Local M-estimation with discontinuous criterion for dependent and limited observations

Abstract

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.

Best subset binary prediction

“…There is a large body of the literature that studies maximum score estimation in various other aspects since the seminal work by Manski (1975Manski ( , 1985. In the context of semiparametric binary response models, advances of the maximum score approach have been made in terms of point identification (Manski, 1988), partial identification (Manski and Tamer, 2002;Komarova, 2013;Blevins, 2015;Chen and Lee, 2015), asymptotic distribution (Kim and Pollard, 1990;Seo and Otsu, 2017), panel data (Manski, 1987;Charlier, Melenberg, and van Soest, 1995;Abrevaya, 2000), time series (Moon, 2004;Guerre and Moon, 2006;de Jong and Woutersen, 2011), dynamic network formation (Graham, 2016), nonparametrically generated regressors (Chen, Lee, and Sung, 2014), and so on. The numerical approach employed in this paper can be adapted to these contexts.…”

Best subset binary prediction

“…There is a large body of the literature that studies maximum score estimation in various other aspects since the seminal work by Manski (1975Manski ( , 1985. In the context of semiparametric binary response models, advances of the maximum score approach have been made in terms of point identification (Manski, 1988), partial identification (Manski and Tamer, 2002;Komarova, 2013;Blevins, 2015;Chen and Lee, 2015), asymptotic distribution (Kim and Pollard, 1990;Seo and Otsu, 2017), panel data (Manski, 1987;Charlier, Melenberg, and van Soest, 1995;Abrevaya, 2000), time series (Moon, 2004;Guerre and Moon, 2006;de Jong and Woutersen, 2011), dynamic network formation (Graham, 2016), nonparametrically generated regressors (Chen, Lee, and Sung, 2014), and so on. The numerical approach employed in this paper can be adapted to these contexts.…”

Testing Stochastic Dominance with Many Conditioning Variables

“…Since the supremum in the theorem is over G β n,δ while it is over G n,δ in this lemma, we first need to establish the relation between different norms. To this end, it was shown by Doukhan et al (1995) and Seo and Otsu (2018) that for bounded random variables obeying the mixing condition of the lemma,…”