Models with multiway fixed effects are frequently used to address selection on unobservables. The data used for estimating these models often contain few observations per value of either indexing variable (sparsely matched data). I show that this sparsity has important implications for inference and propose an asymptotically valid inference method based on subsetting. Sparsity also has important implications for point estimation when covariates or instrumental variables are sequentially exogenous (e.g., dynamic models), and I propose a new estimator for these models. Finally, I illustrate these methods by providing estimates of the effect of class size reductions on student achievement.

Allocating resources without monetary payments is expected to yield inefficient allocations. Theory suggests that introducing rationing when resources are allocated repeatedly over time can mitigate this issue, while the magnitude of the resulting efficiency gains is an empirical question in most settings. We study a dynamic assignment mechanism used by the Michigan Department of Natural Resources to allocate bear hunting licenses and find that it yields a more efficient allocation than static mechanisms, allocating participants to types of resources for which they have a higher value without crowding out participants with a high overall value for hunting. Our empirical analysis also highlights the importance of heterogeneity across participants and across allocated resources for determining the efficiency of a dynamic allocation mechanism.