To show the OLS estimator is unbiased, which of the following CLRM assumptions do we need?

The disturbance term has mean zero.
The disturbance term is homoskedastic.
The disturbance terms in observations are mutually independent.

Question

To show the OLS estimator is unbiased, which of the following CLRM assumptions do we need?

The disturbance term has mean zero.
The disturbance term is homoskedastic.
The disturbance terms in observations are mutually independent.

BlackTom AI · Accepted Answer

To assess when the OLS estimator is unbiased, we start by recalling a key condition for unbiasedness: the expected value of the disturbance term given the regressors must be zero, E[u|X] = 0. This implies that on average, the errors do not systematically bias the estimated coefficients, which is the essence of an unbiased estimator.

Option A: The disturbance term has mean zero. This is directly connected to E[u|X] = 0 in the simple sense of zero mean of the errors. If the errors have zero mean conditional on the regressors, OLS estimators are unbiased with respect to the sample, satisfying the core requirement for unbiasedness.

Option B: The disturbance term is homoskedastic. Homoskedasticity concerns the constancy of error variance across observations. While it matters for efficiency (Gauss–MarkovBLUE) and standard error accuracy, it does not by itself guarantee unbiasedness. You can have unbiased OLS estimates even with heteroskedasticity; they just won’t be efficient.

Option C: The disturbance terms in observations are mutually independent. Independence pertains to the absence of correlation between error terms across observations. Unbiasedness can still hold in the presence of certain types of dependence; however, independence is not a necessary condition for unbiasedness. What matters for unbiasedness is the zero-mean condition conditional on X, not independence alone.

Putting it together, only the mean-zero disturbance assumption (A) is essential for unbiasedness in the CLRM, while B and C are not required for unbiasedness—though they affect efficiency and inference. Therefore, the correct selection is the option that states A only.

To show the OLS estimator is unbiased, which of the following CLRM assumptions do we need? The disturbance term has mean zero. The disturbance term is homoskedastic. The disturbance terms in observations are mutually independent.Single choice

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