Portfolio Sort Versus Regression Evidence

You've found that a characteristic — say, a measure of accruals — seems to predict next month's stock return. Two analysts test it on the same data and reach different verdicts. One sorts stocks into ten buckets by the characteristic and reports the spread between top and bottom bucket; the other runs a monthly cross-sectional regression of returns on the characteristic and reports the slope. If the effect were perfectly linear and evenly spread across all stocks, both methods would agree. In practice they often don't, and the disagreement is not a bug — it tells you something about the shape of the relationship you'd otherwise miss.

An analogy: judging a race by finish order versus by stopwatch

Imagine grading runners two ways. Method one splits them into fast, medium and slow groups and compares average finish times. Method two fits a straight line through everyone's time against hours trained, and reports the slope. If training helps every runner by roughly the same amount per hour, both agree. But if training only helps beginners and does nothing for elites near their limit, the grouped comparison shows a real difference, while the straight-line slope gets diluted by the elites for whom the line is flat. Neither method is "wrong"; they see different things, and a good analyst runs both.

The two estimators, one symbol at a time

Let $r_{i,t}$ be stock $i$'s return in month $t$ and $x_{i,t}$ its characteristic value, both measured at the start of the month. The portfolio sort ranks stocks by $x_{i,t}$ into $K$ groups (deciles are typical) and computes each group's average return; the reported test statistic is usually the spread $\bar r_{K,t} - \bar r_{1,t}$ between the top and bottom group, averaged over time and tested with a $t$-statistic. The Fama-MacBeth regression instead fits, month by month,

$$r_{i,t} = a_t + b_t x_{i,t} + \varepsilon_{i,t},$$

where $a_t$ is that month's intercept, $b_t$ is that month's slope (the "price" the market paid for one unit of the characteristic), and $\varepsilon_{i,t}$ is the leftover return unexplained by the characteristic. The slopes $b_t$ are averaged over time and tested against zero. In plain English: the sort compares the extremes directly and says nothing about the middle, while the regression forces a single straight line through every stock, including those in the middle.

This matters because the regression slope is heavily influenced by outliers — one extreme $x$ value can tilt the whole line — whereas a sort's decile spread only cares which stocks land in the top and bottom groups, making it more robust to outliers and able to reveal a relationship that's strong at the extremes but flat in the middle, a "kinked" pattern the straight line washes out.

Worked example 1: a kinked relationship

Suppose true monthly returns depend on characteristic $x$ (scaled 1–10) as: stocks with $x \le 5$ earn a flat 0.4%, and stocks with $x > 5$ earn $0.4\% + 0.3\%\times(x-5)$. A decile sort shows deciles 1–5 all near 0.4% and deciles 6–10 rising to $0.4\% + 0.3 \times 5 = 1.9\%$ — a clear top-minus-bottom spread near 1.5%, correctly flagging the effect as concentrated at the top. A regression forced straight across all ten deciles instead estimates an average slope of roughly $b \approx 0.15\%$ per unit of $x$ — real, but understated, since half the line bends through a flat region contributing no signal.

Worked example 2: an outlier problem

Now suppose 98 stocks have $x$ between 1 and 10 with no relationship to return (average 0.5% flat), but 2 stocks have extreme $x=500$ (a data error) and returns of 3%. A regression slope is pulled toward these two points — with such leverage, $b_t$ can come out significantly positive purely from the outliers, producing a spuriously significant Fama-MacBeth $t$-statistic. A decile sort instead places those two stocks in the top decile alongside 8 ordinary ones; their average is diluted, and the top-minus-bottom spread stays near zero, correctly showing no real effect.

What this means in practice

Neither method dominates; they answer different questions. A regression gives a single number that's easy to compare across studies and lets you control for other characteristics simultaneously, which a simple sort can't do without double-sorting. A sort is more robust to outliers and reveals nonlinearity, but only within the buckets you chose, and adding controls means moving to double or triple sorts, which get sparse fast. The standard practice in empirical asset pricing is to run both: report decile spreads to show the shape of the effect, and a Fama-MacBeth regression (often with control variables added) to show it survives controlling for known factors. Disagreement between the two is itself informative — it usually means the effect is concentrated in the tails or driven by a handful of extreme observations, not spread evenly across the whole cross-section.