We have focused throughout on the six-month/six-month strategy, because it has become a standard benchmark for evaluating momentum strategies. But of course this is somewhat arbitrary. To provide more information, Figure 2 plots cumulative returns in event time. In so doing, we use the methodology of Table 6—we assign stocks to performance categories based on six months’ prior beta-adjusted returns, and do an independent sort based on the analyst-coverage residuals from Model 1. We then track cumulative beta-adjusted returns on a month-by-month basis, out to 36 months.
In Table 9, we do everything else the same as in Table 4, except that we skip a month between the six-month ranking period and the six-month investment holding period. Jegadeesh and Titman (1993) suggest this approach as a way to check that neither bid-ask bounce nor any other high-frequency phenomenon is coloring any of the results. As it turns out, nothing changes—the numbers are almost identical to those in Table 4.
In Table 8. we again use raw returns, and this time generate the coverage residuals from Model 8 of Table 2. which includes the turnover variables. But before turning to the numbers, we should point out that it is far from clear that it makes economic sense to control for turnover in this way. As noted above, it may well be that the positive correlation of coverage and turnover reflects causality running from the former to the latter: high-coverage stocks have lower adverse-selection costs of trading, and hence attract more trading volume (Brennan and Subrahmanyam 1995). To the extent that this story is true, we should not use Model 8 to generate our residuals—we will just be reducing the exogenous variation in coverage by regressing it on a noisy proxy for itself, thereby weakening the power of our tests.
In Tables 6-9, we redo the analysis of Table 4, using a variety of alternative specifications. First, in Table 6, we depart from Jegadeesh and Titman’s (1993) focus on raw returns. Given that our economic story is all about firm-specific information, it seems sensible to focus on returns adjusted for any market-wide factors. In Table 6 all the returns—both in the pre-formation and post-formation periods—are market-model adjusted, using individual stock betas. As can be seen, the use of this beta adjustment does not significantly alter our central results. The P3-P1 momentum measure for the entire sample actually rises somewhat, to 1.20% per month (it was 0.94% in Table 4), and the difference between the low-coverage SUB1 and the high-coverage SUB3 also goes up a bit, to 0.49%, with a t-stat of 4.04 (it was 0,42% in Table 4). Finally, the LAST strategy, which is long P1/SUB3 and short P1/SUB1, continues to do well—though not quite as well as before-generating an average beta-adjusted return of 0.50% per month (t-stat = 3.64).
Overall, the size disaggregation effort in Table 5 lends further credence to our interpretation of the evidence. It makes it clear that the earlier numbers in Table 4 are not an artifact of imperfect size matching in the full sample. And it is comforting to know that analyst coverage has more of an impact on momentum in precisely those parts of the size distribution where one a priori suspects that gradual information diffusion is likely to be important and where momentum effects are most pronounced to begin with.
Table 5 also helps put into perspective the extent to which firm size and residual coverage might each be capturing something related to the phenomenon of gradual information flow. On the one hand, it is natural to focus most of the attention on residual coverage as a proxy for this phenomenon-it makes for a cleaner test of our hypothesis because it is less likely than size to be bringing in other confounding factors. But in gauging the quantitative significance of the results, it is important to recognize that, if we hold size fixed, we cannot hope to capture the full magnitude of any gradual-information-flow effect.
Two-Way Cuts on Size and Residual Coverage
In Table 5, we disaggregate the analysis of Table 4 by size. The methodology is exactly the same except we look at four separate subsamples. The first includes all stocks between the 20th and 40th NYSE/AMEX percentiles, the second includes those between the 40th and 60th percentiles, and so forth. We have two motivations for doing this disaggregation. First, as a matter of economics, it seems reasonable to conjecture that the marginal importance of coverage will be greater in the smaller stocks, which have fewer analysts on average, and are probably less well-researched in other ways. Second, as a matter of methodology, this approach should give us better size matches across residual coverage classes, since we now will be running the analyst coverage regressions separately for each size-based subsample. Compared to our earlier approach, this is like allowing the analyst-size relationship to be piecewise linear.
We wiil attempt to remedy this deficiency shortly, in Table 5. For the moment, it suffices to say that the imperfect size matching in Table 4 does not color any of the conclusions.
Turning to the returns numbers, two patterns emerge that hold up throughout our subsequent analysis. First, as predicted by the theory, there is more momentum in stocks with low residual coverage. The P3-P1 momentum measure is 1.13% per month in the low-residual-coverage subsample SUB1, and only 0.72% per month in the high-residual-coverage subsample SUB3. The difference of 0.42% between SUB1 and SUB3 in this regard is highly statistically significant, with a t-stat of 3.50. Moreover, the economic magnitude is clearly important–momentum profits are roughly 60% higher in SUB1 than in SUB3.
Cuts on Residual Analyst Coverage
Next we turn to the cuts based on residual analyst coverage. Here, and in everything else that follows, we exclude all stocks that are below the 20th percentile NYSE/AMEX breakpoint. Again, this is because the vast majority of these small stocks simply never have any analysts, so there is no variation to work with. Within this truncated universe, we create three subsamples based on residual analyst coverage, with the residuals coming from month-by-month cross-sectional regressions of log(l 4-ANALYSTS) on log(SIZE) and a NASDAQ dummy, just as in Model 1 of Table 2.
Figure 1 illustrates the results, plotting the relationship between size and the magnitude of the momentum effect. As can be seen, there is a pronounced, inverted U-shape. In the very smallest stocks (which are tiny, with a mean market capitalization of $7 million) momentum is actually negative. By the second size decile, momentum profits are significantly positive, and they reach a peak in the third size decile, where market capitalization averages about $45 million and where the profits are a striking 1.43% per month (t-stat = 6.66), which is almost three times the value for the sample as a whole. After the third size decile, momentum profits decline monotonically, to the point where they are essentially zero in the largest stocks.
Cuts on Raw Size
We begin our analysis of momentum strategies in Table 3. In this table, unlike in any of those that come later, we look at the entire universe of stocks, without dropping those below the 20th NYSE/AMEX percentile. In so doing, we follow the methodology of Jegadeesh and Titman (1993) closely in many respects. In particular, we focus on their preferred six-month/six-month strategy, we couch everything in terms of raw returns, and we equal-weight these returns. But there are also three noteworthy differences. First, our sample period of 1980-1996 is more recent. Second, we do not exclude NASDAQ stocks. And third, our measure of momentum differs from theirs. They sort stocks into ten deciles according to past performance, and then measure the return differential of the most extreme deciles—which they denote by P10-P1. In contrast, we place less emphasis on the tails of the performance distribution. We sort our sample into only three parts based on past performance: PI, which includes the worst-performing 30%; P2 which includes the middle 40%; and P3, which includes the best-performing 30%. Our basic measure of momentum is then P3-P1.