72 1. Streaks
the stock market exhibits streaky behavior (or perhaps other forms
of non-randomness).
One obvious way in which stock prices could be streaky concerns
their daily up-and-down motions. Most stocks have more up days
than down days, so one might model them with coins whose success
probabilities are greater than .5. Using the data, one can compute
the observed success probability and then count the number of success
(or failure) streaks of a given length, or of any length. Then one can
compare these observed values with the theoretical values.
Our data set consists of daily prices, over a period of 14 years, of
439 of the stocks that make up the S&P 500 list. These prices have
been adjusted to take into account dividends and stock splits. For this
reason, many of the stock prices are very low near the beginning of
the data set and hence are subject to rounding errors. We typically
get around this problem by using only the part of the data set in
which a stock’s price is above some cutoff value. Also, we throw out
all days for a given stock on which the stock price was unchanged.
Suppose that we want to compare the expected and the observed
number of pairs of consecutive days in which a given stock’s price
went up on both days. If we have n data points, then we can define
the random variable Xi to equal 1 if the i’th and (i + 1)’st price
changes are both positive, and 0 otherwise. Under the assumption
that the signs of the daily price changes are independent events, the
probability that Xi = 1 is just
p2,
where p is the observed long-range
probability of success for that stock. Thus, it is easy to calculate
the expected number of up-up pairs. There are n 1 daily changes
(since there are n data points), so there are n 2 random variables
Xi, each with the same distribution. Thus, the expected number of
up-up pairs is just
(n
2)p2
.
Unfortunately, the Xi’s are not mutually independent. For example,
if X6 = 1 and X8 = 1, then it is not possible for X7 to equal 0.
Nonetheless, the Xi’s are m-independent, for m = 2. The sequence
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