|
Size: 1686
Comment: Rewrite 4
|
Size: 1667
Comment: Simplify
|
| Deletions are marked like this. | Additions are marked like this. |
| Line 1: | Line 1: |
| ## page was renamed from Econometrics/OrdinaryLeastSquares/UnivariateProof | |
| Line 6: | Line 7: |
The model is fit by a minimization problem: {{attachment:min.svg}} |
|
| Line 47: | Line 52: |
| Expand the formula for correlation as: | {{attachment:beta5.svg}} |
| Line 49: | Line 54: |
| {{attachment:b10.svg}} | Therefore, the regression line is estimated to be: |
| Line 51: | Line 56: |
| 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}} |
Ordinary Least Squares Univariate Proof
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:
