slides_Ch5_W

Unbiasedness does not mean that the estimate we get

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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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