advanced econometrics

Consider the following simple linear population regression model: y = Bo + Biw + u, where y, w, and u are all n x 1 column vectors. (a) Let i denote a n x 1 column matrix of 1's. Let P denote the projection matrix for i, i.e. P = i(i'i)-li'. Calculate P and show that it is both symmetric and idempotent. (b) Let M = I - P denote the residual matrix of i. Denote y* = My and w* = Mw. Calculate y* and w*, respectively. (c) An econometrician estimated the following partitioned regression without a con- stant term: y* = Biw* te*. Show that the least square estimator bj for this partitioned regression is b1 = Et wigi - nwy Er w? - nw2Consider the following simple linear population regression model: y = Bo + Biw + u, where y, w, and u are all n x 1 column vectors. (a) Let i denote a n x 1 column matrix of 1's. Let P denote the projection matrix for i, i.e. P = i(i'i)-li'. Calculate P and show that it is both symmetric and idempotent. (b) Let M = I - P denote the residual matrix of i. Denote y* = My and w* = Mw. Calculate y* and w*, respectively. (c) An econometrician estimated the following partitioned regression without a con- stant term: y* = Biw* te*. Show that the least square estimator bj for this partitioned regression is b1 = Et wigi - nwy Er w? - nw2