Econometrics · properties of an estimator

Unbiasedness and consistency

Two different questions we ask of an estimator β̂. Is it centered on the truth? And does it close in on the truth as data accumulates? The first is unbiasedness; the second is consistency — and they are not the same thing.

Property 01

Unbiasedness

E[β̂] = β

At a fixed sample size, the estimator is right on average: the center of its sampling distribution sits exactly on β. This says nothing about how far any single estimate might land from β.

sample size fixed · n = 30
Sampling distribution of the chosen estimator at a fixed sample size.
true β = 2 average of β̂
average of β̂
2.00
bias = avg − β
0.00
verdict
Unbiased

Unbiasedness is a statement about a fixed sample size — here, n = 30. Look at the ±1 ping-pong estimator: it is unbiased at every sample size, because the +1 and −1 errors cancel on average, yet no single estimate is ever close to β. Being right on average is not the same as being close — which is exactly what the next property fixes.

Property 02

Consistency

Unbiasedness pins the sample size down. Consistency asks what happens as the sample size grows without bound. As the sample size n grows, a consistent estimator's whole sampling distribution collapses onto β — the estimates don't just average to the truth, they crowd around it and stay there.

β̂ →p β   as  n → ∞

Pick any tolerance you like. As the sample size n grows, the probability that β̂ falls within that tolerance of β rises toward 1. Bigger samples don't just shrink the error on average — they make sizeable errors vanishingly unlikely.

5
Histogram showing estimates of β concentrating on the true value as n grows.
standard error σ/√n
0.447
samples drawn
2,000
P( |β̂ − β| < 0.10 )
17%

The dashed line is the true β = 2; the shaded band is a tolerance of ±0.10. Drag n toward infinity and watch the estimates pile into the band.

Putting them together

The two properties are independent — an estimator can have either, both, or neither. The engine behind consistency is the law of large numbers: sample averages converge to population means, and estimators built from averages (the sample mean, OLS under standard assumptions) inherit that behaviour.

Unbiased · Consistent

Sample mean, OLS

Centered on β and collapsing onto it. The ideal case.

Unbiased · Inconsistent

The ±1 ping-pong

Right on average, but the spread never shrinks.

Biased · Consistent

Small-sample bias that fades

Off-center for small n, but the bias vanishes as n → ∞.

Biased · Inconsistent

OLS under omitted-variable bias

Off-center and staying there, no matter how much data.

In applied econometrics, consistency is usually the minimum we insist on — it is the formal reason that "collect more data" is sound advice rather than wishful thinking, and why omitted-variable bias is so damaging: more data cannot save an inconsistent estimator.