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Leave-One-Out Regression

Section 7.22. 7.8 Summary of Covariance Matrix Notation The notation we have introduced may be somewhat confusing so it is helpful to write it down in one place. The exact variance of b (under the assumptions of the linear regression model) and the asymptotic variance of p n  b  (under the more general assumptions of the linear projection model) are V b = var  b j X  = X0X 1 X0DX X0X 1 V = avar p n  b  = Q1 xx Q1 xx: The HC0 estimators of these two covariance matrices are Vb HC0 b = X0X 1 Xn i=1 xix 0 i eb 2 i ! X0X 1 CHAPTER 7. ASYMPTOTIC THEORY FOR LEAST SQUARES 233 and satisfy the simple relationship Vb HC0 = nVb HC0 b : Similarly, under the assumption of homoskedasticity the exact and asymptotic variances simplify to V 0 b = X0X 1  2 V 0 = Q1 xx 2 and their standard estimators are Vb 0 b = X0X 1 s 2 Vb 0 = Qb 1 xxs 2 which also satisfy the relationship Vb 0 = nVb 0 b: The exact formula and estimates are useful when constructing test statistics and standard errors. However, for theoretical purposes the asymptotic formula (variances and their estimates) are more useful, as these reta