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**Unformatted text preview: **241B Lecture Generalized Least Squares One classic assumption (1.4) states that the errors have a spherical conditional covariance matrix, E ( UU j X ) = & 2 I: Today we relax the assumption and allow E ( UU j X ) = & 2 V ( X ) ; (1) where V ( X ) is an n & n nonsingular matrix. Under (1) the errors are (con- ditionally) heteroskedastic and the elements of V ( X ) are, in general, nonlinear functions of X . Di/ering diagonal elements correspond to heteroskedasticity and non-zero o/-diagonal elements correspond to correlation. Consequence of Relaxing Classic Assumption 1.4 There are 3 main consequences for the OLS estimator ¡ The Gauss-Markov Theorem no longer holds, hence the BLUE estimator is not the OLS estimator. ¡ The OLS estimator remains unbiased. ¡ The t test statistic no longer has a t distribution. (Same for the F test statistic.) Homoskedasticity: Conditional versus Unconditional There are three "levels" of homoskedasticity. The simplest case is one in which the error is conditionally homoskedastic...

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