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.