Differences between revisions 2 and 3

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:

Insert the first point into the estimation. This is quickly solved for α.

Insert the second point into the original form.

Now additionally insert the solution for a in terms of b.

Expand all terms to produce:

This can now be eliminated into:

Giving a solution for b:

This solution is trivially rewritten as:

Expand the formula for correlation as:

This can now be eliminated into:

Finally, b can be eloquently written as:

Giving a generic formula for the regression line:

CategoryRicottone

Statistics/OrdinaryLeastSquares/Univariate (last edited 2025-01-10 14:33:54 by DominicRicottone)

-  ⇤ ← Revision 2 as of 2024-06-05 20:41:15 → 
  Size: 1289
  Editor: DominicRicottone
  Comment: Rewrite 1
+   ← Revision 3 as of 2024-06-05 20:57:05 → ⇥
  Size: 1253
  Editor: DominicRicottone
  Comment: Rewrite 2
-Deletions are marked like this.
+Additions are marked like this.
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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
+Insert the first point into the estimation. This is quickly solved for ''α''.
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-{{attachment:regression2.svg}}
+{{attachment:alpha1.svg}}
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-Take the generic equation form of a line:

{{attachment:b01.svg}}

Insert the first point into this form.

{{attachment:b02.svg}}

This can be trivially rewritten to solve for ''a'' in terms of ''b'':

{{attachment:b03.svg}}
+{{attachment:alpha2.svg}}

Diff for "Statistics/OrdinaryLeastSquares/Univariate"

Ordinary Least Squares Univariate Proof