Inference after estimation of breaks

In an important class of econometric problems, researchers select a target parameter by maximizing the Euclidean norm of a data-dependent vector. Examples that can be cast into this frame include threshold regression models with estimated thresholds, and structural break models with estimated breakdates. Estimation and inference procedures that ignore the randomness of the target parameter can be severely biased and misleading when this randomness is non-negligible. This paper proposes conditional and unconditional inference in such settings, reflecting the data-dependent choice of target parameters. We detail the construction of quantile-unbiased estimators and confidence sets with correct coverage, and prove their asymptotic validity under data generating process such that the target parameter remains random in the limit. We also provide a novel sample splitting approach that improves on conventional split-sample inference.