406 The Basics of financial economeTrics
of one single observation might have unbounded effects on the mean, while
the median is insensitive to changes of up to half the sample. Robust meth-
ods assume that there are indeed parameters in the distributions under study
and attempt to minimize the effects of outliers as well as erroneous assump-
tions on the shape of the distribution.
A general definition of robustness is, by nature, quite technical. The
reason is that we need to define robustness with respect to changes in distri-
butions. That is, we need to make precise the concept that small changes in
the distribution, which is a function, result in small changes in the estimate,
which is a number. Therefore, we give only an intuitive, nontechnical over-
view of the modern concept of robustness and how to measure robustness.
Qualitative and Quantitative Robustness
Let’s begin by introducing the concepts of qualitative and quantitative
robustness of estimators. Estimators are functions of the sample data. Given
an N-sample of data X = (x 1 ,... , xN)′ from a population with a cumulative
distribution function (cdf) F(x), depending on parameter θ∞, an estimator
for θ∞ is a function of the data. Consider those estimators that can be writ-
ten as functions of the empirical distribution defined as FN(x) = percentage
of samples whose value is less than x.
For these estimators we can write
θ=ˆ θNN()FMost estimators can be written in this way with probability 1. In general,
when N → ∞ then FN(x) → F(x) almost surely and θ→ˆN θ∞ in probability
and almost surely. The estimator θˆN is a random variable that depends on
the sample. Under the distribution F, it will have a probability distribution
LF(θN). Intuitively, statistics defined as functionals of a distribution are robust
if they are continuous with respect to the distribution. This means that small
changes in the statistics are associated with small changes in the cdf.
Resistant Estimators
An estimator is called resistant if it is insensitive to changes in one single
observation.^2 Given an estimator θ=ˆ θNN()F , we want to understand what
happens if we add a new observation of value x to a large sample. To this
(^2) For an application to the estimation of a stock’s beta, see R. Douglas Martin and
Timothy T. Simin, “Outlier Resistant Estimates of Beta,” Financial Analysts Journal
(September–October 2003): 56–58. We discuss this application in Chapter 8.