Suppose you have some ordered data, for instance average income for some people arranged by age. Suppose that you divide your sample in two subsets, say $P_1=$ people within 50 years old and $P_2=$ people above 50 years old.
If you perform two linear regressions on $P_1$ and $P_2$ you get the coefficients of determination $(R_1)^2$ and $(R_2)^2$, the regression coefficients $\beta_1$ and $\beta_2$ and the the intercept terms $c_1$ and $c_2$. Set also $|P_i|=N_i$ for $i=1,2$.
How to compute, as a function of the eight numbers $R_i$, $\beta_i$, $c_i$, $N_i$ ($i=1,2$), the expected value of the determination coefficient and the regression coefficient of the whole sample $P_1\cup P_2$ when performing a global linear regression?