Post-shrinkage and machine learning strategies in high-dimensional beta regression: application in econometrics and health care data

The Beta Regression (BR) model is useful in the analysis of bounded continuous outcomes such as proportions. It is well-known that for any regression model, the presence of multicollinearity leads to poor performance of the maximum likelihood estimators. Furthermore, when some predictors have insignificant or weak effects on the outcomes, it is desirable to recover as much information as possible from these predictors instead of discarding them altogether. In this paper, we propose ridge-type shrinkage estimators for the BR model that, addressing both multicollinearity and information recovery from weak predictors in low- and high-dimensional settings. For the high-dimensional case, we adopt a two-stage "screen–then–shrink" strategy in the first stage, we apply a machine learning variable selection algorithm known as Boruta, and in the second stage we construct ridge-type shrinkage estimators using the selected variables. We show that, under mild conditions, the Boruta screening step enjoys a sure–screening property and controls the number of false positives. Furthermore, we show that the second-stage ridge-type estimators operate in a low-dimensional regime and achieve oracle efficiency. We illustrate the usefulness of the methods by simulations as well as application to two real data sets, one in economics and the other in health care.