6
CHAPTER 2
TEACHING NOTES
This is the chapter where I expect students to follow most, if not all, of the algebraic derivations.
In class I like to derive at least the unbiasedness of the OLS slope coefficient, and usually I
derive the variance. At a minimum, I talk about the factors affecting the variance. To simplify
the notation, after I emphasize the assumptions in the population model, and assume random
sampling, I just condition on the values of the explanatory variables in the sample.
Technically,
this is justified by random sampling because, for example, E(
u
i

x
1
,
x
2
,…,
x
n
) = E(
u
i

x
i
) by
independent sampling. I find that students are able to focus on the key assumption SLR.4 and
subsequently take my word about how conditioning on the independent variables in the sample is
harmless. (If you prefer, the appendix to Chapter 3 does the conditioning argument carefully.)
Because statistical inference is no more difficult in multiple regression than in simple regression,
I postpone inference until Chapter 4. (This reduces redundancy and allows you to focus on the
interpretive differences between simple and multiple regression.)
You might notice how, compared with most other texts, I use relatively few assumptions to
derive the unbiasedness of the OLS slope estimator, followed by the formula for its variance.
This is because I do not introduce redundant or unnecessary assumptions. For example, once
SLR.4 is assumed, nothing further about the relationship between
u
and
x
is needed to obtain the
unbiasedness of OLS under random sampling.
Incidentally, one of the uncomfortable facts about finitesample analysis is that there is a
difference between an estimator that is unbiased conditional on the outcome of the covariates and
one that is unconditionally unbiased. If the distribution of the
𝑥
𝑖
is such that they can all equal
the same value with positive probability – as is the case with discreteness in the distribution –
then the unconditional expectation does not really exist. Or, if it is made to exist then the
estimator is not unbiased. I do not try to explain these subtleties in an introductory course, but I
have had instructors ask me about the difference.