Bias from Classical and Other Forms of Measurement Error

We consider the implications of a specific alternative to the classical measurement error model, in which the data are optimal predictions based on some information set. One motivation for this model is that if respondents are aware of their ignorance they may interpret the question what is the value of this variable?' as what is your best estimate of this variable?', and provide optimal predictions of the variable of interest given their information set. In contrast to the classical measurement error model, this model implies that the measurement error is uncorrelated with the reported value and, by necessity, correlated with the true value of the variable. In the context of the linear regression framework, we show that measurement error can lead to over- as well as under-estimation of the coefficients of interest. Critical for determining the bias is the model for the individual reporting the mis-measured variables, the individual's information set, and the correlation structure of the errors. We also investigate the implications of instrumental variables methods in the presence of measurement error of the optimal prediction error form and show that such methods may in fact introduce bias. Finally, we present some calculations indicating that the range of estimates of the returns to education consistent with amounts of measurement error found in previous studies. This range can be quite wide, especially if one allows for correlation between the measurement errors