Obtain the daily closing prices of the S\&P 500 Index for the year 2017 and compute the log returns.

(a) Compute the Hill estimator of the lower tail index of the distribution of the daily returns of the index for that period, letting the tuning parameter range from 11 to 100.

(b) Compute an estimate of the lower tail index for these data using tuning parameter values of \(20,30,40\), and 50 .

How do they differ?

Now, use the bootstrap to estimate the bias and the variance for the estimator at each value of the tuning parameter.

What conclusions can you draw about the bias-variance tradeoff?

(c) Compute a basic bootstrap 95\% confidence interval for the the tail index of the distribution of the daily returns of the index for the year 2017.

For this, take the estimate as the mean of Hill's estimate over the same interval in each bootstrap sample, where this interval is chosen (subjectively) by inspection of the Hill plot for the full dataset.