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Notes5 - Lecture Notes 5 Econ 410 Introduction to...

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Lecture Notes 5 Econ 410 – Introduction to Econometrics 1 Linear Regression with One Regressor: Part III Heteroskedasticity and Homoskedasticity The error term i u is homoskedastic if the variance of the conditional distribution of i u given i X is constant for n i , ..... , 1 = and in particular it does not depend on i X . Otherwise, the error term is heteroskedastic. Example : consider the following linear regression model: i i i u X Y + + = 0 0 β β Assume that i Y is the level of education and i X is a binary variable that equals 0 if the i th person is a woman and equals 1 if the i th person is a man. Let consider the cases 1 = i X and 0 = i X separately. If the person is a woman ( 0 = i X ), the linear regression model can be written as: i i u Y + = 0 β and i u is the deviation of the i th woman’s earnings from 0 β , the population mean earnings for women. On the other hand, if the person is a man ( 1 = i X ), the linear regression model can be written as: i i u Y + + = 1 0 β β and i u is the deviation of the i th man’s earnings from 1 0 β β + , the population mean earnings for men. The error term i u is homoskedastic if the variance of the conditional distribution of i u given i X does not depend on i X . So, in this example, the error term i u is homoskedastic if its variance is the same for men as it is for women, that is: ( ) ( ) 1 | 0 | = = = i i i i X u Var X u Var If this is not the case, then the error term is heteroskedastic. Another example : consider the model in question #4 in Problem Set 2. u inc sav + + = 1 0 β β , with e inc u = In part b. of this question you show that ( ) inc inc u Var e 2 | σ = . So, in this model the conditional variance of the error term u depends on the regressor inc . It follows that in this model the error term is heteroskedastic.
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Lecture Notes 5 Econ 410 – Introduction to Econometrics 2 Implications of Homoskedasticity and Heteroskedasticity We know that, if the three least squares assumptions hold, the OLS estimator is unbiased, consistent and asymptotically normal. These properties hold whether the error term is homoskedastic or heteroskedastic.
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