Estimation and Inference for Linear Models with Two-Way Fixed Effects and Sparsely Matched Data

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.