Derivation of sample variance of OLS estimator

Question

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?

Answer 1

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.