A two-part measurement error model to estimate participation in undeclared work and related earnings

In undeclared work research, the estimation of the magnitude of the phenomenon (i.e., the amount of income and/or the percentage of workers involved) is of major interest. This has been done either using indirect methods or by means of ad hoc surveys such as the Eurobarometer special survey on undeclared work, our motivating study. The extent of undeclared work can be measured by means of two different outcomes: the event of working off-the-book (binary variable) and, when the event occurs, the amount of earnings deriving from the undeclared activity (continuous variable). This setup has been typically modeled via the so called two-part model: a binary choice model for the probability of observing a positive-versus-zero outcome and then, conditional on a positive outcome, a regression model for the positive outcome. We propose an extension of the two-part model that goes in two directions. The first regards the measurement error that, given the very nature of undeclared activities, is most likely to affect both the outcomes of interest. The second is that we generalize the linear regression part of the model to allow individual-level means. We also conduct an extensive simulation study to investigate the performance of the proposed model and compare it with traditional approaches.