Unformatted text preview: larger sample
size is usually needed to use the t approximation. 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 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 larger sample
size is usually needed to use the t approximation.
Theorem 5.2 requires LR.1 through LR.5 (in particular,
homoshedasticity is used). 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 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 larger sample
size is usually needed to use the t approximation.
Theorem 5.2 requires LR.1 through LR.5 (in particular,
homoshedasticity is used). ˆ
Additional IMPLICATIONS: asymptotic normality of βj implies that
the F statistics has approximate F distribution in large sample sizes,
thus for testing exclusion restrictions or the other joint hypothesis,
nothing changes from what we have done before. Melissa Tartari (Yale) Econometrics 19 / 27 Asymptotic Normality of the OLS Estimator For notational simplicity we focus on Theorem 5.2 within the SLRM:
under ass.s LR.1LR.5.
pˆ
2
a
1
n β1 β1 jx N 0, σ2
s
x 2
3 2 ˆ
σ is a consistent estimator of σ2
ˆ
( β1 β1 ) jx a
N (0, 1)
ˆ
se ( β1 jx ) Melissa Tartari (Yale) Econometrics 20 / 27 Asymptotic Normality of the OLS Estimator: an intutive
proof
Observe that p Melissa Tartari (Yale) ˆ
n β1 β1 p ∑n=1 (xi x ) ui
¯
ni
2
sx
p n (xi x ) ui
¯
= σ n∑
2
sx
σ
i =1
pn
ui
= σ n ∑ ωi
σ
i =1
= Econometrics 21 / 27 Asymptotic Normality of the OLS Estimator: an intutive
proof
Observe that p ˆ
n β1 β1 p ∑n=1 (xi x ) ui
¯
ni
2
sx
p n (xi x ) ui
¯
= σ n∑
2
sx
σ
i =1
pn
ui
= σ n ∑ ωi
σ
i =1
= if ω i = 1 all i then by a direct application of the CLT we would know
p
d
that n ∑n=1 ui ! N (0, 1); because ω i 6= 1 we have instead
i
σ
pn
d
n ∑i =1 ω i ui ! N (0, s12 ) and the result in (1) is proven.
σ
x Melissa Tartari (Yale) Econometrics 21 / 27 An Exercise in STATA The do …le that implements this exercise is
Slides_Ch5_STATA_exercise.do. Melissa Tartari (Yale) Econometrics 22 / 27 Part I: Normal Disturbances I Let ( βo , β1 ) = (3, 1) and consider the population model = βo + β1 x 2 + u
u
N (0, 0.64)
x
N (2, 1) y Melissa Tartari (Yale) Econometrics 23 / 27 Part I: Normal Disturbances I Let ( βo , β1 ) = (3, 1) and consider the population model = βo + β1 x 2 + u
u
N (0, 0.64)
x
N (2, 1) y In STATA I draw N = 500 samples, each of size M = 100, from the
population. For each I compute the OLS estimates of ( βo , β1 ); then I
sort the estimates and graph them into 2 histograms. Melissa Tartari (Yale) Econometrics 23 / 27 Part I: Normal Disturbances II
Observe that the re...
View
Full
Document
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

Click to edit the document details