Unformatted text preview: estimator because sd βj jx is a function of the unknown σ2 . Melissa Tartari (Yale) Econometrics 12 / 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 estimator because sd βj jx is a function of the unknown σ2 . bh
The natural idea is to replace σ2 with an estimator σ2 with good
i
i
h
ˆ
ˆ jx with se β jx .
properties and, accordingly, replace sd βj
j Melissa Tartari (Yale) Econometrics 12 / 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 estimator because sd βj jx is a function of the unknown σ2 . bh
The natural idea is to replace σ2 with an estimator σ2 with good
i
i
h
ˆ
ˆ jx with se β jx .
properties and, accordingly, replace sd βj
j We introduced such an estimator in Chapter 3, see equation (3.58). Melissa Tartari (Yale) Econometrics 12 / 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 estimator because sd βj jx is a function of the unknown σ2 . bh
The natural idea is to replace σ2 with an estimator σ2 with good
i
i
h
ˆ
ˆ jx with se β jx .
properties and, accordingly, replace sd βj
j We introduced such an estimator in Chapter 3, see equation (3.58).
We will next derive its samplying distribution: this is necessary for
deriving the sampling distribution of the (feasible) estimator Melissa Tartari (Yale) Econometrics ˆ
βj
.
ˆ
se [ βj jx] 12 / 41 The Sampling Distribution When sigma is not Known Theorem 4.2: Under CLM assumptions,
ˆ
βj βj
h
i
ˆ
se βj jx Melissa Tartari (Yale) Econometrics tn K1 (4.3) 13 / 41 The Sampling Distribution When sigma is not Known Theorem 4.2: Under CLM assumptions,
ˆ
βj βj
h
i
ˆ
se βj jx tn K1 (4.3) QUESTION: what is the di¤erence between (4.1 bis) and (4.3) that
motivates the di¤erent result (normal versus tstudent)? Melissa Tartari (Yale) Econometrics 13 / 41 The Sampling Distribution When sigma is not Known Theorem 4.2: Under CLM assumptions,
ˆ
βj βj
h
i
ˆ
se βj jx tn K1 (4.3) QUESTION: what is the di¤erence between (4.1 bis) and (4.3) that
motivates the di¤erent result (normal versus tstudent)?
h
i
ˆ
ANSWER: in (4.1 bis) at the denominator we had sd βj jx while
h
i
ˆ
now in (4.3) we have se βj jx ; WHY? because the unknown σ
h
i
ˆ
ˆ
appearing in sd βj jx has been substituted with the RV σ. Melissa Tartari (Yale) Econometrics 13 / 41 The Sampling Distribution (sigma not Known): “Proof”
From Theorem 4.1, under LR.1LR.6 and conditional on x,
ˆ
βj βj
h
i jx
ˆ
sd βj jx Melissa Tartari (Yale) Econometrics N (0, 1) (4.1 bis) 14 / 41 The Sampling Distribution (sigma not Known): “Proof”
From Theorem 4.1, under LR.1LR.6 and conditional on x,
ˆ
βj βj
h
i jx
ˆ
sd βj jx now write N (0, 1) (4.1 bis) h
i
ˆ
ˆ
ˆ
βj βj
β j β j sd β j j x
z
zj
h
i=r j
h
i=
h
i
=q
wj
2 β jx
ˆ]
se [ j
ˆ
ˆ
ˆ
se βj jx
sd βj jx se βj jx
nK
2 β jx
sd [ ˆ j ] 1 where we let
zj ˆ
βj βj
h
i
ˆ
sd βj jx M elissa Tartari (Yale) and wj Econometrics (n K 1) dˆ
Var βj jx
ˆ
Var βj jx
14 / 41 The Sampling Distribution (sigma not Known): “Proof” cont.
From previous slide:
r
ˆ
βj βj
i = zj /
h
n
ˆ
se βj jx Melissa Tartari (Yale) Econometrics wj
K 1 15 / 41 The Sampling Distribution (sigma not Known): “Proof” cont.
From previous slide: We know that zj jx Melissa Tartari (Yale) r
ˆ
βj βj
i = zj /
h
n
ˆ
se βj jx wj
K 1 N (0, 1). Econometrics 15 / 41 The Sampling Distribution (sigma not Known): “Proof” cont.
From previous slide:
r
ˆ
βj βj
i = zj /
h
n
ˆ
se βj jx We know that zj jx N (0, 1).
We can prove that wj jx χ2n
( Melissa Tartari (Yale) wj
K 1 K 1) : Econometrics 15 / 41 The Sampling Distribution (sigma not Known): “Proof” cont.
From previous slide:
r
ˆ
βj βj
i = zj /
h
n
ˆ
se βj jx We know that zj jx N (0, 1).
We can prove that wj jx χ2n
( wj
K 1 K 1) : dˆ
the intuition for this last result is that Var βj jx is ˆ
σ2 = ∑n=1 ui2 /n K 1 i.e. to the (rescaled) sum of squares of
ˆ
i
independent and identically distributed normal random variables. Melissa Tartari (Yale) Econometrics 15 / 41 The Sampling Distribution (sigma not Known): “Proof” cont.
From previous slide:
r
ˆ
βj βj
i = zj /
h
n
ˆ
se βj jx We know that zj jx N (0, 1).
We can prove that wj jx χ2n
( wj
K 1 K 1) : dˆ
the intuition for this last result is that Var βj jx is ˆ
σ2 = ∑n=1 ui2 /n K 1 i.e. to the (rescaled) sum of squares of
ˆ
i
independent and identically distributed normal random variables. We can show that zj jx and wj jx are independent RVs. Melissa Tartari (Yale) Econometrics 15 / 41 The Sampling Distribution (sigma not Known): “Proof” cont.
From previous slide:
r
ˆ
βj βj
i = zj /
h
n
ˆ
se βj jx We know that zj jx N (0, 1).
We can prove that wj jx χ2n
( wj
K 1 K 1) : dˆ
the intuition for this last result is that Var βj jx is ˆ
σ2 = ∑n=1 ui2 /n K 1 i.e. to the (rescaled) sum of squares of
ˆ
i
independent and identically distributed nor...
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 Fall '10
 DonaldBrown
 Econometrics, Normal Distribution, Yale, Jx, Melissa Tartari

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