Deep Tobit networks: A novel machine learning approach to microeconometrics

Abstract

Tobit models (also called as “censored regression models” or classified as “sample selection models” in microeconometrics) have been widely applied to microeconometric problems with censored outcomes. However, due to their linear parametric settings and restrictive normality assumptions, the traditional Tobit models fail to capture the pervading nonlinearities and thus may be inadequate for microeconometric analysis with large-scale datasets. This paper proposes two novel deep neural networks for Tobit problems and explores machine learning approaches in the context of microeconometric modeling. We connect the censored outputs in Tobit models with some deep learning techniques, which are thought to be unrelated to microeconometrics, and use the rectified linear unit activation and a particularly designed network structure to implement the censored output mechanisms and realize the underlying econometric conceptions. The benchmark Tobit-I and Tobit-II models are then reformulated as two carefully designed deep feedforward neural networks named deep Tobit-I network and deep Tobit-II network, respectively. A novel significance testing method is developed based on the proposed networks. Compared with the traditional models, our networks with deep structures can effectively describe the underlying highly nonlinear relationships and achieve considerable improvements in fitting and prediction. With the novel testing method, the proposed networks enable highly accurate and sophisticated econometric analysis with minimal random assumptions. The encouraging numerical experiments on synthetic and realistic datasets demonstrate the utility and advantages of the proposed method.