  Causal inference is crucial for understanding the true impact of
interventions, policies, or actions, enabling informed decision-making and
providing insights into the underlying mechanisms that shape our world. In this
paper, we establish a framework for the estimation and inference of average
treatment effects using a two-sample empirical likelihood function. Two
different approaches to incorporating propensity scores are developed. The
first approach introduces propensity scores calibrated constraints in addition
to the standard model-calibration constraints; the second approach uses the
propensity scores to form weighted versions of the model-calibration
constraints. The resulting estimators from both approaches are doubly robust.
The limiting distributions of the two sample empirical likelihood ratio
statistics are derived, facilitating the construction of confidence intervals
and hypothesis tests for the average treatment effect. Bootstrap methods for
constructing sample empirical likelihood ratio confidence intervals are also
discussed for both approaches. Finite sample performances of the methods are
investigated through simulation studies.
