This is a problem given as practice for my exam

Y: 3 4 6 8 5 5 9 8 7 5
X1: 1 -1 -1 1 0 0 0 0 0 0
X2: 0 0 0 1 -1 -1 1 0 0 0
X3: 0 0 0 0 0 0 0 1 -1 0
X4: 0 0 0 0 0 0 0 0 1 -1

Part 1: Find LSE of beta_0 and beta_2 in the model Y = beta_0 + beta_1 X1 + beta_2 X2 + beta_3 X3 + beta_4 X4

I did this manually by b = (X'X)^-1 X'Y where X is a matrix composed of columns X1...4 not fun... Couldn't find another way to calculate it though?

My answer: b = (-3/15, 27/15, 4/3, 5/3)

Part 2: Compute SSR(X2, X4 | X1, X3)

So SSR(X2, X4 | X1, X3) = SSE(X1, X3) - SSE(X1, X2, X3, X4)

I tried calculating using SSE = Y'(I - H)Y pbut the 10x10 matrix is too big to do by hand.

I'm guessing that there should be a trick since X2 and X4 are orthogonal, as are X1 and X3. I just don't see how to calculate without manually doing the 10x10 H matrix!