Obtain the VIX index from $2012-01-01$ to $2017-02-01.$ Focus on the series of the daily

closing index. Compute the monthly $log$ return $r_t$ of the VIX index. $($you can use the

function monthlyReturn$).$

getSymbols $("^$VIX", from$=\text{'}2012-01-01\text{'},$ to$=\text{'}2017-02-01\text{'},$ src$=$"yahoo" $)$

$(1)$ Using the command ar to identify an AR mdoel for $r_t.$ Perform estimation and write

down the fitted model, including residual standard deviation.

$(2)$ Make an ACF plot. Propose and fit an MA model based on the ACF plot. Perform

estimation and write down the fitted model, including residual standard deviation. Are

all the MA coefficient significant? Is the fitted model invertible? $($check the roots of the

MA polynomial$)$ Make a tsdiag plot for the model. Judging from the plots, is the fitted

model adequate?

$(3)$ Make a PACF plot. Propose and fit an AR model based on the PACF plot. Perform

estimation and write down the fitted model, including residual standard deviation. Are

all the AR coefficient significant. Is the fitted model stationary? $($check the roots of the

AR polynomial$)$ Make a tsdiag plot for the model. Judging from the plots, is the fitted

model adequate?

$(4)$ Based only on the AIC values of the two models in $(2)$ and $(3),$ which model is better?