Measurement errors

Summary

One explanatory variable with measurement errors

• Given: a random sample $\left( y_1,x_1 \right),…,\left( y_n,x_n \right)$ where $x_i$ is a measurement of $z_i$ with measurement errors. $z_i$ is not directly observed.
• Suppose that the conditional expectation of $y$ depends on the actual value $z$ and not the observed value $x$ :

$$E\left( z,x \right)=β_1+β_2z$$

• then the OLS estimators $b_1$ and $b_2$ from a regression of $y$ on an intercept and $x$ are biased and inconsistent estimators of $β_1$ and $β_2$ .
• Generally, the more measurement errors, the bigger the bias/inconsistency.
• The OLS estimator $b_2$ will attenuated towards zero . This means that in large samples, $b_2$ tend to be smaller than $β_2$ in absolute terms.

Several explanatory variables

• If at least one explanatory variable has measurement errors, then all the OLS estimators $b_1,…,b_k$ are biased and inconsistent estimators of $β_1,…,β_k$ .
• The attenuation towards zero need not apply.

Measurement errors in the dependent variable

• With measurement errors in the dependent variable, the OLS estimators will still be unbiased and consistent (assuming exogeneity).
• Measurement errors in the dependent variable will generally increase the standard errors of $b_1,…,b_k$ widening confidence intervals and decreasing the power in hypothesis tests.