Feb

3

I took closing prices for the 30 DJIA constituents for each day in January 2011 (20 days) and tested 1st and 2nd digits of closing prices for Benford's Law compliance, which is to say I separately compared the 1st and 2nd digit actual distributions vs. Benford-predicted frequencies of 1st and 2nd digits for Chi-Square significance (p=0.05).

On each trading day of January 2011, the actual distribution of the first digit of the combined 30 Dow constituents was NS different than predicted by BL.

On January 5, 6, and 13, the actual distribution of the second digit of the combined 30 Dow constituents was significantly different than predicted by BL.

Ran the following simple linear regressions:

1. p-value of 1st digit BL chi-square (actual vs. expected, 30 DJIA constituents) vs. next day DIA return;

y = -0.0021x + 0.0018
R² = 0.0014

2. p-value of 2nd digit BL chi-square (actual vs. expected, 30 DJIA constituents) vs. next day DIA return;

y = -0.0049x + 0.0019
R² = 0.0471

3. sum of p-values of 1st and 2nd digit BL chi-square (actual vs. expected, 30 DJIA constituents) vs. next day DIA return;

y = -0.0033x + 0.0035
R² = 0.0336  

4. sum of p-values of 1st and 2nd digit BL chi-square (actual vs. expected, 30 DJIA constituents) vs. next day QQQQ return;

y = -0.0041x + 0.004
R² = 0.0144

5. sum of p-values of 1st and 2nd digit BL chi-square (actual vs. expected, 30 DJIA constituents) vs. next day SPY return.

y = -0.003x + 0.0028
R² = 0.0163
 

Not much to see here, thus far.

A mildly interesting aside is that on the three dates in which the distribution of 2nd digits was significantly different than predicted by BL, the number of which there appeared a superfluity was the same, 4; there appeared three times as many as predicted on Jan 5th and 6th, twice as many on the 13th.

As for those underrepresented in the actual distribution of 2nd digits, on Jan 5th and 6th the 2's disappeared. On Jan 13th (an "unlucky" day, at least superstitiously), no 7s (a "lucky" number, again superstitiously) appeared.


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