I'm just reading a book on econometrics and now I'm stuck with a problem:
There is a Theorem on "Orthogonal Partitioned Regression" which says:
"In the multiple linear least squares regression of $y$ on two sets of variables $X_1$ and $X_2$, if the two sets of variables are orthogonal, then the separate coefficient vectors can be obtained by separate regressions of $y$ on $X_1$ alone and $y$ on $X_2$ alone. ..."
Furthermore, the authors says that $X_1$ and $X_2$ are orthogonal if: $$ (X_1)' X_2=0 \, , $$ which confuses me because so far I always thought the condition for orthogonality between two matrices is that $(X_1)' X_2=I$.
The book is: W. Greene, Econometric Analysis, 7th ed. / Theorem 3.1 / p.33
What I found so far is: