Unformatted text preview: ., xK ).
The normality ass. or LR.6 is:
u jx N (E [u jx ] , Var [u jx ]) Observe:
LR.1  LR.6 are called the classical linear model ass.s.
Our LR.6 is slightly di¤erent from the LR.6 of the book; to us LR.6 is
strictly an ass. about the form of the conditional distribution fu jx ; it
does not imply that Var [u jx ] = Var [u ] (i.e it does not imply LR.5);
nor does it imply E [u jx ] = 0 (i.e. it does not imply LR.3). Melissa Tartari (Yale) Econometrics 3 / 41 The Normality Assumption: LR.6
We may use x a shorthand for (x1 , ..., xK ).
The normality ass. or LR.6 is:
u jx N (E [u jx ] , Var [u jx ]) Observe:
LR.1  LR.6 are called the classical linear model ass.s.
Our LR.6 is slightly di¤erent from the LR.6 of the book; to us LR.6 is
strictly an ass. about the form of the conditional distribution fu jx ; it
does not imply that Var [u jx ] = Var [u ] (i.e it does not imply LR.5);
nor does it imply E [u jx ] = 0 (i.e. it does not imply LR.3).
When LR.3, LR.5, and LR. 6 hold, we have what the book calls LR.6:
u jx M elissa Tartari (Yale) N 0, σ2 Econometrics 3 / 41 The Normality Assumption: LR.6
We may use x a shorthand for (x1 , ..., xK ).
The normality ass. or LR.6 is:
u jx N (E [u jx ] , Var [u jx ]) Observe:
LR.1  LR.6 are called the classical linear model ass.s.
Our LR.6 is slightly di¤erent from the LR.6 of the book; to us LR.6 is
strictly an ass. about the form of the conditional distribution fu jx ; it
does not imply that Var [u jx ] = Var [u ] (i.e it does not imply LR.5);
nor does it imply E [u jx ] = 0 (i.e. it does not imply LR.3).
When LR.3, LR.5, and LR. 6 hold, we have what the book calls LR.6:
u jx N 0, σ2 Our de…nition is a “better” de…nition since we can work under LR.6
without LR.5 (see Chp. 8 ).
Melissa Tartari (Yale) Econometrics 3 / 41 The Normality Assumption: An Alternative Rendition Assuming LR.1 through LR.5 + LR.6 implies that
y jx M elissa Tartari (Yale) N βo + β1 x1 + ... + βK xK , σ2 Econometrics 4 / 41 The Normality Assumption: An Alternative Rendition Assuming LR.1 through LR.5 + LR.6 implies that
y jx N βo + β1 x1 + ... + βK xK , σ2 The situation is depicted in Figure 4.1 for the case K = 1. Melissa Tartari (Yale) Econometrics 4 / 41 The Normality Assumption: Figure 4.1 f y x The Simple LRM under Homoschedasticity y x1 x2 E [y  x ]= β o + β1 x x3
x Melissa Tartari (Yale) Econometrics 5 / 41 The Normality Assumption: Justi…ed?
Justi…cation for LR.6
Because u is the sum of many di¤erent unobserved factors a¤ecting
y , we can invoke the CLT to conclude that u has an approximate
normal distribution. Melissa Tartari (Yale) Econometrics 6 / 41 The Normality Assumption: Justi…ed?
Justi…cation for LR.6
Because u is the sum of many di¤erent unobserved factors a¤ecting
y , we can invoke the CLT to conclude that u has an approximate
normal distribution.
Problems with this argument: Melissa Tartari (Yale) Econometrics 6 / 41 The Normality Assumption: Justi…ed?
Justi…cation for LR.6
Because u is the sum of many di¤erent unobserved factors a¤ecting
y , we can invoke the CLT to conclude that u has an approximate
normal distribution.
Problems with this argument:
the factors in u can have very di¤erent distributions in the population;
while the CLT can still hold in such cases, the normal approximation
can be poor depending on how many factors appear in u and how
di¤erent are their distributions; Melissa Tartari (Yale) Econometrics 6 / 41 The Normality Assumption: Justi…ed?
Justi…cation for LR.6
Because u is the sum of many di¤erent unobserved factors a¤ecting
y , we can invoke the CLT to conclude that u has an approximate
normal distribution.
Problems with this argument:
the factors in u can have very di¤erent distributions in the population;
while the CLT can still hold in such cases, the normal approximation
can be poor depending on how many factors appear in u and how
di¤erent are their distributions;
the argument assumes that all unobserved factors a¤ecting y do so in
an additive and separate fashion; nothing guarantees that this is so, in
particular, if u is a complicated function of the unobserved factors,
then the CLT argument does not apply; Melissa Tartari (Yale) Econometrics 6 / 41 The Normality Assumption: Justi…ed?
Justi…cation for LR.6
Because u is the sum of many di¤erent unobserved factors a¤ecting
y , we can invoke the CLT to conclude that u has an approximate
normal distribution.
Problems with this argument:
the factors in u can have very di¤erent distributions in the population;
while the CLT can still hold in such cases, the normal approximation
can be poor depending on how many factors appear in u and how
di¤erent are their distributions;
the argument assumes that all unobserved factors a¤ecting y do so in
an additive and separate fashion; nothing guarantees that this is so, in
particular, if u is a complicated function of the unobserved factors,
then the CLT argument does not apply;
LR.6 is certainly false if y takes on just a few values: in such case it
cannot have anything close to a normal distribution (see chapter 13)
Melissa Tartari (Yale) Econometrics 6 / 41 The Sampling Distribution of the OLS Estimator We derive the sampling distribution of b under two circumnstances :
β 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 Melissa Tartari (Yale) Econometrics...
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 Fall '10
 DonaldBrown
 Econometrics, Normal Distribution, Yale, Jx, Melissa Tartari

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