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
Close<Prior Low
|
Stock
Close Between Prior Low and High
|
Stock
Close>Prior 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
Close<Prior Low
|
Stock
Between Prior Low and High
|
Stock
Close>Prior 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
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