ec20-newlec11-v5 - Multiple Regression Analysis y = 0 1x1...

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1 Multiple Regression Analysis y = β 0 + β 1 x 1 + β 2 x 2 + . . . β k x k + u Heteroskedasticity
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2 What is Heteroskedasticity? Homoskedasticity = constant error variance (conditional on the explanatory variables) Heteroskedasticity = opposite = the variance of u is different for different values of the x ’s Example: estimating returns to education If we think the variance in the errors differs by educational attainment, then we have heteroskedasticity E.g. more variation in ability among college grads than HS dropouts.
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3 . x x 1 x 2 y f( y|x ) Example of Heteroskedasticity x 3 . . E( y | x ) = β 0 + β 1 x
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What’d we might see in data 4 0 1 2 3 4 5 lnwage 0 5 10 15 20 yrsed big error variance  small error variance 
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Other Examples of het. Y data are means e.g., average wages in different countries: will be more precisely estimated in large countries than small countries Dummy dependent variable Always suffers from heteroskedasticity because of bounds on y variation (more variation in the middle than at the extremes) 5
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What are the consequences of  heteroskedasticity? 1. OLS estimates of the slope will be biased 2. OLS is not “efficient”: other methods of  estimation besides OLS will give smaller  standard errors 3. The standard errors will be biased 4. In particular, the OLS standard errors will  be too big 0% 0% 0% 0% Economics 20 - Prof. Anderson 6
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7 Heteroskedasticity: Consequences OLS is still unbiased and consistent But the standard errors are biased So we can not use the usual t statistics or F statistics (or LM statistics) for drawing inferences Today: correction for that OLS is not efficient – standard errors could be smaller Weighted least squares is
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Standard Errors: Bivariate Case Assume SLR.1-SLR.4 hold. To be as general as possible, define: What this really means: error variance a function of individual characteristics (and not nec. X’s that are actually in your regression, but more later…) First, define again “little” variables as variables minus their mean: Purpose: make intercept go away 8 ( 29 2 | var i i i X u σ = X X x Y Y y i i i i - = - = ; ( 29 ( 29 ( 29 i i i i i i i i u x U U X X U X U X Y Y y + = - + - = + + - + + = - = 1 1 1 0 1 0 β β β β β β
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Standard Errors: Bivariate Case In a previous class, we showed OLS slope estimate is So what is ? i.e. how much can we expect our slope estimate to vary under heteroskedasticity? For any random variable U, the variance is defined as mean squared deivation from mean: Var(U) = E[(U-E[U]) 2 ] = E[u 2 ] 9 2 1 1 ˆ + = i i i x u x β β ( 29 1 ˆ var β
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Bivariate Standard Errors In this case: Under SLR.1-4, , which means Or written out… Then squaring this… 10 [ ] 1 1 ˆ β β = E [ ] 2 1 1 1 1 ˆ ˆ ˆ = - = - i i i x u x E β β β β [ ] 2 2 2 2 2 1 1 1 1 ...
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