Zhiqiang Tan recently wrote two papers on the theory of causal inference: see here and here. Here are the abstracts:

From Zhiqiang:

Drawing inferences about the effects of treatments is a common challenge in economics, epidemiology, and other fields. We adopt Rubin’s potential outcomes framework for causal inference, and propose two methods serving complementary purposes. One can be used to estimate average causal effects, assuming no confounding given measured covariates. The other can be used to assess how the estimates might change under various departures from no confounding. Both methods are developed from a nonparametric likelihood perspective. The propensity score plays a central role and is estimated through a parametric model. Under the assumption of no confounding, the joint distribution of covariates and each potential outcome is estimated as a weighted empirical distribution. Expectations from the joint distribution are estimated as weighted averages or, equivalently to first order, regression estimates. The likelihood estimator is at least as efficient and the regression estimator is at least as e±cient and robust as existing estimators. Regardless of the no-confounding assumption, the marginal distribution of covariates times the conditional distribution of observed outcome given each treatment assignment and the covariates is estimated. For a fixed bound on unmeasured confounding, the marginal distribution of covariates times the conditional distribution of counterfactual outcome given each treatment assignment and the covariates is explored to the extreme and then compared with the composite distribution corresponding to observed outcome given the same treatment assignment and covariates. We illustrate the methods by analyzing the data from an observational study on right heart catheterization.

and, also from Zhiqiang:

Recent researches in econometrics and statistics have gained considerable insights into the use of instrumental variables (IV) for causal inference. A basic idea is that instrumental variables serve as an experimental handle, turning of which may change each
individual’s treatment status and, through and only through this effect, also change observed outcome. The average di®erence in observed outcome relative to that in treatment status gives the average treatment effect for those whose treatment status is changed in this hypothetical experiment. We build on the modern IV framework, and develop two estimation methods in parallel to regression adjustment and propensity score weighting in the case of treatment selection based on covariates. The IV assumptions are made explicitly conditional on covariates to allow for the fact that instruments can be related to these background variables. The regression method focuses on the relationship between responses (observed outcome and treatment status jointly) and instruments adjusted for covariates.
The weighting method focuses on the relationship between instruments and covariates in order to balance different instrument groups with respect to covariates. For both methods, modelling assumptions are made directly on observed data and separated from the IV assumptions, while causal e®ects are inferred by combining observed-data models with the IV assumptions through identification results. This approach is straightforward, and flexible enough to host various parametric and semiparametric techniques that attempt to learn associational relationships from observed data. We illustrate the methods by an
application to estimating returns to education.