Regularized maximum likelihood estimation for the random coefficients model

Abstract

. The random coefficients model Y i = β 0 i + β 1 i X 1 i + β 2 i X 2 i + … + β d i X d i , with 𝐗 i , Y i , 𝜷 i i.i.d, and 𝜷 i independent of 𝐗 i is often used to capture unobserved heterogeneity in a population. We propose a quasi-maximum likelihood method to estimate the joint density distribution of the random coefficient model. This method implicitly involves the inversion of the Radon transformation in order to reconstruct the joint distribution, and hence is an inverse problem. To add stability to the solution, we apply Tikhonov-type regularization methods. Nonparametric estimation for the joint density of β i = ( β 0 i , … , β d i ) based on kernel methods or Fourier inversion have been proposed in recent years. Most of these methods assume a heavy tailed design density f 𝐗 . We analyze the convergence of the quasi maximum likelihood method without assuming heavy tails for f 𝐗 and illustrate performance by applying the method on simulated and real data.