After 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 distribution. 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 number of cases (but not always), the parameters that identify a closed-form representation 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 ...