Extended-support beta regression for $[0, 1]$ responses

Abstract: We introduce the XBX regression model, a continuous mixture of extended-support beta regressions for modelling bounded responses with boundary observations. The core building block of XBX regression is the extended-support beta distribution, a censored version of a four-parameter beta distribution with the same exceedance on the left and right of $(0, 1)$. Hence, XBX regression is a direct extension of beta regression. We prove that beta regression and heteroscedastic normal regression with censoring at both $0$ and $1$ -- also known as the heteroscedastic two-limit tobit model in the econometrics literature -- are special cases of extended-support beta regression, depending on whether a single extra parameter is zero or infinity, respectively. To overcome identifiability issues due to the similarity of the beta and normal distributions for certain parameter values, we shrink towards beta regression by letting the extra parameter have an exponential distribution with unknown mean. A Gauss-Laguerre quadrature approximation results in efficient likelihood-based estimation and inference procedures, which the betareg R package implements since version 3.2.0. We use XBX regression to analyze investment decisions in a behavioural economics experiment about the occurrence and extent of loss aversion. In contrast to standard approaches, we capture both the probability of rational behaviour and the mean of loss aversion. Extensive comparisons with alternative approaches illustrate the effectiveness of the new model.