<p>While constructing the F-test with matrix notation,</p>
<p><strong>For the Unrestricted model</strong>, we have:</p>
<blockquote>
<p><span class="math-container">$$ y=X_{1}\cdot b_{1}+X_{2}\cdot b_{2}+e $$</span></p>
</blockquote>
<p><strong>For the Restricted model</strong>;</p>
<blockquote>
<p><span class="math-container">$$ y=\widehat{X}_{1}.b_{1}+e_{R} $$</span></p>
</blockquote>
<p>To create a linear relationship between <span class="math-container">$ e_{R} $</span> AND <span class="math-container">$ e $</span> ,we multiply the unrestricted model with <span class="math-container">$M_{1}$</span> such that:</p>
<p><span class="math-container">$$
 \underbrace{M_{1}\cdot y}_{\text{$e_{R}$}} = \underbrace{M_{1}\cdot X_{1}}_{\text{0}}  b_{1}+M_{1}\cdot X_{2}\cdot b_{2}+ \underbrace{M_{1}\cdot e}_{\text{e}}
$$</span></p>
<p>Therefore,</p>
<p><span class="math-container">$$
e_{R}^{T}\cdot e_{R} = \left( e+M_{1}\cdot X_{2}\cdot b_{2}\right) ^{T} \cdot \left( e+M_{1}\cdot X_{2}\cdot b_{2}\right)
$$</span></p>
<p>After multiplication and by using Normal Equations, we get:</p>
<p><span class="math-container">$$
e_{R}^{T}\cdot e_{R} -e^{T}\cdot e = b_{2}^{T}\cdot X_{2}^{T}.M_{1}\cdot X_{2}\cdot b_{2}
$$</span></p>
<p>And here after solving a partitioned regression we have the value of <span class="math-container">$b_{2} $</span> such that:</p>
<p><span class="math-container">$$
b_{2} = (X_{2}^{T}.M_{1}\cdot X_{2})^{-1} \cdot X_{2}^{T} \cdot  M_{1} \cdot y
$$</span></p>
<p>Inserting <span class="math-container">$ y = X_{1}\cdot \beta_{1} + X_{2}\cdot \beta_{2} $</span> , and after multiptication,
we have such a <span class="math-container">$b_{2}$</span> :</p>
<ul>
<li><span class="math-container">$$  b_{2} = \underbrace{(X_{2}^{T}.M_{1}\cdot X_{2})^{-1} \cdot
   X_{2}^{T} \cdot X_{1} \cdot  \beta_{1}}_{\text{$0$}} +
   \underbrace{(X_{2}^{T}.M_{1}\cdot X_{2})^{-1} \cdot X_{2}^{T} \cdot
   X_{2} \cdot  \beta_{2}}_{\text{$0$}} + ... $$</span></li>
</ul>
<p>and it continues actually but here in the part    <span class="math-container">$ \underbrace{(X_{2}^{T}.M_{1}\cdot X_{2})^{-1} \cdot X_{2}^{T} \cdot X_{2} \cdot  \beta_{2}}_{\text{$0$}}$</span> for this expression to be <span class="math-container">$0$</span> we need to the null hypothesis <span class="math-container">$H_{0}:\beta_{2} = 0$</span></p>
<p><strong>My question is why do we need to impose this? and what happens if we don't?</strong></p>

CC BY-SA 4.0
2022-11-26T20:40:41.400Z
Hi: It's because that's the restriction that the null hypothesis is imposing. It's analogous with a t-test for testing $\mu =0$. When we calculate the t-statistic, we use $\mu = 0$ in the expression for the t-statistic. So, we end up with $ T = \frac{\bar{x}}{\frac{\sigma}{\sqrt{n}}}$.
mark leeds
CC BY-SA 4.0
2022-11-27T15:31:18.473Z
Thank you for the comment then can I ask what happens if we don't impose that? if we can't eliminate the whole term?
Tatanik501
CC BY-SA 4.0
2022-11-27T21:11:20.840Z
Hi:  What happens is that then the statistic you obtain will be incorrect because it's not assuming that the null hypothesis is true. The steps of a hypothesis test ( neyman pearson ) require that the test- statistic is such that the null hypothesis is true when it is calculated. The idea is to create a  pivotal quantity that has a known distribution when the null is true.  Then, if this pivotal quantity ends up being extreme in terms of where it falls in the distribution, then this is evidence that the null is not true.
mark leeds
CC BY-SA 4.0
2022-11-27T21:11:39.313Z
You'll understand the answer more clearly if you take a class or look at book on statistical hypothesis testing.
mark leeds
CC BY-SA 4.0
2022-11-28T12:47:45.140Z
Thank you very much for your time.
Tatanik501
CC BY-SA 4.0
2022-11-28T18:14:30.057Z
I hope that it made sense. Lehmann is one book for statistical hypothesis testing. Another one might be Casella and Berger.
mark leeds
CC BY-SA 4.0
2022-11-29T15:15:09.233Z
Thanks, I realized I need to refresh my knowledge and I guess I need to know about projection matrices more.
Tatanik501
CC BY-SA 4.0
2022-11-30T17:02:45.010Z
Hi: I'm not sure that your question is so much about projection matrices but it won't hurt to understand projection in regression. I think a more key issue is what is termed "the pivotal quantity". This is what we want to use as a part of the test statistic, because, when the null hypothesis is true, the distribution of the test-statistic is then independent of the value of the estimate being used to test the hypothesis. ( in a t-test, it's $\bar{x}$ ).
mark leeds
CC BY-SA 4.0
2022-11-30T17:40:59.573Z
yes, I am trying to understand that "independency" part but also maybe not be directly related but I felt I learned projection matrices halfway only. These 2 subjects that I am having difficulty with. And may I ask for any well-organized and more illuminating source for those? currently, we are using Greene's Econometric Analysis and Davidson and MacKinnon - Econometric Theory and Methods. Thank you.
Tatanik501
CC BY-SA 4.0
2022-12-01T19:00:56.827Z
you are reading good econometrics books but your question is more related to mathematical statistics-hypothesis testing which won't be covered in them.  I would look at wilkes ( math stat ) or cramer or any fairly advanced math stat book.  Casella and Berger is a good one. You need a text that discusses the pivotal quantity concept. Maybe look at Rao's 1973 text.
mark leeds
CC BY-SA 4.0
2022-12-01T19:06:08.853Z
Also, I didn;t mean to imply that projections are a waste of time. It's good to understand all of it but your question is really related to pivotal quantity which is why I was de-emphasizing the projection material. But yes, in regression, one can view the whole procedure as a projecton.  It's great to see what you want to understand everything. The problem with taking an economerics course straight away is a lot of the text writers are quite well versed in mathematical  statistics  but they avoid the details. A good math stat class, IMHO, should be a pre-requisite for an econometrics class.
mark leeds
CC BY-SA 4.0
2022-12-02T06:14:41.943Z
Thank you for your suggestions sir I will check that pivotal quantity concept which seems vital to understand the essence of it. This is an MSc course actually. I have taken Introductory Econometrics by using Wooldridge only so the theoric and abstract parts arent easy for now to me.
Tatanik501
CC BY-SA 4.0
2022-12-03T15:37:41.093Z
Hi: Sounds good. But see if your school offers a mathematical statistics class either through the stat or math department.  A lot of things used in econometrics have underlying theory based on math-stat.
mark leeds
CC BY-SA 4.0
2022-12-03T18:48:23.093Z
Thanks for your time really. I need more solid background in that. even not this year, for the next year I will take both again if necessary and will learn what is going on in this field. Have a great day sir.
Tatanik501
CC BY-SA 4.0
2022-12-05T00:43:35.930Z
take it slow. if you know what you like ( statistics, econometrics ), then that's half the battle. look for a math-stat class from math or stat department this will give you some useful background.
mark leeds
CC BY-SA 4.0
2022-12-05T16:15:03.787Z
Thanks a lot for the advice and your time.
Tatanik501
<econometrics><regression><hypothesis-testing>
Construction of F-Test
Tatanik501
# Construction of F-Test

(Asked by: username_0 on 2022-11-25)

While constructing the F-test with matrix notation,

**For the Unrestricted model**, we have:

> $$ y=X\_{1}\cdot b\_{1}+X\_{2}\cdot b\_{2}+e $$

**For the Restricted model**;

> $$ y=\widehat{X}\_{1}.b\_{1}+e\_{R} $$

To create a linear relationship between $ e\_{R} $ AND $ e $ ,we multiply the unrestricted model with $M\_{1}$ such that:

$$
\underbrace{M\_{1}\cdot y}\_{\text{$e\_{R}$}} = \underbrace{M\_{1}\cdot X\_{1}}\_{\text{0}} b\_{1}+M\_{1}\cdot X\_{2}\cdot b\_{2}+ \underbrace{M\_{1}\cdot e}\_{\text{e}}
$$

Therefore,

$$
e\_{R}^{T}\cdot e\_{R} = \left( e+M\_{1}\cdot X\_{2}\cdot b\_{2}\right) ^{T} \cdot \left( e+M\_{1}\cdot X\_{2}\cdot b\_{2}\right)
$$

After multiplication and by using Normal Equations, we get:

$$
e\_{R}^{T}\cdot e\_{R} -e^{T}\cdot e = b\_{2}^{T}\cdot X\_{2}^{T}.M\_{1}\cdot X\_{2}\cdot b\_{2}
$$

And here after solving a partitioned regression we have the value of $b\_{2} $ such that:

$$
b\_{2} = (X\_{2}^{T}.M\_{1}\cdot X\_{2})^{-1} \cdot X\_{2}^{T} \cdot M\_{1} \cdot y
$$

Inserting $ y = X\_{1}\cdot \beta\_{1} + X\_{2}\cdot \beta\_{2} $ , and after multiptication,
we have such a $b\_{2}$ :

* $$ b\_{2} = \underbrace{(X\_{2}^{T}.M\_{1}\cdot X\_{2})^{-1} \cdot
  X\_{2}^{T} \cdot X\_{1} \cdot \beta\_{1}}\_{\text{$0$}} +
  \underbrace{(X\_{2}^{T}.M\_{1}\cdot X\_{2})^{-1} \cdot X\_{2}^{T} \cdot
  X\_{2} \cdot \beta\_{2}}\_{\text{$0$}} + ... $$

and it continues actually but here in the part $ \underbrace{(X\_{2}^{T}.M\_{1}\cdot X\_{2})^{-1} \cdot X\_{2}^{T} \cdot X\_{2} \cdot \beta\_{2}}\_{\text{$0$}}$ for this expression to be $0$ we need to the null hypothesis $H\_{0}:\beta\_{2} = 0$

**My question is why do we need to impose this? and what happens if we don't?**

**Tags:** econometrics, regression, hypothesis-testing

> **Comments:**
> **username_1**: Hi: It's because that's the restriction that the null hypothesis is imposing. It's analogous with a t-test for testing $\mu =0$. When we calculate the t-statistic, we use $\mu = 0$ in the expression for the t-statistic. So, we end up with $ T = \frac{\bar{x}}{\frac{\sigma}{\sqrt{n}}}$.  
> **username_0**: Thank you for the comment then can I ask what happens if we don't impose that? if we can't eliminate the whole term?  
> **username_1**: Hi:  What happens is that then the statistic you obtain will be incorrect because it's not assuming that the null hypothesis is true. The steps of a hypothesis test ( neyman pearson ) require that the test- statistic is such that the null hypothesis is true when it is calculated. The idea is to create a  pivotal quantity that has a known distribution when the null is true.  Then, if this pivotal quantity ends up being extreme in terms of where it falls in the distribution, then this is evidence that the null is not true.  
> **username_1**: You'll understand the answer more clearly if you take a class or look at book on statistical hypothesis testing.  
> **username_0**: Thank you very much for your time.  
> **username_1**: I hope that it made sense. Lehmann is one book for statistical hypothesis testing. Another one might be Casella and Berger.  
> **username_0**: Thanks, I realized I need to refresh my knowledge and I guess I need to know about projection matrices more.  
> **username_1**: Hi: I'm not sure that your question is so much about projection matrices but it won't hurt to understand projection in regression. I think a more key issue is what is termed "the pivotal quantity". This is what we want to use as a part of the test statistic, because, when the null hypothesis is true, the distribution of the test-statistic is then independent of the value of the estimate being used to test the hypothesis. ( in a t-test, it's $\bar{x}$ ).  
> **username_0**: yes, I am trying to understand that "independency" part but also maybe not be directly related but I felt I learned projection matrices halfway only. These 2 subjects that I am having difficulty with. And may I ask for any well-organized and more illuminating source for those? currently, we are using Greene's Econometric Analysis and Davidson and MacKinnon - Econometric Theory and Methods. Thank you.  
> **username_1**: you are reading good econometrics books but your question is more related to mathematical statistics-hypothesis testing which won't be covered in them.  I would look at wilkes ( math stat ) or cramer or any fairly advanced math stat book.  Casella and Berger is a good one. You need a text that discusses the pivotal quantity concept. Maybe look at Rao's 1973 text.  
> **username_1**: Also, I didn;t mean to imply that projections are a waste of time. It's good to understand all of it but your question is really related to pivotal quantity which is why I was de-emphasizing the projection material. But yes, in regression, one can view the whole procedure as a projecton.  It's great to see what you want to understand everything. The problem with taking an economerics course straight away is a lot of the text writers are quite well versed in mathematical  statistics  but they avoid the details. A good math stat class, IMHO, should be a pre-requisite for an econometrics class.  
> **username_0**: Thank you for your suggestions sir I will check that pivotal quantity concept which seems vital to understand the essence of it. This is an MSc course actually. I have taken Introductory Econometrics by using Wooldridge only so the theoric and abstract parts arent easy for now to me.  
> **username_1**: Hi: Sounds good. But see if your school offers a mathematical statistics class either through the stat or math department.  A lot of things used in econometrics have underlying theory based on math-stat.  
> **username_0**: Thanks for your time really. I need more solid background in that. even not this year, for the next year I will take both again if necessary and will learn what is going on in this field. Have a great day sir.  
> **username_1**: take it slow. if you know what you like ( statistics, econometrics ), then that's half the battle. look for a math-stat class from math or stat department this will give you some useful background.  
> **username_0**: Thanks a lot for the advice and your time.  

---

## 0 answers

