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SECOND ORDER EFFECTS IN STOCK PRICES

Turning now to the question of how satisfactory the first order Markov model is for describing the underlying process of price movements, we seek to determine the effect, if any, of ДК(_* on AYt. To this end, we present the joint distribution of ДК*_2, ДК|-1, and Д Yt in Table III. For simplicity in presenta �tion and analysis, we have reduced the price movements to just five classes. We have combined into one class the changes of +2/8 and +3/8, and into another class the changes of —2/8 and —3/8. The arrangement of Table III is designed to permit analysis of the relation between ДК(_г and AYt with AYt-i held constant. Each row in each of the five tables shows the estimated probability distribution of AYt for one combination of Д Yi~i and ДУм. The right hand margin contains the total frequencies upon which the estimate is based.

For example, we learn from Table III-D that after a decline of 1/8 followed by a rise of 1/8, the relative frequency of rises of 1/8 on the next transaction is .108. Similarly, the entry in the second column and fourth row of Table III-E discloses that after a rise of 1/8 followed by a rise of +2/8 or greater, the rela �tive frequency of declines of 1/8 was .190. The approximate significance levels are indicated at the bottom of each table.

Tables III A-III E consist of five 5 by 5 contingency tables. To test indepen �dence between Р” Yt and AYt-i, Anderson and Goodman [l ] propose that X2 be calculated for each table, with the sum of the 5 X2 values (with the appropriate degrees of freedom) serving as the test statistic for the null hypothesis.

By scrutinizing selected differences between proportions (or differences be �tween differences between proportions), we can observe certain interesting properties of the price movements. For example, in Table III-C, a negative change followed by a change of 0/8 indicates that a rise on the next transaction is more likely than a decline, and a positive change followed by a zero change indicates a decline is more likely. That is

These probabilities are almost identical to the comparable first-order prob �abilities as derived from Table II. That is,

In this second-order pattern then, it appears that an issuo behaved just as if the change of 0/8 had not occurred. In other words, the movement in price seems to be governed by the change which occurred before the zero change.

Results similar to this lead us to focus attention on continuations and re �versals. The data of Table РЁРђ-РЁР• may be incorporated into a 4X2 table by deleting the no-change row and no-change column and combining all posi �tive changes and negative changes into two classes as follows.

Notice that a negative change is 1.32 times as likely after two consecutive negative changes as after a positive change followed by a negative change (.337 vs .256). In addition, a positive change is 1.32 times as likely after two positive changes as after a negative change followed by a positive change (.319 vs .241).

Although these tables reemphasize the preponderant tendency for stock price movements to reverse direction, they indicate that the probability of reversal is not constant, but depends on the direction of previous movements. A reversal is more probable after a previous reversal than after a continuation; a continuation is more probable after a previous continuation than after a reversal.

An obvious next step in this analysis is to check whether an advance (de �cline) is more probable after three consecutive advances (declines) than after two advances (declines). For the six stocks in our Dow Jones sample, a con �tinuation is approximately 11/2 times as likely after three consecutive con �tinuations as after two continuations. Furthermore, after four continuations, a subsequent continuation is 1.27 times as likely as after three. Unfortunately, a paucity of data (only 65 occurrences of three consecutive continuations) prevents us from pursuing this line of analysis here.

These results came as a surprise to several readers who saw them in pre �liminary form.3 In the next section, however, we hope to show that they are the natural consequence of the mechanics of trading on the stock exchanges.