1. The regression model in matrix form
The least squares normal equations take the form:
and we can calculate the OLS estimator as:
under the assumption that (X’X) is invertible.

2. Distribution of the OLS estimator
This depends on our assumptions about the errors and the
variance-covariance matrix of the errors.
This is a matrix with the variances of the errors on the diagonal
and the covariances off the diagonal.
e.g. if N = 3.
Note that this matrix is symmetric by construction.

3. The Gauss-Markov assumptions in matrix form
and finally,
5. the errors follow a normal distribution.
These assumptions are essentially the same as for the bivariate
regression model.

4. Assumptions 2 and 3 mean that the variance-covariance matrix
of the errors takes the form:
If the GM assumptions hold then we can show that OLS is BLUE
i.e the Best Linear Unbiased Estimator.

5. OLS is unbiased under the GM assumptions
Proof:
GM4 states that the X matrix is non-stochastic and GM1
states that the error has expectation zero, using these we have:

6. The variance of the OLS estimator can be derived as follows:
By GM4 the X matrix is exogenous. Therefore we have:
By GM2 and GM3 we can write:

7. Therefore:
This is the variance-covariance matrix of the OLS estimator.
It contains the variances of the coefficient estimates on the
diagonal and the covariances off the diagonal.
This matrix will be k x k where k is the number of coefficients
estimated (including any constant term).

8. Example: The bivariate regression model.
The Gauss-Markov Theorem shows that, under the GM
assumptions, the variance-covariance matrix of any linear unbiased
estimator differs from the OLS v-cov matrix by a positive semi-
definite matrix.
Hence OLS is BLUE.

9. The sample variance-covariance matrix is obtained by
replacing the unknown error variance with an unbiased estimator.
This is the expression we use to calculate the coefficient
standard errors which are reported in the regression output.