OLS question

3.) (Ordinary Least Squares.) (a) For a given n-by-k matrix X, the residual maker matrix associated with X is defined as Mx = I-X(X'X)-1X, where I is a n-by-n identity matrix. Nowlet & be a n-by-1 matrix of ones (all the entries are 1). Let M, be the residual maker matrix associated with c. Argue that M, is a de-meaning operator, i.e. Mex replaces the observations in the column vector x by its deviations from the column mean. (b) Let y = XB + e be a regression model (using the matrix notation). The uncentered R-squared is a measure of goodness of fit, defined as Riz = I P xy|2 where Px is the projection matrix Px = X(X X)-1XT. Suppose we add a Tly | |2 constant a to the dependent variable and run the regression y + at = X B +E. Show that when X includes a constant as a regressor, R2 = Pxytall2 llytal|2 , hence Ra can be arbitrarily close to 1 for very large a. (c) The centered R-squared is defined as R2 = IM.Pxyl|2 Show that the R2 is not affected by adding an arbitrary constant to y. (d) Show that Re always lies between 0 and 1. Hint: show that Show that if X contains a regressor of ones, then the sum of the fitted residuals is always zero, that is, Me = E. Hence we have My = M. XB te