Effect on beta one

**statisticsisawesome** · May 10th 2010, 09:11 PM

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?

**Anonymous1** · May 10th 2010, 09:17 PM

Originally Posted by statisticsisawesome

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 $x_i:$

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

So...

**statisticsisawesome** · May 10th 2010, 09:38 PM

Is there a way to prove that mathematically?

**Anonymous1** · May 10th 2010, 09:56 PM

Originally Posted by statisticsisawesome

Is there a way to prove that mathematically?

Unless I am mistaken, this is trivially true since:

$y = \beta_0 + \beta_1 x \implies \beta_1 = \frac{y-\beta_0}{x}$

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

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