slides_Ch5_W

# The exact normality of j hinges crucially on the

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

## This note was uploaded on 02/13/2014 for the course ECON 350 taught by Professor Donaldbrown during the Fall '10 term at Yale.

Ask a homework question - tutors are online