Best affine unbiased response decomposition

Given two linear regression models y1=X1β1+u1 and y2=X2β2+u2 where the response vectors y1 and y2 are unobservable but the sum y=y1+y2 is observable, we study the problem of decomposing y into components ŷ1 and ŷ2, intended to be close to y1 and y2, respectively. We develop a theory of best affine unbiased decomposition in this setting. A necessary and sufficient condition for the existence of an affine unbiased decomposition is given. Under this condition, we establish the existence and uniqueness of the best affine unbiased decomposition and provide an expression for it.