A researcher estimates the return to education using self-reported years of schooling, which is known to contain random reporting errors. Compared to a perfect measure of education, how will the OLS estimate of the return be affected?
AIt will be biased upward — measurement error inflates estimated effects
BIt will be unbiased — random errors cancel out in large samples
CIt will be biased downward toward zero — measurement error attenuates the coefficient
DIt will have larger standard errors but remain consistent
Classical measurement error in a regressor causes attenuation bias: the OLS coefficient is biased toward zero, not away from it. The noisy measure x = x* + u creates endogeneity because the true x* is embedded in the composite error, making the observed x correlated with the disturbance. The attenuation factor is Var(x*) / [Var(x*) + Var(u)], always between 0 and 1. The researcher's estimated return will be smaller than the true return — the more noise, the more severe the downward bias.
Question 2 Multiple Choice
A study measures household income perfectly but measures household consumption with random error. Which of the following best describes the effect on OLS estimates of income's effect on consumption?
AThe coefficient on income is biased toward zero due to attenuation bias
BThe coefficient on income is unbiased, but standard errors are larger than they would be without measurement error
CBoth the coefficient and standard errors are unaffected because the error is random
DThe coefficient on income is biased upward because noise inflates variation in the outcome
Measurement error in the outcome variable Y (consumption here) does not bias OLS coefficients when it is classical (uncorrelated with X and the structural error). It does increase variance in the residuals, which inflates standard errors and reduces precision — but the estimates remain consistent. This asymmetry is critical: researchers often worry too much about mismeasured outcomes and too little about mismeasured regressors, when the regressor case is far more damaging.
Question 3 True / False
Measurement error in the dependent variable (Y) causes the same attenuation bias as measurement error in a regressor.
TTrue
FFalse
Answer: False
This is the key asymmetry in measurement error analysis. Classical measurement error in the outcome Y adds noise to the residuals but does not bias OLS coefficients — the estimates remain consistent, though less precise. Measurement error in a regressor X creates endogeneity (because the observed X is correlated with the composite error), causing attenuation bias that shrinks coefficients toward zero. The two cases have completely different consequences for estimation.
Question 4 True / False
A second independent measurement of the same mismeasured variable can serve as a valid instrumental variable to correct for attenuation bias.
TTrue
FFalse
Answer: True
Under the classical measurement error assumption, two independent measurements of the same true variable share the signal (x*) but not the noise — the measurement errors are uncorrelated. This means the second measure is correlated with the first through their common true component (relevance condition) and uncorrelated with the first's measurement error (exclusion condition). It therefore qualifies as a valid instrument that can recover a consistent estimate of the structural coefficient via 2SLS.
Question 5 Short Answer
Why does measurement error in a regressor create endogeneity, and what direction does the resulting bias go?
Think about your answer, then reveal below.
Model answer: When you observe x = x* + u instead of the true x*, the regression model becomes y = β·x + (ε − β·u). The composite error (ε − β·u) contains u, and u is part of x, so the regressor is correlated with the error — the definition of endogeneity. OLS interprets this correlation as evidence that x explains less of y than it actually does, producing a coefficient that is biased toward zero. The attenuation factor Var(x*)/[Var(x*) + Var(u)] quantifies the shrinkage: the noisier the measure, the closer the estimated coefficient is to zero.
This is the intuitive core of attenuation bias: the noisy x is a contaminated version of the true signal, so the OLS regression attributes some of what the true x explains to 'unexplained' residual variation. The bias always goes toward zero for a single regressor under classical errors. In multiple regression, coefficients on other variables can be biased in any direction, depending on their correlations with the mismeasured variable.