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 avail ble data. Then for each split the sample regression residuals were com outed 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 calculations. 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 j are based on all months during the 1926-60 period for price relatives are available for the security. For the 940 splits the sm m 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 300 months of data are available. Thus in the vast majority 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 Rjt) and the approximate monthly rates of return on the market portfolio (i.e, loge Ly. 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 TIMATED COEFFICIENTS FOR THE DIFFERENT SPLIT SECURITIES Statistic MeanMedian Mean absolute StandardExtreme deviatio Skewness 00000.001 0.004 0.007-0.06, 0.04 Slightly left 08940.880 0.242 0 0.10*, 1.95 Slightly right 0182-0.04,0.91 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 s those reported in the text and certainly support the same conclusions. 1o The sample average or mean absolute deviation of the random variable a is de- fined N where a is the sample mean of the a'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 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 re- sidual return in month t+1 versus estimated residual return in month t have been prepared, along with(c) 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 noted by Fama [3] for the distributions of returns themselves. Fama(following Timken Detroit Axle 0 0.03-0.02-0.0100.01002003004 Regression residuals-Uit FiGure 1 NORMAL PROBABILITY PLOT OF RESIDUALS* left and upper right corners of the graph represent the most extreme sample For clarity, only every tenth point is plotted in the central portion of the
ADJUSTMENT OF STOCK PRICES 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 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 are not "efficient, for most members of the stable Paretian family they provide estimates which are unbiased and consistent. Thus, given our large samples east 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 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 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 e. In this study we do not attempt to determine the effects of splits for ividual companies. Rather we are concerned with whether the process of splitting is in general associated with specifie types of return behavior. T 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 surrounding split dates Some additional definitions. The procedure is as follows: For a given split, 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 ore than one month 0).1 Month 1 is then defined as the month immediately Essentially, this is due to the fact that in computing the variance of a sam 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 fa 13 About a third of the securities in the master file split. About a third of these split more than once