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Math Help - Effect on beta one by changing x variables

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

    Hi,

    please help me with this proof,

    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?

    thanks in advance!!
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  2. #2
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    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?
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  3. #3
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    Yeah, the x-values for the x variable, i know coz I tried it out on excel....
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  4. #4
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    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.
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  5. #5
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    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
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  6. #6
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    oh wait i mininterpreted your question, my bad. if you're asking about the transformation:

    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)
    Last edited by SpringFan25; May 15th 2011 at 04:10 AM.
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  7. #7
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    Hi,

    I checked on the formula sheet and that

    β_1= \frac{∑(X-\bar{X})(Y-\bar{Y}}{∑(X-\bar{X})^2}
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  8. #8
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    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.
    Last edited by SpringFan25; May 16th 2011 at 11:18 AM.
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  9. #9
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    thank you. however, how could you write up a proof to show that beta zero will change when x values have changed?
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  10. #10
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    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.
    Last edited by SpringFan25; May 17th 2011 at 07:39 AM.
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  11. #11
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    thank you very much
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