FRM Part I · FRM Exam Part I · Linear Regression
A researcher estimates Y = a + bX + e, but the true X is measured with random noise, so the observed regressor equals true X plus an independent error. Compared with the true slope, what is the asymptotic effect on the OLS slope estimator, and why?
The OLS slope is biased toward zero, called attenuation bias. Measurement noise in the regressor makes it correlated with the error term, violating exogeneity. The estimate converges to the true slope scaled by the share of true variance in observed variance, which is below one.
- AIt is biased toward zero because the noisy regressor is correlated with the composite error, a result known as attenuationCorrect
- BIt is biased away from zero because noise inflates the variance of X
- CIt is unbiased because the measurement error has mean zero
- DIt is unbiased but has a larger standard error only
Explanation
Classical measurement error in X makes the observed regressor correlated with the regression error, violating exogeneity. The probability limit of the slope equals the true slope times var(X)/(var(X)+var(noise)), which is less than one, so the estimate is biased toward zero and inconsistent. Zero-mean noise does not rescue unbiasedness.
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