Nonparametric Identification and Semiparametric Estimation of Classical Measurement Error Models Without Side Information

Abstract

Virtually all methods aimed at correcting for covariate measurement error in regressions rely on some form of additional information (e.g., validation data, known error distributions, repeated measurements, or instruments). In contrast, we establish that the fully nonparametric classical errors-in-variables model is identifiable from data on the regressor and the dependent variable alone, unless the model takes a very specific parametric form. This parametric family includes (but is not limited to) the linear specification with normally distributed variables as a well-known special case. This result relies on standard primitive regularity conditions taking the form of smoothness constraints and nonvanishing characteristic functions’ assumptions. Our approach can handle both monotone and nonmonotone specifications, provided the latter oscillate a finite number of times. Given that the very specific unidentified parametric functional form is arguably the exception rather than the rule, this identification result should have a wide applicability. It leads to a new perspective on handling measurement error in nonlinear and nonparametric models, opening the way to a novel and practical approach to correct for measurement error in datasets where it was previously considered impossible (due to the lack of additional information regarding the measurement error). We suggest an estimator based on non/semiparametric maximum likelihood, derive its asymptotic properties, and illustrate the effectiveness of the method with a simulation study and an application to the relationship between firm investment behavior and market value, the latter being notoriously mismeasured. Supplementary materials for this article are available online.

Keywords:

• Errors-in-variables
• Higher-order moments
• Measurement error

Acknowledgments

The authors thank Daniel Wilhelm and Xavier d’Haultfoeuille for useful comments and Jiaxiong Yao for excellent research assistance. S. M. Schennach acknowledges support from NSF grants SES-0752699 and SES-1061263/1156347.

Notes

NOTE: We use the specification m(x*, θ) = θ₁ x* + θ₂(x*)² with x* ∼ N(0.08, 0.4), Δx ∼ N(0, 0.41), Δy ∼ N(0, 0.9), and θ₁ = 1. We consider a range of values of θ₂ and calculate the corresponding standard errors of estimates of θ₁. Note how the latter increase drastically as we reach the nonidentified case (θ₂ = 0). The sample size is 3000 while the number of replications used to compute the standard errors is 400.