Let U be an n × 1 vector with 1 as its first element and 0s elsewhere. Consider computing the regression of U on an n × p′ full rank matrix X. As usual, let H = X(X′X) −1 X′ be the hat matrix with...

Let U be an n × 1 vector with 1 as its first element and 0s elsewhere. Consider computing the regression of U on an n × p′ full rank matrix X. As usual, let H = X(X′X) −1 X′ be the hat matrix with elements hij.

Show that the elements of the vector of fitted values from the regression of U on X are the

h_1j, j = 1, 2, . . ., n.

Show that the first element of the vector of residuals is 1 − h₁₁, and the other elements are

−h_1j, j > 1.