slides_Ch5_W[1]

N 0 1 because i 6 1 we have instead i pn d n i 1 i

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Unformatted text preview: σ i =1 pn ui = σ n ∑ ωi σ i =1 = Econometrics 21 / 27 Asymptotic Normality of the OLS Estimator: an intutive proof Observe that p ˆ n β1 β1 p ∑n=1 (xi x ) ui ¯ ni 2 sx p n (xi x ) ui ¯ = σ n∑ 2 sx σ i =1 pn ui = σ n ∑ ωi σ i =1 = if ω i = 1 all i then by a direct application of the CLT we would know p d that n ∑n=1 ui ! N (0, 1); because ω i 6= 1 we have instead i σ pn d n ∑i =1 ω i ui ! N (0, s12 ) and the result in (1) is proven. σ x Melissa Tartari (Yale) Econometrics 21 / 27 An Exercise in STATA The do …le that implements this exercise is Slides_Ch5_STATA_exercise.do. Melissa Tartari (Yale) Econometrics 22 / 27 Part I: Normal Disturbances I Let ( βo , β1 ) = (3, 1) and consider the population model = βo + β1 x 2 + u u N (0, 0.64) x N (2, 1) y Melissa Tartari (Yale) Econometrics 23 / 27 Part I: Normal Disturbances I Let ( βo , β1 ) = (3, 1) and consider the population model = βo + β1 x 2 + u u N (0, 0.64) x N (2, 1) y In STATA I draw N = 500 samples, each of size M = 100, fr...
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