May

3

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:

- In these cases, is it legitimate to use a one-tailed test? After all, we suspected UP was followed by UP.

- 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.

- 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


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