Question

Using scalar notation where ? is the y-intercept and ? is the slope coefficient, derive OLS estimators for alpha-hat and beta-hat.

Answer 1

In the method of OLS estimation, we try to minimize the sum of square of errors as:

Note that the summation is from i = 1 to N where N is the number of points here.

Where ? is the y-intercept and ? is the slope coefficient

Now the minimization is done by differentiating the above expression with respect to both ? first and then ? and equating them to 0 to get:

From the first equation above, we get:

Dividing the whole equation by N, we get:

Now this can be put in the second equation to obtain the estimator for ? as:

This is the required OLS estimator for ?

This is the required OLS estimator for ?