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5.5 The Theoretical Foundations of Ordinary Least Squares 209

As this example of modeling earnings illustrates, heteroskedasticity arises in
many econometric applications. At a general level, economic theory rarely gives
any reason to believe that the errors are homoskedastic. It therefore is prudent to
assume that the errors might be heteroskedastic unless you have compelling rea-
sons to believe otherwise.

Practical implications. The main issue of practical relevance in this discussion is
whether one should use heteroskedasticity-robust or homoskedasticity-only stan-
dard errors. In this regard, it is useful to imagine computing both, then choosing
between them. If the homoskedasticity-only and heteroskedasticity-robust stan-
dard errors are the same, nothing is lost by using the heteroskedasticity-robust
standard errors; if they differ, however, then you should use the more reliable
ones that allow for heteroskedasticity. The simplest thing, then, is always to use
the heteroskedasticity-robust standard errors.

For historical reasons, many software programs report homoskedasticity-
only standard errors as their default setting, so it is up to the user to specify the
option of heteroskedasticity-robust standard errors. The details of how to imple-
ment heteroskedasticity-robust standard errors depend on the software package
you use.

All of the empirical examples in this book employ heteroskedasticity-robust
standard errors unless explicitly stated otherwise.1

*5.5 The Theoretical Foundations

of Ordinary Least Squares

As discussed in Section 4.5, the OLS estimator is unbiased, is consistent, has a
variance that is inversely proportional to n, and has a normal sampling distribu-
tion when the sample size is large. In addition, under certain conditions the OLS
estimator is more efficient than some other candidate estimators. Specifically, if
the least squares assumptions hold and if the errors are homoskedastic, then the
OLS estimator has the smallest variance of all conditionally unbiased estimators
that are linear functions of Y1, c, Yn. This section explains and discusses this
result, which is a consequence of the Gauss–Markov theorem. The section concludes

1In case this book is used in conjunction with other texts, it might be helpful to note that some text-
books add homoskedasticity to the list of least squares assumptions. As just discussed, however,
this additional assumption is not needed for the validity of OLS regression analysis as long as
heteroskedasticity-robust standard errors are used.
*This section is optional and is not used in later chapters.