Robust Econometrics

a HBP method such as LMS or LTS is used to estimate the model and it
eliminates a large portion of observations from the calculation (e.g., one half
of them), the remaining data could easily contain only income of men or only
income of women, and consequently, the mean income of one of the groups
could not be then identified. Even though this seems unlikely in our simple
example, it becomes more pronounced as the number of discrete variables
grows, see Hubert and Rouseeuw (1997) for an example.

A common strategy employs a robust estimator with a HBP for a model
with only continuous variables, and using this initial estimate, the model with
all variables is estimated by an M -estimator. Such a combined procedure
preserves the breakdown point of the HBP estimator: even though a mis-
classified values of categorical explanatory variables can bias the estimates,
this bias will be bounded in common models as the categorical variables are
bounded as well. See Hubert and Rousseeuw (1997) and Maronna and Yohai
(2000), who combine an initial S-estimator with an M -estimator.

3.2 Time series

In time series, there are several issues not addressed by the standard the-
ory of robust estimation because of time-dependency of observations. First,
the asymptotic behavior of various robust methods has to be established;
see Koenker and Machado (1999), Koenker and Xiao (2002) for L₁ regres-
sion, KUnsch (1984) and Bai (1997) for M-estimators and Sakata and White

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