# Sheet - 1 Stationarity and ACF 1.1 1.3 Find Formula a. Find...

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1Stationarity and ACF1.2Problem 1(MA Models) For each of the following, state if it is a stationaryprocess. If so, give the mean and the autocovariance functions.Assume{Wt}is i.i.d. N(0,1). List here:(a)Xt=Wt-Wt-3(b)Xt=W3(c)Xt=t+W3(d)Xt=W2t(e)Xt=WtWt-2Problem 3FindACFforARmodels,sayXt=φXt-1+wtwithwtWN(0, σ2).The solution is:γ(0) =Cov(Xt, Xt) =Cov(φXt-1+wt-1, φXt-1+wt-1) =φ2γ(0) +σ2w. This solvesforγ(0) =σ2w[Solution]1.3Problem 2Finda. Find the ACF of the time seriesXt=Wt+5c. Which of the MA models in a and b is invertible?[Solution]a. Compute three cases: since theWtare uncorrelated, we canignore any-terms of the formEWsWtwhens6=t. Thenγ(0)=EW2t+251.4Problem 3FindACFforARmodels,sayXt=φXt-1+wtwithwtWN(0, σ2).The solution is:γ(0) =Cov(Xt, Xt) =Cov(φXt-1+wt-1, φXt-1+wt-1) =φ2γ(0) +σ2w. This solvesforγ(0) =σ2w

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Term
Spring
Professor
Abolfazl Safikhani
Tags
Autocorrelation, ACF, Recurrence relation, Xt
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