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Heteroskedasticity-Consistent Standard Errors

Heteroskedasticity-Consistent Standard Errors -...

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Heteroskedasticity-Consistent Standard Errors Review Questions For the scalar regression in deviation-from-means form y t = °x t + u t write the variance for the OLSE of for each of the follow- ing cases 1. V ar ( u t j X ) = ± 2 for t 6 = s Cov ( u t ; u s j X ) = 0 2. V ar ( u t j X ) = ± 2 t for t 6 = s Cov ( u t ; u s j X ) = 0 3. V ar ( u t j X ) = ± 2 t for t 6 = s Cov ( u t ; u s j X ) = ± 2 j t ° s j
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Goal Given y t = x 0 t ° + u t accurately estimate V ar ° ^ ° j X ± : ° X x t x 0 t ± ° 1 X u 2 t x t x 0 t ° X x t x 0 t ± ° 1 = 1 n ² 1 n X x t x 0 t ³ ° 1 1 n X u 2 t x t x 0 t ² 1 n X x t x 0 t ³ ° 1 ± 1 n S ° 1 xx S S ° 1 xx
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Eicker-White Estimator Need to estimate S = 1 n X u 2 t x t x 0 t Eicker-White estimator ^ S = 1 n X ^ u 2 t x t x 0 t ^ u t = y t ° x 0 t ^ ° ^ ° consistent for ° (e.g. OLSE) Heteroskedasticity-consistent standard errors c se = s 1 n S ° 1 xx ^ S S ° 1 xx
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Finite-Sample Accuracy Test H 0 : ° k = 0 Statistic ^ ° k c se Size Pr ( reject H 0 j H 0 true ) in practice: over-rejection problem nominal size 5%,empirical size > 5% reason: estimated standard error is too small ^ u 2 t is a downward biased estimator of u 2 t
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