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:

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Statistics/OrdinaryLeastSquares/Univariate

Ordinary Least Squares Univariate Proof