ADJUSTMENT OF STOCK PRICES residuals are apparently non-zero. The exclusion procedure was as follows First, the parameters of (1)were estimated for each security using all avail able data. Then for each split the sample regression residuals were com- puted for a number of months preceding and following the split. When the number of positive residuals in any month differed substantially from the number of negative residuals, that month was excluded from subsequent alculations This criterion caused exclusion of fifteen months before the split for all securities and fifteen months after the split for splits followed by dividend decreases° Aside from these exclusions, however, the least squares estimates Bi for security 3 are based on all months during the 1926-60 period for price relatives are available for the security. For the 940 splits the smallest effective sample size is 14 monthly observations. In only 46 cases is the ample size less than 100 months, and for about 60 per cent of the splits more than 800 months of data are available. Thus in the vast ma jority of cases the samples used in estimating a and P in(1)are quite large Table 1 provides summary descriptions of the frequency distributions of the estimated values of ai, Bi, and ri, where ri is the correlation between monthly rates of return on security j(i.e, loge Rit)and the approximate monthly rates of return on the market portfolio (i.e., loge Lt). The table indicates that there are indeed fairly strong relationships between the market and monthly returns on individual securities; the mean value of the i, is 0.682 with an average absolute deviation of 0. 106 about the mean o TABLE 1 SUMMARY OF FREQUENCY DISTRIBUTIONS OF ESTIMATED COEFFICIENTS FOR THE DIFFERENT SPLIT SECURITIES Statistic Mean Median Mean absolute Standard reme 0.007-0.06, 0.04 Slightly left 08940.880 0.305-0.10*, 1.95 Slightly right 0. 132-0.04*, 0.91 Slightly left Only negative value in distribution. A Moreover, the estimates of equation (1) for the different securities conform irly well to the assumptions of the linear regression model. For example 9 admittedly the exclusion criterion is arbitrary. As a check, however the analysis of regression residuals discussed later in the paper has been carried out using the regression estimates in which no data are excluded The results were much the same as those reported in the text and certainly support the same conclusions 10 The sample average or mean absolute deviation of the random variable a is de fined as N where I is the sample mean of the as and N is the sample size
ADJUSTMENT OF STOCK PRICES 5 residuals are apparently non-zero. The exclusion procedure was as follows: First, the parameters of (1) were estimated for each security using all available data. Then for each split the sample regression residuals were computed for a number of months preceding and following the split. When the number of positive residuals in any month differed substantially from the number of negative residuals, that month was excluded from subsequent calcu'lations. This criterion caused exclusion of fifteen months before the split for a11 securities and fifteen months after the split for splits followed by dividend decreases9. Aside from these exclusions, however, the least squares estimates 8j and jj for security j are based on all months during the 1926-60 period for which price relatives are available for the security. For the 940 splits the smallest effective sample size is 14 monthly observations. In only 46 cases is the sample size less than 100 months, and for about 60 per cent of the splits more than 800 months of data are available. Thus in the vast majority of cases the samples used in estimating cu and P in (1)are quite large. Table f provides summary descriptions of the frequency distributions of the estimated values of aj, pi, and ri, where rj is the correlation between monthly rates of return on security j (i.e., log, Rjt) and the approximate monthly rates of return on the market portfolio (i.e., log, Lt). The table indicates that there are indeed fairly strong relationships between the market and monthly returns on individual securities; the mean value of the qi is 0.632 with an average absolute deviation of 0.106 about the mean.1° TABLE 1 SUMMARY OF FREQUENCY DISTRIBUTIONS OF ESTIMATED COEFFICIENTS FOR THE DIFFERENT SPLIT SECURITIES Statistic Mean PI'Iedian Mean absolute Standard I / deviation deviation Extreme Skewness values 1--1 a 0.000 0.001 0.004 0.007 -0.06, 0.04 Slightly left ,: 0.894 0.880 0.242 0.305 -0.10*, 1.95 Slightly right T 0.632 0.655 0.106 0.132 -0.04*, 0.91 Slightly left * Only negative value in distribution. Moreover, the estimates of equation (1)for the different securities conform fairly well to the assumptions of the linear regression model. For example, 9 Admittedly the exclusion criterion is arbitrary. As a check, however, the analysis of regression residuals discussed later in the paper has been carried out using the regression estimates in which no data are excluded. The results were much the same as those reported in the text and certainly support the same conclusions. 10 The sample average or mean absolute deviation of the random variable x is defined as where Z is the sample mean of the x's and N is the sample size
FAMA, FISHER, JENSEN AND ROLL the first order auto-correlation coefficient of the estimated residuals from (1) has been computed for every twentieth split in the sample(ordered al phabetically by security). The mean(and median) value of the forty-seven oefficients is-0.10, which suggests that serial dependence in the residuals not a serious problem For these same forty-seven splits scatter diagrams of (a) monthly security return versus market return, and(b)estimated re sidual return in month t+1 versus estimated residual return in month t have been prepared, along with(e) normal probability graphs of estimated residual returns. The scatter diagrams for the individual securities support very well the regression assumptions of linearity, homoscedasticity, and serial independence. It is important to note, however, that the data do not conform well to the normal, or Gaussian linear regression model. In particular, the distributions of the estimated residuals have much longer tails than the Gaussian. The typical normal probability graph of residuals looks much like the one shown for Timken Detroit Axle in Figure 1. The departures from normality in the distributions of regression residuals are of the same sort as those note by Fama [3] for the distributions of returns themselves. Fama(following Timken Detroit Axle -0.030.02-0.010 010020.03004 FIGURE 1 NORMAL PROBABILITY PLOT OF RESIDUALS* of the graph rer For clarity, only every tenth point is plotted in the central portion
6 FAMA, FISHER, JENSEN AND ROLL the first order auto-correlation coefficient of the estimated residuals from (1) has been computed for every twentieth split in the sample (ordered a1- phsbetically by security). The mean (and median) value of the forty-seven coefficients is -0.10, which suggests that serial dependence in the residuals is not a serious problem. For these same forty-seven splits scatter diagrams of (a) monthly security return versus market return, and (b) estimated residual return in month t + l versus estimated residual return in month t have been prepared, along with (c) normal probability graphs of estimated residua1 returns. The scatter diagrams for the individual securities support very well the regression assumptions of linearity, homoscedasticity, and serial independence. It is important to note, however, that the data do not conform well to the normal, or Gaussian linear regression model. In particular, the distributions of the estimated residuals have much longer tails than the Gaussian. The typical normal probability graph of residuals looks much like the one shown for Timken Detroit Axle in Figure 1. The departures from normality in the distributions of regression residuals are of the same sort as those noted by Fama [3] for the distributions of returns themselves. Fama (following * Regression residuals- Ujt FIGURE1 NORMAL PROBABILITY PLOT OF RESIDUALS* * The lower left and upper right corners of the graph represent the most extreme sample points. For clarity, only every tenth point is plotted in the central portion of the figure
ADJUSTMENT OF STOCK PRICES Mandelbrot [12])argues that distributions of returns are well approximate by the non-Gaussian (i.e. infinite variance) members of the stable Paretian family. If the stable non-Gaussian distributions also provide a good descrip- tion of the residuals in (1), then, at first glance, the least squares regression model would seem inappropriate Wise [19] has shown, however, that although least square estimates "efficient, "for most members of the stable Paretian family they provide estimates which are unbiased and consistent. Thus, given our large samples least squares regression is not completely inappropriate. In deference to the stable Paretian model, however, in measuring variability we rely primarily on the mean absolute deviation rather than the variance or the standard deviation. The mean absolute deviation is used since, for long-tailed distri butions, its sampling behavior is less erratic than that of the variance or the standard deviation! In sum we find that regressions of security returns on market returns over time are a satisfactory method for abstracting from the effects of gen eral market conditions on the monthly rates of return on individual secu rities. We must point out, however, that although (1)stands up fairly well to the assumptions of the linear regression model, it is certainly a gros over-simplified model of price formation; general market conditions alone do not determine the returns on an individual security. In (1)the effects of these"omitted variables"are impounded into the disturbance term u. In particular, if a stock split is associated with abnormal behavior in returns during months surrounding the split date, this behavior should be reflected in the estimated regression residuals of the security for these months. The re mainder of our analysis will concentrate on examining the behavior of the estimated residuals of split securities in the months surrounding the splits 3. "EFFECTS OF SPLITS ON RETURNS: EMPIRICAL RESULTS In this study we do not attempt to determine the effects of splits for in- dividual companies. Rather we are concerned with whether the process of splitting is in general associated with specific types of return behavior. To abstract from the eccentricities of specific cases we can rely on the simple process of averaging; we shall therefore concentrate attention on the behavior of cross-sectional averages of estimated regression residuals in the months b a. Some additional definitions. The procedure is as follows: For a given olit, define month 0 as the month in which the effective date of a split occurs. (Thus month 0 is not the same chronological date for all securities, and indeed some securities have been split more than once and hence have more than one month 0). 12 Month 1 is then defined as the month immediately Essentially, this is due to the fact that in computing the variance of a sample large deviations are weighted more heavily than in computing the mean absolute deviation, For empirical evidence concerning the reliability of the mean absolute deviation relative to the variance or standard deviation see Fama [3,(94-8) split more than once
ADJUSTMENT OF STOCK PRICES 7 Mandelbrot [12]) argues that distributions of returns are well approximated by the non-Gaussian (i.e., infinite variance) members of the stable Paretian family. If the stable non-Gaussian distributions also provide a good description of the residuals in (I), then, at first glance, the least squares regression model would seem inappropriate. Wise [I91 has shown, however, that although least square estimates are not "efficient," for most members of the stable Paretian family they provide estimates which are unbiased and consistent. Thus, given our large samples, least squares regression is not completely inappropriate. In deference to the stable Paretian model, however, in measuring variability we rely primarily on the mean absolute deviation rather than the variance or the standard deviation. The mean absolute deviation is used since, for long-tailed distributions, its sampling behavior is less erratic than that of the variance or the standard deviation1'. In sum we find that regressions of security returns on market returns over time are a satisfactory method for abstracting from the effects of genera,l market conditions on the monthly rates of return on individual securities. We must point out, however, that although (1)stands up fair!y well to the assumptions of the linear regression model, it is certainly a grossly over-simplified model of price formation; general market conditions alone do not determine the returns on an individual security. In (1)the effects of these "omitted variables" are impounded into the disturbance term u. In particular, if a stock split is associated with abnormal behavior in returns during months surrounding the split date, this behavior should be reflected in the estimated regression residuals of the security for these months. The remainder of our analysis will concentrate on examining the behavior of the estimated residuals of split securities in the months surrounding the splits. 3. "EFFECTS" OF SPLITS ON RETURNS: EMPIRICAL RESULTS In this study we do not attempt to determine the effects of splits for individual companies. Rather we are concerned with whether the process of splitting is in general associated with specific types of return behavior. To abstract from the eccentricities of specific cases we can rely on the simple process of averaging; we shall therefore concentrate attention on the behavior of cross-sectional averages of estimated regression residuals in the months surrounding split dates. a. Some additional definitions. The procedure is as follows: For a given split, define month 0 as the inonth in which the effective date of a split occurs. (Thus month 0 is not the same chronological date for all securities, and indeed some securities have been split more than once and hence have more than one month 0).12 Month 1is then defined as the month immediately " Essentially, this is due to the fact that in computing the variance of a sample, large deviations are weighted more heavily than in computing the mean absolute deviation. For empirical evidence concerning the reliability of the mean absolute deviation relative to the variance or standard deviation see Fama [3, (94-8)]. '2 About a third of the securities in the master file split. About a third of these split more than once
FAMA, FISHER, JENSEN AND ROLL following the split month, while month-1 is the month preceding, etc. Now define the average residual for month m(where m is always measured rela- tive to the split month)as there Aim is the sample regression residual for security j in month m and nm is the number of splits for which data are available in month m. 3 Our principal tests will involve examining the behavior of um for m in the in- terval-29 s m s 30, i.e., for the sixty months surrounding the split month. We shall also be interested in examining the cumulative effects of abnormal return behavior in months surrounding the split month. Thus we define the cumulative average residual Um as The average residual um can be interpreted as the average deviation (in month m relative to the split month) of the returns of split stocks from their normal relationships with the market. Similarly, the cumulative average residual Um can be interpreted as the cumulative deviation(from month-29 to month it shows the cumulative effects of the wanderings of the re- turns of split stocks from their normal relationships to market movements Since the hypothesis about the effects of splits on returns expounded in Section 2 centers on the dividend behavior of split shares, in some of the tests to follow we examine separately splits that are associated with increased dividends and splits that are associated with decreased dividends. In addi- tion, in order to abstract from general changes in dividends across the market, "increased and "decreased"dividends will be measured relative to the average dividends paid by all securities on the New york Stock Exchange during the relevant time periods. The dividends are classified as follows: Define the dividend change ratio as total dividends (per equivalent unsplit share) paid in the twelve months after the split, divided by total dividends paid during the twelve months before the split. 4 Dividend"increases"are then defined as cases where the dividend change ratio of the split stock is greater than the ratio for the Exchange as a whole while dividend"decreases include cases of relative dividend decline. s We then define u+, u- and U+ 13 Since we do not consider splits of c that were not on the New York Stock Exchange for at least a year be a year after a split, nm will be 940 for -11 s m s 12. For other months, a dividend day the security trades ex-dividend on the excha 15 When dividend"increase"and "decrease"are defined relative to the market it turns out that dividends were never unchanged. " That is, the dividend change ratios of split securities are never identical to the corresponding ratios for the ex (Continued on neat page
8 FAMA, FISHER, JENSEN AND ROLL following the split month, while month -1 is the month preceding, etc. Now define the average residual for month m (where m is always measured relative to the split month) as where iij, is the sample regression residual for security j in month m and n,, is the number of splits for which data are available in month m.13 Our principal tests will involve examining the behavior of u, for m in the interval -29 5 m 5 30, i.e., for the sixty months surrounding the split month. We shall also be interested in examining the cumulative effects of abnormal return behavior in months surrounding the split month. Thus we define the cumulative average residual U, as The average residual u, can be interpreted as the average deviation (in month m relative to the split month) of the returns of split stocks from their normal relationships with the market. Similarly, the cumulative average residual Urn can be interpreted as the cumulative deviation (from month -29 to month m); it shows the cumulative effects of the wanderings of the returns of split stocks from their normal relationships to market movements. Since the hypothesis about the effects of splits on returns expounded in Section 2 centers on the dividend behavior of split shares, in some of the tests to follow we examine separately splits that are associated with increased dividends and splits that are associated with decreased dividends. In addition, in order to abstract from general changes in dividends across the market, "increased" and "decreased" dividends will be measured relative to the average dividends paid by all securities on the New York Stock Exchange during the relevant time periods. The dividends are classified as follows: Define the dividend change ratio as total dividends (per equivalent unsplit share) paid in the twelve months after the split, divided by total dividends paid during the twelve months before the split.I4 Dividend "increases" are then defined as cases where the dividend change ratio of the split stock is greater than the ratio for the Exchange as a whole, while dividend "decreases" include cases of relative dividend decline.15 We then define ud, u; and Uk, '3 Since we do not consider splits of companies that were not on the New York Stock Exchange for at least a year before and a year after a split, rzm will be 940 for -11 5 m 5 12. For other months, however, nm. < 940. '4 A dividend is considered "paid" on the first day the security trades ex-dividend on the Exchange. '5 When dividend "increase" and "decrease" are defined relative to the market, it turns out that dividends were never "unchanged." That is, the dividend change ratios of split securities are never identical to the corresponding ratios for the Exchange as a whole. (Continued on next page)