Evidence on the Validity of Cross-sectional and Longitudinal Labor Market Data
Evidence on the Validity of Cross-sectional and Longitudinal Labor Market Data (DOI: https://doi.org/10.1086/298348) was written by John Bound, Charles Brown, Greg J. Duncan, and Willard L. Rodgers. It was published in the Journal of Labor Economics (1994).
Building off of Fuller, the authors specify two further cases or errors.
If the measurement error uj is correlated with the true Xj, there is a downward bias. This bias is theoretically equivalent to regressing the error term on all other predictors. If uj and Xj are negatively correlated, the bias can be lower than the conventional Fuller estimate, or even negative.
If a true Y is measured as v with a correlated measurement error v* (i.e. v = δY + v*), then modeling y while estimating with v results in regression coefficients that are biased proportional to δ.
These bias terms are estimated in the Panel Study of Income Dynamics (PSID) Validation Study.
"[A]nnual earnings are fairly reliably reported".
Errors in earnings reporting tends to be negatively correlated with the earnings. This reduces bias when using earnings as an independent variable, but introduces another bias effect when using earnings as a dependent variable.