model yt = ρyt−1 + t t = 1, 2, . . . , T; with |ρ| <1 and t ∼ IIN(0, σ2 )

(a) Show that if yo ∼ N(0, σ2 /1−ρ2), then E(yt) = 0 for all t and var(yt) = σ2 /(1−ρ2) so that the mean and variance are independent of t. Note that if ρ = 1 then var(yt) is ∞. If |ρ| > 1 then var(yt) is negative!

(b) Show that cov(yt, yt−s) = ρsσ2 which is only dependent on s, the distance between the two time periods. Conclude from parts

(a) and

(b) that this AR(1) model is weakly stationary.

(c) Generate the above AR(1) series for T = 250, σ2 = 0.25 and various values of ρ = ±0.9, ±0.8, ±0.5, ±0.3 and ±0.1. Plot the AR(1) series and the autocorrelation function ρs versus s.