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Highs and Lows, by Victor Niederhoffer

Whenever I have spent a few days of qualitative thinking, I feel like I am going to knock the hat off the next macro theorist I meet unless I take out the pencil and paper. I have spent too much time looking at the undue pessimism engendered by the weakness last December to look at the moves in the rest of the year after a strong first two weeks, as well as the tendencies to reversal or continuation in performance of the strong and weak performers among individual stocks, other than noting that the best 10 in the S&P 500 the first two weeks of 2006 are Engelhard, Corning, JDS, PMC, Broadway, Apple, Network Appliance, Marathon Oil, Gateway and Janus Capital, and the worst 10 are Maytag, Dupont, Amazon, Navistar, Tyco, Wellpoint, Apollo, McGraw-Hill, Dollar General and Compuware. Okay, it doesn't look too good for the value stocks so far this year, but do the early winners tend to wax or wane?

I wanted to try something new. Something I don't like to look at because it's usually mumbo in the sense that you can't tell it happened until after it happened. And then it's descriptive and not predictive. And it's based on ephemeral rather than stable numbers. Nevertheless, I thought I'd look at The Predictive Properties of New Highs and New Lows, just because it's so alien to anything I have considered or encouraged.

I looked at the runs in new highs and new lows during the six months ended 01/15/06. I thought it apt to start by considering the distribution of the number of runs in higher highs and lower lows, considered separately. For example, the lows in S&P futures for the 10 trading days beginning 12/23/2005 were: 1273.10, 1262.70, 1263.40, 1258.50, 1251.50, 1251.70, 1274.20, 1276.00, 1281.50, 1290.80. Disregarding the days where the lows weren't lower than the previous, there was a run of length 1 ending at 1262.70, and a run of length 2 ending at 1251.50. I performed the same calculations for runs of higher highs. Here is the distribution:

Length of  Runs of  Chance of     Runs of  Chance of  
Run        Highs    Continuation  Lows     Continuation

1          13       60%           12       58%
2          13       38%            5       64%
3           2       70%            6       33%
4           1       75%            3        0%
5           3        0%            0      N/A

I note a tendency for positive runs to stop at 2, for negative runs to stop at 3 or more, and for runs of 1 either way to continue.

Many questions emerge. On the first day of a higher high or higher low, are the expectations positive for fixed intervals following long trades at the close and short trades at the close? What about trades entered when a run of highs or lows is started, and held until the run is broken? How do different periods work out? What are the joint predictive properties of highs and lows when considered together rather than separately? What happens when a long run of highs or lows is broken? How do magnitudes of the changes in lows and highs enter into the picture? What is the best way to incorporate information on the days that there is not a run of length 1 or more in lower lows, i.e. when the current low is higher than the previous? What's the best forecasting equation, using a multiple regression with independent variables the lengths of the runs of higher highs and lower lows, and dependent variable the next day's price change? What moving average autoregressive predictive scheme might work? How do the cycles change when the system sellers uncover a seeming regularity in the past data? Such questions are only the beginning and I believe that certain fish might be eaten for many days with such a study.

James Sogi comments:

Being from Hilo, Hawaii, I was interested in Chair's post when he asked, "What are the joint predictive properties of highs and lows when considered together rather than separately."

The Senator's book Long Term Secrets to Short Term Trading, page 16, describes a three-day "ringed high" as "a high with lower highs on either side," derived from Henry Wheeler Chase's work in the 1930s. I would add Chair s query and add the qualification of lower lows on either side. Looking at this on intraday bars portrays the much maligned head and shoulders formation. December 10-12 SP CME shows the general idea as it bounced off the round number, but did not qualify for the following test. Testing for the last 10 years shows the next several days significantly bearish after 36 occurrences satisfying a variant of the head and shoulders. Stats after 2 days N=36, average -8.60,down 64.8%, T= -3.31. Oddly, the reversed bottom formation is significantly bearish as well.

From "Minister" Kim Zussman:

Looking at intra-day highs and lows of SPY from 1993 non-overlapping
10day periods:

1. If the high of the last 10 days > high of prior 10 days, comparing
return (cl/cl) of next 10days to all non-overlapping 10 day returns:


t-Test: Two-Sample Assuming Equal Variances

10d ret after H>H    all 10d
Mean 1.005338111 1.00436805
Variance 0.000616324 0.000922253
Observations 200 326
Pooled Variance 0.00080607
Hypothesized Mean Difference 0
df 524
t Stat 0.380402873
P(T<=t) one-tail 0.351900267
t Critical one-tail 1.647766763
P(T<=t) two-tail 0.703800534
t Critical two-tail 1.964501432

The 10 day periods following higher highs were higher (sounds like
1969), but the difference wasn't significant.

2. If the low of the last 10 days < low of prior 10 days, comparing
return (cl/cl) of next 10days to all non-overlapping 10 day returns:

t-Test: Two-Sample Assuming Equal Variances

10d ret after L<L    all 10d
Mean 1.003910496 1.00436805
Variance 0.001278941 0.000922253
Observations 130 326
Pooled Variance 0.001023603
Hypothesized Mean Difference 0
df 454
t Stat -0.137871583
P(T<=t) one-tail 0.445201518
t Critical one-tail 1.648216848
P(T<=t) two-tail 0.890403036
t Critical two-tail 1.965202892

Returns were a tad lower but also not significant.

3. What if the high of the last 10d > high of the prior 10d, AND the
low of the last 10d < low of the prior?


t-Test: Two-Sample Assuming Equal Variances

10d ret L<L+H>H    all 10d
Mean 1.006847546 1.00436805
Variance 0.000972999 0.000922253
Observations 35 326
Pooled Variance 0.000927059
Hypothesized Mean Difference 0
df 359
t Stat 0.457824484
P(T<=t) one-tail 0.323677757
t Critical one-tail 1.649109151
P(T<=t) two-tail 0.647355514
t Critical two-tail 1.966593866

These 35 cases were slightly higher than the mean 10d returns, and the
highest of the three, but again the difference wasn't significant.

From Russel Sears:

The problem with working with higher highs, and lower lows, in streaks is that I suspect it changes the distribution to non-normal. It seems to me that: Highs and Lows are a type measures of second moment, "Higher highs" and "lower lows" are a measure of when this is extreme, therefore a type measure of 3 moment and finally streaks would be a type of 4th moment measure.

While going beyond the 2nd moment is intimidating to my scant knowledge of stats. With all this it is not likely that the normal distribution assumption will hold. Rather, I would suggest that you must look at the distribution, to catch these fish.

Further, I would suggest you need more data. and because you are trying to measure skew and such. This is also why, I used the LN return from close to close. While using such long term data may not give to accurate picture of where the fish currently are, perhaps such a survey will tell you what you can expect to catch.

Defining a higher high as a high greater than the prior 10 days high and likewise for the lower low on the SPY Then I looked at the next days return:

All days 3265, next day positive 1694, average 0.03%, stdev 1.10%


Lower Lows   Count Next Day Average Stdev.
in a row     positive % %
1   224 135 0.06 1.29
2   119 74 0.04 1.46
3   46 30 0.04 1.4
4   23 12 0.06 2.25
Higher Highs   Count Next Day Average Stdev.
in a row     positive % %
1   339 156 0.03 0.86
2   180 83 0.02 0.71
3   93 39 0.02 0.73
4   43 19 0.02 0.68

The averages do not appear to be statistically significant. However, the distributions show some interesting frequencies. I will leave it to the reader to piece together the frequency distributions. But for the 2nd day in row of days lower lows total 119:

frequency of "0 to + 1stdev" = 60
frequency of "0 to -1 stdev" = 32

Also, 1 day higher high have fat tails.

Perhaps, the catch is better if the fish are fresh, but the day old fish appear to be small fries.

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