6. Runs in the Stock Market 75

Pattern Average z-Value Standard Deviation Units Percent Positive

UU -0.010 -0.21 50.1

DD -0.074 1.56 51.7

UD 0.005 0.10 51.0

DU 0.002 0.04 51.0

UUU -0.035 -0.74 44.9

UUUU -0.020 -0.41 46.2

DDD 0.064 1.34 50.8

DDDD -0.060 -1.26 43.7

Table 5. Simulated z-values for various patterns in weekly

price changes

The results in these tables should be compared with simulated

data from the model in which weekly changes for a given stock are

mutually independent events. For each stock in our set, we used the

observed probabilities of a positive and a negative weekly change in

the stock price to create a simulated set of stock prices. Then, using

the same algorithms as were used above, we calculated the average

z-values for various patterns over our set of stocks. The results are

shown in Table 5.

The simulated data from this model is much different than the actual

data. This supports the observation that both daily and weekly stock

prices are anti-streaky.

In their book [29], Andrew Lo and Craig MacKinlay discuss a

parameter they call the variance ratio. Here is a description of this

concept. Suppose that {Xi}i=0

n

is a sequence of n logarithms of a

stock’s price. The time increment between successive values might be

days, or weeks, or even something as small as minutes. The log price

increments are the values {Xi+1 −

Xi}i=01. n−

A central question that

concerns this sequence of log price increments is whether it can be