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Math Help - Design matrices problem

  1. #1
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    Design matrices problem

    I'm a bit rusty on my linear algebra and was given this problem in my advanced linear models class.

    Consider the use of two design matrices X and Z for estimating a curve based on the two models y = Xb + e1 and y = Za + e2. Let H = X(X'X)^-1*X' denote the hat matrix for the first model. Recall that hat matrices are projection matrices. Consider the case of projecting the design matrix Z onto the column space of X using the transformation Zhat = HZ.

    Let bhat = (X'X)^-1*X'y denote the least squares estimator of b in the first model. Let a2 denote the least squares estimator of a using the design matrix Zhat. Find an expression for a2 in terms of bhat.

    Any help would be greatly appreciated!!
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  2. #2
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    Sep 2007
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    This is how I started:

    a2 = (Zhat'*Zhat)^-1*Zhat'*y

    = ((HZ)'(HZ))^-1*(HZ)'*y

    = (Z'H'HZ)^-1*Z'H'y

    = (Z'HZ)^-1*Z'Hy

    = (Z'HZ)^-1*Z'Xb

    = (Z'X(X'X)^-1*X'Z)^-1*Z'Xb

    = (X'Z)^-1*(X'X)*(Z'X)^-1*Z'Xb

    = (X'Z)^-1*(X'X)b

    Can I simplify this further?

    Thanks!!
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