Effect on beta one

In linear regression, If the value of the x-variable changes, how do I provide a proof that the value of beta one will be affected?
 
Nov 2009
517
130
Big Red, NY
In linear regression, If the value of the x-variable changes, how do I provide a proof that the value of beta one will be affected?
Well the estimator is dependent on \(\displaystyle x_i:\)

\(\displaystyle \hat\beta_1 = \frac{\sum(x_i - \bar x)y_i}{\sum(x_i - \bar x)^2}\)

So...
 
Nov 2009
517
130
Big Red, NY
Is there a way to prove that mathematically?
Unless I am mistaken, this is trivially true since:

\(\displaystyle y = \beta_0 + \beta_1 x \implies \beta_1 = \frac{y-\beta_0}{x}\)

Can you think of a situation in which \(\displaystyle x\) would change and \(\displaystyle \beta_1\) wouldn't?