This paper presents a test of the exogeneity of a single explanatory variable in a multivariate model. It does not require the exogeneity of the other regressors or the existence of instrumental variables. The fundamental maintained assumption is that the model must be continuous in the explanatory variable of interest. This test has power when unobservable confounders are discontinuous with respect to the explanatory variable of interest, and it is particularly suitable for applications in which that variable has bunching points. An application of the test to the problem of estimating the effects of maternal smoking in birth weight shows evidence of remaining endogeneity, even after controlling for the most complete covariate specification in the literature.

We study the effect of maternal labor supply in the first three years of life on early childhood cognitive skills. We pay particular attention to heterogeneous effects by the skill of the mother, by the intensity of her labor supply, and by her pre-birth wages. We correct for selection using a control function approach which uses the fact that many mothers are bunched at zero working hours – skill variation in the children of these bunched mothers is informative about the effect of unobservables on skills. We find that maternal labor supply typically has a significant, negative effect on children’s early cognitive skills, with more negative effects for higher-skill mothers. By contrast, we do not find significant heterogeneity depending on the pre-birth wage rate of the mother. These findings suggest that there may be more scope to avoid short-term, unintended consequences of maternal labor supply through policies that promote more flexible work arrangements rather than through policies that increase the financial rewards to working.