Regularized Maximum Likelihood Estimation for the Random Coefficients Model

Abstract

The random coefficients model $Y_i={\beta_0}_i+{\beta_1}_i {X_1}_i+{\beta_2}_i {X_2}_i+\ldots+{\beta_d}_i {X_d}_i$, with $\mathbf{X}_i$, $Y_i$, $\mathbf{\beta}_i$ i.i.d, and $\mathbf{\beta}_i$ 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. Nonparametric estimation for the joint density of $\mathbf{\beta}_i=({\beta_0}_i,\ldots, {\beta_d}_i)$ based on kernel methods or Fourier inversion have been proposed in recent years. Most of these methods assume a heavy tailed design density $f_\mathbf{X}$. To add stability to the solution, we apply regularization methods. We analyze the convergence of the method without assuming heavy tails for $f_\mathbf{X}$ and illustrate performance by applying the method on simulated and real data. To add stability to the solution, we apply a Tikhonov-type regularization method.