Tobias Econ 472

A Tale of Two Estimators:Unbiased and Consistent?

Tobias Econ 472

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.”

Tobias Econ 472

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.

Tobias Econ 472

Formalizing the result

• The golfing “joke” is analogous to the following estimator of a parameter :

• This estimator is unbiased since

Tobias Econ 472

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.

Tobias Econ 472

Another estimator

• Now, consider a different estimator of the parameter :

• This estimator is clearly biased since:

• This bias, however, does vanish as n ! 1

Tobias Econ 472

Is this second estimator consistent?

• The following slides illustrate what is happening to the sampling distributions as n ! 1

Tobias Econ 472

Tobias Econ 472

Tobias Econ 472

Tobias Econ 472

Tobias Econ 472

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