Seasonal adjustment via SSM. The file CAURN.csv contains the monthly unemployment rates of California from January 1976 to August 2017. The rates are not seasonally adjusted. Let yt be the unemployment rate and consider the model yt = Tt + St + It, where Tt is the trend component following the local trend model, St denotes the seasonal component following the model

Σ12????=1 St−????+1 = ????t, where ????t is a Gaussian white noise series with mean zero and variance ????2

???? , and It is the irregular component following an AR(5) model.

(a) Write down an SSM for yt. What is the dimension of the state vector?

(b) Fit the specified SSM model and write down the parameter estimates.

(c) Obtain a smoothed seasonal component St. Denote the series by ̂St.

(d) Let zt = yt − ̂St be the seasonally adjusted series of California monthly unemployment rate. Let ????12 be the lag-12 autocorrelation of zt. Test H0 : ????12 = 0 versus Ha : ????12 ≠ 0 and draw a conclusion using the 5%

type-I error.