Page 213 -

212 Chapter 5 Regression with a Single Regressor: Hypothesis Tests and Confidence Intervals

Although theoretically elegant, the practical problem with weighted least squares
is that you must know how the conditional variance of ui depends on Xi, some-
thing that is rarely known in econometric applications. Weighted least squares is
therefore used far less frequently than OLS, and further discussion of WLS is
deferred to Chapter 17.

The least absolute deviations estimator. As discussed in Section 4.3, the OLS

estimator can be sensitive to outliers. If extreme outliers are not rare, then other

estimators can be more efficient than OLS and can produce inferences that are

more reliable. One such estimator is the least absolute deviations (LAD) estima-

tor, in which the regression coefficients b0 and b1 are obtained by solving a mini-
mization problem like that in Equation (4.6) except that the absolute value of the

prediction “mistake” is used instead of its square. That is, the LAD estimators of
n
b0 and b1 are the values of b0 and b1 that minimize g i= 1 0 Yi - b0 - b1Xi 0 . The

LAD estimator is less sensitive to large outliers in u than is OLS.

In many economic data sets, severe outliers in u are rare, so use of the LAD

estimator, or other estimators with reduced sensitivity to outliers, is uncommon

in applications. Thus the treatment of linear regression throughout the remainder

of this text focuses exclusively on least squares methods.

*5.6 Using the t-Statistic in Regression

When the Sample Size Is Small

When the sample size is small, the exact distribution of the t-statistic is compli-
cated and depends on the unknown population distribution of the data. If, how-
ever, the three least squares assumptions hold, the regression errors are
homoskedastic, and the regression errors are normally distributed, then the OLS
estimator is normally distributed and the homoskedasticity-only t-statistic has a
Student t distribution. These five assumptions—the three least squares assump-
tions, that the errors are homoskedastic, and that the errors are normally distrib-
uted—are collectively called the homoskedastic normal regression assumptions.

The t-Statistic and the Student t Distribution

Recall from Section 2.4 that the Student t distribution with m degrees of freedom
is defined to be the distribution of Z> 2W>m, where Z is a random variable with
a standard normal distribution, W is a random variable with a chi-squared distribution

*This section is optional and is not used in later chapters.