hw4 - Homework 4 Time series Due day March 10th 1(Linear...

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Homework 4. Time series. Due day: March 10thFebruary 28, 20171. (Linear prediction). Suppose that{Xt}is anAR(1) process with mean 0.Xt=λ1Xt-1+WtwithWtN(0,1) and|λ1| 6= 1. Suppose we have observedX1andX3and we would like to estimate the missing valueX2. Find the best linearpredictor ofX2givenX1andX3.
2. We consider in this problem a forecasting model on ARMA processes. LetXtbe an ARMA(p, q) process, andXtt+1be the one-step ahead linearforecasting.(a) LetX= (X1, . . . , Xn) be a random vector andYa random variable.LetP(Y|X) be the projection operator o the best linear predictorofYgivenX, given byP(Y|X) =μy+φT(X-μX)) , withφ=cov(X, X)-1cov(X, Y) . Show thatP(Y|X) is linear inY, i.e. thatP(α1Y1+α2Y2|X) =α1P(Y1|X)+α2P(Y2|X) for any pair of randomvariablesY1andY2and any constantsα1andα2.(b) Suppose first thatp= 1 andq= 0 and assume thatμ= 0. ShowthatXtt+1=ρX(1)XtHind: Use the above result sinceXtt+1=P(Xt+1|X1, . . . , Xt) and wecan use the above result forY=Xt+1andX= (X1, . . . , Xt).Question: Does the prediction get better astincreases? Interpret thisfact. Hint: What is the prediction error in this caseE|Xt+1-Xtt+1|2?(c) Let us consider now theMA(1) case (p= 0 andq= 1), withXt=(1 +θB)Wtand|θ|<1. Show ?rst that the invertible representation
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