Unformatted text preview: 7 / 41 The Sampling Distribution of the OLS Estimator We derive the sampling distribution of b under two circumnstances :
β
CASE 1: σ2 is known
CASE 2: σ2 is not known Melissa Tartari (Yale) Econometrics 7 / 41 The Sampling Distribution of the OLS Estimator We derive the sampling distribution of b under two circumnstances :
β
CASE 1: σ2 is known
CASE 2: σ2 is not known As you might suspect, σ2 is typically not known; however, it is
instructive to start by assuming that we know σ2 and then explore the
implications of relaxing such assumption. Melissa Tartari (Yale) Econometrics 7 / 41 The Sampling Distribution of the OLS Estimator We derive the sampling distribution of b under two circumnstances :
β
CASE 1: σ2 is known
CASE 2: σ2 is not known As you might suspect, σ2 is typically not known; however, it is
instructive to start by assuming that we know σ2 and then explore the
implications of relaxing such assumption.
(Notation: I will use sigma in place of σ2 in the title of the slides b/c
σ2 is not accepted by the editor.) Melissa Tartari (Yale) Econometrics 7 / 41 The Sampling Distribution (sigma known)
Normality of u (LR.6) translates into normality of the OLS estimator. Melissa Tartari (Yale) Econometrics 8 / 41 The Sampling Distribution (sigma known)
Normality of u (LR.6) translates into normality of the OLS estimator.
Theorem 4.1 Under LR.1LR.6, conditional on the sample values of
the eVars x, Melissa Tartari (Yale) Econometrics 8 / 41 The Sampling Distribution (sigma known)
Normality of u (LR.6) translates into normality of the OLS estimator.
Theorem 4.1 Under LR.1LR.6, conditional on the sample values of
the eVars x,
ˆ
the distribution of βj is normal:
h
i
h
i
ˆ
ˆ
N E βj jx , Var βj jx ˆ
βj jx , Melissa Tartari (Yale) ˆ
βj βj
h
i
ˆ
sd βj jx Econometrics N (0, 1) (4.1)
(4.1 bis) 8 / 41 The Sampling Distribution (sigma known)
Normality of u (LR.6) translates into normality of the OLS estimator.
Theorem 4.1 Under LR.1LR.6, conditional on the sample values of
the eVars x,
ˆ
the distribution of βj is normal:
h
i
h
i
ˆ
ˆ
N E βj jx , Var βj jx ˆ
βj jx , ˆ
βj βj
h
i
ˆ
sd βj jx ˆ0
any linear combination of the βj s is Melissa Tartari (Yale) Econometrics N (0, 1) (4.1)
(4.1 bis) N. 8 / 41 The Sampling Distribution (sigma known)
Normality of u (LR.6) translates into normality of the OLS estimator.
Theorem 4.1 Under LR.1LR.6, conditional on the sample values of
the eVars x,
ˆ
the distribution of βj is normal:
h
i
h
i
ˆ
ˆ
N E βj jx , Var βj jx ˆ
βj jx , ˆ
βj βj
h
i
ˆ
sd βj jx N (0, 1) (4.1)
(4.1 bis) ˆ0
any linear combination of the βj s is N .
ˆ0
any collection of βj s has a joint distribution that is multiv. N . Melissa Tartari (Yale) Econometrics 8 / 41 The Sampling Distribution (sigma known): “Proof”
Consider the simple RM yi = βo + β1 xi + ui ; we showed that
¯
¯
∑n (xi x ) (yi y )
ˆ
β 1 = i =1 n
2
¯
∑i =1 (xi x ) Melissa Tartari (Yale) Econometrics 9 / 41 The Sampling Distribution (sigma known): “Proof”
Consider the simple RM yi = βo + β1 xi + ui ; we showed that
¯
¯
∑n (xi x ) (yi y )
ˆ
β 1 = i =1 n
2
¯
∑i =1 (xi x )
Substitute yi = βo + β1 xi + ui and y = βo + β1 x + u to obtain
¯
¯
∑n=1 (xi
i x ) (( βo + β1 xi + ui ) ( βo + β1 x + u ))
¯
¯
2
sx
n
¯
¯
∑i =1 (xi x ) ( β1 (xi x ) + (ui u ))
=
2
sx
¯
¯
¯
∑n (xi x ) (xi x ) ∑n=1 (xi x ) ui
= β 1 i =1
+i
= ...
2
2
sx
sx ˆ
β1 = Melissa Tartari (Yale) Econometrics 9 / 41 The Sampling Distribution (sigma known): “Proof”  cont.
From previous slide ...
2 s
ˆ
β1 = β1 x +
2
sx ¯
∑n=1 (xi x ) ui
i
2
sx
n = β1 + ∑ wi ui i =1  {z } where wi (xi x)
¯
2
sx . linear combination of n iidRV N (0 ,σ2 ) Melissa Tartari (Yale) Econometrics 10 / 41 The Sampling Distribution (sigma known): “Proof”  cont.
From previous slide ...
2 s
ˆ
β1 = β1 x +
2
sx ¯
∑n=1 (xi x ) ui
i
2
sx
n = β1 + ∑ wi ui where wi i =1  {z } (xi x)
¯
2
sx . linear combination of n iidRV N (0 ,σ2 ) Recall that any linear combination of iid Normal RVs is also a normal
RV; thus, we conclude
ˆ
β1 jx M elissa Tartari (Yale) Normal Econometrics 10 / 41 The Sampling Distribution (sigma known): “Proof”  cont.
From previous slide ...
2 s
ˆ
β1 = β1 x +
2
sx ¯
∑n=1 (xi x ) ui
i
2
sx
n = β1 + ∑ wi ui where wi i =1  {z } (xi x)
¯
2
sx . linear combination of n iidRV N (0 ,σ2 ) Recall that any linear combination of iid Normal RVs is also a normal
RV; thus, we conclude
ˆ
β1 jx Normal An aside: verify that ∑n=1 wi = 0 (recall the proof of the
i
GMtheorem). Which ui are attached to the largest (in abs value)
weights? How would you prove that e.g. the sum of two normal
random variables is a normal random variable? Recall you statistics
class.
Melissa Tartari (Yale) Econometrics 10 / 41 The Sampling Distribution (sigma known): Preview We will show that the normality of the OLS estimators is still
approximately true in large samples even without LR.6. Melissa Tartari (Yale) Econometrics 11 / 41 The Sampling Distribution When sigma is not Known When σ2 is not known, ˆ
βj
ˆ
sd [ β jx]
j i
h
ˆ
is not an...
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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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