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EMET2007 Lecture 7 - EMET2007/6007 Econometrics I...

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EMET2007/6007 Econometrics I: Econometric Methods Professor Rodney W. Strachan, ANU heteroscedasticity 24 th April 2013 Lecture 7 (heteroscedasticity) EMET2007/6007 24 th April 2013 1 / 34
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RODNEY: TURN ON THE RECORDING! Lecture 7 (heteroscedasticity) EMET2007/6007 24 th April 2013 2 / 34
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Mid Semester Exam We expect to complete marking in about one week Some students are yet to sit the exam, so I cannot discuss it this week I will discuss the answers and release the marking guide/answers next week The marking will adjust for overall performance if necessary (i.e., rescaling) Lecture 7 (heteroscedasticity) EMET2007/6007 24 th April 2013 3 / 34
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Heteroscedasticity (or Heteroskedasticity) Introductory Econometrics: A Modern Approach by Je/rey M. Wooldridge, 4e This will be primarily Chapter 8 Lecture 7 (heteroscedasticity) EMET2007/6007 24 th April 2013 4 / 34
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Heteroscedasticity What is it? What are the implications of having it? How can I tell if I have it? Anecdotal evidence Formal evidence from testing What can I do about it? Adjust the data to remove it Adjust the variances to account for it Lecture 7 (heteroscedasticity) EMET2007/6007 24 th April 2013 5 / 34
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De°nition of Heteroscedasticity In the MLR model y i = β 0 + β 1 x 1 , i + β 2 x 2 , i + ° ° ° + β k x k , i + ε i we usually assume Var ( ε i j x i ) = Var ( ε i ) = E ° ε 2 i ± = σ 2 That is, we assume the variance of the error is always the same, σ 2 Here we are explicitly assuming the variance is NOT a function of x i This assumption is called Homoscedasticity The assumptoin of Homoscedasticity is sometimes not reasonable and we have the situation in which the variance changes as one or more of the x i change This the case where Var ( ε i j x i ) = E ° ε 2 i j x i ± = σ 2 ( x i ) Thus the variance is a function of one or more of the x i This is a violation of the assumption Homoscedasticity This is called Heteroscedasticity Lecture 7 (heteroscedasticity) EMET2007/6007 24 th April 2013 6 / 34
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Graphical illustration of homoskedasticity The variability of the unobserved influences does not dependent on the value of the explanatory variable Lecture 7 (heteroscedasticity) EMET2007/6007 24 th April 2013 7 / 34
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An example for heteroskedasticity: Wage and education Thevariance of the unobserved determinants of wages increases with the level of education Lecture 7 (heteroscedasticity) EMET2007/6007 24 th April 2013 8 / 34
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Consequences of heteroscedasticity for OLS Some things do not change: OLS still unbiased and consistent under heteroscedasticty! Also, interpretation of R-squared is not changed R 2 t 1 ± σ 2 ε σ 2 y σ 2 ε ± Unconditional error variance is una/ected by heteroscedasticity (which refers to the conditional error variance) However, some things do change: Heteroscedasticity invalidates variance formulas for OLS estimators The usual F-tests and t-tests are not valid under heteroscedasticity Under heteroscedasticity, OLS is no longer the best linear unbiased estimator (BLUE); there may be more e¢ cient linear estimators Lecture 7 (heteroscedasticity) EMET2007/6007 24 th April 2013 9 / 34
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How can I tell if I have it? Anecdotal evidence
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