Repeat Example 19.3 using the current 30 stocks of the Dow Jones Industrial Average and the daily returns for the most recent 10 years. (Historical stock price data is available online; in particular, adjusted closing prices can be downloaded from the Yahoo! finance site.)

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Example 19.3 (Dow Jones Industrial Index). The time series data of this example are the daily returns of the 30 stocks of the Dow Jones Indus- trial Index (DJIA), from from August 30, 1993 through August 29, 2003. (The 30 stocks correspond to the composition of the index on Septem- ber 1, 2003.) Pairs of stocks are tested for independence of their daily re- turns. The DJIA closing prices data for this example were from the R pack- age VaR [Daniyarov, 2004] with the source attributed to www.nasdaq.com. Each series is identified by a stock ticker symbol; see the VaR package for the corresponding list of company names. (The package is archived at https://cran.r-project.org/src/contrib/Archive/VaR/.) The time se- ries we analyze are log(returns); that is, the sequence of differences of the logarithms of prices. A plot of log(returns) (not shown) suggests that the data approximately satisfy the strong stationarity condition. There are 2520 values of log(returns). We set p = 100 and n = 50 (see Proposition 19.3). The t statistics (Tn) here are approximately standard nor- mal under the independence hypothesis, and a statistic greater than 1.645 is significant at the 5% level, indicating dependence. Most of the pairs of stocks had significant t statistics. The statistics for 20 pairs of stocks are shown in Table 19.2. Large positive values of Tn are significant. Instead of a table of 435 statistics we prepared a graphical summary, using a dendrogram to summarize the 435 pairwise R* statistics. This figure can be found in Szkely and Rizzo [2013a] (Figure 4) along with a detailed explanation of how the clustering was obtained. From the cluster dendrogram, one can observe, for example, that the financial stocks (AXP, C, JPM) cluster together, and the technology stocks (HPC, IBM, INTC, MSFT) also cluster together, but these two clusters are well separated and not merged until the second to last step when there are three clusters. One can also observe that five manufacturers (AA, CAT, DD, GM, IP) cluster together, as do two retail industry (HD, WMT), two drug (JNJ, MRK), and two telecommunications stocks (SBC, T).