The Generalized Causal Dantzig: A Unified Approach to Instruments and Environments

Many recent works have proposed regression models which are invariant across data collection environments [Rothenh\"ausler et al., 2019, Peters et al., 2016, Heinze-Deml et al., 2018, Meinshausen, 2018, Gimenez andRothenh\"ausler, 2021]. Under conditions on the environments and type of invariance imposed, these estimators often have a causal interpretation. One recent example is the Causal Dantzig (CD). In this work we derive the CD as generalized method of moment (GMM) estimator. In this form, the environment plays a role nearly identical to the instrument in classical estimators such as Two Stage Least Squares (TSLS), illustrating a close connection between the concepts of instruments, environments, and invariance. We show that several of the conceptual motivations behind environments such as do--interventions can be modeled with instrumental variables. This unified treatment of environments and instruments produces many practical results including: 1) immediate generalization of the Causal Dantzig to problems with continuous instruments/environments 2) straightforward asymptotic results based on GMM theory and 3) new hybrid estimators which have properties superior to CD or TSLS alone. We illustrate these results in simulations and an application to a Flow Cytometry data set.