Econometrics Notation

Data

The number of observations is n.

The outcome variable is y. For observation i, the outcome value is y_i.

The treatment variable is x₁. For observation i, the treatment value is x_1i.

The control variables are x₂ through x_k (up to k - 1 control variables). For observation i, a control value might be x_2i.

Statistics

The average outcome is:

The variance is:

The standard deviation is:

The covariance between the treatment and outcome is:

The correlation between the treatment and outcome is:

Regression

A regression line passes through 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:

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