Extended-support beta regression for [0, 1] responses
Ioannis Kosmidis, Achim Zeileis
Abstract
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 four-parameter beta distribution with same exceedance on the left and right of (0,1). Hence, XBX regression directly extends beta regression. We prove that beta regression and heteroscedastic normal regression with censoring at 0 and 1—also known as heteroscedastic two-limit tobit in the econometrics literature—are special cases of extended-support beta regression, when a parameter is zero or infinity, respectively. To overcome identifiability issues from the potential similarity of the beta and normal distributions, we shrink towards beta regression by letting that parameter have an exponential distribution with unknown mean. A Gauss–Laguerre quadrature approximation results in efficient likelihood-based estimation and inference, which the betareg R package implements. We analyse investment decisions in a behavioural economics experiment about the occurrence and extent of loss aversion. In contrast to standard approaches, XBX regression captures both the probability of rational behaviour and the mean of loss aversion. Extensive comparisons with alternative approaches illustrate the effectiveness of the new model.