<p>I don't like the blue sentences. More rigorously, it should state that, <strong>if denominator is fixed</strong>, an increase in numerator gives a larger variance.</p>
<p>That is, if <span class="math-container">$\sum_{i=1}^{n}x_{i}^{2}$</span> increases, meanwhile, <span class="math-container">$\bar{x}$</span> increases too so that the denominator remains the same, then variance becomes smaller. Without any statement about the denominator (more precisely, about <span class="math-container">$\bar{x}$</span>), you don't know whether the variance becomes larger or smaller. What if <span class="math-container">$\bar{x}=0$</span>, will increase in <span class="math-container">$\sum_{i=1}^{n}x_{i}^{2}$</span> affect the variance？</p>

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2021-10-11T07:32:20.387Z
2021-10-11T07:32:20.387Z
Q9y5
<p>You should be able to confirm the claim after dividing the numerator and denominator by <span class="math-container">$\sum x_i^2$</span>. Then, <em>ceteris paribus</em>, a greater <span class="math-container">$\sum x_i^2$</span> increases the denominator and reduces the variance of the random variable <span class="math-container">$b_1$</span>.</p>

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2021-10-11T07:21:23.723Z
2021-10-11T13:08:02.910Z
2021-10-11T13:08:02.910Z
Bertrand
<p><a href="https://i.sstatic.net/Q0VmD.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/Q0VmD.png" alt="enter image description here" /></a></p>
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<p>Above is the capture from the Econometrics slide from &quot;Hill,Griths and Lim (2018) Principles of Econometrics&quot;.</p>
<p>I have no problem the see the blue font sentence separately, it's all about the numerator and denominator issues. But, if I think deeply about its intuitions, it bugs me.</p>
<p>The first blue sentence could interpret as <strong>more coverage of <span class="math-container">$x_i$</span> can reduce the variance</strong></p>
<p><strong>then I expand the equation, larger <span class="math-container">$\sum(x_i-\bar{x})^2=\sum x_i^2-n\bar{x}^2$</span> means Larger the value of<span class="math-container">$\sum x_i^2$</span>. It's reduce variance, and this contradict to second blue sentence</strong> Clearly second blue is right, so what's go wrong with my reasoning about the larger <span class="math-container">$\sum x_i^2$</span>
.</p>
<p>Please enlighten me, Thank you so much</p>

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2021-10-11T13:27:12.963Z
Welcome to economics.se, please next time do not include picture of texts or equations you should rewrite them so the whole question is searchable
1muflon1
<econometrics><regression>
intuitively explain the equation of $Var(b_1)$ in OLS model
LJNG
# intuitively explain the equation of $Var(b_1)$ in OLS model

(Asked by: username_0 on 2021-10-11)

[![enter image description here](https://i.sstatic.net/Q0VmD.png)](https://i.sstatic.net/Q0VmD.png)

[![enter image description here](https://i.sstatic.net/F3Cnj.png)](https://i.sstatic.net/F3Cnj.png)

Above is the capture from the Econometrics slide from "Hill,Griths and Lim (2018) Principles of Econometrics".

I have no problem the see the blue font sentence separately, it's all about the numerator and denominator issues. But, if I think deeply about its intuitions, it bugs me.

The first blue sentence could interpret as **more coverage of $x\_i$ can reduce the variance**

**then I expand the equation, larger $\sum(x\_i-\bar{x})^2=\sum x\_i^2-n\bar{x}^2$ means Larger the value of$\sum x\_i^2$. It's reduce variance, and this contradict to second blue sentence** Clearly second blue is right, so what's go wrong with my reasoning about the larger $\sum x\_i^2$
.

Please enlighten me, Thank you so much

**Tags:** econometrics, regression

> **Comments:**
> **username_1**: Welcome to economics.se, please next time do not include picture of texts or equations you should rewrite them so the whole question is searchable  

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## 2 answers

### Answer by username_3 on 2021-10-11. Score: 3 | Accepted answer

You should be able to confirm the claim after dividing the numerator and denominator by $\sum x\_i^2$. Then, *ceteris paribus*, a greater $\sum x\_i^2$ increases the denominator and reduces the variance of the random variable $b\_1$.


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### Answer by username_2 on 2021-10-11. Score: 3

I don't like the blue sentences. More rigorously, it should state that, **if denominator is fixed**, an increase in numerator gives a larger variance.

That is, if $\sum\_{i=1}^{n}x\_{i}^{2}$ increases, meanwhile, $\bar{x}$ increases too so that the denominator remains the same, then variance becomes smaller. Without any statement about the denominator (more precisely, about $\bar{x}$), you don't know whether the variance becomes larger or smaller. What if $\bar{x}=0$, will increase in $\sum\_{i=1}^{n}x\_{i}^{2}$ affect the variance？


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