Differences between revisions 1 and 2

Ordinary Least Squares Univariate Proof

The model is constructed like:

This is estimated as:

As a starting point, the regression line must pass through these two points:

and

Take the generic equation form of a line:

Insert the first point into this form.

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

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:

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

-  ⇤ ← Revision 1 as of 2023-10-28 16:16:10 → 
  Size: 1172
  Editor: DominicRicottone
  Comment: Extracted this content from main page for OLS
+   ← Revision 2 as of 2024-06-05 20:41:15 → ⇥
  Size: 1289
  Editor: DominicRicottone
  Comment: Rewrite 1
-Deletions are marked like this.
+Additions are marked like this.
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+The model is constructed like:

{{attachment:model1.svg}}

This is estimated as:

{{attachment:model2.svg}}

Diff for "Statistics/OrdinaryLeastSquares/Univariate"