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.