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.