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| This reduced form can be quickly solved for ''β''. |
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| Expand the formula for correlation as: | Because the correlation coefficient can be expressed in terms of covariance and standard deviations... |
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| {{attachment:b10.svg}} | {{attachment:correlation.svg}} |
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| This can now be eliminated into: | ...the solution for ''β'' can be further reduced. |
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| {{attachment:b11.svg}} | {{attachment:beta5.svg}} |
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| Finally, ''b'' can be eloquently written as: | Therefore, the regression line is estimated to be: |
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| {{attachment:b12.svg}} Giving a generic formula for the regression line: {{attachment:b13.svg}} |
{{attachment:regression.svg}} |
Ordinary Least Squares Univariate Proof
The model is constructed like:
This is estimated as:
This line must pass through the mean and the slope of the line must be the marginal change in Y given a unit change in X. In other words, the line must pass through two points:
where:
X‾ is the sample mean of X (estimating μX)
Y‾ is the sample mean of Y (estimating μY)
sX is the sample standard deviation of X (estimating σX)
sY is the sample standard deviation of Y (estimating σY)
and rXY is the sample correlation coefficient between X and Y (estimating ρXY)
Insert the first point into the estimation. This is quickly solved for α.
Insert the second point and the solution for α into the estimation.
This reduced form can be quickly solved for β.
Because the correlation coefficient can be expressed in terms of covariance and standard deviations...
...the solution for β can be further reduced.
Therefore, the regression line is estimated to be:
