## Still more on the Heckman Curve!

Saw your blog post on the Heckman Curve. I went through Heckman’s response that you linked, and it seems to be logically sound but terribly explained, so I feel like I need to explain why Rea+Burton is great empirical work, but it doesn’t actually measure the Heckman curve.

The Heckman curve just says that, for any particular person, there exists a point where getting more education isn’t worth it anymore because the costs grow as you get older, or equivalently, the benefits get smaller. This is just trivially true. The most obvious example is that nobody should spend 100% of their life studying, since then they wouldn’t get any work done at all. Or, more tellingly, getting a PhD isn’t worth it for most people, because most people either don’t want to work in academia or aren’t smart enough to complete a PhD. (Judging by some of the submissions to PPNAS, I’m starting to suspect most of academia isn’t smart enough to work in academia.)

The work you linked finds that participant age doesn’t predict the success of educational programs. I have no reason to suspect these results are wrong, but the effect of age on benefit:cost ratios for government programs does not measure the Heckman curve.

To give a toy model, imagine everyone goes to school as long as the benefits of schooling are greater than the costs for them, then drops out as soon as they’re equal. So now, for high school dropouts, what is the benefit:cost ratio of an extra year of school? 1 — the costs roughly equal the benefits. For college dropouts, what’s the benefit:cost ratio? 1 — the costs roughly equal the benefits. And so on. By measuring the effects of government interventions on people who completed x years of school before dropping out, the paper is conditioning on a collider. This methodology would only work if when people dropped out of school was independent of the benefits/costs of an extra year of school.

(You don’t have to assume perfect rationality for this to work: If everyone goes to school until the benefit:cost ratio equals 1.1 or 0.9, you still won’t find a Heckman curve. Models that assume rational behavior tend to be robust to biases of this sort, although they can be very vulnerable in some other cases.)

Heckman seems to have made this mistake at some points too, though, so the authors are in good company. The quotes in the paper suggest he thought an individual Heckman curve would translate to a downwards-sloping curve for government programs’ benefits, when there’s no reason to believe they would. I’ve made very similar mistakes myself.

Sincerely,

An econ undergrad who really should be getting back to his Real Analysis homework

Interesting. This relates to the marginal-or-aggregate question that comes up a lot in economics. It’s a common problem that we care about marginal effects but the data more easily allow us to estimate average effects. (For the statisticians in the room, let me remind you that “margin” has opposite meanings in statistics and economics.)

But one problem that Parada doesn’t address with the Heckman curve is that the estimates of efficacy used by Heckman are biased, sometimes by a huge amount, because of selection on statistical significance; see section 2.1 of this article. All the economic theory in the world won’t fix that problem.

P.S. In an amusing example of blog overlap, Parada informs us that he also worked on the Minecraft speedrunning analysis. It’s good to see students keeping busy!