Theory for sparse probit estimators meeting high-dimensional rate conditions

Establish convergence-rate results for sparse probit estimators, such as post-lasso probit used to estimate the propensity score e(X) in high-dimensional settings, that verify the multiplicative rate conditions in Assumption 2 needed for uniformly valid inference with the augmented inverse probability weighting estimator. These results should be analogous to existing post-lasso linear regression guarantees used for the outcome model m(X).

Background

The paper derives uniformly valid inference for augmented inverse probability weighting (AIPW) under high-dimensional nuisance estimation and unobserved confounding, requiring rate conditions on the nuisance estimators specified in Assumption 2.

For the outcome regression m(X), existing results (e.g., post-lasso linear regression) provide rates that satisfy these conditions under sparsity. However, for the propensity score e(X), which is estimated with a probit model after variable selection, the authors note the absence of comparable theoretical guarantees for sparse probit estimators.

They nonetheless use post-lasso probit empirically in simulations, highlighting a gap in theory: a need for formal results ensuring that sparse probit estimators achieve the required rates to justify uniformly valid inference.