LinearRegression4

The p value against a two sided alternative is about

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Unformatted text preview: an see that the coefficient on marr becomes 0.3453 (SE = 0.229). Therefore, holding experience and position fixed, a married man has about one-third more assist per game. The p-value against a two-sided alternative is about 0.13, which is stronger, but not overwhelming, evidence that married men are more productive when it comes to assists. VER. 10/23/2012. © P. KOLM 29 Multiple Regression Analysis: Heteroscedasticity VER. 10/23/2012. © P. KOLM 30 Recall: What Does Homoscedasticity Mean? VER. 10/23/2012. © P. KOLM 31 Recall: What Does Heteroscedasticity Mean? VER. 10/23/2012. © P. KOLM 32 Heteroscedasticity • Recall: In the “homoscedasticity assumption” [MLR.5] we assume that the conditional variance is constant Var (u | x 1,..., x k ) = σ 2 • If this is not true, that is if the variance of u is different for different values of the x ’s, that is Var (u | x 1,..., x k ) = σ 2 ⋅ h(x 1,..., x k ) then the errors are heteroscedastic VER. 10/23/2012. © P. KOLM 33 Examples of Heteroscedasticity Heteroscedasticity often arises in cross-sectional data: • Estimating wage as a function of education: Higher educational attainment is associated with a large number of career opportunities (resulting in a higher dispersion in wages) • Estimating savings as a function of income: Higher income households have more discretion over what to do with their money In finance we may also have time varying heteroscedasticity called autoregressive conditional heteroscedasticity (ARCH) • You will learn more about this when you study time series models in other classes VER. 10/23/2012. © P. KOLM 34 Why Worry About Heteroscedasticity? • OLS estimators are still consistent even if we do not assume homoscedasticity [We need MLR.1-MLR.4 to be valid] • However: o The standard errors of the estimators are biased if we have heteroscedasticity o If the standard errors are biased, we cannot use the usual t - or F- statistics for inferences (as the statistics are no longer t- and Fdistributed) VER. 10/23/2012. © P. KOLM 35 Variance with Heteroscedasticity For the simple case, x − x )ui ˆ = β + ∑( i β1 , 1 2 ∑ (x i − x ) so ˆ Var (β1 ) ∑ (x = − x ) σi2 2 i 2 x SST , where SSTx = ∑ (x i − x ) 2 A valid estimator for this when σi2 ≠ σ 2 is ∑ (x ˆ − x ) ui2 2 i 2 x SST , ˆ where ui are the OLS residuals VER. 10/23/2012. © P. KOLM 36 Variance of the OLS Estimators with Heteroscedasticity (1/3) Recall: Under homoscedasticity the sample variance of the OLS estimators are given by ˆ Var (β j ) = ≈ σ2 ( SSTj 1 − Rj2 s2 ( SSTj 1 − R 2 j ) ) s2 = SSRj where n • SSTj = ∑ (x ij − x j ) 2 i =1 • R is the R-squared from regressing x j on all other independent variables 2 j (all other x ’s) n 1 1 ˆ • We estimate σ by s = ∑ ui2 = df SSR n − k − 1 i =1 2 VER. 10/23/2012. © P. KOLM 2 37 Variance of the OLS Estimators with Heteroscedasticity (2/3) () ˆ Under heteroscedasticity a valid estimator of Var βj is giv...
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This document was uploaded on 02/17/2014 for the course COURANT G63.2751.0 at NYU.

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