I am trying to understand the answer from whuber. I think I know how to derive the standard error $$s = \sqrt{\frac{(n_1-p) s_1^2 + (n_2 - p)s_2^2}{n_1 +n_2 - 2p}}.$$ To be specific, $s_1^2 = \frac{1}{n_1 -p} \sum_{i=1}^{n_1} \hat{\epsilon}_i^2$, and $s_2^2 = \frac{1}{n_2 -p} \sum_{i=n_1+1}^{n_1 + n_2} \hat{\epsilon}_i^2$. This implies that $\sum_{i=1}^{n_1 + n_2} \hat{\epsilon}_i^2 = s_1^2(n_1 -p) + s_2^2 (n_2-p)$. This gives us the result as the residual for the combined regression is the same. However, I am struggling to derive the second equation: $$SE(b) = s\sqrt{(SE(b_1)/s_1)^2 + (SE(b_2)/ s_2)^2}.$$ Could you explain how to derive this equation (if possible, in detail)?
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