157CHAPTER
8robust regressions
A
fter reading this chapter you will understand:■ (^) Under what conditions standard regression parameter estimates are
sensitive to small changes in the data.
■ (^) The concept of robust estimates of regression parameters.
■ (^) How to construct robust regression estimators.
■ (^) How to apply robust regressions to problems in finance.
Broadly speaking, statistics is the science of describing and analyzing data
and making inferences on a population based on a sample extracted from
the same population. An important aspect of statistics is the compression of
the data into numbers that are descriptive of some feature of the distribu-
tion. Classical statistics identifies several single-number descriptors such as
mean, variance, skewness, kurtosis, and higher moments. These numbers
give a quantitative description of different properties of the population.
Classical statistics chooses single-number descriptors that have nice
mathematical properties. For example, if we know all the moments of a
probability distribution, we can reconstruct the same distribution. In a num-
ber of cases (but not always), the parameters that identify a closed-form rep-
resentation of a distribution correspond to these descriptive concepts. For
example, the parameters that identify a normal distribution correspond to
the mean and to the variance. However, in classical statistics, most of these
descriptive parameters are not “robust.” Intuitively, robustness means that
small changes in the sample or small mistakes in identifying the distribution
do not affect the descriptive parameters.
Robust statistics entails a rethinking of statistical descriptive concepts;
the objective is to find descriptive concepts that are little affected by the
choice of the sample or by mistakes in distributional assumptions. Robust
statistics is not a technical adjustment of classical concepts but a profound
rethinking of how to describe data. For example, robust statistics identifies