A note on the sample selection (switching regression) model and treatment effects for a log-transformed outcome variable, in the context of residential self-selection

This study examines the residential self-selection issue from conceptual and methodological aspects, with an empirical application. The study was motivated by the challenge of interpreting the results of an endogenous switching model and the lack of convenient/proper equations for calculating treatment effects when the dependent variable is log-transformed. We classify expected returns from living in a certain kind of area (urban vs. non-urban) into location-related versus transportation-related, and economic versus lifestyle, aspects. From that, we note that the outcome variable of interest to this study among many others—vehicle-miles driven (VMD)—may not correspond to the returns an individual is seeking to optimize when choosing where to live. Previous studies tend to expect negative error correlations between selection and outcome equations, on the presumption that urbanite people would desire to drive less (so, unobserved traits increasing the utility of an urban neighborhood should also tend to decrease VMD). However, given the variety of possible expected returns, we argue that error correlations (and thus the directions of selection corrections) could be either positive or negative. As an empirical application, we apply the endogenous switching model to modeling VMD, where the treatment is living in an urban area (vs. living in a less urban area). In addition, the paper provides conceptual and graphical illustrations to improve the understanding of treatment effects and their mathematical mechanisms.