EMET2007 Lecture 4 - EMET2007/6007 Econometrics I...

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EMET2007/6007 Econometrics I: Econometric Methods Professor Rodney W. Strachan, ANU econometrics and simple linear regression 13 th March 2013 Lecture 4 (econometrics and simple linear regression) EMET2007/6007 13 th March 2013 1 / 50
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RODNEY: TURN ON THE RECORDING! Lecture 4 (econometrics and simple linear regression) EMET2007/6007 13 th March 2013 2 / 50
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Simple Linear Regression: Inference Introductory Econometrics: A Modern Approach by Je/rey M. Wooldridge, 4e This will be a bit of Chapter 2 and a bit of Chapter 4 but purely for the simple linear regression model Lecture 4 (econometrics and simple linear regression) EMET2007/6007 13 th March 2013 3 / 50
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Statistical inference in the regression model Hypothesis tests about population parameters Construction of con°dence intervals Conducting hypothesis tests requires we know something of the distribution of the estimators Lecture 4 (econometrics and simple linear regression) EMET2007/6007 13 th March 2013 4 / 50
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Examples: Does x have an e/ect on the mean of y ? Does x have a positive e/ect on the mean of y ? Recall the mean of y depends upon x as E ( y j x ) = β 0 + β 1 x So these questions are asking: Is β 1 = 0 ? (Against β 1 6 = 0) Is β 1 > 0 ? (Against β 1 ° 0) Is β 1 = a ? (Against β 1 6 = a ) These will become our (null or alternative) hypotheses Lecture 4 (econometrics and simple linear regression) EMET2007/6007 13 th March 2013 5 / 50
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Sampling distributions of the OLS estimators The OLS estimators are random variables We already know their expected values and their variances However, for hypothesis tests we need to know their distribution In order to derive their distribution we need additional assumptions Assumption about distribution of errors: normal distribution Lecture 4 (econometrics and simple linear regression) EMET2007/6007 13 th March 2013 6 / 50
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Assumption SLR.6 (Normality of error terms) ε i s N ° 0 , σ 2 ± independently of x i which implies y i s N ° β 0 + β 1 x i , σ 2 ± It is assumed that the unobserved factors are normally distributed around the population regression function. The form and the variance of the distribution does not depend on any of the explanatory variables. Lecture 4 (econometrics and simple linear regression) EMET2007/6007 13 th March 2013 7 / 50
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Discussion of the normality assumption The error term is the sum of ±many² di/erent unobserved factors Sums of independent factors are normally distributed (CLT) Problems: How many di/erent factors? Number large enough? Possibly very heterogeneous distributions of individual factors How independent are the di/erent factors? The normality of the error term is an empirical question At least the error distribution should be ±close² to normal In many cases, normality is questionable or impossible by de°nition Lecture 4 (econometrics and simple linear regression) EMET2007/6007 13 th March 2013 8 / 50
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Discussion of the normality assumption (cont.) Examples where normality cannot hold: Interest rate (nonnegative; also currently at zero in many countries) Number of arrests (takes on a small number of integer values) Unemployment (indicator variable, takes on only 1 or 0)
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