Inference after estimation of breaks

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