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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?

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  • $\begingroup$ The condition |x is missing in the two variances in the middle of your equalities $\endgroup$
    – Bertrand
    Sep 25, 2022 at 13:17

1 Answer 1

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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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