The file m-ppiaco4709 txt contains year, month, day, and U.S. producer price index (PPI) from January 1947 to November 2009. The index is for all commodities and not seasonally adjusted. Let \(z_{t}=\ln \left(Z_{t}\right)-\ln \left(Z_{t-1}\right)\), where \(Z_{t}\) is the observed monthly PPI. It turns out that an \(\operatorname{AR}(3)\) model is adequate for \(z_{t}\) if the minor seasonal dependence is ignored. Let \(y_{t}\) be the sample mean-corrected series of \(z_{t}\).

(a) Fit an \(\mathrm{AR}(3)\) model to \(y_{t}\) and write down the fitted model.

(b) Suppose that \(y_{t}\) has independent measurement errors so that \(y_{t}=x_{t}+e_{t}\), where \(x_{t}\) is a zero-mean \(\mathrm{AR}(3)\) process and \(\operatorname{Var}\left(e_{t}\right)=\sigma_{e}^{2}\). Use a statespace form to estimate parameters, including the innovational variances to the state and \(\sigma_{e}^{2}\). Write down the fitted model and obtain a time plot of the smoothed estimate of \(x_{t}\). Also, show the time plot of filtered response residuals of the fitted state-space model.