# Runs and Reversals of Length X, from Victor Niederhoffer

May 11, 2007 |

At the founding of American scientific work on market movements in the 1910s, the standard way of looking at non-random properties was to compute the distribution of positive runs and negative runs of exactly 1, 2, 3, 10, et al. This approach when combined with expectations and comovements and lifetimes, a subject generally considered under the subject of survival statistics today (and well covered for example in the highly recommended book Regression Modeling Strategies by Frank Harrell in his chapter Parametric Survival Models) has always held great appeal to me.

Among its other virtues, the approach leads to some ready validations and generalizations of point and figure analysis. It also leads to models of changing probabilities of continuations as the length of the run varies. This has always been partial to me, as some 50 years ago in the JASA, Osborne and I found changing probabilities based on same, and posited this as a general phenomenon.

The one problem I have with all the distribution of runs studies, which are so much more meaningful and useful than the completely ad hoc work on power law distributions and fat tails, which make for great coffee table conversation (and opportunities to feather the nest and appear erudite and garner consulting fee, but in actuality have nothing to do with whether risk is properly priced) in my opinion is that there has been little attention paid to the distributions of reversals of length x. For example, everyone looks at the probability of a rise after two rises, but hardly anyone looks at the probability of a rise after a decline followed by a rise followed by a decline followed by a rise. That's what I would call a positive reversal of length four. Many of these considerations were in my mind yesterday as the Dow rocketed down 150 points and the S&P index rocketed down 21 points after huge continuous rises.

Without providing any meals for a day rather than a lifetime, here's how I've been looking at the Dow.

Runs of 100 against reversals of 100 considering round numbers only:

 Date New Reference Achieved Level Current Position 3/13 12100 12075 Neg reversal 1 3/19 12200 12226 Pos reversal 2 3/21 12400 12447 Pos sequence 2 4/03 12500 12510 Pos sequence 3 4/13 12600 12612 Pos sequence 4 4/16 12700 12720 Pos sequence 5 4/18 12800 12803 Pos sequence 6 4/20 12900 12962 Pos sequence 7 4/25 13000 13089 Pos sequence 8 4/26 13100 13136 Pos sequence 9 5/2 13200 13211 Pos sequence 10 5/7 13300 13312 Pos sequence 11 Present: 13215

There are many ways to analyze such data with pencil and paper or computer. I like a fuzzy augmentation of it, and I like it with smaller increments with such things as the S&P. However, one can say that this positive sequence of length 11 is the longest in history; the durations between the consecutive sequences are smaller and smaller. And one can wonder what a negative reversal of length one when and if it inevitably happen after such a move will augur.

## From Steve Ellison:

I counted moves up and down in the S&P 500 futures since June 13, 2005 between predefined price points at 1% intervals. Presumably the probability of such a small move being up is close to 0.5. For each move, I tabulated the length of the run in progress and whether the move continued the run or reversed it.

Run
length  Stopped   Continued     p
1           45            47           0.46
2           25            21           0.33
3           11            10           0.50
4            7              4            0.27
5            3              1            0.31
6            0              1
7            1              0

## From Michael Cohn:

The Chair recommended Regression Modeling Strategies. Mine just arrived today. I noted that many of the examples are in SPLUS. I recommend Robust Statistics: Theory and Methods, (Wiley Series in Probability and Statistics) by Ricardo A. Maronna, Douglas R. Martin, and Victor J. Yohai. The book comes with a very nice license for SPLUS for 1 year. Kind of an arbitrage of knowledge!