1. Chapter 4.
The Normality Assumption: CLassical Normal Linear Regression Model (CNLRM)

2. The Probability Distribution of Disturbances, ui
Recall that error disturbances
have zero expectations
are uncorrelated
have a constant variance
In the regression context, it is usually assumed that the u’s follow the normal distribution.

3. The Normality Assumption
CNLRM assumes that each ui is distributed normally with
Mean: E(ui) = 0
Variance: E(u2i) = 2
Cov(ui, uj): E(ui, uj) = i  j
Which can also be stated as
ui  N(0, 2)
Where,
 Stands for “distributed as” and N stands for normal distribution.
0 stands for zero mean.

4. Properties of OLS Estimators under the Normality Assumption
OLS Estimators are:
With the assumption of normality, OLS estimators have the following statistical properties:
They are unbiased
2. They have minimum variance. Combined with 1, this means they are minimum-variance unbiased, or efficient estimators.
3. Consistency, that is, as the sample size increases indefinitely, the estimators converge to their true population values.

5. Assumptions

6. Assumptions

7. Figure 4.1 and Figure 4.2

8. Assumptions

9. The Method of Maximum Likelihood (ML)
A method of point estimation with some stronger theoretical properties than the method of OLS is the method of maximum likelihood (ML).
If ui is assumed to be normally distributed, the ML and OLS estimators of the regression coefficients, the ’s, are identical. The ML estimator of 2 is u2i/n. This estimator is biased, whereas the OLS estimator of 2 is u2i/(n-2), which is unbiased.
However, as sample size, n, gets larger, two estimators of 2 tend to be equal, therefore as n gets larger, the ML estimator of 2 is also unbiased.