The expectation of Yt is given by E(Yt)=limE(p+y0t+y1t-1+y2t-2+……+gret-r) 一 The variance of Yt is E(Y1-p)2 lim E(oEt +P1Et-1+92Et-2+..+PTEt-T im(y+92+2+…+1)o Fori>0 (Yt-p)(Y-) (9390+9+191+y+292+9+393+….)2 k=0 Thus, E(Y and %j are both finite and independent of t. The MA(oo) process with absolute-summable coefficients is weakly-stationary 1.3.3 Check Ergodicity Proposition The absolute summability of the moving average coefficients implies that the pro- Proof: Recall the autocovariance of an M A(o)isThe expectation of Yt is given by E(Yt) = lim T→∞ E(µ + ϕ0εt + ϕ1εt−1 + ϕ2εt−2 + .... + ϕT εt−T ) = µ The variance of Yt is γ0 = E(Yt − µ) 2 = lim T→∞ E(ϕ0εt + ϕ1εt−1 + ϕ2εt−2 + .... + ϕT εt−T ) 2 = lim T→∞ (ϕ 2 0 + ϕ 2 1 + ϕ 2 2 + .... + ϕ 2 T )σ 2 . For j > 0, γj = E(Yt − µ)(Yt−jµ) = (ϕjϕ0 + ϕj+1ϕ1 + ϕj+2ϕ2 + ϕj+3ϕ3 + ....)σ 2 = σ 2X∞ k=0 ϕj+kϕk. Thus, E(Yt) and γj are both finite and independent of t. The MA(∞) process with absolute-summable coefficients is weakly-stationary. 1.3.3 Check Ergodicity Proposition: The absolute summability of the moving average coefficients implies that the pro￾cess is ergodic. Proof: Recall the autocovariance of an MA(∞) is γj = σ 2X∞ k=0 ϕj+kϕk. 6