OLS Univariate Derivation
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: