# Math Help - Effect on beta one by changing x variables

1. ## Effect on beta one by changing x variables

Hi,

I had to do a linear regression of y on x, where the x variables are 1851 to 1900 inclusive.
I was asked to change x varaibles to 1 to 50 inclusive and whether that would change beta one.

i know beta one would stay the same, but how would I be able to prove it?

2. do you mean x values instead of x variables? or did you really have 50 x variables in your first model?

why do you think it would stay the same?

3. Yeah, the x-values for the x variable, i know coz I tried it out on excel....

4. write down the formula defining your model (ie, say what beta 1 is).

I wouldn't expect that to happen except by chance, unless your model has no random error term.

5. nah it kinda make sense coz gradient equals rise/run yeah, and the run is still 50.... but that's a dodgy proof.... trying to find a better one lolz

x' = x - 1850

You know (i hope!) the formula for beta 1, so just show that Var(X) and Cov(X,Y) dont change.

ie, show that
Cov(X',Y) = Cov(X,Y)
Var(X') = Var(X)

7. Hi,

I checked on the formula sheet and that

β_1= \frac{∑(X-\bar{X})(Y-\bar{Y}}{∑(X-\bar{X})^2}

8. i'll use "b" instead of beta 1 to save typing

$b = \frac{Cov(X,Y)}{Var(X)} = \frac{ \sum (X - \bar{X})(Y - \bar{Y})}{\sum (X - \bar{X})(X - \bar{X})}$

Define X' = X - 1850

If we run a regression using X', we will get

$b' = \frac{ \sum (X' - \bar{X'})(Y - \bar{Y})}{\sum (X' - \bar{X'})(X' - \bar{X'})}$

substitute X' = X - 1850

$b' = \frac{\sum (X - 1850 - \bar{X'})(Y - \bar{Y})}{\sum (X - 1850 - \bar{X'})(X - 1850 - \bar{X'})}$

You should know or be able to prove that $\bar{X'} = \bar{X} - 1850$. Substitute that in:

$b' = \frac{\sum (X - 1850 - \left[\bar{X} - 1850 \right])(Y - \bar{Y})}{\sum (X - 1850 - \left[\bar{X} - 1850 \right])(X - 1850 - \left[\bar{X} - 1850 \right])}$

$b' = \frac{\sum (X - 1850 + 1850 - \bar{X})(Y - \bar{Y})}{\sum (X - 1850 +1850 - \bar{X} )(X - 1850 +1850 - \bar{X} )}$

$b' = \frac{\sum (X - \bar{X})(Y - \bar{Y})}{\sum (X - \bar{X} )(X - \bar{X} )}$

$b' = b$

Which is the required result.

9. thank you. however, how could you write up a proof to show that beta zero will change when x values have changed?

10. start from the formula for b0'

$b'_0 = \bar{y} - b_1 \bar{X'}$

and substitute $\bar{X'}=\bar{X} - 1850$

compare the result to the formula for $b0$.

11. thank you very much