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Unformatted text preview: Contents 1 Heteroscedasticity 1 1.1 Definition and implications . . . . . . . . . . . . . . . . . . . . . 1 1.2 Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2 Solutions 5 2.1 Weighted regression . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Nonlinear models . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1 Heteroscedasticity Contents 1.1 Definition and implications Contents • Recall assumption A.4 of homoscedasticity from earlier: σ 2 u i = σ 2 u for all i • In other words, the variability of the error term is the same for each observation • In this topic we consider the implications of relaxing that assumption, and allowing for heteroscedasticity (‘differing dispersion’) Definition Heteroscedasticity occurs when the variance of the error term differs across observations, i.e. σ 2 u i is not the same for each observation i Y = β 1 + β 2 X 2 X 2 Y X 2 a X 2 b X 2 c X 2 d • Homoscedasticity: each error u i is drawn from a probability distribution with the same variance σ 2 u Y = β 1 + β 2 X X Y X a X b X c X d • Heteroscedasticity: the errors u i are drawn from probability distributions with different variances σ 2 u i Causes Y = β 1 + β 2 X + u ∼ estimated model • Influence of omitted variables; measurement errors? • Both of these are ‘dumped’ in the errors u i • Both likely to vary systematically with X – e.g. if omitted variables tend to be large when X large – e.g. if measurement error (of X ) is proportional to X • σ 2 u i will also vary systematically with X i Implications • OLS estimators still unbiased • Standard errors reported by STATA are wrong – computed assuming homoscedasticity – probably too small ⇒ overstated significance from t & Ftests • OLS is inefficient : there are other unbiased estimators that have lower variances 2 – we just have to find them • Why is OLS inefficient? • In other words, why is a more efficient estimation procedure achievable? • Answer: a more efficient estimator would be one that found a way to put more weight on the low σ 2 u i observations than on the high  σ 2 u i ones Y = β 1 + β 2 X obs. #1 obs. #2 X Y X a X b X c X d • An efficient estimator will place more weight on obs. #1 than on obs. #2 • obs. #1 is a better guide to the location of the true line Y = β 1 + β 2 X 1.2 Detection Contents Many different tests for heteroscedasticity, falling into two categories: 1. Tests that make a priori assumptions on the nature of the heteroscedas ticity • e.g. GoldfeldQuandt test 2. Tests that make no such assumptions • e.g. White test GoldfeldQuandt test • Most widelyused test for heteroscedasticity • Makes following a priori assumptions: 3 1.1....
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This note was uploaded on 01/14/2012 for the course ECON 201 taught by Professor Witte during the Spring '08 term at Northwestern.
 Spring '08
 Witte
 Macroeconomics

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