Robust estimation of style analysis coefficients 169Fig. 2.Comparison of the median estimators through quantile regression (QR) forT=250,
T =500 andT =1000. The different subpanels of the two plots refer to the portfolio
constituents (rows) and to the different cases of the presence of outliers (columns). In particular
the first column depicts the situation with no outlying observation, the second and third columns
refer, respectively, to the presence of outliers in portfolio returns and outliers in constituent
returns, while the last column depicts the case when outliers are considered both in portfolio
returns and in constituent returns. Theboxplots depict the sampling distributions of the QR
estimators forT=250 (left boxplot),T=500 (middle boxplot) andT=1000 (right boxplot)
dian regression, showing some empirical results for efficiency and consistency of
the robust estimators. The results of the simulation study encourage us to further
investigate this approach. A topic deserving further attention is a formal study of the
robustness of the constrained median regression estimator in the presence of outliers
in theXseries based on the influence function of the constrained robust estimator.
It is worthwile to point out that further gain in estimator efficiency can be obtained
as the median regression has been estimated through quantile regression. Such a