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 statespace 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.