<p>The conditioning is on <span class="math-container">$x$</span>, where <span class="math-container">$x$</span> represents all independent variables for all observations. Thus <span class="math-container">$x$</span> is treated as a constant throughout the derivation.</p>
<p>This is the standard method of deriving the variance of estimates, it is done conditional on the exogenous regressors.</p>

CC BY-SA 4.0
2022-09-25T10:23:55.247Z
2022-09-25T10:23:55.247Z
Michael Gmeiner
<p>Consider a Simple Linear Regression with the following assumptions:</p>
<ol>
<li><p>The dependent variable is related to the independent variable and the error term like: <span class="math-container">$y = \beta_0 + \beta _1 x + u$</span></p>
</li>
<li><p>We have a random sample of size <span class="math-container">$n$</span> following the population model in assumption #1</p>
</li>
<li><p>The sample outcomes on <span class="math-container">$x$</span> are not all the same value</p>
</li>
<li><p><span class="math-container">$E[u|x] = 0$</span> is true</p>
</li>
</ol>
<p>I am reading on the derivation of the sample variance for <span class="math-container">$\beta _1$</span>. <strong>However, I don't understand why they assume that <span class="math-container">$\Sigma _ {i=1} ^ n x_i - \bar x$</span> is treated like a constant. Why is this true?</strong></p>
<p><a href="https://i.sstatic.net/b0fQF.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/b0fQF.png" alt="enter image description here" /></a></p>

CC BY-SA 4.0
2022-09-25T13:17:29.473Z
The condition |x is missing in the two variances in the middle of your equalities
Bertrand
<econometrics><regression>
Derivation of sample variance of OLS estimator
hu234
# Derivation of sample variance of OLS estimator

(Asked by: username_0 on 2022-09-24)

Consider a Simple Linear Regression with the following assumptions:

1. The dependent variable is related to the independent variable and the error term like: $y = \beta\_0 + \beta \_1 x + u$
2. We have a random sample of size $n$ following the population model in assumption #1
3. The sample outcomes on $x$ are not all the same value
4. $E[u|x] = 0$ is true

I am reading on the derivation of the sample variance for $\beta \_1$. **However, I don't understand why they assume that $\Sigma \_ {i=1} ^ n x\_i - \bar x$ is treated like a constant. Why is this true?**

[![enter image description here](https://i.sstatic.net/b0fQF.png)](https://i.sstatic.net/b0fQF.png)

**Tags:** econometrics, regression
> **Comments:**
> **username_1**: The condition |x is missing in the two variances in the middle of your equalities  

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## 1 answers

### Answer by username_2 on 2022-09-25. Score: 1 | Accepted answer

The conditioning is on $x$, where $x$ represents all independent variables for all observations. Thus $x$ is treated as a constant throughout the derivation.

This is the standard method of deriving the variance of estimates, it is done conditional on the exogenous regressors.


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