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Sampling distribution of b under two circumnstances

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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.1-LR.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.1-LR.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.1-LR.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.1-LR.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 GM-theorem). 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.

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