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reversals in dow stocks

Our purpose here is to examine the correspondence between the movements of ticker prices and the predictions of the random walk hypothesis. The data consist of the complete set of ticker prices of six of the first seven stocks in the Dow Jones Industrial Averages for the twenty-two trading days of October, 1964. (See Table I.)2 Although these six stocks represent 0.5% of the average number of issues traded on a given day, they account for some 2.5% of all transactions during the period. However, the additional data reported in sec �tion 5 indicate that qualitatively the results apply to almost all traded issues.

Preliminary examination of a small segment of the entire sample suggests some interesting properties. In Figure 1, which contains data for Allied Chemi �cal Corporation for the fourth day of the sample period, 29 of the 33 changes in price were less than 1/4 of a point away from the preceding transaction. This is consistent with Securities and Exchange Commission reports that 85% to 95% of all transactions in active stocks on the Exchange are less than 1/4 of a point removed from each other (15, p. 378).

Though the number of transactions is small, Figure 1 suggests another phe �nomenon which has been mentioned in the literature, the "stickiness of even eighths." All sixteen of the zero changes occurred at the even eighths, even though there were three odd and two even eighth positions in the total sample.

Finally we observe a striking feature which pervades the entire sample of price movements analyzed in this study. Most of the non-zero changes in price were opposite in direction to the preceding non-zero change: twelve in the op �posite direction versus four in the same direction. When the signs of two non �zero consecutive changes are unlike each other, this pattern will be named a reversal, and when they are in the same direction, the pattern will be called a continuation. Thus, we have 12 reversals and four continuations in the price movements of Allied Chemical on October 6, 1964. Of these sixteen, only the 1/8 reversals and continuations are marked on Figure 1 (see section 5).

Considering the entire sample of transactions for six stocks during October 1964, we present the joint frequency distribution ot consecutive pairs of changes in Table I and the estimated transition matrix derived from these changes in Table II. In row 5 of Table I, for example, the figure 2156 in the right margin is the total number of rises of 1/8 and the figure 709 is the number of these 2156 rises that were followed by a decline of 1/8. Thus, in Table И the ratio 709/2156 = 0.329 appears in row 5 indicating the fraction of all rises of 1/8 that were followed by a decline of 1/8. In standard notation,

The tendency for stock price movements to reverse direction shows up in Table II as negative correlation between Д7.-1 and ДУ,. Notice how the entries in the diagonal from lower left to upper right are all, except for the com �mon one, larger than the corresponding entries in the diagonal from upper left to lower right. If the changes were truly independent—as assumed in a random walk model—both diagonals should be the same within the limits of random error. In addition, there should be no significant variation in the con �ditional distribution of &Yt over the tabulated values of AK �_i. That is, all these conditional distributions should be the same as the marginal distribu �tion, within the limits of random error.

A formal test for independence in transition matrices has been proposed by Anderson and Goodman [l]. Applied to Table II, this test has exactly the same form as a chi-square test for independence in a 7X7 contingency table. The chances of finding deviations from independence at least as large as those observed are approximated by Р {С…СЉ> 1147.9| 36), an exceedingly small number. (The 99.99999999th percentile of the x2 statistic with 36 d.f. is 106.) The varia �tions in Table II cannot reasonably be attributed to chance.

To highlight this tendency toward reversal, we have abridged Table I by eliminating the no-change row and the no-change column and then consolidat �ing the remaining entries into four classes; the result is a 2X2 table as follows:

It is apparent that two changes in opposite directions occur approximately three times as often as changes in the same direction.

A consequence of independence ot successive price changes is that all subsets of price changes have the same frequency distribution. For example, the price changes following a change of —3/8 would have the same distribution (hence expected value) as the price changes after a +3/8 change. But this is not true.

From row 1 of Table II we can see that after a change of —3/8,14.3% of the next changes were declines of 1/8,19.0% were advances of 2/8, and 9.5% were rises of 3/8. In other words, after a change of —3/8, the expected value of the next transaction is 0.67 eighths of a point, i.e., the sum of (0.143)( —1/8) + (0.143) (1 /8) + (0.190) (2/8) + (0.095) (3/8).

Similarly, we have calculated the average price change at transaction t corresponding to each of the other six changes at transaction t— 1. These aver �age changes are given below in eighths:

As measured by a Kruskal-Wallis one-way analysis of variance, or otherwise, the tendency for the average to decrease as Р”/Vi increases is obvious.

Finally, the data suggest that large changes tend to be followed by large changes. For example, 21.2% of the changes of 2/8 or more were followed by changes of 2/8 or more in absolute value, as were 22% of the changes of —2/8 or less. After moves of —1/8, 0.8, and 1/8, the percentages of subsequent changes which were at least 2/8 in absolute value were respectively 5.4%, 4.8% and 4.9%. These results may be exaggerated slightly by the possibility that a large change, followed by a large change in the opposite direction, may be a printing error on the ticker. But this eventuality is very unlikely because the degree of accuracy of the ticker is very high. For example, Lcffler and Farwell report that on a day in which 30,000 transactions occur there is an average of only 10 printing errors on the ticker (7, p. 158).