For the MA(1) model yt = t + t1 t = 1, 2, . . . , T; with t IIN(0, 2 ) (a)

For the MA(1) model yt = t + θt−1 t = 1, 2, . . . , T; with t ∼ IIN(0, σ2 )

(a) Show that E(yt) = 0 and var(yt) = σ2 (1+θ2) so that the mean and variance are independent of t.

(b) Show that cov(yt, yt−1) = θσ2 and cov(yt, yt−s) = 0 for s > 1 which is only dependent on s, the distance between the two time periods. Conclude from parts

(a) and

(b) that this MA(1)

model is weakly stationary.

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