A Tale of Two Estimators: Unbiased and Consistent?

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A Tale of Two EstimatorsUnbiased and Consistent?
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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.

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Why is this funny?

• Well, its not funny, actually.
• But what it does illustrate is the idea of
  unbiasedness on average, they are performing
  well.

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Formalizing the result

• The golfing joke is analogous to the following
  estimator of a parameter ?
• This estimator is unbiased since

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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.

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Another estimator

• Now, consider a different estimator of the
  parameter ?
• This estimator is clearly biased since
• This bias, however, does vanish as n ! 1

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Is this second estimator consistent?

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

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Consistency, continued

• This estimator is consistent since its sampling
  distribution is collapsing around ? as n ! 1.
• That is, for any ? gt 0, there is an n
  sufficiently large such that all of the mass of
  the sampling density is within ? of ?