<p>What I mean in the title is that when we regress <span class="math-container">$y$</span> on <span class="math-container">$X_1$</span> and by using projection matrix <span class="math-container">$M_{X1}$</span>
<strong>how can I proceed with that:</strong></p>
<p>the model is
<span class="math-container">$$
y = X_1 \cdot \beta_1 + X_2 \cdot \beta_2 + u
$$</span></p>
<p>or as estimated version :</p>
<p><span class="math-container">$$
y = X_1 \cdot b_1 + X_2 \cdot b_2 + e
$$</span>
and residual vector comes from:
<span class="math-container">$$
\tilde{y} = M_{X1} \cdot y
$$</span></p>
<p>So to find that SSR of <span class="math-container">$\tilde{y}$</span> on  <span class="math-container">$X_1$</span> and <span class="math-container">$X_2$</span> is equal to SSR of <span class="math-container">${y}$</span> on  <span class="math-container">$X_1$</span> and <span class="math-container">$X_2$</span></p>
<p><em><strong>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.</strong></em></p>
<p>Thanks.</p>

CC BY-SA 4.0
2022-11-29T11:35:21.863Z
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.
chan1142
CC BY-SA 4.0
2022-11-29T15:18:49.560Z
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.
Tatanik501
CC BY-SA 4.0
2022-11-29T15:25:09.967Z
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?
chan1142
CC BY-SA 4.0
2022-11-29T15:30:24.940Z
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.
chan1142
CC BY-SA 4.0
2022-11-29T15:55:39.533Z
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.
Tatanik501
<econometrics><regression><applied-econometrics>
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
Tatanik501
# 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

(Asked by: username_0 on 2022-11-27)

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.

**Tags:** econometrics, regression, applied-econometrics

> **Comments:**
> **username_1**: 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.  
> **username_0**: 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.  
> **username_1**: 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?  
> **username_1**: 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.  
> **username_0**: 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.  

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## 0 answers

