The bias of an estimator is the difference between the estimator’s expected value and the true parameter.
$$$$
If (\hat{\theta}) estimates (\theta), then:
$$
\operatorname{Bias}(\hat{\theta}) = E(\hat{\theta}) - \theta
$$
If that expression equals zero, the estimator is unbiased.
Why bias matters
Bias tells you whether an estimator systematically misses the truth in one direction. An estimator can be very stable from sample to sample and still be wrong on average if it is biased.
Econometricians care because model misspecification, omitted variables, measurement error, and sample selection can all produce biased estimates.
Bias versus overall accuracy
Low bias is desirable, but it is not the whole story. An estimator with a little bias and much lower variance can sometimes outperform an unbiased estimator in mean squared error terms.
That is why empirical work often studies the bias-variance trade-off rather than treating unbiasedness as the only criterion.
Knowledge Check
### What does estimator bias measure?
- [x] The systematic difference between an estimator's expected value and the true parameter
- [ ] The total sample size of a study
- [ ] The number of regressors in a model
- [ ] The slope of a demand curve
> **Explanation:** Bias is about average systematic error, not about one particular sample draw.
### Can an estimator have low variance and still be biased?
- [x] Yes
- [ ] No
- [ ] Only in time-series models
- [ ] Only if the sample is large
> **Explanation:** Precision and centeredness are different properties. An estimator can be tightly clustered around the wrong value.
### Why do economists often discuss bias together with variance?
- [x] Because overall estimator quality depends on both, not just on unbiasedness
- [ ] Because bias automatically determines variance
- [ ] Because variance matters only in experimental economics
- [ ] Because unbiased estimators always have zero variance
> **Explanation:** Mean squared error depends on both squared bias and variance, so empirical work usually weighs both.