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Homoskedasticity and Heteroskedasticity

5.14 ConÖdence Intervals for Error Variance We can also construct a conÖdence interval for the regression error variance  2 using the sampling distribution of s 2 from Theorem 5.16, which states that in the normal regression model (n k) s 2  2   2 nk : (5.14) Let F(u) denote the  2 nk distribution function, and for some set c1 = F 1 ( =2) and c2 = F 1 (1 =2) (the =2 and 1 =2 quantiles of the  2 nk distribution). Equation (5.14) implies that P  c1  (n k) s 2  2  c2  = F(c2) F(c1) = 1 : Rewriting the inequalities we Önd P (n k) s 2 =c2   2  (n k) s 2 =c1  = 1 : This shows that an exact 1 conÖdence interval for  2 is C =  (n k) s 2 c2 ; (n k) s 2 c1  : (5.15) Theorem 5.20 In the normal regression model, (5.15) has coverage probability P  2 2 C  = 1 . The conÖdence interval (5.15) for  2 is asymmetric about the point estimate s 2 , due to the latterís asymmet