Testing and Correcting for Endogeneity with Discontinuities and No Exclusion Restriction

Applied microeconomists, like us, spend a lot of our time thinking (…erm… worrying) about the bias from endogeneity embedded in our empirical estimates. That is why the work of Carolina Caetano (and co-authors), in methodological papers published in Econometrica and the Journal of Econometrics seems so exciting to us.

Caetano’s original work takes advantage of bunching to test for exogeneity and is best explained with an illustrative example, in Caetano’s own words.

“Consider the problem of estimating the marginal effect of the amount a woman smokes during pregnancy on the baby’s birth weight. The variable “average number of cigarettes per day” is naturally prone to endogeneity, given that there are many pre-existing selection factors associated with both smoking and birth weight. Examples of such factors include the mother’s education level, marital status, age, etc. The question is whether the amount smoked is exogenous after controlling for the observable factors available in the data.”

 The test can be illustrated, using the smoking example, as follows:

“The fundamental maintained assumption of the exogeneity test is that the structural function must be continuous in the explanatory variable of interest. In the smoking example, it means that the mother’s smoking amount must have a continuous effect on the baby’s birth weight. Suppose that this is indeed true, and then consider [panel a in the figure above], which illustrates the expected birth weight for each amount smoked. If the expected birth weight conditional on the amount smoked is discontinuous, this discontinuity cannot be due to smoking, since smoking has a continuous effect on birth weight. However, this discontinuity may be due to selection on observables, since the observable mother’s characteristics may be discontinuous at zero cigarettes. Next, consider [panel b in the figure above] which depicts the expected birth weight for each amount smoked for a subgroup of mothers who share the same observable characteristics. If the birth weight per amount smoked is still discontinuous, this discontinuity cannot be caused by smoking (by assumption) nor by selection on observables (because the observable mother’s characteristics are held fixed). The only explanation for this discontinuity is that there is at least one confounder that was not included in the structural equation, and thus smoking is endogenous.”

This method maintains a few requirements. The most fundamental of which is that a variable of interest is both discontinuously distributed at some observable value (in other words bunched at some value such as 0) but the true causal effect of that variable on the outcome is continuous. The Econometica paper, of course, lays out all of these requirements and assumptions in wonderful detail. Interested readers should read that entire paper, if they have not done so already.

This method, at the very least, seems beneficial in providing another sensitivity test (similar to Oster’s coefficient stability and unobservable selection test) for applications where it could be challenging to convincingly demonstrate exogeneity.

However, the intuition of this method can also be used to correct estimates for endogeneity driven by selection on unobservables. In a follow-up paper, Carolina Caetano, Gregorio Caetano, and Eric Nielsen use this method to estimate the effect of educational enrichment activities—such as reading, homework, and extracurricular studies on academic achievement. The authors address the issue of endogeneity by exploiting the fact that many children spend zero hours reading, working on homework or in extracurricular activities. Confounding variables vary for the children who spend zero hours in these activities, but importantly latent enrichment does not vary. This provides information about the effect of confounders on academic skills.

The main identifying assumption is that the unobservables enter the equation linearly. The figure below illustrates this assumption for the unosbservable confounder “ability.” The figure shows that high ability children desire high levels of enrichment activities and low ability children desire low levels of enrichment activities. Although some children actually desire negative enrichment time, all of these children end up being bunched at zero time in enrichment activities. This creates a discontinuity in the distribution. The conclusion at this point is that there must be an omitted variable that is causing enrichment activities to be endogenous.

This test and its application to correct for endogeneity strike us as powerful and quite useful in a variety of scenarios. Many variables have discontinuous distributions because of either of some sort of natural law or legal thresholds. Caetano lists examples such as:

• Wages in the problem of the estimation of Engel curves (bunching at the minimum wage).
• Schooling in the problem of the estimation of Mincer equations (bunching at the minimum attendance laws’ thresholds).
• Weekly hours of work in the problem of the estimation of the effects of hours worked by the mother on the child’s test scores (bunching at zero hours of work).

Many other examples seem potentially relevant and thus warrant careful thought. We look forward to seeing how this neat method is used in applied work!

This post represents a collaboration with my excellent co-author and friend Mo Alloush.