A Kernelization-Based Approach to Nonparametric Binary Choice Models
Guo Yan
TL;DR
The paper tackles nonparametric binary choice modeling by introducing a kernelized nonparametric (KNP) estimator that treats the systematic function and error distribution nonparametrically, achieving computational scalability via RKHS-based sieves and spectral cut-off regularization. It proves consistency, convergence rates, and asymptotic normality of weighted average derivative functionals (APE and cAPE) for the CCP, while providing practical, gradient-based algorithms and closed-form CDFs to enable feasible estimation in high dimensions. Simulation studies show that KNP improves finite-sample performance under misspecification with only mild efficiency losses when correctly specified, and the empirical application uncovers a nonlinear, bell-shaped relationship between outdoor temperature and judges’ grant decisions, with heterogeneous effects across temperature ranges. Overall, the approach offers a robust, scalable framework for fully nonparametric BCMs with accessible inference for policy-relevant averages and derivatives, extending sieve estimation into RKHS-based regimes.
Abstract
We propose a new estimator for nonparametric binary choice models that does not impose a parametric structure on either the systematic function of covariates or the distribution of the error term. A key advantage of our approach is its computational scalability in the number of covariates. For instance, even when assuming a normal error distribution as in probit models, commonly used sieves for approximating an unknown function of covariates can lead to a large-dimensional optimization problem when the number of covariates is moderate. Our approach, motivated by kernel methods in machine learning, views certain reproducing kernel Hilbert spaces as special sieve spaces, coupled with spectral cut-off regularization for dimension reduction. We establish the consistency of the proposed estimator and asymptotic normality of the plug-in estimator for weighted average partial derivatives. Simulation studies show that, compared to parametric estimation methods, the proposed method effectively improves finite sample performance in cases of misspecification, and has a rather mild efficiency loss if the model is correctly specified. Using administrative data on the grant decisions of US asylum applications to immigration courts, along with nine case-day variables on weather and pollution, we re-examine the effect of outdoor temperature on court judges' ``mood'', and thus, their grant decisions.