Estimation of Conditional Random Coefficient Models using Machine Learning Techniques

Abstract

Nonparametric random coefficient (RC)-density estimation has mostly been considered in the marginal density case under strict independence of RCs and covariates. This paper deals with the estimation of RC-densities conditional on a (large-dimensional) set of control variables using machine learning techniques. The conditional RC-density allows to disentangle observable from unobservable heterogeneity in partial effects of continuous treatments adding to a growing literature on heterogeneous effect estimation using machine learning. %It is also informative of the conditional potential outcome distribution. This paper proposes a two-stage sieve estimation procedure. First a closed-form sieve approximation of the conditional RC density is derived where each sieve coefficient can be expressed as conditional expectation function varying with controls. Second, sieve coefficients are estimated with generic machine learning procedures and under appropriate sample splitting rules. The $L_2$-convergence rate of the conditional RC-density estimator is derived. The rate is slower by a factor then typical rates of mean regression machine learning estimators which is due to the ill-posedness of the RC density estimation problem. The performance and applicability of the estimator is illustrated using random forest algorithms over a range of Monte Carlo simulations and with real data from the SOEP-IS. Here behavioral heterogeneity in an economic experiment on portfolio choice is studied. The method reveals two types of behavior in the population, one type complying with economic theory and one not. The assignment to types appears largely based on unobservables not available in the data.