Define the consistency of an estimator and describe why the idea is useful.
The idea
An estimator is consistent if it converges in probability to the true parameter as the sample size grows without limit. For any tolerance, the chance of missing by more than that tolerance tends to zero. A sufficient condition is that both the bias and the variance go to zero as n increases. Consistency is useful because it justifies estimators that are biased in small samples, such as the divisor-n variance or the sample standard deviation, so long as the data set is large. It is not the same as unbiasedness. The first observation alone is an unbiased but inconsistent estimator of the mean, since its variance never shrinks. Consistency also tells you nothing about how accurate the estimate is in any particular finite sample.
A practice question
Why is the idea of consistency useful in practice?
- It guarantees that the estimator is accurate even in a very small sample.Why not: Consistency is only an asymptotic guarantee; small-sample accuracy is not implied.
- It guarantees that the estimator is unbiased.Why not: A consistent estimator can be biased; bias just has to vanish as n grows.
- It justifies using estimators biased in small samples, since with enough data they get arbitrarily close to the truth. - correct
- It guarantees that the estimator has a smaller variance than any other estimator.Why not: Consistency compares nothing against other estimators; it describes the estimator's own limit.
Many useful estimators (such as the divisor-n variance) are biased in finite samples. Consistency tells us they still converge to the truth, which gives a basis for using them with the large samples typical in risk management.
A harder one
To estimate the mean daily return, an analyst uses only the first observation of each sample, however many observations n are available. Which description is correct?
- The estimator is consistent but biased, because it converges to the mean as n grows.Why not: It is not biased (the first observation has the population mean as expectation) and it does not converge.
- The estimator is both unbiased and consistent, since it is unbiased.Why not: Unbiasedness does not ensure consistency; the estimator ignores new data and its variance stays at sigma².
- The estimator is neither unbiased nor consistent, because it discards information.Why not: Discarding data costs variance, not bias; the first observation still has expected value equal to the mean.
- The estimator is unbiased but not consistent, because its variance does not shrink as n grows. - correct
E[X1] = mu, so it is unbiased, but Var(X1) = sigma² for every n, so the estimate never concentrates around mu. Unbiased and not consistent.
The hardest tier
The sample variance s² (n - 1 divisor) is an unbiased and consistent estimator of the population variance. Which statement about the sample standard deviation s as an estimator of sigma is correct?
- It is unbiased and consistent, because s² is unbiased and consistent.Why not: The square root is nonlinear, so unbiasedness is lost even though consistency is retained.
- It is biased downward in finite samples but consistent, since the square root preserves convergence in probability. - correct
- It is biased and therefore inconsistent.Why not: Bias that fades as n grows is compatible with consistency.
- It is consistent only if returns are normally distributed.Why not: Consistency of s needs only that s² converges to sigma² (finite fourth moment suffices), not normality.
s² converges in probability to sigma² and the square root is continuous, so s converges to sigma. By Jensen's inequality E[s] < sigma, so s is biased in finite samples. Biased but consistent.
In the app: 6 practice questions on this outcome, spaced review that brings it back before you forget it.
Start free →More in Sample Moments
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