# NONPARAMETRIC LIKELIHOOD AND FURTHER DEVELOPMENT OF INVERSE WEIGHTING AND G-ESTIMATION FOR MARGINAL AND NESTED STRUCTURAL MODELS

#### Abstract

Drawing inferences about treatment effects are commonly of interest in economics, epidemiology, and other fields. We consider Robins' marginal and nested structural models, and propose corresponding likelihood and regression estimators under the assumption of no confounding. The propensity score plays a central role and is estimated through a parametric model. The likelihood estimator is derived from a profile likelihood in which the joint distributions of potential outcomes and covariates are profiled out by ignoring part of the information about these distributions. The profile log-likelihood ratio like a parametric log-likelihood ratio has an asymptotic chi-squared distribution with the degrees of freedom equal to the number of parameters. Furthermore, two regression estimators algebraically similar to the augmented inverse weighting or G-estimator are suggested as first-order approximations to the likelihood estimator, and one of them is deliberately constructed to be doubly robust, i.e. remain consistent and asymptotically normal if the propensity score model is misspecified but the outcome regression model is correct. The likelihood estimator is at least as efficient and the doubly robust regression estimator is at least as efficient and robust as existing estimators. We illustrate the methods by analyzing the data from an observational study on right heart catheterization.

*This paper has been withdrawn.*