Problem Set 1

Question 1

Let diffbp be the changes (that is, differences) in the variable bp, the U.S. dollar to British pound exchange rate, which is in the Garch data set of R’s Ecdat package.

a. (20 points)

Create a \(3\times2\) matrix of normal plots of diffbp and in each plot add a reference line that goes through the \(p\) and \((1 − p)\) quantiles, where \(p = 0.25,0.1,0.05,0.025,0.01\), and \(0.0025\), respectively, for the six plots. Create a second set of six normal plots using \(n\) simulated \(N(0,1)\) random variables, where \(n\) is the number of changes in bp plotted in the first figure. Discuss how the reference lines change with the value of \(p\) and how the set of six different reference lines can help detect nonnormality.

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b. (15 points)

Create a third set of six normal plots using changes in the logarithm of bp. Do the changes in \(log(\textrm{bp})\) look closer to being normally distributed than the changes in bp?

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Question 2

Download daily prices for Twitter stock (TWTR) from http://finance.yahoo.com for December 2016. Using the adjusted closing prices, compute daily returns by calculating the differences in the natural logarithm of prices: \[\begin{align} r_t & = \log(p_t) − \log(p_{t−1}). \end{align}\]

a. (5 points)

What is the sample mean, sample median and sample standard deviation of daily Twitter returns?

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b. (6 points)

Create a normal probability plot of Twitter returns. Do they look normally distributed?

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c. (12 points)

Using a solid curve, plot a kernel density estimate of the returns. Using a dashed curve, superimpose a normal density with the same mean and standard deviation as the sample. Do the two estimated densities look similar? Describe how they differ.

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d. (12 points)

Repeat part (c), but with the mean and standard deviation equal to the median and MAD. Do the two densities appear more or less similar compared to the two densities in part (a)?

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Question 3

(30 points) The file mystery.txt contains 6,875 data observations drawn from an unknown distribution. Using the tools covered in the course so far, analyze the data both numerically, graphically and with words. What could have generated this data?

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