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# week8 - Econ 508 Heteroskedasticity and Autocorrelation...

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Econ 508: Heteroskedasticity and Autocorrelation Juan Fung MSPE April 8, 2011 Juan Fung (MSPE) Econ 508: Heteroskedasticity and Autocorrelation April 8, 2011 1 / 23

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Heteroskedasticity What is heteroskedasticity? Recall the linear regression model, Y i = β 0 + β 1 X 1 i + ··· + β k X ki + ± i What is assumed about the error term ± i ? ± N (0 , σ 2 I ) Heteroskedasticity is one violation of this assumption, namely E ( ± 2 i ) = σ 2 i , that the variance is not constant . Juan Fung (MSPE) Econ 508: Heteroskedasticity and Autocorrelation April 8, 2011 2 / 23
Heteroskedasticity Why it occurs Typically a cross-sectional phenomenon: Cross-section data concerns diﬀerent members of a population, at a given point in time. Some members may be of diﬀerent sizes (e.g., small vs. large ﬁrms, low vs. high income families, etc.) Time-series data often concern a particular member of a population across time (e.g., China’s GDP, US unemployment rate, etc.) Observations may be of same magnitude. Question : Are errors always constant in time-series? Juan Fung (MSPE) Econ 508: Heteroskedasticity and Autocorrelation April 8, 2011 3 / 23

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Heteroskedasticity Problems? Recall Gauss-Markov Theorem: Under the classical assumptions, the OLS estimator ˆ β is BLUE. What if one of the assumptions fails? If E ( ± i ± 0 i ) = σ 2 Ω, where Ω ij = σ 2 i , i = j , and Ω ij = 0 , i 6 = j then 1. ˆ β is biased? unbiased : E [ ˆ β ] = E [( X 0 X ) - 1 X 0 Y ] = E [( X 0 X ) - 1 X 0 ( X β + ± )] = E [( X 0 X ) - 1 X 0 X β + ( X 0 X ) - 1 X 0 ± ] = β + ( X 0 X ) - 1 X 0 E [ ± ] = β 2. Also consistent and asymptotically normal. Juan Fung (MSPE) Econ 508: Heteroskedasticity and Autocorrelation April 8, 2011 4 / 23
Heteroskedasticity Problems? If E ( ± i ± 0 i ) = σ 2 Ω, where Ω ij = σ 2 i , i = j , and Ω ij = 0 , i 6 = j then 3. ˆ β is ineﬃcient (why?), with variance V ( ˆ β ) = E [( ˆ β - β )( ˆ β - β ) 0 ] = E [( X 0 X ) - 1 X 0 ±± 0 X ( X 0 X ) - 1 ] =( X 0 X ) - 1 X 0 E [ ±± 0 ] X ( X 0 X ) - 1 = σ 2 ( X 0 X ) - 1 X 0 Ω X ( X 0 X ) - 1 . Compare to V ( ˆ β ) when all assumptions hold. 4. Variance estimator, [ V ( ˆ β ) = s 2 ( X 0 X ) - 1 , is biased . This results in bad inference. Estimated standard errors not valid for conﬁdence intervals; t (resp. F ) stats do not have t (resp. F ) distribution. Juan Fung (MSPE) Econ 508: Heteroskedasticity and Autocorrelation April 8, 2011 5 / 23

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Heteroskedasticity Detecting heteroskedasticity For the null H 0 : no heteroskedasticity, several tests can be used: 1. White test. Suppose k = 2, i.e., Y i = β 0 + β 1 X 1 i + β 2 X 2 i + ± i . Estimate by OLS, save residuals, and estimate auxiliary regression e 2 i = γ 0 + γ 1 X 1 i + γ 2 X 2 i + γ 3 X 2 1 i + γ 4 X 4 i + γ 5 X 1 i X 2 i + u i , then either (a) test for joint signiﬁcance of the γ ’s, ( F test) or (b) compute the LM statistic, nR 2 χ 2 j , where j is the number of regressors in the auxiliary regression (in this case j = 5). 2.
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week8 - Econ 508 Heteroskedasticity and Autocorrelation...

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