Problem4_71 - Problem 4.71 STA 6934 Sato Klein The AR(1)...

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Unformatted text preview: Problem 4.71 STA 6934 Sato Klein The AR(1) model Xk=βXk-1+εk, k=0,1,...n, where the εk’s are iid N(0,1), X0 is distributed N(0,σ2), and β is an unknown parameter satisfying β<1. The Xk is independent from Xk-2,Xk-3,...conditionally on Xk-1. The Xk’s have marginal normal distributions with mean zero. (a)The variance of Xk satisfies var(Xk)=βvar(Xk-1)+1, and var(Xk)=σ2. βvar(Xk-1)+1=σ2 σ2=1/(1-β2) (b)E(Xkx0)=βkx0 x0=x0, and E(x0x0)=E(x0)=0 X1=βx0+ε1, and E(X1x0)=E(βx0+ε1x0)=E(βx0+ε1)=βx0 X2=βX1+ε2, and E(X2x0)=E(βX1+ε2x0)=βE(X1x0)=βE(βx0)=β2x0 M M Similarly, Xk=βXk-1+εk, and E(Xkx0)=E(βXk-1+εkx0)=βE(Xk-1x0)=βE(βk-1x0)=βkx0 (c)cov(X0,Xk)=βk/(1-β2) cov(X0,Xk)=EX0Xk-µ0µk=EX0Xk=E[X0E(XkX0)]=E(X0βkX0)=βkE(X02)=βkσ2 By (a) σ2=1/(1-β2), cov(X0,Xk)=βk/(1-β2) ...
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