How to Determine Whether an Estimator Is Good
Statisticians and econometricians typically require the estimators they use for inference and prediction to have certain desirable properties. For statisticians, unbiasedness and efficiency are the two most-desirable properties an estimator can have.
An estimator is unbiased if, in repeated estimations using the method, the mean value of the estimator coincides with the true parameter value. An estimator is efficient if it achieves the smallest variance among estimators of its kind. In some instances, statisticians and econometricians spend a considerable amount of time proving that a particular estimator is unbiased and efficient.
Sometimes statisticians and econometricians are unable to prove that an estimator is unbiased. In that case, they usually settle for consistency. An estimator is consistent if it approaches the true parameter value as the sample size gets larger and larger. For this reason, consistency is known as an asymptotic property for an estimator; that is, it gradually approaches the true parameter value as the sample size approaches infinity.
In practical situations (that is, when you’re working with data and not just doing a theoretical exercise), knowing when an estimator has these desirable properties is good, but you don’t need to prove them on your own. You simply want to know the result of the proof (if it exists) and the assumptions needed to carry it out.
Besides unbiasedness and efficiency, an additional desirable property for some estimators is linearity. An estimator has this property if a statistic is a linear function of the sample observations.
This property isn’t present for all estimators, and certainly some estimators are desirable (efficient and either unbiased or consistent) without being linear. The linearity property, however, can be convenient when you’re using algebraic manipulations to create new variables or prove other estimator properties.