A Sparse Grid Approach for the Nonparametric Estimation of High-Dimensional Random Coefficient Models

TL;DR

This paper introduces a scalable nonparametric estimator for high-dimensional random coefficient models using a truncated tensor product basis, effectively addressing the curse of dimensionality and enabling feasible estimation.

Contribution

It proposes a novel truncated tensor product basis approach that reduces parameter growth, making nonparametric estimation of high-dimensional models computationally feasible.

Findings

Estimator performs well in Monte Carlo simulations

Application to air pollution data demonstrates practical utility

Method significantly reduces computational complexity

Abstract

A severe limitation of many nonparametric estimators for random coefficient models is the exponential increase of the number of parameters in the number of random coefficients included into the model. This property, known as the curse of dimensionality, restricts the application of such estimators to models with moderately few random coefficients. This paper proposes a scalable nonparametric estimator for high-dimensional random coefficient models. The estimator uses a truncated tensor product of one-dimensional hierarchical basis functions to approximate the underlying random coefficients' distribution. Due to the truncation, the number of parameters increases at a much slower rate than in the regular tensor product basis, rendering the nonparametric estimation of high-dimensional random coefficient models feasible. The derived estimator allows estimating the underlying distribution with constrained least squares, making the approach computationally simple and fast. Monte Carlo experiments and an application to data on the regulation of air pollution illustrate the good performance of the estimator.