164 M. La Rocca and D. Vistocco
Following Horst et al. [12], we distinguish three types of style models:i. weak style analysis: the coefficients are estimated using an unconstrained regres-
sion model;
ii. semi-strong style analysis: the coefficients are imposed to be positive;
iii.strong style analysis: the coefficients are imposed to be positive and to sum up to
one.
The three types of style model are typically estimated as regression through the origin.
The use of the double constraint (strong style analysis) and the absence of the
intercept allow the interpretation of the regression coefficients in terms of compo-
sition quotas and the estimation of the internal composition of the portfolio [8, 9].
Notwithstanding, classical inferential procedures should be interpreted with caution,
due to the imposition of inequality linear constraints [13]. Some general results are
available for the normal linear regression model [11]; a different approach based on
Bayesian inference is formulated in [10].
In the framework of style analysis, a commonly applied solution is the approx-
imation proposed by Lobosco and Di Bartolomeo [21]. These authors obtain an ap-
proximate solution for the confidence intervals of style weights using a second-order
Taylor approximation. The proposed solution works well except when the param-
eters are on the boundaries, i.e., when one or more parameters are near 0 and/or
when a parameter falls near 1. Kim et al. proposes two approximate solutions for this
special case [14] based on the method of Andrews [1] and on the Bayesian method
proposed by Geweke [11]. A different Bayesian approach is instead discussed by
Christodoulakis [6, 7].
As they are essentially based on a least-squares estimation procedure, common
solutions for the estimation of the style analysis coefficients suffer from the presence
of outliers. In this paper we investigate the use of quantile regression [18] to estimate
style coefficients. In particular we compare the classical solution for the strong style
model with robust estimators based on constrained median regression. Different sets
of outliers have been simulated both in constituent returns and in portfolio returns. The
estimators are then compared with respect to efficiency and some considerations on the
consistency of the median regression estimator is provided too. The use of the quantile
regression approach allows a further gain in efficiency as an L-estimator [15, 19] can
be easily obtained using linear combinations of quantile estimators, i.e., for different
conditional quantiles.
The paper is organised as follows: in the next section the classical Sharpe-style
model is briefly introduced along with the basic notation. In Section 3 the quantile
regression approach to style analysis is described. The simulation schema and the
main results are discussed in Section 4. Finally, some concluding remarks and possible
further developments are provided in Section 5.