Causal Inference with Corrupted Data: Measurement Error, Missing Values, Discretization, and Differential Privacy

The 2020 US Census will be published with differential privacy, implemented by injecting synthetic noise into the data. Controversy has ensued, with debates that center on the painful trade-off between the privacy of respondents and the precision of economic analysis. Is this trade-off inevitable? To answer this question, we formulate a semiparametric model of causal inference with high dimensional data that may be noisy, missing, discretized, or privatized. We propose a new end-to-end procedure for data cleaning, estimation, and inference with data cleaning-adjusted confidence intervals. We prove consistency, Gaussian approximation, and semiparametric efficiency by finite sample arguments. The rate of Gaussian approximation is $n^{-1/2}$ for semiparametric estimands such as average treatment effect, and it degrades gracefully for nonparametric estimands such as heterogeneous treatment effect. Our key assumption is that the true covariates are approximately low rank, which we interpret as approximate repeated measurements and validate in the Census. In our analysis, we provide nonasymptotic theoretical contributions to matrix completion, statistical learning, and semiparametric statistics. We verify the coverage of the data cleaning-adjusted confidence intervals in simulations. Finally, we conduct a semi-synthetic exercise calibrated to privacy levels mandated for the 2020 US Census.