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The exact normality of j hinges crucially on the

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Unformatted text preview: artari (Yale) Econometrics 16 / 27 Large Sample Inference in the MLRM I ˆ We know that under LR.1 - LR.6, βj jx N. This distributional result was the basis for deriving the t and F distributions of the t and F statistics used to test hypothesis about the βj0 s . ˆ The exact normality of βj hinges crucially on the normality of the unobservable u in the population: if ui iid from some distribution other than N , ˆ βj N , and the t and the F statistics will NOT have the t and F distributions. ˆ Recall however that LR.6 plays no role in the unbiasedness of βj , nor ˆ does it a¤ect the conclusion that βj is BLUE under LR.1-LR.5. It is just exact inference based on t and the F statistics that requires LR.6. Melissa Tartari (Yale) Econometrics 16 / 27 Large Sample Inference in the MLRM II How can we perform inference when LR.6 fails? Should we abandon the t and the F statistics? What shall we use in their place? Melissa Tartari (Yale) Econometrics 17 / 27 Large Sample Inference in the MLRM II How can we perform inference when LR.6 fails? Should we abandon the t and the F statistics? What shall we use in their place? Even though the ui0 s (hence the yi0 s ) are not normal when LR.6 does ˆ not hold, we can invoke the CLT and conclude that βj are approximately normally distributed, at least in large samples. Melissa Tartari (Yale) Econometrics 17 / 27 Asymptotic Normality of the OLS Estimator Theorem 5.2: Under ass.s LR.1 through LR.5, ˆ βj βj a ˆ se βj Melissa Tartari (Yale) Econometrics N (0, 1) (5.7) 18 / 27 Asymptotic Normality of the OLS Estimator Theorem 5.2: Under ass.s LR.1 through LR.5, ˆ βj βj a ˆ se βj N (0, 1) (5.7) IMPLICATIONS: Melissa Tartari (Yale) Econometrics 18 / 27 Asymptotic Normality of the OLS Estimator Theorem 5.2: Under ass.s LR.1 through LR.5, ˆ βj βj a ˆ se βj N (0, 1) (5.7) IMPLICATIONS: since as n " the t distribution approaches the normal distribution it is legitimate to write ˆ βj βj a tn K 1 ˆ se βj Melissa Tartari (Yale) Econometrics 18 / 27 Asymptotic Normality of the OLS Estimator Theorem 5.2: Under ass.s LR.1 through LR.5, ˆ βj βj a ˆ se βj N (0, 1) (5.7) IMPLICATIONS: since as n " the t distribution approaches the normal distribution it is legitimate to write ˆ βj βj a tn K 1 ˆ se βj the t testing and the construction of CI are carried out exactly as under LR.6. Melissa Tartari (Yale) Econometrics 18 / 27 Asymptotic Normality of the OLS Estimator WARNING: 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 Melissa Tartari (Yale) Econometrics N. 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...
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