Identification and Estimation of Semiparametric Multilayered Sample Selection Models

Many selection problems are multilayered: agents first decide whether to participate and then sort among ordered or unordered categories. This paper shows that the sorting layer changes the geometry of identification. Unlike binary selection, in which selection bias can be summarized by a scalar control function, ordered and multinomial sorting generally produce multi-index control functions whose dimension determines the continuous covariate variation needed for identification. I establish matched non-identification and point-identification results for both architectures, showing how nonlinearity in the selection structure can substitute for excluded variables. I also show how additional structural restrictions reduce the control-function dimension and make estimation practical. I propose root-n-consistent two-step sieve plug-in estimators and apply the framework to gender wage gaps among Korean college graduates. Accounting for sorting reshapes the entry-level gap along the firm-size margin, where the corrected female coefficient turns positive for large-firm employment.