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Asymptotic Local Power

ly associated. What we found through this example is that in the presence of heteroskedasticity there is no simple relationship between the correlation of the regressors and the correlation of the parame CHAPTER 7. ASYMPTOTIC THEORY FOR LEAST SQUARES 228 β1 β2 −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 Figure 7.4: Contours of Joint Distribution of ( b 1 ; b 2 ); Homoskedastic Case We can extend the above analysis to study the covariance between coe¢ cient sub-vectors. For example, partitioning x 0 i = (x 0 1i ; x 0 2i ) and 0 = 0 1 ; 0 2  ; we can write the general model as yi = x 0 1i 1 + x 0 2i 2 + ei and the coe¢ cient estimates as b 0 =  b 0 1 ; b 0 2  : Make the partitions Qxx =  Q11 Q12 Q21 Q22  ; =  11 12 21 22  : From (2.41) Q1 xx =  Q1 112 Q1 112Q12Q1 22 Q1 221Q21Q1 11 Q1 221  where Q112 = Q11 Q12Q1 22 Q21 and Q221 = Q22 Q21Q1 11 Q12. Thus when the error is homoskedastic, cov  b 1 ; b 2  =  2Q1 112Q12Q1 22 which is a matrix generalization of the two-regressor case. In the general case, you can show that (Exercise 7.5) V =  V 11 V 12 V 21 V 22  (7.13) where V 11 = Q1 112 11 Q12Q1 22 21 12Q1 22 Q21 + Q12Q1 22 22Q1 22 Q21 Q1 112 (7.14) V 21 = Q1 221 21 Q21Q1 11 11 22Q1 22 Q21 + Q21Q1 11 12Q1 22 Q21 Q1 112 (7.15) V 22 = Q1 221 22 Q21Q1 11 12 21Q1 11 Q12 + Q21Q1 11 11Q1 11 Q12 Q1 221 (7.16) Unfortunately, the CHAPTER 7. ASYMPTOTIC THEORY FOR LEAST SQUARES 229 β1 β2 −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 Figure 7.5: Contours of Joint Distribution of b1 and b2; Heteroskedastic Case