A recent working paper by David Price and Nicholas Bloom of Stanford University, Fatih Guvenen of the University of Minnesota, and Jae Song of the Social Security Administration argues that nearly the entire rise in earnings inequality in the U.S. labor market between 1980 and 2012 is accounted for by rising inequality in average wages across firms. In other words, it isn’t that well-paid chief executives are pulling away from their employees, but rather that the salaries at some firms are pulling away from their competitors—even within the same industry.

The working paper, “Firming Up Inequality,” got a lot of attention because it conflicts with research that shows rising inequality is due in large part to skyrocketing compensation by “supermanagers,” a position advanced by Thomas Piketty of the Paris School of Economics in his book “Capital in the 21st Century” and in separate research by Piketty, Emmanuel Saez at the University of California-Berkeley, and Stefanie Stantcheva at Harvard University, in their 2014 *American Economic Journal: Policy * paper “Optimal Taxation of Top Labor Incomes: a Tale of Three Elasticities.” Other analysis of extraordinary CEO pay comes courtesy of the Economic Policy Institute.

My new research note, however, shows that the sampling procedure in “Firming Up Inequality” is biased in two distinct ways. Together, these two statistical biases reduce the scale of rising earnings inequality and hence minimize the very phenomenon the paper seeks to investigate. Importantly, both sources of bias get worse the more inequality grows, which is exactly what happened over the period studied in the paper.

The first problem is that the paper analyzes only a single 1/16^{th} random sample of the distribution of labor earnings in the United States over the full period studied. Normally taking such a large sample of a population wouldn’t bias the outcomes, but it does when the variable of interest is very unequal, as is the case with labor earnings. Analyzing a 1/16^{th} sample biases inferences about inequality because by its very nature a random sampling misses some observations—and the point of inequality is that a small number of observations matter a great deal.

For simplicity, imagine an extreme case with a population of 16 people in which 15 earn nothing and only one person has any earnings. If you select one person at random from this population to estimate the average earnings of all 16 people, then the result will be biased downward (to zero in this case). On average, in 15 out of 16 cases, the estimate of average earnings for the group will be zero, which is too low. Of course, in 1 out of 16 cases—when the highest earner is chosen—the estimate of the average wage of the population will be too high.

Critically, the higher the income of the one person who earns anything, the more biased the result. Continuing with the simple example, the difference between the average wage estimate of zero and the true average wage would be larger.

The second problem is that the paper “Winsorizes,” or caps, the earnings of the top 0.001 percent of earners. The reason why capping top earnings introduces bias is obvious—it eliminates information about the earnings of the very highest earners. The larger share of total earnings they control, the more bias that procedure introduces. The paper does not report the exact number of capped earners, but public data from the U.S. Social Security Administration suggests that in 2013 this would exclude about 1,500 people, who collectively earn at least $40 billion. As a result, the procedure greatly reduces the degree of measured inequality because earnings disparities are so extreme at the very top.

In the note, I conclude that the first source of bias (the small sample) alone is probably not large enough to affect the results, given the current actual level of inequality. But in combination with the second bias from capping top earnings, the results change significantly, especially when “Firming Up Inequality” makes inferences about whether and how much CEO pay contributes to rising inequality.

The most important point here is not biased sampling in this one paper, but rather that inequality inherently introduces a number of methodological concerns that wouldn’t matter if income and wealth were distributed more equally. In “Capital in the 21^{st} Century,” Piketty reports that the share of income of the top one percent was 8 percent in 1979, rising to 20 percent in 2012. If the top 1 percent share were still 8 percent, then the statistics in “Firming Up Inequality” wouldn’t be biased. Because it’s 20 percent, they probably are.