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Use the real

returns from the data file attached

Asymmetry is important because it may tell us something about whether stock prices fall more dramatically than they rise. (There are more extreme crashes.) To solve this we have two different measures.

a.       First estimate the skewness on the entire series. Use the rret for this problem.

b.       Now try to simulate a distribution for this estimate using the bootstrap. With 10,000 bootstrap simulations of the annual returns estimate the probability a getting a skewness estimate greater than zero.

c.       Now estimate a 95 percent confidence band using the percentile method. Is zero inside this interval?

d.       A more complicated measure of skewness measures the differences between two extreme quantiles and the median. A version of this is

z=(q 0.5 −q 0.05 )−(q 0.95 −q 0.5 ) z=(q0.5−q0.05)−(q0.95−q0.5)

Estimate this for the data. Is it positive or negative? How is this measure trying to tell you something about the relative shapes of the left versus right tails? If the distribution were symmetric about the median, what value should this take?

e.       Estimate a 95 percent confidence band for this measure using the previous bootstrap procedure. Use 10,000 bootstrap replications, and the percentile method to determine the confidence bands.

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