The first autocovariance is E(Y-p(Yt-1-u)= E(Et+BEt-1(Et-1+8Et-2) E(EtEt-1+0E_1+0EtEt-2+62Et-1Et-2 0+602+0+0 Higher autocovariances are all zero D=E(Y-u)(Yi-i-u=E(Et+0Et-1(Et-i+8Et-j-1)=0 forj>1 Since the mean and the autocovariances are not functions of time, an MA(1) process is weakly-stationary regardless of the value of 6 1.1.2 Check Ergodicity It is clear that the condition ∑h=(1+62)+|21< is satisfied. Thus the MA(1) process is ergodic. 1.1.3 The Dependence Structure The jth autocorrelation of a weakly-stationary process is defined as its jth autocovariance divided by the variance By Cauchy-Schwarz inequality, we have Ir,l s 1 for all From above results, the autocorrelation of an MA(1) process is whenj=0 when 3 The autocorrelation r, can be plotted as a function of 3. This plot is usually called autocogram See the plots of pThe first autocovariance is E(Yt − µ)(Yt−1 − µ) = E(εt + θεt−1)(εt−1 + θεt−2) = E(εtεt−1 + θε 2 t−1 + θεtεt−2 + θ 2 εt−1εt−2) = 0 + θσ 2 + 0 + 0 = θσ 2 . Higher autocovariances are all zero: γj = E(Yt − µ)(Yt−j − µ) = E(εt + θεt−1)(εt−j + θεt−j−1) = 0 for j > 1. Since the mean and the autocovariances are not functions of time, an MA(1) process is weakly-stationary regardless of the value of θ. 1.1.2 Check Ergodicity It is clear that the condition X∞ j=0 |γj | = (1 + θ 2 ) + |θσ 2 | < ∞ is satisfied. Thus the MA(1) process is ergodic. 1.1.3 The Dependence Structure The jth autocorrelation of a weakly-stationary process is defined as its jth autocovariance divided by the variance rj = γj γ0 . By Cauchy-Schwarz inequality, we have |rj | ≤ 1 for all j. From above results, the autocorrelation of an MA(1) process is rj =    1 when j = 0 θσ 2 (1+θ 2)σ2 = θ (1+θ 2) when j = 1 0 when j > 1 . The autocorrelation rj can be plotted as a function of j. This plot is usually called autocogram. See the plots of p.50. 2