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Extrema of Quadratic Forms

Clustered Dependence In Section 4.21 we introduced clustered dependence. We can also use the methods of clustered dependence for 2SLS estimation. Recall, the g th cluster has the observations yg = (y1g; :::; yngg) 0 , Xg = (x1g; :::; xngg) 0 ; and Zg = (z1g; :::; zngg) 0 . The structural equation for the g th cluster can be written as the matrix system yg = Xg + eg: Using this notation the centere 2SLS estimator can be written as b 2sls =  X0Z Z 0Z 1 Z 0X 1 X0Z Z 0Z 1 Z 0e =  X0Z Z 0Z 1 Z 0X 1 X0Z Z 0Z 1 0 @ X G g=1 Z 0 geg 1 A : The cluster-robust covariance matrix estimator for b 2sls thus takes the form Vb =  X0Z Z 0Z 1 Z 0X 1 X0Z Z 0Z 1 Sb Z 0Z 1 Z 0X  X0Z Z 0Z 1 Z 0X 1 with Sb = X G g=1 Z 0 gbegbe 0 gZg and the clustered residuals beg = yg Xg b 2sls: The di§erence between the heteroskedasticity-robust estimator and the cluster-robust estimator is the