Round Numbers, by Victor Niederhoffer & Alex Castaldo
Today's break through the 1500 level by the S&P index is the first break through of a round number since Nov-17-2006 (1401.2). It is also the seventh consecutive movement upwards through an hundred level without a fall through an hundred level since Nov-01-2002 (900.96).
A chart of the S&P shows a relatively continuous movement up from 100 in 1980 to 1100 in 1998, then a little backing and filling, and then a rise to 1500 (3/22/2000). This was then followed by a precipitous decline back to 800 (7/23/2002).
This raises all sorts of questions about randomness, continuity, tendency for long runs, the drift, and gravitation.
We thought we'd start by looking at a few of the more obvious ones.
SPX Index Daily Data, 100 point box size
| Date | Reference | Close | Dist. frm Ref. | Move |
| 1/2/1980 | 105.76 | |||
| 11/21/1985 | 100 | 201.41 | 101.41 | UP |
| 3/23/1987 | 200 | 301.16 | 101.16 | UP |
| 12/26/1991 | 300 | 404.84 | 104.84 | UP |
| 3/24/1995 | 400 | 500.97 | 100.97 | UP |
| 11/17/1995 | 500 | 600.07 | 100.07 | UP |
| 10/4/1996 | 600 | 701.46 | 101.46 | UP |
| 2/12/1997 | 700 | 802.77 | 102.77 | UP |
| 7/2/1997 | 800 | 904.03 | 104.03 | UP |
| 2/2/1998 | 900 | 1001.27 | 101.27 | UP |
| 3/24/1998 | 1000 | 1105.65 | 105.65 | UP |
| 8/31/1998 | 1100 | 957.28 | -142.72 | DOWN |
| 11/2/1998 | 1000 | 1111.6 | 111.6 | UP |
| 12/21/1998 | 1100 | 1202.84 | 102.84 | UP |
| 3/15/1999 | 1200 | 1307.26 | 107.26 | UP |
| 7/9/1999 | 1300 | 1403.28 | 103.28 | UP |
| 8/9/1999 | 1400 | 1297.8 | -102.2 | DOWN |
| 11/16/1999 | 1300 | 1420.03 | 120.03 | UP |
| 3/22/2000 | 1400 | 1500.64 | 100.64 | UP |
| 4/14/2000 | 1500 | 1356.56 | -143.44 | DOWN |
| 7/14/2000 | 1400 | 1509.98 | 109.98 | UP |
| 10/10/2000 | 1500 | 1387.02 | -112.98 | DOWN |
| 12/20/2000 | 1400 | 1264.74 | -135.26 | DOWN |
| 3/12/2001 | 1300 | 1180.16 | -119.84 | DOWN |
| 5/21/2001 | 1200 | 1200 | 1312.83 | UP |
| 7/6/2001 | 1300 | 1190.59 | -109.41 | DOWN |
| 9/7/2001 | 1200 | 1085.78 | -114.22 | DOWN |
| 9/20/2001 | 1100 | 984.54 | -115.46 | DOWN |
| 10/25/2001 | 1000 | 1100.09 | 100.09 | UP |
| 6/21/2002 | 1100 | 989.14 | -110.86 | DOWN |
| 7/18/2002 | 1000 | 881.56 | -118.44 | DOWN |
| 7/23/2002 | 900 | 797.7 | -102.3 | DOWN |
| 7/30/2002 | 800 | 902.78 | 102.78 | UP |
| 10/7/2002 | 900 | 785.28 | -114.72 | DOWN |
| 11/1/2002 | 800 | 900.96 | 100.96 | UP |
| 6/16/2003 | 900 | 1010.74 | 110.74 | UP |
| 12/29/2003 | 1000 | 1109.48 | 109.48 | UP |
| 12/14/2004 | 1100 | 1203.38 | 103.38 | UP |
| 3/15/2006 | 1200 | 1303.02 | 103.02 | UP |
| 11/17/2006 | 1300 | 1401.2 | 101.2 | UP |
| 5/3/2007 | 1400 | 1501.31 | 101.31 | UP |
| TODAY | 1500 |
We noticed a tendency for UP's to be followed by UP's (and vice versa) so we tested this with a two by two contingency table (previous move listed at the side):
Transition Matrix
| UP | DOWN | |
| UP | 19 | 7 |
| DOWN | 7 | 6 |
Fisher's exact test p=0.19
Although the tendency is there, it does not seem to be statistically significant.
Bruno Ombreux writes:
A classic runs test yields the same p-value in the 2-tailed case. The one-tailed test has obviously half the p-value. That's 0.097, which, as Tukey would say, "is leaning in the right direction". That's not significant but warrants further exploration.
I have a few questions:
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In these cases, is it legitimate to use a one-tailed test? After all, we suspected UP was followed by UP.
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We are testing on the data used to formulate the hypothesis. It is not good practice but in such a long-term study, there is no choice, is there? Not enough data.
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Aren't we wasting our time anyway, UP followed by UP just being an artifact of the positive drift everybody already knows about? What I mean is that the whole exercise is assuredly non-predictive but raises an interesting philosophical question: one-tailed or two-tailed?
Runs Test
Data: S&P Standard Normal = -1.301, p-value = 0.193 alternative hypothesis: two-sided <==> non-random
Data: S&P Standard Normal = -1.301, p-value = 0.097 alternative hypothesis: less <==> trending