Pay equity regression

Why log. A grade step at Apex Digital and a grade step at Apex Retail are different sums of money but similar percentages, and a model in percentages is one model rather than four. On the log scale a coefficient of -0.04 on sex reads as "about 4 percent less, other things in the model held equal", and the conversion back is one minus e to the coefficient.

Why least squares. It is the fit that leaves the smallest sum of squared residuals, it has a closed form that numpy solves in one call, and its coefficients have the interpretation everybody wants: the change in log pay for a one-unit change in the variable, holding the others in the model fixed. The statistics course teaches it from scratch; here it is a tool, and the tool's limits matter more than its mechanics.

What the coefficient on sex means. It is the average gap between women and men who share the same grade and tenure values. It is not the raw gap, which is bigger whenever women sit lower on the ladder, and it is not "discrimination", because a coefficient is a difference that the other variables in the model did not account for, and the model only contains what somebody chose to put in it. Change the variables and the coefficient changes. Part 3 is about that choice.

Three habits. Print the unit. A coefficient on tenure is per year; a coefficient on grade is per band, and only if the bands are evenly spaced does one number describe the ladder, which is why the build-along treats grade as four separate rungs in the homework. Print the raw and the adjusted gap side by side, because the distance between them is the part the measured variables absorbed, and that distance is the finding, not the adjusted number alone. Print the residual spread. A within-band coefficient of 4 percent with a residual standard deviation of 12 percent means the individual variation dwarfs the average difference, and any single pair of colleagues is a poor test of the average.

Fit log pay on grade and tenure for men, giving coefficients b_m, and for women, giving b_w. Least squares passes through the means, so each group's mean log pay is its mean variables times its coefficients. The gap between the means is then (x_m minus x_w) times b_m, plus x_w times (b_m minus b_w). The first term prices the difference in average grade and tenure at men's coefficients; the second applies the difference in coefficients to women's averages. They add to the gap exactly, and an assert in the code says so.

The hidden choice is which group's coefficients price the first term. Priced at women's coefficients, the same data give a different explained share, and the difference between the two versions is a measure of how far the two pay structures diverge. Report both, or report one and say which; a decomposition that names only "explained 70 percent" without the reference group is not reproducible. On the constructed table the two versions sit close, because the generator used one structure with a shift; on a real table they may not.

Goldin 2014 (American Economic Review 104(4) 1091-1119) argued that the remaining gap is driven by occupations that reward long and particular hours: jobs where being available at specific times pays disproportionately, and where the penalty for flexibility falls more often on women. Hours is therefore a variable that belongs in a pay model when it is measured, and a variable the UK filing cannot see at all. Nor can the filing see occupation, tenure or part-time status. That is why the filing's 2.3-point residual and a regression's coefficient on sex are not the same number and should never be compared as if they were: the first is a gap the pyramid did not account for; the second is a gap that a specific list of variables did not account for. Neither is a measure of intent, and both move when the list changes.

The clearest bad control is what the person was paid before: last year's salary, or the salary at the previous employer. It predicts current pay almost perfectly, so the fit improves, and it already contains whatever gap existed last year, so the coefficient on sex falls toward zero. Nothing was fixed. The gap was moved from the variable you were watching to a variable you were not. Step 7 of the build-along plants exactly that: a "prior pay" column that carries the same shift, then refits, and the coefficient on sex shrinks while the pay itself has not changed by a cent.

Grade is the honest borderline. For the question "are two people in the same band paid the same", grade is the right control and the coefficient on sex answers it. For the question "is the whole pay system fair", grade cannot be a control, because who reaches each grade is part of what is being asked. The same table, the same code, two questions; the analyst's job is to say which one was asked. A pay equity report that answers the first and is read as answering the second is the most common way a true number misleads.

Fit the model on Apex HR, 76 people, and the coefficient on sex will have a standard error wide enough that almost any value is consistent with the data. That is not a finding of no gap; it is a finding of not enough people to say. The statistics course covers the standard error properly; the habit here is to print the count under every coefficient the way foundations session 2 printed the base under every rate, and to refuse a sentence about a group whose count would not support it. The anonymity threshold that deep-dive session 7 applies to survey groups applies here too, for a second reason: with five people in a band, the coefficient on sex identifies individuals.