This article studies an asymptotic framework for conducting inference on parameters of the form φ( heta_0), where φ is a known directionally differentiable function and heta_0 is estimated by $\hat heta_n$. In these settings, the asymptotic distribution of the plug-in estimator $φ(\hat heta_n)$ can be derived employing existing extensions to the Delta method. We show, however, that (full) differentiability of φ is a necessary and sufficient condition for bootstrap consistency whenever the limiting distribution of $\hat heta_n$ is Gaussian. An alternative resampling scheme is proposed that remains consistent when the bootstrap fails, and is shown to provide local size control under restrictions on the directional derivative of φ. These results enable us to reduce potentially challenging statistical problems to simple analytical calculations—a feature we illustrate by developing a test of whether an identified parameter belongs to a convex set. We highlight the empirical relevance of our results by conducting inference on the qualitative features of trends in (residual) wage inequality in the U.S.

In the US, both taxes and old-age social security benefits depend on one’s marital status and tend to reduce the labour supply of the secondary earner. To what extent are these provisions holding back the female labour supply? We estimate a rich dynamic life-cycle model of labour supply and savings for couples and singles using the Method of Simulated Moments for the 1945 and 1955 birth cohorts. Our model matches well the life-cycle profiles of labour market participation, hours, and savings for married and single people, and generates plausible elasticities of labour supply. It implies that eliminating these marriage-related provisions would drastically increase the participation of married women over their entire life cycle, reduce the participation of married men after age 60, and increase savings. If the resulting government surplus were used to lower income taxation, there would be large welfare gains for the vast majority of the population. These results hold for both cohorts, including the later one, which has participation similar to that of more recent generations.