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Unformatted text preview: mples, we would obtain θ . Unbiasedness does
not mean that the estimate we get with any particular sample is equal
to θ or even close to it! Unbiasedness only ensures that the sampling
distribution of W has mean value equal to the parameter it is
supposed to be estimating.
Melissa Tartari (Yale) Econometrics 3 / 27 Large sample properties When we refer to properties of an estimator that hold as the sample
size grows without bound we are talking of asymptotic or large
sample properties. Melissa Tartari (Yale) Econometrics 4 / 27 Large sample properties When we refer to properties of an estimator that hold as the sample
size grows without bound we are talking of asymptotic or large
sample properties.
EXAMPLE: Consistency Melissa Tartari (Yale) Econometrics 4 / 27 Consistency: A Review I De…nition: let Wn be an estimator of θ based on fY1 , ..., Yn g. Then
Wn is a consistent estimator of θ if 8ε > 0
P ( j Wn Melissa Tartari (Yale) θ j > ε) ! 0 as n ! ∞ Econometrics (C.7) 5 / 27 Consistency: A Review I De…nition: let Wn be an estimator of θ based on fY1 , ..., Yn g. Then
Wn is a consistent estimator of θ if 8ε > 0
P ( j Wn θ j > ε) ! 0 as n ! ∞ (C.7) Meaning: consistency concerns how far Wn is likely to be from the
parameter it is supposed to be estimating as we let the sample size "
inde…nitely. (C.7) means that the distribution of Wn becomes more
and more concentrated about θ which roughly means that for large n,
Wn is less and less likely to be very far from θ . Melissa Tartari (Yale) Econometrics 5 / 27 Consistency: A Review II
Consistency: Sampling Distribution of Wn for 3 sample sizes fWn ( w)
p limWn = θ
n=40 n=16 n=4 w θ
Melissa Tartari (Yale) Econometrics 6 / 27 Consistency: A Review III Consistency is a property of point estimators. Melissa Tartari (Yale) Econometrics 7 / 27 Consistency: A Review III Consistency is a property of point estimators.
If an estimator is not consistent (inconsistent) it does not help us
learn about θ even with an unlimited amount of data (for this reason
consistency is a minimal requirement of an estimator). Melissa Tartari (Yale) Econometrics 7 / 27 Consistency: A Review III Consistency is a property of point estimators.
If an estimator is not consistent (inconsistent) it does not help us
learn about θ even with an unlimited amount of data (for this reason
consistency is a minimal requirement of an estimator).
Consistency involves the behavior of the sampling distribution of the
estimator as n ". Melissa Tartari (Yale) Econometrics 7 / 27 Consistency: A Review IV Unbiased estimators are not necessarily consistent. Melissa Tartari (Yale) Econometrics 8 / 27 Consistency: A Review IV Unbiased estimators are not necessarily consistent.
EXAMPLE: we have shown that Bias [Y1 ] = 0; however Var [Y1 ] = σ2
and so Var [Y1 ] 6= 0 and ? n. This estimator uses only the …rst
observation. Y1 is an unbiased but inconsistent estimator of µ. Melissa Tartari (Yale) Econometrics 8 / 27 Consistency: A Review IV Unbiased...
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 Fall '10
 DonaldBrown
 Econometrics

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