The file aa-rv-20m txt contains the realized daily volatility series of Alcoa stock returns from January 2, 2003, to May 7, 2004; see the example in Section 11.1. The volatility series is constructed using 20 -minute intradaily log returns.

(a) Fit an \(\operatorname{ARIMA}(0,1,1)\) model to the log volatility series and write down the model.

(b) Estimate the local trend model in Eqs. (11.1) and (11.2) for the \(\log\) volatility series. What are the estimates of \(\sigma_{e}\) and \(\sigma_{\eta}\) ? Obtain time plots for the filtered and smoothed state variables with pointwise \(95 \%\) confidence interval.

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Yt=lt + et, e, ~ N(0,), (11.1) M+1 = M + nt, n ~ N(0, ), (11.2)