TAILIEUCHUNG - Class Notes in Statistics and Econometrics Part 13

CHAPTER 25 Variance Estimation: Should One Require Unbiasedness? There is an imperfect analogy between linear estimation of the coefficients and quadratic estimation of the variance in the linear model. This chapter sorts out the principal commonalities and differences, a task obscured by the widespread but unwarranted imposition of the unbiasedness assumption. | CHAPTER 25 Variance Estimation Should One Require Unbiasedness There is an imperfect analogy between linear estimation of the coefficients and quadratic estimation of the variance in the linear model. This chapter sorts out the principal commonalities and differences a task obscured by the widespread but unwarranted imposition of the unbiasedness assumption. It is based on an unpublished paper co-authored with Peter Ochshorn. We will work in the usual regression model y X 3 e 675 676 25. VARIANCE ESTIMATION UNBIASEDNESS where y is a vector of n observations X is nonstochastic with rank r n and the disturbance vector e satisfies E e o and E eeT a21. The nonstochastic vector and scalar a2 0 are the parameters to be estimated. The usual estimator of a2 is s2 y My 1 n E 4 n r i 1 1 n r where M I X XTX -XT and e My. If X has full rank then e y X 3 where is the least squares estimator of . Just as is the best minimum mean square error linear unbiased estimator of it has been shown in Ati62 see also Seb77 pp. 52 3 that under certain additional assumptions s2 is the best unbiased estimator of a2 which can be written in the form yTAy with a nonnegative definite A. A precise formulation of these additional assumptions will be given below they are for instance satisfied if X i the vector of ones and the j are . But they are also satisfied for arbitrary X if e is normally distributed. In this last case s2 is best in the larger class of all unbiased estimators. This suggests an analogy between linear and quadratic estimation which is however by no means perfect. The results just cited pose the following puzzles Why is s2 not best nonnegative quadratic unbiased for arbitrary X-matrix whenever the j are . with zero mean What is the common logic behind 25. VARIANCE ESTIMATION UNBIASEDNESS 677 the two disparate alternatives that either restrictions on X or restrictions on the distribution of the j can make s2 optimal It comes as a surprise that again under

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