2.1.2 The Dependence Structure The expectation of Yt is given by E(Y) = E( C et+et-1+a2et-2+… 1- The variance of Yt is E(Y-F E(et+et-1+a2et-2+…)2 (1+2+a4+…,lg2 1-φ For,>0 =E(Y-p)(Y--p) BI +2=-y++et-1-1++2t-y-2+…) (et-j+et--1+2et-j-2+…,) (1+02+4+… It follows that the autocorrelation function which follows a pattern of geometric decay as the plot on p 50 2.1.3 An Alternative Way to Calculate the Moments of a Stationary AR(1)PI Assume that the AR(1) process under consideration is weakly-stationary, then taking expectation on both side we have E(Y=c+oE(Y-1)+E(Et)2.1.2 The Dependence Structure The expectation of Yt is given by E(Yt) = E( c 1 − φ + εt + φ 1 εt−1 + φ 2 εt−2 + ...) = c 1 − φ = µ. The variance of Yt is γ0 = E(Yt − µ) 2 = E(εt + φ 1 εt−1 + φ 2 εt−2 + ....) 2 = (1 + φ 2 + φ 4 + ....)σ 2 =  1 1 − φ 2  σ 2 . For j > 0, γj = E(Yt − µ)(Yt−j − µ) = E(εt + φ 1 εt−1 + φ 2 εt−2 + .... + φ j εt−j + φ j+1εt−j−1 + φ j+2εt−j−2 + ....) × (εt−j + φ 1 εt−j−1 + φ 2 εt−j−2 + ....) = (φ j + φ j+2φ j+4 + ...)σ 2 = φ j (1 + φ 2 + φ 4 + ....)σ 2 =  φ j 1 − φ2  σ 2 . It follows that the autocorrelation function rj = γj γ0 = φ j , which follows a pattern of geometric decay as the plot on p.50. 2.1.3 An Alternative Way to Calculate the Moments of a Stationary AR(1) Process Assume that the AR(1) process under consideration is weakly-stationary, then taking expectation on both side we have E(Yt) = c + φE(Yt−1) + E(εt). 9