Abstract :
Linearity in a causal relationship between a dependent variable and a set of regressors
is a common assumption throughout economics. In this paper we consider the
case when the coefficients in this relationship are random and distributed independently
from the regressors. Our aim is to identify and estimate the distribution of
the coefficients nonparametrically.We propose a kernel-based estimator for the joint
probability density of the coefficients. Although this estimator shares certain features
with standard nonparametric kernel density estimators, it also differs in some important
characteristics that are due to the very different setup we are considering. Most
importantly, the kernel is nonstandard and derives from the theory of Radon transforms.
Consequently, we call our estimator the Radon transform estimator (RTE).
We establish the large sample behavior of this estimator—in particular, rate optimality
and asymptotic distribution. In addition, we extend the basic model to cover
extensions, including endogenous regressors and additional controls. Finally, we analyze
the properties of the estimator in finite samples by a simulation study, as well
as an application to consumer demand using British household data.
