158 The Basics of financial economeTrics
parameters that represent the center or the spread of a distribution and that
are robust with respect to outliers and to small changes in the distributions.
Robust statistics seeks descriptive concepts that are optimal from the point
of view of being insensitive to small errors in the data or assumptions.
The notion of robust statistics carries over to statistical modeling. Sta-
tistical models such as regression models are theoretically elegant but not
robust. That is, small errors in distributional assumptions or small data
contamination might have unbounded effects on the overall model. Robust
statistics is a technique to find models that are robust (i.e., to find models
that yield approximately the same results even if samples change or the
assumptions are not correct). For example, robust regressions are not very
sensitive to outliers.
In Appendix F we provide a more detailed explanation of robust sta-
tistics, providing the basic concepts used in this chapter. In this chapter, we
cover robust regression estimators and robust regression diagnostics.
Robust Estimators of Regressions
Let’s begin by applying the concept of robust statistics described in Appendix
F to the estimation of regression coefficients that are insensitive to outliers.
Identifying robust estimators of regressions is a rather difficult problem.
In fact, different choices of estimators, robust or not, might lead to radically
different estimates of slopes and intercepts. Consider the following linear
regression model:
yxii
iN
=+ +
=ββ 0 ∑ ε
1If data are organized in matrix form as usual,
YX=
=
Y
Y
XX
T XX
NTTN111111
1
=
=
βεεεββ01
NT
then the regression equation takes the form,
Y = Xβ + ε (8.1)
The standard nonrobust least squares (LS) estimation of regression param-
eters minimizes the sum of squared residuals,