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.

This paper considers a binary response model with a partially linear latent equation, where ϕ is an unknown function and β is a finite-dimensional parameter of interest. Using the principle of smoothed maximum score estimation (Horowitz, 1992; Econometrica 60(3), 505–531), a consistent and asymptotically normal (C.A.N.)estimator for β is proposed under the restriction that the median of the error conditional on the covariates is equal to 0. Furthermore, the rate of convergence in probability is close to the parametric rate, if certain functions admit enough derivatives. This method neither restricts the form of heteroskedasticity in the error term nor suffers from the curse of dimensionality whenever ϕ is multivariate. Some Monte Carlo experiments suggest that this estimator performs well compared with conventional estimators.

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