Questions: Estimator Properties: Consistency, Unbiasedness, and Efficiency

Question 1

An estimator β̂ is unbiased (E[β̂] = β) but its variance remains constant at 0.5 regardless of sample size. How should this estimator be classified?

• A. Unbiased and consistent — unbiasedness guarantees that estimates are centered on the true value, which implies convergence
• B. Consistent but biased — a fixed variance is acceptable for large-sample properties
• C. Unbiased but inconsistent — without variance shrinking toward zero, the estimator never converges in probability to β
• D. Efficient, since it is unbiased and has a well-defined variance

Answer: (C)

Unbiasedness (E[β̂] = β) and consistency (plim β̂ = β as n → ∞) are logically independent properties. Unbiasedness says estimates are centered on β on average, but says nothing about how spread out they are as n grows. Consistency requires the sampling distribution to collapse onto β — which means variance must shrink to zero. If variance stays constant at 0.5 forever, collecting more data never narrows the distribution of estimates. This is the canonical counterexample to the misconception that 'unbiased implies consistent.'

Question 2

A researcher runs OLS on a dataset where the key regressor is correlated with the error term. After collecting ten times more observations, what happens to the OLS estimate?

• A. It becomes unbiased, because the larger sample reduces sampling error toward zero
• B. It becomes more precise — the variance shrinks — but it converges toward a biased limit rather than the true parameter
• C. It improves toward the true value because consistency holds even under endogeneity
• D. Sample size has no effect on the estimate when endogeneity is present

Answer: (B)

When E[u | x] ≠ 0 (endogeneity), OLS is neither unbiased nor consistent. More data makes the estimate more precise — variance shrinks — but the estimate converges to a biased limit, not the true β. This is the crucial practical implication of inconsistency: no amount of data can fix a violation of the identifying assumption. The only solutions address the endogeneity directly (instruments, fixed effects, natural experiments). More data with a broken design gives you a very precise wrong answer.

Question 3

A consistent estimator should also be unbiased, since convergence to the true value in large samples implies there is no systematic error.

• T. True
• F. False

Answer: False

Consistency and unbiasedness are independent. A consistent estimator can have finite-sample bias that vanishes as n → ∞. The maximum likelihood estimator of variance (dividing by n instead of n−1) is biased in finite samples but consistent. In econometrics, OLS under contemporaneous exogeneity (E[uᵢ | xᵢ] = 0 but not strict exogeneity) is technically biased in finite samples but consistent. The bias is negligible in large samples — which is precisely what makes consistency the operative guarantee in practice.

Question 4

Consistency is often considered more practically important than unbiasedness in applied econometrics because it guarantees a useful answer given enough data, while unbiasedness alone makes no such guarantee.

• T. True
• F. False

Answer: True

This is the practical hierarchy in the explainer. Unbiasedness guarantees no systematic error at any sample size, which sounds strong, but if variance doesn't shrink, more data never helps — you remain permanently imprecise. Consistency guarantees convergence: with enough data, you can get as close to the true value as you want. In applied research with large samples, consistency is the operative guarantee. The worst case is inconsistency under endogeneity: collecting more data produces a precise but wrong answer that you cannot distinguish from a correct one.

Question 5

In your own words, explain why consistency is often more practically valuable than unbiasedness in applied econometrics, even though unbiasedness sounds like the stronger guarantee.

Model answer: Unbiasedness means E[β̂] = β — no systematic error on average — but it says nothing about what happens as you collect more data. An unbiased estimator with non-shrinking variance stays imprecise regardless of sample size. Consistency means that as n → ∞, β̂ converges to the true β — given enough data, you recover the true value. In practice, we work with large samples where the asymptotic guarantee of consistency matters most. An unbiased but inconsistent estimator is useless at scale; a slightly biased but consistent estimator improves reliably and predictably.

The deeper issue is the failure mode of inconsistency: under endogeneity, OLS converges to a wrong limit. More data narrows the distribution around a biased target. No statistical technique can fix a design flaw — only a better identification strategy (instruments, randomization, fixed effects) restores consistency. This is why applied econometrics is fundamentally about identification: getting the consistency condition right is the prerequisite for any valid inference, regardless of sample size.