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Dr. Alex Castaldo

5/12/2005
Alfred Cowles on Sequences and Reversals, from Dr. Alex Castaldo

Alfred Cowles (1944):
The author and the late Herbert E. Jones once made an investigation of the evidence as to the element of inertia in stock prices as follows: (6)

In a penny-tossing series there is a probability of one-half that a reversal will occur. If the stock market rises for one hour, day, week, month, or year, is there a probability of one-half that it will decline in the succeeding comparable unit of time?

In a attempt to answer this question, sequences and reversals, as defined in footnote 5, were counted. A study of the ratio of sequences to reversals will probably disclose structure as previously defined, if it exists within the series, and the significance of this structure can be investigated by ordinary statistical methods.

For instance, the probability can be determined that any ratio occurred by chance, from a random population of possible price series. Also, from the frequency distribution of these ratios one can estimate the probabilities of success in forecasting a rise or decline in stock prices. Samples of adequate length, were available, were examined, the intervals between observations being successively 20 minutes, 1 hour, 1 day, 1, 2, and 3 weeks, 1, 2, 3, ... , 11 months and 1, 2, 3, ... , 10 years.

It was found that for every series with intervals between observations of from 20 minutes up to and including 3 years, the sequences outnumbered the reversals.

As a result of various considerations it appeared that a unit of 1 month was the most promising from a forecasting viewpoint. In the case of the 100-year monthly series of common-stock prices from 1836 to 1935, a total of 1200 observations, there were 748 sequences and 450 reversals. That is, the estimated probability was 0.625 that, if the market had risen in any given month, it would rise in the succeeding month or, if it had fallen, that it would continue to decline for another month.

The standard deviation for such a long series constructed by random penny tossing would be 17.3 ; therefore the deviation of 149 from the expected value 599 is in excess of 8 times the standard deviation. The probability of obtaining such a result in a penny-tossing series is infinitesimal. An investigation of the average amount the stock market moved in each month, a consideration of brokerage costs, and determination of the degree of consistency revealed by the data, were used to supplement the information as to the ratio of sequences to reversals. This further analysis indicated an average net gain of 6.7 per cent a year with a probability of a net loss in 1 year out of 3.

(5) The word "sequence" is used here to denote when a rise follows a rise, or a decline a decline. A "reversal" is when a decline follows a rise, or a rise a decline.
(6) "Some A Posteriori Probabilities in Stock Market Action," by Alfred Cowles and Herbert E. Jones, ECONOMETRICA, Vol. 5, July, 1937, pp. 280-294.

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