Nonparametric Errors in Variables Models with Measurement Errors on both sides of the Equation∗

Measurement errors are often correlated, as in surveys where respondent’s biases or tendencies to err affect multiple reported variables. We extend Schennach (2007) to identify moments of the conditional distribution of a true Y given a true X when both are measured with error, the measurement errors in Y and X are correlated, and the true unknown model of Y given X has nonseparable model errors. We also provide a nonparametric sieve estimator of the model, and apply it to nonparametric Engel curve estimation. In our application measurement errors on the expenditures of a good Y are by construction correlated with measurement errors in total expenditures X. This feature of most consumption data sets has been ignored in almost all previous demand applications. We find accounting for this feature casts doubt on Hildenbrand’s (1994) “increasing dispersion” assumption.