Semi‐linear mode regression

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.