# ie_Slide05 - Introductory Econometrics ECON2206/ECON3209...

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Introductory Econometrics ECON2206/ECON3209 Slides05 Lecturer: Minxian Yang ie_Slides05 my, School of Economics, UNSW 1

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5. Multiple Regression Model: Asymptotics (Ch5) 5. Multiple Regression Model: Asymptotics • Lecture plan – Why large-sample properties (asymptotics) – Consistency of the OLS estimators – Asymptotic normality of the OLS estimators ie_Slides05 my, 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. c) The normality leads to the exact distributions of the t-stat and the F-stat, which are a basis for inference. – 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 my, 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. – The strategy will work if • the asymptotic distribution is available (usually true) and • the sample size n is large. – The strategy does work for the OLS estimators under MLR1-5. ie_Slides05 my, School of Economics, UNSW 4
5. Multiple Regression Model: Asymptotics (Ch5) • Consistency – Let be an estimator for parameter β j , from a sample of size n . is consistent for β j if and only if tends to zero as n goes infinity. (see also Appendix C) – Consistency comes from the LLN (law of large numbers). ie_Slides05 my, School of Economics, UNSW 5 j ˆ j ˆ ) from differes ˆ ( j j P

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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 . – 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. • MLR4′ is not sufficient for “unbiasedness”, and for properly defining PRF. ie_Slides05 my, School of Economics, UNSW 6 j ˆ . ,..., for ) , cov( and ) ( k j x u u E j 1 0 0
5. Multiple Regression Model: Asymptotics (Ch5) • Consistency – In the simple regression model, – Multiply both numerator and denominator by 1/ n . The LLN indicates that

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ie_Slide05 - Introductory Econometrics ECON2206/ECON3209...

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