Control charts are also useful when you look at financial data. These charts help you recognize that some processes are not very simple: Their features change over time. X-bar and S-charts are convenient to plot means and standard deviations, and the control limits supply a point of reference for deciding whether something unusual is going on.
For this exercise, consider the daily prices of stock in Apple during 2004–2011 (2,015 rows).
Motivation
(a) Explain why a sequence of prices of a stock will not appear to be in control. Confirm your explanation by looking at the timeplot of the prices of Apple stock during these years.
(b) Explain whether it would be useful to monitor both the mean and the variance of the sequence of daily returns (percentage changes) on a stock.
Method
(c) If you observe a period of stable behavior, suggest how you can use these data to set the mean and SD for monitoring the future returns.
(d) How many days should you collect data in order to compute a mean and standard deviation for tracking in the control chart? For example, if you use 10 days (about two weeks) will you be able to rely on the Central Limit Theorem when building a control chart for the daily percentage changes? How can you tell?
Mechanics
(e) Use the daily percentage changes during 2004–2007 to choose values for μ and σ. Does the series of returns form a simple time series that can be summarized in a histogram, or do you find patterns?
(f) Do the percentage changes in 2004–2007 appear to be normally distributed? How large must the batch size be in order to meet the sample size condition?
(g) Generate control charts for 2008–2011 (30 days at a time) using values for μ and σ chosen in part (e). Are the returns under control?
Message
(h) Explain your findings in language that an investor will understand. Avoid statistical jargon (e.g., Type I error, a, and the like).