results could be affected by transaction costs of small-cap stocks and nonsynchronous trading for
small-firm stocks.
The second statistical test of independence is the runs test.5 Given a series of price changes,
each price change is either designated a plus (+) if it is an increase in price or a minus (–) if it is
a decrease in price. The result is a set of pluses and minuses as follows: +++–+––++––++. A run
occurs when two consecutive changes are the same; two or more consecutive positive or negative
price changes constitute one run. When the price changes in a different direction, such as when a
negative price change is followed by a positive price change, the run ends and a new run may
begin. To test for independence, you would compare the number of runs for a given series to the
number in a table of expected values for the number of runs that should occur in a random series.
Studies that have examined stock price runs have confirmed the independence of stock price
changes over time. The actual number of runs for stock price series consistently fell into the
range expected for a random series. Therefore, these statistical tests of stocks on the NYSE and
on the OTC market have likewise confirmed the independence of stock price changes over time.
Although short-horizon stock returns have generally supported the weak-form EMH, several
studies that examined price changes for individual transactions on the NYSE found significant
serial correlations. Notably, none of these studies attempted to show that the dependence of
transaction price movements could be used to earn above-average risk-adjusted returns after considering
the trading rule’s substantial transactions costs.
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