Unformatted text preview: of
n is the true β .
fβ
ˆ
j
j n
ˆ
If βj is also consistent, then fβ becomes more and more tightly
ˆ
j n
distributed around βj as n grows; as n ! ∞, fβ collapses to the
ˆ
j single point βj . Consistency means that we can make our estimator arbitrarily close
to βj if we can collect as much data as we want.
Melissa Tartari (Yale) Econometrics 9 / 27 Consistency of the OLS Estimator: Theorem 5.1 Conveniently, the same ass.s imply both unbiasedness and consistency
ˆ
of OLS: Theorem 5.1 states that under LR.1LR.4, βj is (unbiased
and) consistent. Melissa Tartari (Yale) Econometrics 10 / 27 Consistency of the OLS Estimator: Theorem 5.1 Conveniently, the same ass.s imply both unbiasedness and consistency
ˆ
of OLS: Theorem 5.1 states that under LR.1LR.4, βj is (unbiased
and) consistent.
Actually, consistency obtains under a less stringent version of LR.3
which we call LR.3’or Zero Mean and Zero Correlation (ZC):
E [u ] = 0 and Cov (u , xk ) = 0 for k = 1, ..., K . Melissa Tartari (Yale) Econometrics 10 / 27 Consistency of the OLS Estimator: Theorem 5.1 Conveniently, the same ass.s imply both unbiasedness and consistency
ˆ
of OLS: Theorem 5.1 states that under LR.1LR.4, βj is (unbiased
and) consistent.
Actually, consistency obtains under a less stringent version of LR.3
which we call LR.3’or Zero Mean and Zero Correlation (ZC):
E [u ] = 0 and Cov (u , xk ) = 0 for k = 1, ..., K .
(Recall: we showed LR .3 ) LR .30; in general LR .30 ; LR .3). Melissa Tartari (Yale) Econometrics 10 / 27 Consistency of the OLS Estimator: Theorem 5.1 Conveniently, the same ass.s imply both unbiasedness and consistency
ˆ
of OLS: Theorem 5.1 states that under LR.1LR.4, βj is (unbiased
and) consistent.
Actually, consistency obtains under a less stringent version of LR.3
which we call LR.3’or Zero Mean and Zero Correlation (ZC):
E [u ] = 0 and Cov (u , xk ) = 0 for k = 1, ..., K .
(Recall: we showed LR .3 ) LR .30; in general LR .30 ; LR .3). Just as correlation between the disturbance and the regressors causes
ˆ
bias in βj , it also generally causes inconsistency (we then speak of
asymptotic bias). Melissa Tartari (Yale) Econometrics 10 / 27 Asymptotic Normality: Why relevant? Consistency does not allow us to perform statistical inference: simply
ˆ
knowing that βj is getting closer to βj does not allow us to test
hypothesis about βj . Melissa Tartari (Yale) Econometrics 11 / 27 Asymptotic Normality: Why relevant? Consistency does not allow us to perform statistical inference: simply
ˆ
knowing that βj is getting closer to βj does not allow us to test
hypothesis about βj .
ˆ
For testing we need the sampling distribution of β .
j Melissa Tartari (Yale) Econometrics 11 / 27 Asymptotic Normality: Why relevant? Consistency does not allow us to perform statistical inference: simply
ˆ
knowing that βj is getting closer to βj does not allow us to test
hypothesis about βj .
ˆ
For testing we need the sampling distribution of β .
j In Chapter 4 w...
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This note was uploaded on 02/13/2014 for the course ECON 350 taught by Professor Donaldbrown during the Fall '10 term at Yale.
 Fall '10
 DonaldBrown
 Econometrics

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