Forecasting Heavy‐Tailed Densities with Positive Edgeworth and Gram‐Charlier Expansions*
| Author | Javier Perote,Trino‐Manuel Ñíguez |
| DOI | http://doi.org/10.1111/j.1468-0084.2011.00663.x |
| Date | 01 August 2012 |
| Published date | 01 August 2012 |
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©Blackwell Publishing Ltd and the Department of Economics, University of Oxford 2011. Published by Blackwell Publishing Ltd,
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OXFORD BULLETIN OF ECONOMICS AND STATISTICS, 74, 4 (2012) 0305-9049
doi: 10.1111/j.1468-0084.2011.00663.x
Forecasting Heavy-Tailed Densities with Positive
Edgeworth and Gram-Charlier Expansions*
Trino-Manuel ˜
N´
iguez† and Javier Perote‡
†Department of Economics and Quantitative Methods, Westminster Business School,
University of Westminster, 35 Marylebone Road, London NW1 5LS. UK
(e-mail: T.M.Niguez@wmin.ac.uk)
‡Department of Economics, University of Salamanca, Campus Miguel de Unamuno (Edif. FES),
Salamanca 37007. Spain (e-mail: perote@usal.es)
Abstract
This article presents a new semi-nonparametric (SNP) density function, named Positive
Edgeworth-Sargan (PES). Weshow that this distribution belongs to the family of (positive)
Gram-Charlier (GC) densities and thus it preserves all the good properties of this type of
SNP distributions but with a much simpler structure. The in- and out-of-sample perfor-
mance of the PES is compared with symmetric and skewed GC distributions and other
widely used densities in economics and finance. The results confirm the PES as a good
alternative to approximate financial returns distribution, specially when skewness is not
severe.
I. Introduction
The Edgeworth and Type A Gram-Charlier (GC hereafter) series were introduced in the
statistical literature by Edgeworth (1896, 1907) and Charlier (1906) to expand and approx-
imate probability distributions (see Gram, 1883; and Chebyshev, 1890; for seminal results
on polynomial expansions). Since then, GC series have been investigated from many
perspectives and disciplines. Sargan (1975) brought them into semi-nonparametric (SNP
hereafter) econometrics. During the last decades, SNP densities have been extensively
developed by authors such as: Jarrow and Rudd (1982), Gallant and Nychka (1987),
Maule´on and Perote (2000), Jondeau and Rockinger (2001), Velasco and Robinson (2001)
ÅWe thank a co-editor, two anonymous referees, Gianni Amisano, Gabriele Fiorentini, Jes´us Gonzalo, Javier
Hidalgo, ´
Angel Le´on, Ignacio Maule ´on,Antonio Rubia, Stefan Sperlich, Carlos Velasco, and participants at the 26th
International Symposium on Forecasting in Santander, Spain, June 2006, XXX Symposium of Economic Analysis
in Murcia, Spain, December 2005, XIII Forum of Finance in Madrid, Spain, November 2005, and Journal of Applied
Econometrics Annual Conference in Venice, Italy, May 2005, for comments and suggestions that have helped to
improve the paper.All remaining errors are ours. Financial support from the Spanish Ministry of Science and Educa-
tion under grants SEJ2006-06104/ECON and SEJ2007-66592-CO3-03 is gratefully acknowledged. The first stages
of this article were developed while the authors were research scholars at the London School of Economics and
Political Science.
JEL Classification numbers: C16, C53, G12.
The Positive Edgeworth-Sargan density 601
and Le´on, Men´cia and Sentana (2009), among others. These papers provide studies of SNP
densities theoretical properties and apply them to deal with a wide variety of problems,
which range from asymptotic theory to economic modelling.
The SNP densities empirical applications require the truncation of their polynomial
expansion, which results in a truncated function that cannot be strictly considered as a
probability density function (pdf hereafter) since it may be negative for combinations of
its parameter values in the parametric space. This is a well-known feature of SNP densities,
firstly highlighted by Barton and Dennis (1952), which has been tackled in the literature
in different ways depending on the end-use of the density: On the one hand, Maule´on and
Perote (2000) considered the strategy of carefully selecting initial values for the optimi-
zation algorithms in models estimation, while Jondeau and Rockinger (2001) proposed
parametric constraints and, more recently, Le´on, Rubio and Serna (2005) and Le´on et al.
(2009) have proposed pdf transformations based on the methodology of Gallant and
Nychka (1987). The first method is appropriate for in-sample analysis, whilst the sec-
ond and the third may also be more appropriate for out-of-sample analysis. Nonetheless, it
is also known that parametric constraints may lead to sub-optimization, which in turn may
cause losses in model flexibility to fit the data, while reformulations render theoretically
less tractable specifications.
In this article, we present a well-defined SNP density, that we call Positive Edgeworth
Sargan (PES hereafter). The PES pdf is defined by means of a type of reformulation of the
truncated GC expansion which shares the same philosophy of Gallant and Nychka’s (1987)
method. Wealso define a family of (positive) GC densities that embodies the PES and those
obtained by means of the Gallant and Nychka’s (1987) methodology. We study the PES
theoretical properties and illustrate its empirical performance through an experiment to fit
and forecast the conditional density of several asset returns series. The goodness-of-fitof
the PES density is assessed through a comparative analysis with densities widely used in
the literature for asset returns data, namely: the Normal (Bollerslev, 1986), the Student’s t
(Bollerslev, 1987), the Normal Inverse Gaussian (NIG hereafter) (Jensen and Lunde, 2001)
and (positive) GC densities (Le´on et al., 2005, 2009).
In particular, the relative empirical performance of the models considered is assessed
through a statistical analysis that focuses on out-sample forecasting. First, the models
are assessed in relation to their ability to forecast the conditional variance of the returns
series by using the models confidence set (MCS hereafter) and superior predictive ability
(SPAhereafter) tests of Hansen, Lunde and Nason (2003) and Hansen (2005). Second, the
models are compared in relation to their performance for full density forecasting using
the probability integral transform (PIT hereafter) paradigm (Rosenblatt, 1952; Diebold,
Gunther and Tay 1998; among others). Finally, the models comparison focuses on their
ability to forecasts Value-at-Risk, for what we use the following criteria: statistical functions
for exceptions (L´opez, 1999), likelihood ratio (LR) tests (Kupiec, 1995; Christoffersen,
1998), HIT test of Engle and Manganelli (2004) and strictly proper scoring rules (Gneiting
and Raftery, 2007).
The remainder of the article is organized as follows. Section 2 introduces the PES
density analysing their theoretical properties in a SNP framework. Section 3 provides an
empirical application to asset returns data to illustrate the relative performance of the PES
density and section 4 summarizes the main conclusions.
©Blackwell Publishing Ltd and the Department of Economics, University of Oxford 2011
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