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 z_t = ln(Z_t) - ln(Z_t-1), where Z_t is the observed monthly PPI. It turns out that an AR(3) model is adequate for y_t if the minor seasonal dependence is ignored. Let y_t be the sample-mean corrected series of z_t.

(a) Fit an 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 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 x_t. Also, show the time plot of filtered response residuals of the fitted state-space model.