Nonparametric errors in variables models with measurement errors on both sides of the equation

Abstract

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.

Measurement systems

“…Instruments are widely used in economic applications and can be seen as special cases of indicators, often exhibiting bias or Berkson errors (Newey (2001), Schennach (2004b), Schennach (2007a), Wang and Hsiao (2011), Nadai and Lewbel (2016)) Time series (   ) and panel data (  ) often naturally provide either repeated measurements, instruments or more general indicators (Griliches and Hausman (1986), Hausman and Taylor (1981), Evdokimov (2009), Horowitz and Markatou (1996), Wilhelm (2015), Evdokimov and Zeleneev (2020a)).…”

“…As conditional expectations are not necessarily integrable, their Fourier transform need to be interpreted as generalized functions (Lighthill (1962), Schwartz (1966), Temple (1963)). Nadai and Lewbel (2016) have extended this approach by (i) allowing for some forms of correlations between the errors in  and , (ii) considering multiplicative errors and (iii) providing identification results for polynomial moments of  .…”

The Nutrient-Income Elasticity in Ultra-Poor Households: Evidence from Kenya

“…When total expenditure is combined with caloric intake or availability in a regression model this may lead to the common measurement error problem, which arises if total expenditure and caloric intake are derived from the same data source and as a result measurement error in one could correlate with measurement error in the other. This phenomenon is likely to occur in survey data where tendencies for error affect series of selfreported variables (De Nadai and Lewbel, 2016). Bouis and Haddad (1992) were the first to indicate that the common measurement error leads to a positive bias in estimated calorieincome elasticities.…”

Misclassification and the Hidden Silent Rivalry

“…see Hausman, Abrevaya, and Scott-Morton (1998); Lewbel (2000) and Meyer and Mittag (2017) for binary choice models, Hsiao and Sun (1998) for multinomial models, Abrevaya and Hausman (1999) for duration models and Li, Trivedi, and Guo (2003) for count models. In the continuous setting, Lewbel (1996); De Nadai and Lewbel (2016) investigate the measurement errors on both sides of regression. Unlike in the linear models where measurement errors on the left hand side cause only efficiency loss, there is sizable distortion of econometric analysis of nonlinear models with measurement errors on the dependent variable.…”