Extreme values in volatile time series

When you take a long series of iid data, such as several thousand observations, and select a small subset of the most extreme observations, like less than 100, then these extremes appear at random and the spaces or gaps between the extremes follow a distribution that is very close to exponential. When we carry out the same exercise for a volatile financial log-return series then the extremes appear in clusters during periods of high volatility. This is another feature of real log-return data that we need to take account of when building models.

In this exercise, you will investigate the irregular time series djx_extremes which contains the 100 most extreme negative log-returns of the Dow Jones index between 1985 and 2015. You will compare it with iid_extremes which contains the 100 most extreme values in an iid series of the same length as djx_extremes. To do this, you will use the object exp_quantiles, which contains 100 theoretical quantiles of the standard exponential distribution. These can be used to construct a Q-Q plot of each dataset against the exponential reference distribution.

The djx_extremes, iid_extremes, and exp_quantiles objects are available in your workspace.

This exercise is part of the course

Quantitative Risk Management in R

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Exercise instructions

Partition the plotting area into a row of 3 panels (this has been done for you)
Use plot() and type = "h" to plot djx_extremes.
Use time() and diff() in succession to compute the spaces between the dates of the extremes and assign them to djx_spaces.
Use hist() and as.numeric() in succession to make a histogram of djx_spaces after coercing the data to numeric values.
Use the appropriate function to make a Q-Q plot of djx_spaces against the exponential quantiles in exp_quantiles.
Carry out the previous 4 steps for iid_extremes: plot the raw data with same type parameter, compute the spaces as iid_spaces, make a histogram, and make a Q-Q plot.

Hands-on interactive exercise

Have a go at this exercise by completing this sample code.

# Partition plotting area into 3 pieces
par(mfrow = c(1, 3))

# Plot djx_extremes
___(___)

# Compute the spaces between the times of the extremes
djx_spaces <- ___

# Make a histogram of these spaces
___(___)

# Make a Q-Q plot of djx_spaces against exp_quantiles
___(exp_quantiles, djx_spaces)

# Carry out the previous 4 steps for iid_extremes
___(___, ___)
iid_spaces <-
___(___(___))
___(exp_quantiles, iid_spaces)

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