We study inference in structural models with a jump in the conditional density, where location and size of the jump are described by regression curves. Two prominent examples are auction models, where the bid density jumps from zero to a positive value at the lowest cost, and equilibrium job-search models, where the wage density jumps from one positive level to another at the reservation wage. General inference in such models remained a long-standing, unresolved problem, primarily due to nonregularities and computational difficulties caused by discontinuous likelihood functions. This paper develops likelihood-based estimation and inference methods for these models, focusing on optimal (Bayes) and maximum likelihood procedures. We derive convergence rates and distribution theory, and develop Bayes and Wald inference. We show that Bayes estimators and confidence intervals are attractive both theoretically and computationally, and that Bayes confidence intervals, based on posterior quantiles, provide a valid large sample inference method.

We use instrumental quantile regression approach to examine the effects of 401(k) plans on wealth using data from the Survey of Income and Program Participation. Using 401(k) eligibility as an instrument for 401(k) participation, we estimate the quantile treatment effects of participation in a 401(k) plan on several measures of wealth. The results show the effects of 401(k) participation on net financial assets are positive and significant over the entire range of the asset distribution, and that the increase in the low tail of the assets distribution appears to translate completely into an increase in wealth. However, there is significant evidence of substitution from other forms of wealth in the upper tail of the distribution.