Semi‐linear mode regression

Abstract

In this paper, I estimate the slope coefficient parameter β of the regression model Y = X β + φ(V ) + e, where the error term e satisfies Mode(e|X, V ) = 0 almost surely and φ is an unknown function. It is possible to achieve n −2/7 -consistency for estimating β when φ is known up to a finite-dimensional parameter. I present a consistent and asymptotically normal estimator for β, which does not require prescribing a functional form for φ, let alone a parametrization. Furthermore, the rate of convergence in probability is equal to at least n −2/7 , and approaches n −1/2 if a certain density is sufficiently differentiable around the origin. This method allows both heteroscedasticity and skewness of the distribution of e|X, V . Moreover, under suitable conditions, the proposed estimator exhibits an oracle property, namely the rate of convergence is identical to that when φ is known. A Monte Carlo study is conducted, and reveals the benefits of this estimator with fat-tailed and/or skewed data. Moreover, I apply the proposed estimator to measure the effect of primogeniture on economic achievement.

Modal regression for fixed effects panel data

“…Due to such reasons, we have experienced much development of modal regression recently. Especially, with the distinguished characteristics of modal regression (such as robustness and better prediction performance (shorter prediction interval)), the idea of linear modal regression was subsequently extended by many researchers such as Yao and Xiang (2016), Zhou and Huang (2016), Chen et al (2016), Krief (2017), Chen (2018), Li and Huang (2019), Ota et al (2019), Kemp et al (2020), among others.…”

Bootstrap inference for quantile-based modal regression

“…Starting from the pioneering work of Sager and Thisted (1982), there is now a large literature on modal regression. There are two major approaches to estimating the conditional mode comparable to our method; one is linear modal regression where the conditional mode is assumed to be linear in covariates (Lee, 1989(Lee, , 1993Kemp and Santos-Silva, 2012;Yao and Li, 2014), and the other is nonparametric estimation (Yao et al, 2012;Chen et al, 2016); see also Lee and Kim (1998); Manski (1991); Einbeck and Tutz (2006); Sasaki et al (2016); Ho et al (2017); Khardani and Yao (2017); Krief (2017) for alternative methods including semiparametric and Bayesian estimation. Lee (1989Lee ( , 1993 assume symmetry of the error distribution to derive limit theorems for their proposed estimators, but the symmetry assumption implies that the conditional mean, median, and mode coincide, thereby significantly reducing the complexity of estimating the conditional mode.…”

Modal regression using kernel density estimation: A review

“…Parametric unimodal regression forms estimators using Equation ( 7) or its generalizations (Kemp & Silva, 2012;Khardani & Yao, 2017;Krief, 2017;Lee, 1989Lee, , 1993Lee & Kim, 1998;Manski, 1991;Yao & Li, 2014). Parameters estimated through the maximizing criterion in Equation ( 7) are equivalent to those obtained through maximum likelihood estimation.…”

“…Lee (1989) proposed a linear modal regression that combined a smoothed 0-1 loss with a maximum likelihood estimator (see Equation ( 7) for how these two ideas are connected). The idea proposed by Lee (1989) was subsequently modified in many studies; see, for example, Lee (1989Lee ( , 1993, Manski (1991), Lee and Kim (1998), Kemp and Silva (2012), Yao and Li (2014), Krief (2017).…”