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| = Ordinary Least Squares Univariate Proof = | = OLS Univariate Derivation = |
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| The model is fit by a minimization problem: {{attachment:min.svg}} |
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| As a starting point, the regression line must pass through these two points: | 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: |
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| {{attachment:regression1.svg}} | {{attachment:model3.svg}} |
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| and | where: |
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| {{attachment:regression2.svg}} | * ''X‾'' is the sample mean of ''X'' (estimating ''μ,,X,,'') * ''Y‾'' is the sample mean of ''Y'' (estimating ''μ,,Y,,'') * ''s,,X,,'' is the sample standard deviation of ''X'' (estimating ''σ,,X,,'') * ''s,,Y,,'' is the sample standard deviation of ''Y'' (estimating ''σ,,Y,,'') * and ''r,,XY,,'' is the sample correlation coefficient between ''X'' and ''Y'' (estimating ''ρ,,XY,,'') |
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| Take the generic equation form of a line: | Insert the first point into the estimation. This is quickly solved for ''α''. |
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| {{attachment:b01.svg}} | {{attachment:alpha1.svg}} |
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| Insert the first point into this form. | {{attachment:alpha2.svg}} |
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| {{attachment:b02.svg}} | Insert the second point and the solution for ''α'' into the estimation. |
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| This can be trivially rewritten to solve for ''a'' in terms of ''b'': | {{attachment:beta1.svg}} |
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| {{attachment:b03.svg}} | {{attachment:beta2.svg}} |
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| Insert the second point into the original form. | {{attachment:beta3.svg}} |
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| {{attachment:b04.svg}} | This reduced form can be quickly solved for ''β''. |
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| Now additionally insert the solution for ''a'' in terms of ''b''. | {{attachment:beta4.svg}} |
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| {{attachment:b05.svg}} | Because the correlation coefficient can be expressed in terms of covariance and standard deviations... |
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| Expand all terms to produce: | {{attachment:correlation.svg}} |
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| {{attachment:b06.svg}} | ...the solution for ''β'' can be further reduced. |
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| This can now be eliminated into: | {{attachment:beta5.svg}} |
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| {{attachment:b07.svg}} | Therefore, the regression line is estimated to be: |
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| Giving a solution for ''b'': {{attachment:b08.svg}} This solution is trivially rewritten as: {{attachment:b09.svg}} Expand the formula for correlation as: {{attachment:b10.svg}} This can now be eliminated into: {{attachment:b11.svg}} Finally, ''b'' can be eloquently written as: {{attachment:b12.svg}} Giving a generic formula for the regression line: {{attachment:b13.svg}} |
{{attachment:regression.svg}} |
OLS Univariate Derivation
The model is constructed like:
The model is fit by a minimization problem:
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:
