Question

Classify each of the following statements as True, False, or Uncertain, and if False or Uncertain briefly explain:

a) If the dependent variable Y has measurement error, the standard errors are biased but the coefficient estimates are not.
b) If an independent variable X has measurement error, coefficient estimates are biased but consistent.
c) If an independent variable X has measurement error, the bias will be small if the measurement error variance is small compared with the variance of X.

Answer 1

Measurement error can introduce bias and affect the standard errors of coefficient estimates in regression analysis. However, the estimated effects of independent variables on the dependent variable can still be accurate, although the uncertainty around those estimates may be larger due to the measurement error.

Step-by-step explanation:

a) True. If the dependent variable Y has measurement error, the standard errors are biased but the coefficient estimates are not. This means that the estimated effect of the independent variable on the dependent variable may still be accurate, but the uncertainty or standard errors around that estimated effect will be larger.

b) True. If an independent variable X has measurement error, coefficient estimates are biased but consistent. This means that the estimated effect of the independent variable on the dependent variable will be consistently off in the same direction, but the magnitude of the bias will decrease as the sample size increases.

c) True. If an independent variable X has measurement error, the bias will be small if the measurement error variance is small compared with the variance of X. This means that if the measurement error is relatively small compared to the natural variation in the independent variable, the bias introduced by the measurement error will be negligible.