13. In this problem, you will fit a t-distribution to daily log returns of Siemens.

You will estimate the degrees-of-freedom parameter graphically and then by maximum likelihood. Run the following code, which produces a 3 ×

2 matrix of probability plots. If you wish, add reference lines as done in Sect. 4.10.1.

library(evir)

data(siemens)

n = length(siemens)

par(mfrow = c(3, 2))

qqplot(siemens, qt(((1 : n) - 0.5) / n, 2), ylab = "t(2) quantiles", xlab = "data quantiles")

qqplot(siemens,qt(((1:n)-.5)/n,3),ylab="t(3) quantiles", xlab="data quantiles")

qqplot(siemens,qt(((1:n)-.5)/n,4),ylab="t(4) quantiles", xlab="data quantiles")

qqplot(siemens,qt(((1:n)-.5)/n,5),ylab="t(5) quantiles", xlab="data quantiles")

qqplot(siemens,qt(((1:n)-.5)/n,8),ylab="t(8) quantiles", xlab="data quantiles")

qqplot(siemens,qt(((1:n)-.5)/n,12),ylab="t(12) quantiles", xlab="data quantiles")

R has excellent graphics capabilities—see Chap. 12 of An Introduction to R for more about R graphics and, in particular, pages 67 and 72 for more information about par and mfrow, respectively.

(a) Do the returns have lighter or heavier tails than a t-distribution with 2 degrees of freedom?

(b) Based on the QQ plots, what seems like a reasonable estimate of ν?

(c) What is the MLE of ν for the Siemens log returns?