【Econometrics】Estimator

As "econometrics" means to "measure" "economics", our goal is to derive the best estimate on an economic relationship. Then, what is the "best" estimator?

 

Properties of Estimator

1. Unbiased (불편; 편향되지 않은)

Equation 1

$$
E(\bar{X}) = \mu
$$

 

• We prefer that estimator be "accurate" or "unbiased" : meaning the estimator will reflect the population parameter as closely as possible.

• Estimator reflects the population parameter when expected value of the sample ($E(\bar{X})$) is the population parameter ($\mu$).

• If this property holds, then we say an estimator is "unbiased".

 

2. Efficient (유효)

• There could be many unbiased estimators for a given parameter. Then, which estimators do we choose?

• Generally, we prefer the estimators with smaller variance.

• If the estimator's distribution is more concentrated around the target parameter, the estimator is considered "efficient".

 

 

3. Consistent (일치)

Equation 2
$$

plim\bar{X} = \mu
$$

• In case we expect the result to vary depending on number of observation (i.e., coin tosses tends to decrease in Head-to-Tail ratio from 100% -> 50% through repetition), we would want an estimator that converges to the population's parameter as sample size increases.

• Formally, if an estimator converges in probability to its target parameter as sample size $n$ goes to infinity, estimator is considered "consistent".

• Hence, "if an estimator is unbiased and its variance tends (coverges) to zero as the number of observation goes to infinity, the the estimator is consistent".