Cluster Weighted Beta Regression

The analysis of data assuming values in the real open interval (0;1) is a common issue in quantitative research when the effect of selected variables on the conditional expectation of a percentage or rate is considered. In the literature, various alternative methods to model ratios and percentage data have been proposed (see e.g. Papke and Wooldridge, 1996 and Kieschnick and McCullough, 2003). A possible solution is to transform the dependent variable y, for instance using a logit or a probit transformation, so that it assumes values on the whole real line, and then model the mean of the transformed response as a linear predictor based on a set of covariates applying OLS (Demsez Lehn, 1985) to obtain the parameter estimates. This approach, however, has drawbacks, one of them being the fact that the model parameters cannot be easily interpreted in terms of the average of the original outcome but in terms of the transformed response. Furthermore the assumptions of OLS regression are often not met despite the transformation of the data. An alternative is to use a regression model that assumes that the response variable follows a beta distribution on the interval (0;1), namely ( ):