## Sunday, December 3, 2017

### The Chi-Square Test and Kendall’s Tau

The Chi-Square Test and Kendall’s Tau

The post here is answer to question two on my list of contingency table problems.   The complete list is found here.

Question Two:   Test whether there is a significant relationship between stock price movement categories and S&P 500 movement categories for these two firm?   How do the Kendall's Tau statistics differ for these two companies?

Discuss differences in Pearson’s chi square and Kendall’s Tau for these two companies.

Discuss how Pearson’s chi-square and Kendall’s Tau are calculated.

Data:  The Contingency Tables summarizing the movement between company stock prices and market stock prices are below:

 Co=movement of Stock Price with S&P 500 Company One Stock ClosePrior High S&P Close < Prior Low 12 18 4 S&P Close Between Prior Low & High 26 80 28 S&P Close > Prior High 5 31 48 Company Two Stock ClosePrior High S&P Close < Prior  Low 10 21 3 S&P Close  Between Prior Low & High 19 68 47 S&P Close > Prior High 14 48 22

Results of Pearson Chi-Square Test and Kendall’s Tau B test from STATA

 Nonparametric Tests on Association Between  Company and Market Returns Company One Company Two Pearson Chi Square 44.7 11.3 p value for chi square test 0.000 0.024 Kendall's Tau B 0.367 0.060

Discussion of results; Company one is Apple, a high-beta company.   Company two is Duke Power a low beta company.    The Pearson’ chi square test reveals a significant association between company and index stock movements for both companies; although, the difference is larger for Apple than for Duke Power.

Kendall Tau’s B is over 6 times higher for Apple than for Duke, which suggests these non-parametric statistics are good measures of systematic risk.

Discussion of calculations of Pearson’s Chi-square in Excel.

Below is a presentation of the calculation of Pearson’s Chi-square statistic for independence in Excel.

 Calculation of Pearson's Chi-Square Statistic for Company One in Excel Observed Expected (O-E)2/e A 12 5.802 6.62 B 18 17.405 0.02 C 4 10.794 4.28 D 26 22.865 0.43 E 80 68.595 1.90 F 28 42.540 4.97 G 5 14.333 6.08 H 31 43.000 3.35 I 48 26.667 17.07 Sum 44.71

Discussion of calculation of Kendall’s Tau

Kendall’s Tau B is difficult to calculate in Excel.   The formula for Kendall’s Tau A is a lot simpler.   (Kendall’s Tau B incorporates information on ties.    Kendall’s Tau A does not do so.)

The formula for Kendall’s Tau A is (C-D)/(C+D) where C is total concordant pairs and D is total discordant pairs.

The observations that are concordant to a pair are down and to the right of the pair.  Below is a picture of the cells that are concordant to the top right cell of a contingency matrix.

 12 80 28 31 48

The total number of pairs that are concordant to the top right cell is 12 (80+28+31+48)  = 2244

There are three other cells with concordant cells

Number of concordant pairs for top row middle column is.18*(28+48) or 1368

Number of concordant pairs for middle row right column cell is 26*(31+48) or 2054.

Number of concordant cells for middle row center column cell is 80*48 or 3840

The total number of concordant cells is 5894.

The observations that are discordant to a cell in a contingency table are down and left to the cell.   Here is a picture of cells that are discordant to the top left cell.

 4 26 80 5 31

The number of discordant cells from the top left is 4*(80+26+5+31) or 568

Discordant pairs for the other three cells with discordant pairs are:

Middle column center top row 18 *(26+5) or 558.

Left column center row 28 x (5+31) or 1008.

Middle column center row is 80 x 5 or 400.

The total number of discordant cells is 1408.

Our estimate of Tau A for company one is (5894-1408)/(5804+1408) or 0.614.

I’ll leave it to the reader to find Pearson’s chi square and Tau A for the second company.

At this point of time I really appreciate my STATA program.

Go back to this file for more contingency table problems