Econ418_Ch8_Heteroskedasticity

# Econ418_Ch8_Heteroskedasticity - ECON 418 Introduction to...

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ECON 418 Introduction to Econometrics Anna Breman Univeristy of Arizona Part 3: Regression analysis Chapter 8: Heteroskedasticity Anna Breman (Univeristy of Arizona) ECON 418 Fall 2008 1 / 33

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Chapter 8 I Heteroskedasticity Outline 1 What is Heteroskedasticity and why should we care about it? 2 Repetition of variance of the OLS estimator, var ( β j ) and standard errors 3 The Lagrange Multiplier test 4 Heteroskedasticity-robust standard errors 5 Test for heteroskedasticity Anna Breman (Univeristy of Arizona) ECON 418 Fall 2008 2 / 33
Homoskedasticity means that the error u has the same variance given any value of the explanatory variables. Why do we care about homoskedasticity? 1 that the OLS estimator is BLUE. 2 It is crucial assumption to test hypotheses about the OLS estimates 3 If it fails, the t-test and the F-test are not valid! Anna Breman (Univeristy of Arizona) ECON 418 Fall 2008 3 / 33

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MLR5 : Homoskedasticity The error u has the same variance given any value of the explanatory variables. In other words, Var ( u j x 1 , x 2 , ..., x k ) = σ 2 u . Example: wage = β 0 + β 1 educ + β 2 exp er + β 3 tenure + u The homoskedasticity assumption says that Var ( u j educ , exper , tenure ) = σ 2 . Note: When the variance of Var ( u j x ) depends on x , the error term is said to to be heteroskedastic. Because Var ( u j x ) = Var ( y j x ) , heteroskedasticity is present whenever Var ( y j x ) is a function of x . Anna Breman (Univeristy of Arizona) ECON 418 Fall 2008 4 / 33
Anna Breman (Univeristy of Arizona) ECON 418 Fall 2008 5 / 33

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Consequences of Heteroskedasticity 1 The OLS estimator is still unbiased and consistent under heteroskedasticity 2 R 2 and adjusted R 2 are both una/ected by the presence of heteroskedasticity 3 The var ( β j ) are biased without the homoskedasticity assumption. Since, the OLS standard errors are based directly on these variances, and F-tests. 4 The OLS statistics does not have a t distribution. 5 The problem of heteroskedasticity is not solved by using large sample sizes! Conclusion : we can still run the OLS regressions and get unbiased and consistent estimates of the coe¢ cients, but we cannot test whether they Anna Breman (Univeristy of Arizona) ECON 418 Fall 2008 6 ± 33
. Sampling Variances of the OLS Slope Estimators Var ( b β j ) = σ 2 u SST j ( 1 R 2 j ) for j = 1 , 2 , ..., k , where SST j = n i = 1 ( x ij x j ) 2 is the total sample variation in x j and R 2 j is the R-squared from regressing x j on all other independent variables (and including the intercept). Anna Breman (Univeristy of Arizona)

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## This note was uploaded on 05/03/2009 for the course ECON 418 taught by Professor Breman during the Spring '08 term at University of Arizona- Tucson.

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Econ418_Ch8_Heteroskedasticity - ECON 418 Introduction to...

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