Consider the weekly crude oil price from January 3, 1986 to October 6, 2017. The data are available from FRED and are also in the file wcoilwtico.

csv. Let yt be the first differenced series of the weekly crude oil price. Part of the series was analyzed in Example 1.2 of Chapter 1. Apply a two-state HMM to the series yt.

(a) Obtain parameter estimates, including estimates of the transition probability matrix.

(b) Show the time plot of smoothed probabilities of state 1.

(c) Obtain the most likely state path.

(d) Compare with the analysis of Example 1.2.