170 M. La Rocca and D. Vistocco
technique allows a simple extension toward L-estimators (defined as weighted linear
combinations of different quantiles) in order to gain an increase in efficiency [15, 19].
Moreover, many other robust estimators have been proposed and studied for linear
regression models. However a comparison of their relative merits in the framework
considered here is beyond the scope of this paper.
A further extension of the proposed approach concerns the use of quantile regres-
sion to draw inferences on style coefficients. The presence of inequality constraints
in the style model, indeed, requires some caution in drawing inferences. Among the
different proposals appearing in the literature, the Lobosco–Di Bartolomeo approx-
imation [21] for computing corrected standard errors is widespread and it performs
well for regular cases, i.e., when parameters are not on the boundaries of the param-
eter space. This proposal, indeed, is a convenient method for estimating confidence
intervals for style coefficients based on a Taylor expansion. Nevertheless, as it is es-
sentially based on a least-squares estimation procedure, the Lobosco–Di Bartolomeo
solution also suffers from the presence of outliers. A possible solution could relate to
a joint use of quantile regression and subsampling theory [23]. Subsampling was first
introduced by Politis and Romano [22] and can be considered as the most general
theory for the construction of first-order asymptotically valid confidence intervals or
regions. The basic idea is to approximate the sampling distribution of the statistic
of interest through the values of the statistic (suitably normalised) computed over
smaller subsets of the data. Subsampling has been shown to be valid under very weak
assumptions and, when compared to other resampling schemes such as the bootstrap,
it does not require that the distribution of the statistic is somehow locally smooth
as a function of the unknown model. Indeed, the subsampling is applicable even in
situations that represent counterexamples to the bootstrap. These issues are still un-
der investigation and beyond the scope of this paper. Here it is worth highlighting
that preliminary results appear promising and encourage us to further investigate this
approach: confidence intervals based on the joint use of QR and subsampling show
better performance with respect to both coverage error and length of the intervals.
The next step should concern an empirical analysis with real financial series.
Acknowledgement.The authors wish to thank the two anonymous referees for their helpful
comments and suggestions on a previous draft of the paper; they helped to improve the final
version of the work.
All computations and graphics were done in the R language [24], using the basic packages
and the additional packages: fGarch [29], ggplot2 [27], mgcv [28] and quantreg [17].
The research work of Michele La Rocca benefits from the research structures of the STAT-
LAB at the Department of Economics and Statistics, University of Salerno and of CFEPSR,
Portici (Na). The research work of Domenico Vistocco is supported by Laboratorio di Calcolo
ed Analisi Quantitative, Department of Economics, University of Cassino.
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