Local M-estimation with Discontinuous Criterion for Dependent and Limited Observations

TL;DR

This paper develops a general theoretical framework for local M-estimators with discontinuous criteria under dependence and limited data, enabling valid inference in complex econometric models.

Contribution

It introduces broad conditions covering many estimators, allowing for discontinuities, dependence, and high-dimensional nuisance parameters, with new asymptotic results and inference methods.

Findings

Derived three nonparametric cube root convergence rates.

Established maximal inequalities for weakly dependent data.

Validated the theory on examples like the Hough transform and maximum score estimators.

Abstract

This paper examines asymptotic properties of local M-estimators under three sets of high-level conditions. These conditions are sufficiently general to cover the minimum volume predictive region, conditional maximum score estimator for a panel data discrete choice model, and many other widely used estimators in statistics and econometrics. Specifically, they allow for discontinuous criterion functions of weakly dependent observations, which may be localized by kernel smoothing and contain nuisance parameters whose dimension may grow to infinity. Furthermore, the localization can occur around parameter values rather than around a fixed point and the observation may take limited values, which leads to set estimators. Our theory produces three different nonparametric cube root rates and enables valid inference for the local M-estimators, building on novel maximal inequalities for weakly dependent data. Our results include the standard cube root asymptotics as a special case. To illustrate the usefulness of our results, we verify our conditions for various examples such as the Hough transform estimator with diminishing bandwidth, maximum score-type set estimator, and many others.