Econometric Theory



Stefan Hoderleina1 c1, Jussi Klemeläa2 c2 and Enno Mammena3 c3

a1 Brown University

a2 University of Oulu

a3 University of Mannheim


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.


c1 Address correspondence to Stefan Hoderlein, Department of Economics, Robinson Hall 302C, Brown University, Providence, RI 02912, USA; e-mail:

c2 Jussi Klemelä, University of Oulu, Department of Mathematical Sciences, P. O. Box 3000, 90014 University of Oulu, Finland; e-mail:

c3 Enno Mammen, University of Mannheim, Department of Economics, L7, 3-5, 68131 Mannheim, Germany; e-mail:


We have received helpful comments from seminar participants in Copenhagen, Göttingen, Frankfurt, Mannheim, Oberwolfach, UCL, Northwestern, and Wien (ESEM), as well as from Jim Heckman and Arthur Lewbel. We would like to thank two referees for a very careful check of the paper and for very helpful suggestions. In particular, we would like to thank one of the referees for comments and hints to references on spherical kernel density estimators. Financial support by Landesstiftung Baden-Württemberg's Eliteförderungsprogramm is gratefully acknowledged.