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# ie_Slide05 - Introductory Econometrics ECON2206/ECON3209...

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Introductory Econometrics ECON2206/ECON3209 Slides05 Lecturer: Rachida Ouysse ie_Slides05 RO, School of Economics, UNSW 1

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5. Multiple Regression Model: Asymptotics (Ch5) 5. Multiple Regression Model: Asymptotics • Lecture plan hy large- ample properties ( symptotics Why large sample properties (asymptotics) – Consistency of the OLS estimators symptotic normality of the OLS estimators – Asymptotic normality of the OLS estimators ie_Slides05 RO, School of Economics, UNSW 2
5. Multiple Regression Model: Asymptotics (Ch5) • What we need for inference – We need the sampling distribution of the OLS estimators a) MLR1-4 imply the OLS estimators are unbiased. b) MLR1-6 (CLM) imply the OLS estimators are normally distributed. ) he normality leads to the xact istributions of the c) The normality leads to the exact distributions of the t-stat and the F-stat, which are a basis for inference. LR6 ( ~ ormal is often too strong an MLR6 ( u iid Normal ) is often too strong an assumption in practice. • Without MLR6, the results in b) and c) no longer hold. • But they hold approximately for large samples . • Inference will be based on large-sample approximation. ie_Slides05 RO, School of Economics, UNSW 3

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5. Multiple Regression Model: Asymptotics (Ch5) • Asymptotic (large-sample) analysis – Reluctant to assume MLR6, we proceed as follows. • find the asymptotic distribution of the estimators (the sampling distribution when n goes to infinity). • use the asymptotic distribution to approximate the sampling distribution of the estimators. he strategy will work if – The strategy will work if • the asymptotic distribution is available (usually true) nd and • the sample size n is large. – The strategy does work for the OLS estimators under MLR1-5. ie_Slides05 RO, School of Economics, UNSW 4
5. Multiple Regression Model: Asymptotics (Ch5) • Consistency – Let be an estimator for parameter β j , from a sample j ˆ of size n . –i s consistent for β j j ˆ if and only if ) from differes ˆ ( j j P tends to zero as n goes infinity. (see also Appendix C) – Consistency comes from the LLN (law of large numbers). ie_Slides05 RO, School of Economics, UNSW 5

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5. Multiple Regression Model: Asymptotics (Ch5) • Consistency Theorem 5.1 (consistency of OLS) Under MLR1 to MLR4, the OLS estimator is consistent for β j , for all j = 0,1,. .., k . j ˆ – In fact, the consistency holds under an assumption weaker than MLR4 (ZCM). MLR4 (zero mean and zero correlation) • MLR4 implies MLR4 . Not-MLR4 implies Not-MLR4. . ,..., for ) , cov( and ) ( k j x u u E j 1 0 0 •MLR4 is not sufficient for “unbiasedness”, and for properly defining PRF. ie_Slides05 RO, School of Economics, UNSW 6
5. Multiple Regression Model: Asymptotics (Ch5) • Consistency – In the simple regression model, LLN: for a random sample, .

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ie_Slide05 - Introductory Econometrics ECON2206/ECON3209...

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