Poisson Regression under Multivariate Sample Selection

This paper develops a Poisson regression model with multivariate sample selection, in which the outcome is observed only when several potentially correlated selection conditions are satisfied. To the best of our knowledge, this is the first Poisson sample selection model that allows for an arbitrary number of selection equations. We derive the conditional mean of the observed outcome under joint normality of the outcome and selection errors and obtain a multivariate selection-correction term. We prove identification of the model parameters and show that, under suitable support and rank conditions, the outcome parameters can be identified without an exclusion restriction. We also propose two-step estimation procedures based on nonlinear least squares and Poisson pseudo-maximum likelihood. To the best of our knowledge, this paper is the first to apply PPML to a Poisson regression model with sample selection. The consistency of both estimators is established, and a robust two-step sandwich covariance matrix is proposed to account for the estimation error from the first-step selection model. In addition, factorial moments are used to recover the variance of the latent outcome error and the correlations between the outcome and selection errors. Monte Carlo simulations show that ignoring sample selection leads to persistent bias when the outcome and selection errors are correlated, while the proposed PPML estimator substantially reduces this bias and is more stable than nonlinear least squares, especially under moderate and strong selection dependence.