The problem is:
Suppose we fit a model Y = XA βA + ε. However, the true model is Y = XA βA + XB βB + ε. A is kA x 1 and B is kB x 1. Show that the OLS estimates of βA will still be equal if XTAXB = 0.
My attempt at a solution:
Under OLS, the estimate of β is (XTX)-1XTY. So I was thinking for the first model, we just have X = XA, but in the second model, we have X = XA + XB. So I was trying to expand (XTX)-1XTY but subbing in the previous values for X, and then seeing if anything stood out, but I wasn't having any luck.
Any help or tips would be appreciated