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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.

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    $\begingroup$ 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. $\endgroup$
    – chan1142
    Nov 29, 2022 at 11:35
  • $\begingroup$ 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. $\endgroup$
    – Tatanik501
    Nov 29, 2022 at 15:18
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    $\begingroup$ 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? $\endgroup$
    – chan1142
    Nov 29, 2022 at 15:25
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    $\begingroup$ 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. $\endgroup$
    – chan1142
    Nov 29, 2022 at 15:30
  • $\begingroup$ 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. $\endgroup$
    – Tatanik501
    Nov 29, 2022 at 15:55

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