  Cellwise outliers are likely to occur together with casewise outliers in
modern data sets with relatively large dimension. Recent work has shown that
traditional robust regression methods may fail for data sets in this paradigm.
The proposed method, called three-step regression, proceeds as follows: first,
it uses a consistent univariate filter to detect and eliminate extreme cellwise
outliers; second, it applies a robust estimator of multivariate location and
scatter to the filtered data to down-weight casewise outliers; third, it
computes robust regression coefficients from the estimates obtained in the
second step. The three-step estimator is shown to be consistent and
asymptotically normal at the central model under some assumptions on the tail
distributions of the continuous covariates. The estimator is extended to handle
both numerical and dummy covariates using an iterative algorithm. Extensive
simulation results show that the three-step estimator is resilient to cellwise
outliers. It also performs well under casewise contaminations when comparing
with traditional high breakdown point estimators.
