# How to show SSR of the residual vector from regression of y on X1 and X2 is equal to SSR of y on X1 and X2

What I mean in the title is that when we regress $$y$$ on $$X_1$$ and by using projection matrix $$M_{X1}$$ how can I proceed with that:

the model is $$y = X_1 \cdot \beta_1 + X_2 \cdot \beta_2 + u$$

or as estimated version :

$$y = X_1 \cdot b_1 + X_2 \cdot b_2 + e$$ and residual vector comes from: $$\tilde{y} = M_{X1} \cdot y$$

So to find that SSR of $$\tilde{y}$$ on $$X_1$$ and $$X_2$$ is equal to SSR of $${y}$$ on $$X_1$$ and $$X_2$$

How should I start? should I use Normal Equations ? or Should I create a linear relationship between some variables? or trying to multiply something with the inverse? I appreciate it if you can help or can give me a hint.

Thanks.

• What is the regression for the first SSR again? Is the regressand the residuals from the regression of $y$ on what? And the regressors for that regression is $X_1$ and $X_2$ or just $X_2$ or something else? Please clarify them. Nov 29, 2022 at 11:35
• we regress y on x1 and x2 and find residuals and SSR of regression of those residuals is equal to SSR of the regression of y on x1 and x2. Nov 29, 2022 at 15:18
• What do you mean by "SSR of regression of those residuals" above? Regression on what? Do you mean the regression of the residuals on x1 and x2 again? Nov 29, 2022 at 15:25
• If you mean (i) SSR from the regression of $\hat{u}$ on x1 and x2 versus (ii) SSR from the regression of y on x1 and x2, then you can use MM=M, where $M=I-P_X$. The residuals from (i) are $MMy$ and the residuals from (ii) are $My$, so they are identical. Nov 29, 2022 at 15:30
• Sorry for not being clear about the notation. I started to take this course new but That is the thing I was looking for I guess. Nov 29, 2022 at 15:55