Semi-linear mode regression



In this paper, I estimate the slope coefficient parameter β of the regression model math formula, where the error term e satisfies math formula almost surely and ϕ is an unknown function. It is possible to achieve math formula-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 math formula, and approaches math formula if a certain density is sufficiently differentiable around the origin. This method allows both heteroscedasticity and skewness of the distribution of math formula. 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.