15 Implications of Conditional Homoskedasticity

15 Implications of Conditional Homoskedasticity - x tk x tj...

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Implications of Conditional Homoskedasticity Hayashi 2.6 Goal: behavior of test statistics under conditional ho- moskedasticity Conditional Homoskedasticity (Assumption 2.7) E u 2 t j x t ± = 2 > 0 Unconditional Homoskedasticity E u 2 t ± = 2 > 0 Conditional , Unconditional?
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Relation Between Conditional and Unconditional Conditional ) Unconditional E u 2 t ± = E X h E u 2 t j x t ±i = 2 Unconditional ; Conditional Example u t = x t v t v t independent of x t Ev 2 t = 2 Ex 2 t = ! 2 E u 2 t ± = E x 2 t ± E v 2 t ± = ! 2 2 E u 2 t j x t ± = x 2 t E v 2 t j x t ± = x 2 t 2
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Estimation of S Need to estimate S = E u 2 t x t x 0 t ± Conditional homoskedasticity estimator ^ S = ^ 2 X x t x 0 t ^ u t = y t x 0 t ^ ± ^ 2 = 1 n P ^ u 2 t ^ S consistent for S
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Consistency of S S = E x E u 2 t x t x 0 t j x t ± = E h x t x 0 t E u 2 t j x t ±i = E x t x 0 t 2 ± = 2 E x t x 0 t ± Implications 1) A2.4 redundant A 2.5 S nonsingular ) 2 > 0 ) E ² x t x 0 t ³ nonsingular (A2.4)
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Assumption Implications A 2.4 redundant and 1 n P t x t x 0 t p ! E x t x 0 t ± (ergodic stationarity A2.2) 1 n P t ^ u 2 t p ! 2 (A2.1-A2.4) no need for A2.6 E ²
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Unformatted text preview: x tk x tj 2 < 1 Asymptotic Behavior Proposition 2.5 Under A 2.1-2.5, 2.7 a) (asymptotic distribution of b ) The OLSE b of & is consistent and asymptotically normal with Avar ( b ) = 2 h E & x t x t i & 1 b) (consistent estimation of asymptotic variance) The quantity Avar ( b ) is consistently estimated by \ Avar ( b ) = ^ 2 P x t x t & 1 Asymptotic Behavior continued c) (asymptotic distribution of test statistics) Under H : & = & , t statistic is asymptotically distributed as N (0 ; 1) Under H : R& = r , # r & F is asymptotically distributed as 2 (# r )...
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15 Implications of Conditional Homoskedasticity - x tk x tj...

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