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Unformatted text preview: artari (Yale) Econometrics 16 / 27 Large Sample Inference in the MLRM I
ˆ
We know that under LR.1  LR.6, βj jx N. This distributional result was the basis for deriving the t and F
distributions of the t and F statistics used to test hypothesis about
the βj0 s .
ˆ
The exact normality of βj hinges crucially on the normality of the
unobservable u in the population: if ui
iid from some distribution
other than N ,
ˆ
βj
N , and
the t and the F statistics will NOT have the t and F distributions. ˆ
Recall however that LR.6 plays no role in the unbiasedness of βj , nor
ˆ
does it a¤ect the conclusion that βj is BLUE under LR.1LR.5. It is
just exact inference based on t and the F statistics that requires
LR.6.
Melissa Tartari (Yale) Econometrics 16 / 27 Large Sample Inference in the MLRM II How can we perform inference when LR.6 fails? Should we abandon
the t and the F statistics? What shall we use in their place? Melissa Tartari (Yale) Econometrics 17 / 27 Large Sample Inference in the MLRM II How can we perform inference when LR.6 fails? Should we abandon
the t and the F statistics? What shall we use in their place?
Even though the ui0 s (hence the yi0 s ) are not normal when LR.6 does
ˆ
not hold, we can invoke the CLT and conclude that βj are
approximately normally distributed, at least in large samples. Melissa Tartari (Yale) Econometrics 17 / 27 Asymptotic Normality of the OLS Estimator
Theorem 5.2: Under ass.s LR.1 through LR.5,
ˆ
βj βj a ˆ
se βj Melissa Tartari (Yale) Econometrics N (0, 1) (5.7) 18 / 27 Asymptotic Normality of the OLS Estimator
Theorem 5.2: Under ass.s LR.1 through LR.5,
ˆ
βj βj a ˆ
se βj N (0, 1) (5.7) IMPLICATIONS: Melissa Tartari (Yale) Econometrics 18 / 27 Asymptotic Normality of the OLS Estimator
Theorem 5.2: Under ass.s LR.1 through LR.5,
ˆ
βj βj a ˆ
se βj N (0, 1) (5.7) IMPLICATIONS:
since as n " the t distribution approaches the normal distribution it is
legitimate to write
ˆ
βj βj a
tn K 1
ˆ
se βj Melissa Tartari (Yale) Econometrics 18 / 27 Asymptotic Normality of the OLS Estimator
Theorem 5.2: Under ass.s LR.1 through LR.5,
ˆ
βj βj a ˆ
se βj N (0, 1) (5.7) IMPLICATIONS:
since as n " the t distribution approaches the normal distribution it is
legitimate to write
ˆ
βj βj a
tn K 1
ˆ
se βj
the t testing and the construction of CI are carried out exactly as under
LR.6. Melissa Tartari (Yale) Econometrics 18 / 27 Asymptotic Normality of the OLS Estimator WARNING: Melissa Tartari (Yale) Econometrics 19 / 27 Asymptotic Normality of the OLS Estimator WARNING:
if n is not very large, then the t distribution can be a poor
approximation to the distribution of the t statistics when u Melissa Tartari (Yale) Econometrics N. 19 / 27 Asymptotic Normality of the OLS Estimator WARNING:
if n is not very large, then the t distribution can be a poor
approximation to the distribution of the t statistics when u N .
the quality of the approximation depends not just on n, but on the
df = n K 1: with more indep vars in the model, a...
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 Fall '10
 DonaldBrown
 Econometrics

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