Patricia Arquette's speech about the gender pay gap during Sunday night's Oscars has the subject back on the public agenda, and with it the controversy over whether it's really true that women earn 23 percent less than men. Max Ehrenfreund at the Washington Post writes that "some of that gap is due to women's own decisions about whether to spend time at home with children and which fields to work in."

Conservatives often cite these considerations and observe that when you fully control for both hours worked and occupational category, the pay gap ends up being considerably smaller than the raw figure. This is both true and important to understand, but it's also important to understand that this statistical work doesn't make the social problem go away. After all, for many purposes gender itself is an important demographic control to apply to statistical analyses. And when you control for gender, the gender pay gap goes away entirely!

But that's trivial. The point of doing a more sophisticated statistical analysis of the gender pay gap is that it tells us how it works.

Discrimination is real

Evan Soltas recently did a bunch of statistical tests and found that with controls "the gender pay gap was 12.6 percent in 1990 and 4.5 percent in 2013."

He was looking at hourly wages, which the Pew Research Center recently estimated have a gender pay gap of 16 percent. That figure is smaller than the 23-percent figure because women tend to work fewer hours than men. So by Soltas' math, about a fourth of the 23 cent gap can be explained by what looks like intra-firm discrimination. Bosses are reluctant to pay women as much as they would pay men for the same job. That could be because they perceive women as less competent, or it could be a form of "statistical discrimination" having something to do with the impact of motherhood on women's pay. Conservative pundits often seem more upset by the fact that 4.5 percent is less than 23 percent than by the fact that 4.5 percent is more than zero percent. But I'd be pretty pissed if someone cut my pay 4.5 percent. And I'd be really pissed if they did it because I'm a man rather than because of something related to my job performance. The narrowly political point that Barack Obama somewhat oversimplifies the situation is interesting, but the thousands of dollars a year that the typical working woman in American can expect to lose to discrimination is more interesting.

Hours worked

The pay gap seems to shrink from about 77 percent to around 84 percent when you consider the difference in hours worked. Of course you can't blame employers for paying more to people who work longer hours. But it's equally silly to act like this is just some crazy coincidence. Women work shorter hours because as a society we hold women to a higher standard of housekeeping, and because they tend to be assigned the bulk of childcare responsibilities.

These aren't necessarily issues that it's reasonable for an employer to address, but as a society we should acknowledge that the domestic expectations of women have a significant economic impact. After all, it's not as if women are working shorter hours because they're loafing around. Men have more leisure time than women, since the gap in time spent doing non-market work is bigger than the gap in hours spent on paid work.

Occupational discrimination

Besides hours worked, the big factor that people like to throw into regression models to make the wage gap go away is occupational choice. Some jobs pay more than others. Christina Hoff Sommers says the upshot of this is that women are just picking the wrong college majors. But if you look at the geographical structure of the wage gap, you'll see that places like DC and Maryland with tons of white collar workers tend to have the smallest wage gap. The big gaps are in states such as North Dakota, West Virginia, and Wyoming where an unusually large share of working class men are able to obtain unusually remunerative jobs in natural resource extraction.

The question to ask about this is what should we conclude from the fact that male-dominated occupations tend to be higher paying than woman-dominated ones? Are women simply en masse failing to notice the wage structure of the American labor market and "choosing" in droves to not make money? Or are they being pushed out of more remunerative fields by discrimination? Are they being discouraged from even considering these careers? And how is it that the male-dominated fields came to have structurally higher pay scales in the first place?

It's true that these problems aren't amenable to super-simple legislative fixes. But that doesn't mean they're not real problems.

The bottom line

Life is complicated. Any summary statistic is, by definition, going to be an effort to simplify that reality. And it is absolutely true to say that pay discrimination on the part of employers between the women they employ and the men they employ only accounts for a minority of the gap. But the statistical controls that reveal that don't make the problem of the wage gap go away. They help us identify where it exists. Some of it exists inside the companies where women work. Some of it exists inside household dynamics and broad social expectations of how family life should work. And some of it exists at the level of occupations, where women's job opportunities are structured in an economically unhelpful way.