context of models with discrete explanatory variables, models with time-
dependent observations, and models involving multiple equations. Later,
robust alternatives to general estimation principles, such as MLE and gen-
eralized method of moments (GMM), are discussed. Before doing so, let
us mention that dangers of data contamination are not only studied only
from the theoretical point of view. There is a number of studies that check
the presence of outliers in real data and their influence on estimation meth-
ods. For example, there is evidence of data contamination and its adverse
effects on LS and MLE in the case of macro economic time series (Balke and
Fomby, 1994; Atkinson, Koopman, and Shephard, 1997), in financial time
series (Sakata and White, 1998; Franses, van Dijk, and Lucas, 2004), mar-
keting data (Franses, Kloek, and Lucas, 1999), and many other areas. These
adverse effects include biased estimates, masking of structural changes, and
creating seemingly nonlinear structures, for instance.
3.1 Discrete variables
To achieve a HBP, many robust methods such as LMS often eliminate a
large portion of observations from the calculation of their objective function.
This can cause non-identification of parameters associated with categorical
variables. For example, having data on income {yi}in=1 of men and women,
where gender is indicated by {di}in=1 ∈ {0, 1}, one can estimate the mean
income of men and women by a simple regression model yi = a + bdi . If
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