slides_Ch5_W

The quality of the approximation depends not just on

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Unformatted text preview: larger sample size is usually needed to use the t approximation. Melissa Tartari (Yale) Econometrics 19 / 27 Asymptotic Normality of the OLS Estimator WARNING: if n is not very large, then the t distribution can be a poor approximation to the distribution of the t statistics when u N . the quality of the approximation depends not just on n, but on the df = n K 1: with more indep vars in the model, a larger sample size is usually needed to use the t approximation. Theorem 5.2 requires LR.1 through LR.5 (in particular, homoshedasticity is used). Melissa Tartari (Yale) Econometrics 19 / 27 Asymptotic Normality of the OLS Estimator WARNING: if n is not very large, then the t distribution can be a poor approximation to the distribution of the t statistics when u N . the quality of the approximation depends not just on n, but on the df = n K 1: with more indep vars in the model, a larger sample size is usually needed to use the t approximation. Theorem 5.2 requires LR.1 through LR.5 (in particular, homoshedasticity is used). ˆ Additional IMPLICATIONS: asymptotic normality of βj implies that the F statistics has approximate F distribution in large sample sizes, thus for testing exclusion restrictions or the other joint hypothesis, nothing changes from what we have done before. Melissa Tartari (Yale) Econometrics 19 / 27 Asymptotic Normality of the OLS Estimator For notational simplicity we focus on Theorem 5.2 within the SLRM: under ass.s LR.1-LR.5. pˆ 2 a 1 n β1 β1 jx N 0, σ2 s x 2 3 2 ˆ σ is a consistent estimator of σ2 ˆ ( β1 β1 ) jx a N (0, 1) ˆ se ( β1 jx ) Melissa Tartari (Yale) Econometrics 20 / 27 Asymptotic Normality of the OLS Estimator: an intutive proof Observe that p Melissa Tartari (Yale) 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 = 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, from the population. For each I compute the OLS estimates of ( βo , β1 ); then I sort the estimates and graph them into 2 histograms. Melissa Tartari (Yale) Econometrics 23 / 27 Part I: Normal Disturbances II Observe that the re...
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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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