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# Or lr6 is u jx n e u jx var u jx observe lr1

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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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