Accurate inference on a parameter of interest via parametric bootstrap in two-way crossed fixed effects models with sparse discrete data

Inference on a scalar parameter in the presence of high-dimensional nuisance parameters is challenging, especially when their dimension increases with the sample size. This issue is exacerbated in models with crossed fixed effects and sparse discrete data. In this scenario, we show that standard likelihood-based methods, such as the signed likelihood root or profile likelihood, often fail to provide reliable results due to limited information. Parametric bootstrap offers an alternative to analytical corrections but it has been less explored in the context of sparse discrete data and crossed fixed effects, and its theoretical properties have yet to be fully demonstrated. For these settings, this study shows that hybrid approaches combining analytical corrections with parametric bootstrap yield superior inferential accuracy. Simulation studies indicate that constrained bootstrap from penalized estimates outperforms standard bootstrap approaches, particularly in logistic regression models with severe sparsity.