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

Abstract

Even the most carefully curated economic data sets have variables that are noisy, missing, discretized, or privatized. The standard workflow for empirical research involves data cleaning followed by data analysis that typically ignores the bias and variance consequences of data cleaning. We formulate a semiparametric model for causal inference with corrupted data to encompass both data cleaning and data analysis. We propose a new end-to-end procedure for data cleaning, estimation, and inference with data cleaning-adjusted confidence intervals. We prove √ n-consistency, Gaussian approximation, and semiparametric efficiency for our estimator of the causal parameter by finite sample arguments. Our key assumption is that the true covariates are approximately low rank. 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.

Causal Matrix Completion

“…Such a condition is necessary as well for generalization, e.g., if every entry of E[S (k) | E] is equal to 0, then no meaningful model β (k) can learned. Such an assumption has also been explored in Agarwal et al (2021a,b); Agarwal and Singh (2021). In particular, in Agarwal et al (2021b) the authors provide a data-driven hypothesis test to verify when such a condition holds.…”

“…In Section 3, we propose a causal framework for matrix completion that draws inspiration from the rich and growing literature in econometrics on panel data and matrix completion; some relevant works include Amjad et al (2018Amjad et al ( , 2019 Agarwal and Singh (2021). As is common in matrix completion, these works impose a (approximate) low-rank factor model on the signal matrix (i.e., A), also known as an interactive fixed effects model, to capture structure across units and time (i.e., the rows and columns of the matrix, respectively).…”

“…To ensure the linear fit is appropriately regularized, a spectral sparsity constraint is imposed on S (k) , which is parameterized by λ (k) . This constrained regression is known in the literature as principal component regression (PCR) (see Agarwal et al (2021c,a,b); Agarwal and Singh (2021)). We note that in lieu of requiring that there exist K close neighbouring rows as in KNN, SNN requires that the i-th row lies in the linear span of the rows in AR (k) ; that is, given Assumption 3 holds, we require |AR (k) |≥ µ.…”

“…Another recent exciting line of work that we build upon is that of panel data and matrix completion, see Amjad et al (2018Amjad et al ( , 2019; Arkhangelsky et al (2019); Bai and Ng (2019); Fernández-Val et al (2020); Athey et al (2021); Agarwal et al (2021c,a,b); Agarwal and Singh (2021). Some of these works allow for MNAR data and entries of a matrix to be deterministically missing.…”

A Finite Sample Theorem for Longitudinal Causal Inference with Machine Learning: Long Term, Dynamic, and Mediated Effects

“…Despite the additional complexity of the longitudinal setting, I recover equally strong results. In particular, [54,52,14,28,101,35,31,51,29,2] study static local parameters and [81,50,40,29] study static proximal parameters. Like the latter, I arrive at mean square rates and projected mean square rates of confounding bridges (also called nonparametric instrumental variable regressions).…”