A Tale of Two Estimators: Unbiased and Consistent?

1. A Tale of Two Estimators:Unbiased and Consistent? Econ 472

2. A Motivating “Joke”: • Consider the following joke: • Please bear in mind that economists (especially econometricians!) are not all that funny: • “Three econometricians go golfing. The first golfer shanks her drive 30 feet to the left of the fairway. The second one shanks her drive 30 feet to the right. The third one then jumps up and down in celebration of how well they are performing.” Econ 472

3. Why is this funny? • Well, it’s not funny, actually. • But what it does illustrate is the idea of unbiasedness – on average, they are performing well. Econ 472

4. Formalizing the result • The golfing “joke” is analogous to the following estimator of a parameter : • This estimator is unbiased since Econ 472

5. Consistent? • Is the estimator described in the pervious slide a consistent estimator of ? Clearly not. The sample size n has no impact whatsoever on the estimator. As the sample size grows, the sampling distribution is always the same and places no mass on  itself. Econ 472

6. Another estimator • Now, consider a different estimator of the parameter : • This estimator is clearly biased since: • This bias, however, does vanish as n !1 Econ 472

7. Is this second estimator consistent? • The following slides illustrate what is happening to the sampling distributions as n !1 Econ 472

8. Econ 472

9. Econ 472

10. Econ 472

11. Econ 472

12. Consistency, continued • This estimator is consistent since its sampling distribution is collapsing around  as n !1. • That is, for any  > 0, there is an n sufficiently large such that all of the mass of the sampling density is within  of  Econ 472