At the start of this chapter is good time to remind students that a specific error distribution
played no role in the results of Chapter 3.
That is because only the first two moments were
derived under the full set of Gauss-Markov assumptions.
Nevertheless, normality is needed to
obtain exact normal sampling distributions (conditional on the explanatory variables).
emphasize that the full set of CLM assumptions are used in this chapter, but that in Chapter 5 we
relax the normality assumption and still perform approximately valid inference.
One could argue
that the classical linear model results could be skipped entirely, and that only large-sample
analysis is needed.
But, from a practical perspective, students still need to know where the
distribution comes from because virtually all regression packages report
statistics and obtain
values off of the
I then find it very easy to cover Chapter 5 quickly, by just saying
we can drop normality and still use
statistics and the associated
-values as being
Besides, occasionally students will have to analyze smaller data sets,
especially if they do their own small surveys for a term project.
It is crucial to emphasize that we test hypotheses about unknown population parameters.
my students that they will be punished if they write something like H
= 0 on an exam or,
even worse, H
: .632 = 0.
One useful feature of Chapter 4 is its illustration of how to rewrite a population model so that it
contains the parameter of interest in testing a single restriction.
I find this is easier, both
theoretically and practically, than computing variances that can, in some cases, depend on
numerous covariance terms.
The example of testing equality of the return to two- and four-year
colleges illustrates the basic method, and shows that the respecified model can have a useful
Of course, some statistical packages now provide a standard error for linear
combinations of estimates with a simple command, and that should be taught, too.
One can use an
test for single linear restrictions on multiple parameters, but this is less
transparent than a
test and does not immediately produce the standard error needed for a
confidence interval or for testing a one-sided alternative.
The trick of rewriting the population
model is useful in several instances, including obtaining confidence intervals for predictions in
Chapter 6, as well as for obtaining confidence intervals for marginal effects in models with
interactions (also in Chapter 6).
The major league baseball player salary example illustrates the difference between individual
and joint significance when explanatory variables (
in this case) are highly
I tend to emphasize the
-squared form of the
statistic because, in practice, it is
applicable a large percentage of the time, and it is much more readily computed.
I do regret that
this example is biased toward students in countries where baseball is played.