Quantile Regression

Here’s a prayer for you. Got a pencil? . . . ‘Protect me from knowing what I don’t need to know. Protect me from even knowing that there are things to know that I don’t know. Protect me from knowing that I decided not to know about the things I decided not to know about. Amen.’ There’s another prayer that goes with it. ‘Lord, lord, lord. Protect me from the consequences of the above prayer.’

Douglas Adams, Mostly Harmless (1995)

Rightly or wrongly, 95 percent of applied econometrics is concerned with averages. If, for example, a training program raises average earnings enough to offset the costs, we are happy. The focus on averages is partly because obtaining a good estimate of the average causal effect is hard enough. And if the dependent variable is a dummy for something like employment, the mean describes the entire distribution. But many variables, like earnings and test scores, have continuous distributions. These distributions can change in ways not revealed by an examination of averages, for example, they can spread out or become more compressed. Applied economists increasingly want to know what’s happening to an entire distribution, to the relative winners and losers, as well as to averages.

Policy-makers and labor economists have been especially concerned with changes in the wage distribution. We know, for example, that flat average real wages are only a small part of what’s been going on in the labor market for the past 25 years. Upper earnings quantiles have been increasing, while lower quantiles have been falling. In other words, the rich are getting richer and the poor are getting poorer. But that’s not all – recently, inequality has grown asymmetrically; for example, among college graduates, it’s mostly the rich getting richer, with wages at the lower decile unchanging. The complete story of the changing wage distribution is fairly complicated and would seem to be hard to summarize.

Quantile regression is a powerful tool that makes the task of modeling distributions easy, even when the underlying story is complex and multi-dimensional. We can use this tool to see whether participation in a training program or membership in a labor union affects earnings inequality as well as average earnings. We
can also check for interactions, like whether and how the relation between schooling and inequality has been changing over time. Quantile regression works very much like conventional regression: confounding factors can be held fixed by including covariates; interaction terms work the same as with regular regression, too. And sometimes we can even use instrumental variables methods to estimate causal effects on quantiles when a selection-on-observables story seems implausible.

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