1.2 The q-th Order Moving Average Process A stochastic process Yt, tE TI is said to be a moving average process of order q(MA(q) if it can be expressed in this form Y=p+et+61et-1+62et-2+…+6=t-q where A, 01, 02,. 0a are constants and Et is a white-noise process 1.2.1 Check Stationarity The expectation of Yt is given by E(Y)=E(H+et+61et-1+62=-2+…+6=t-q) A+E(Et)+01E(Et-1)+B2E(Et-2)++0gE(Et-g)=u, for alltET The variance of Yt is 0=E(Y-p)2=E(t+01=-1+02=1-2+…+04=t-9)2 Since Ets are uncorrelated, the variance is 70=02+6m2+B2+…+2=(1+6++…+)2 Forj=1,2,…,q, =E[(Yt-1)(Y--p E[(et+61et-1+62t-2+…+6t-9)×(et-+61t-1-1+62et-1-2+…+64=t--9) E0=2-1+b+(=2=1-1+61+22=2-2+…+20-ye2- Terms involving Es at different dates have been dropped because their product has expectation zero, and bo is defined to be unity. For i>g, there are no as with common dates in the definition of %j, and so the expectation is zero. Thus, 3+6+161+63+22+…+b,04-2forj=1,2,…,q 0>91.2 The q-th Order Moving Average Process A stochastic process {Yt , t ∈ T } is said to be a moving average process of order q (MA(q)) if it can be expressed in this form Yt = µ + εt + θ1εt−1 + θ2εt−2 + ... + θqεt−q, where µ, θ1, θ2, ..., θq are constants and εt is a white-noise process. 1.2.1 Check Stationarity The expectation of Yt is given by E(Yt) = E(µ + εt + θ1εt−1 + θ2εt−2 + ... + θqεt−q) = µ + E(εt) + θ1E(εt−1) + θ2E(εt−2) + ... + θqE(εt−q) = µ, for all t ∈ T . The variance of Yt is γ0 = E(Yt − µ) 2 = E(εt + θ1εt−1 + θ2εt−2 + ... + θqεt−q) 2 . Since εt ’s are uncorrelated, the variance is γ0 = σ 2 + θ 2 1σ 2 + θ 2 2σ 2 + .... + θ 2 qσ 2 = (1 + θ 2 1 + θ 2 2 + ... + θ 2 q )σ 2 . For j = 1, 2, ..., q, γj = E[(Yt − µ)(Yt−j − µ)] = E[(εt + θ1εt−1 + θ2εt−2 + ... + θqεt−q) × (εt−j + θ1εt−j−1 + θ2εt−j−2 + ... + θqεt−j−q)] = E[θjε 2 t−j + θj+1θ1ε 2 t−j−1 + θj+2θ2ε 2 t−j−2 + .... + θqθq−jε 2 t−q ]. Terms involving ε’s at different dates have been dropped because their product has expectation zero, and θ0 is defined to be unity. For j > q, there are no ε’s with common dates in the definition of γj , and so the expectation is zero. Thus, γj =  [θj + θj+1θ1 + θj+2θ2 + .... + θqθq−j ]σ 2 for j = 1, 2, ..., q 0 for j > q . 3