Measurement Error Without the Proxy Exclusion Restriction

This article studies the identification of the coefficients in a linear equation when data on the outcome, covariates, and an error-laden proxy for a latent variable are available. We maintain that the measurement error in the proxy is classical and relax the assumption that the proxy is excluded from the outcome equation. This enables the proxy to directly affect the outcome and allows for differential measurement error. Without the proxy exclusion restriction, we first show that the effects of the latent variable, the proxy, and the covariates are not identified. We then derive the sharp identification regions for these effects under any configuration of three auxiliary assumptions. The first weakens the assumption of no measurement error by imposing an upper bound on the noise-to-signal ratio. The second imposes an upper bound on the outcome equation coefficient of determination that would obtain had there been no measurement error. The third weakens the proxy exclusion restriction by specifying whether the latent variable and its proxy affect the outcome in the same or the opposite direction, if at all. Using the College Scorecard aggregate data, we illustrate our framework by studying the financial returns to college selectivity and characteristics and student characteristics when the average SAT score at an institution may directly affect earnings and serves as a proxy for the average ability of the student cohort.