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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 onethird more assist per game. The pvalue against
a twosided 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 crosssectional 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.1MLR.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 Rsquared 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.
 Fall '14

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