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The file m-ppiaco.txt contains year, month, day, and U.S. Producer Price Index (PPI) from January 1947 to August 2004. The index is for all commodities and not seasonally adjusted. Let zt = ln(Zt) - ln(Zt-1), where Zt is the observed monthly PPI. It turns out that an AR(3) model is adequate for yt if the minor seasonal dependence is ignored. Let yt be the sample-mean corrected series of zt.
(a) Fit an AR(3) model to yt and write down the fitted model.
(b) Suppose that yt has independent measurement errors so that yt = xt + et, where xt is an AR(3) process and Var(et) =. Use a state-space form to estimate parameters, including the innovational variances to the state and .
Write down the fitted model and obtain a time plot of the smoothed estimate of xt. Also, show the time plot of filtered response residuals of the fitted state-space model.
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