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Unformatted text preview: i ji = 1, ..., ng be a random sample where each Yn has mean µ
and variance σ2 . Then, the CTL states that p Yn
n Melissa Tartari (Yale) µ a σ Econometrics N (0, 1) 14 / 27 Asymptotic Normality: the Central Limit Theorem I Let fYi ji = 1, ..., ng be a random sample where each Yn has mean µ
and variance σ2 . Then, the CTL states that p Yn
n µ a σ N (0, 1) That is, the average from any population, when standardized (recall
2
that Var Y n = σ ), has an asymptotic standard normal distribution.
n Melissa Tartari (Yale) Econometrics 14 / 27 Asymptotic Normality: the Central Limit Theorem II
p To use the de…nition of asymptotic normality let Zn
each n so that you have a sequence fZn jn = 1, 2...g = M elissa Tartari (Yale) p Y1
1 Econometrics σ µ p Y2
,2 µ
σ n Yn µ
σ for , ... 15 / 27 Asymptotic Normality: the Central Limit Theorem II
p To use the de…nition of asymptotic normality let Zn
each n so that you have a sequence fZn jn = 1, 2...g = p Y1
1 σ µ p Y2
,2 µ
σ n Yn µ
σ for , ... Then, the CLT says that for any z the following sequence of real
numbers converges to Φ (z ), that is, to the real number
corresponding to the area below the density of a standard normal and
to the left of z : fPr ob (Z1 Melissa Tartari (Yale) z ) , Pr ob (Z2 Econometrics z ) , ...g . 15 / 27 Large Sample Inference in the MLRM I
ˆ
We know that under LR.1  LR.6, βj jx Melissa Tartari (Yale) Econometrics N. 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 . Melissa Tartari (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 , Melissa Tartari (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 Melissa Tartari (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. Melissa T...
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 Fall '10
 DonaldBrown
 Econometrics

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