Obtain the weekly log returns of the S&P 500 index for the period January 1, 1978 through

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Obtain the weekly log returns of the S\&P 500 index for the period January 1, 1978 through December 31, 2017, and compute the weekly \(\log\) returns. 

(a) Test for an ARCH effect in this time series.

(b) Fit an ARIMA \((1,1,1)+\operatorname{GARCH}(1,1)\) model to the weekly log returns of the S\&P 500 index for the period January 1, 1978 through December 31, 2017.

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