Estimating the dose-response function through the GLM approach

How effective are policy programs with continuous treatment exposure? Answering this question essentially amounts to estimating a dose-response function as proposed in Hirano and Imbens (2004). Whenever doses are not randomly assigned but are given under experimental conditions, estimation of a dose-response function is possible using the Generalized Propensity Score (GPS). Since its formulation, the GPS has been repeatedly used in observational studies, and ad hoc programs have been provided for Stata users (doseresponse and gpscore, Bia and Mattei 2008). However, many applied works remark that the treatment variable may not be normally distributed. In this case, the Stata programs are not usable because they do not allow for different distribution assumptions other than the normal density. In this paper, we overcome this problem. Building on Bia and Mattei's (2008) programs, we provide doseresponse2 and gpscore, which allow one to accommodate different distribution functions of the treatment variable. This task is accomplished through by the application of the generalized linear models estimator in the first step instead of the application of maximum likelihood. In such a way, the user can have a very versatile tool capable of handling many practical situations. It is worth highlighting that our programs, among the many alternatives, take into account the possibility to consistently use the GPS estimator when the treatment variable is fractional, the flogit case by Papke and Wooldridge (1998), a case of particular interest for economists.