Identification and Estimation with Contaminated Data: When Does Covariate Data Sharpen Inference?

When data contain errors, parameters of interest typically are not identified without imposing strong assumptions. However, in many cases, bounds on these parameters can be constructed under relatively weak assumptions. This paper addresses under what conditions variables in addition to the one of interest, covariate data, tighten these bounds and how to optimally incorporate that information. In particular, covariate data are unable to sharpen inference without imposing some exogenous knowledge about the distribution of errors conditional on the covariates. For example, knowing that the probability of erroneous data is either orthogonal to a covariate or monotonically increasing in a covariate is typically sufficient to sharpen inference. The identification region for the distribution of the variable of interest is constructed and used to develop bounds both on probabilities and on parameters of this distribution that respect stochastic dominance. For the case of bounding parameters that respect stochastic dominance, the necessary and sufficient conditions for covariate data to sharpen inference are derived.