Unformatted text preview: 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. Unbiased estimators whose variance ! 0 as n ! ∞ are consistent.
Unbiased estimators that use the entire sample will usually satisfy this
requirement. 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. Unbiased estimators whose variance ! 0 as n ! ∞ are consistent.
Unbiased estimators that use the entire sample will usually satisfy this
requirement.
¯
EXAMPLE: we have shown that Bias [Yn ] = 0; since
2
¯
¯
Var [Yn ] = σ ! 0 as n ! ∞ then we conclude that Yn is
n
consistent for µ E [Yi ]. Melissa Tartari (Yale) Econometrics 8 / 27 Consistency of the OLS Estimator
Can we gain an intuitive understanding behind consistency of OLS
estimators? Melissa Tartari (Yale) Econometrics 9 / 27 Consistency of the OLS Estimator
Can we gain an intuitive understanding behind consistency of OLS
estimators?
ˆ
Let βj be the OLS estimator for βj . For each sample size n, we saw
ˆ
that β has a certain pdf, denote this distribution f n for clarity
ˆ
βj j (notice the indexing to the sample size ). Melissa Tartari (Yale) Econometrics 9 / 27 Consistency of the OLS Estimator
Can we gain an intuitive understanding behind consistency of OLS
estimators?
ˆ
Let βj be the OLS estimator for βj . For each sample size n, we saw
ˆ
that β has a certain pdf, denote this distribution f n for clarity
ˆ
βj j (notice the indexing to the sample size ).
ˆ
Under LR.1 to LR.4 βj is unbiased, then it must be that the mean of
n is the true β .
fβ
ˆ
j
j Melissa Tartari (Yale)
Econometrics 9 / 27 Consistency of the OLS Estimator
Can we gain an intuitive understanding behind consistency of OLS
estimators?
ˆ
Let βj be the OLS estimator for βj . For each sample size n, we saw
ˆ
that β has a certain pdf, denote this distribution f n for clarity
ˆ
βj j (notice the indexing to the sample size ).
ˆ
Under LR.1 to LR.4 βj is unbiased, then it must be that the mean 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 . Melissa Tartari (Yale) Econometrics 9 / 27 Consistency of the OLS Estimator
Can we gain an intuitive understanding behind consistency of OLS
estimators?
ˆ
Let βj be the OLS estimator for βj . For each sample size n, we saw
ˆ
that β has a certain pdf, denote this distribution f n for clarity
ˆ
βj j (notice the indexing to the sample size ).
ˆ
Under LR.1 to LR.4 βj is unbiased, then it must be that the mean...
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 Fall '10
 DonaldBrown
 Econometrics, Normal Distribution, Yale, ols estimator, Melissa Tartari

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