Before revision:  We demonstrate that particular instances of regression discontinuity design and average treatment effect may be problematic for inference with differential privacy because their estimators can only be ensured to converge weakly with their asymptotic limit remaining random and, thus, may not be estimated consistently. This result is clearly supported by our simulation evidence.
After revision:  We demonstrate that particular instances of regression discontinuity design and average treatment effect may be problematic for inference with differential privacy as parameters turn out to be neither point nor partially identified. The set of differentially private estimators converges weakly to a random set.