Consider the monthly U.S. unemployment rates from January 1, 1950 till June 30, 2019. Data are available from FRED and is listed as UNRATE. Let be the unemployment at time t.

(a.) Fit the seasonal model as below m1 <- arima(UNRATE,order=c(4,0,1),seasonal=list(order=c(1,0,1),period=12)) Perform model checking and write down the fitted model, including model checking.

(b.) Fit an AR(11) model to the time series.

(c.) Compare the two models. Which model is preferred? Why?

(d.) Use a GARCH(1,1) model for the volatility for the original time series, and the de-seasonalized series. Which one is better? Why?

(e.) Use the fitted model to obtain 1-step to 5-step ahead predictions series (forecast origin is the last data point). Also, compute the corresponding 95% interval forecasts. Perform the predictions separately for the return and volatility models, for both the original time series and the de-seasonalized time series.