chapter 7 [article]

# chapter 7 [article] - Contents 1 Heteroscedasticity 1.1...

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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

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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 - & F -tests 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. Goldfeld-Quandt test 2. Tests that make no such assumptions e.g. White test Goldfeld-Quandt test Most widely-used test for heteroscedasticity Makes following a priori assumptions: 3

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1. σ u i X i the standard deviation of the error term in each observation is pro- portional to the value X i of the regressor in that observation 2. Assumptions A.1, A.2, A.3, A.5 and A.6 are satisfied
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