Forecasting Heavy‐Tailed Densities with Positive Edgeworth and Gram‐Charlier Expansions*

Abstract

This article presents a new semi-nonparametric (SNP) density function, named Positive Edgeworth-Sargan (PES). We show 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 performance 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.Å We thank a co-editor, two anonymous referees,

Portfolio Risk Assessment under Dynamic (Equi)Correlation and Semi-Nonparametric Estimation: an Application to Cryptocurrencies

“…In order to solve potential positivity problems of the truncated GC series, which is particularly important when applying backtesting techniques, positive transformations can be directly implemented. For instance, the transformation provided by [27] may be considered by replacing equation 7by…”

“…This research is focused on assessing the performance of the multivariate positive GC distribution [27], that implies a symmetric distribution being positive in the whole domain for capturing the risk of highly volatile assets. To this end, we analyze portfolios of cryptocurrencies, which are modeled with a multivariate AR-GJR-GARCH [28] that considers asymmetric conditional variances and a covariance structure consistent to either DCC or DECO.…”

Uncertainty in electricity markets from a semi-nonparametric approach

“…In the same line, Gallant, Rossi, and Tauchen (1992) used SNP modeling to describe the comovements of prices and volumes of the stock market of the United States (US) in the period from 1928 to 1987. Mauleon and Perote (2000) proposed the use mixtures of SNP distribution to model the stock market of the US and the United Kingdom and Ñíguez and Perote (2012) implemented positive SNP transformations to evaluate the stock performance in US. Other works that adopt SNP approaches to the modeling of heavy-tailed model series are those by Cortés, Mora-Valencia and Perote (2016), who measure the productivity of researchers worldwide, and Cortés, Mora-Valencia, and Perote (2017), who estimate the size distribution of US firms in the period from 2004 to 2015.…”

Multivariate approximations to portfolio return distribution

“…Among the latter approach one of the most interesting and fruitful alternative has been the semi-nonparametric (SNP hereafter) methodology developed by authors such as, Sargan (1975), Jarrow and Rudd (1982), Gallant and Nychka (1987), Gallant and Tauchen (1989), Corrado and Su (1997), Mauleón and Perote (2000), Nishiyama and Robinson (2000), Jondeau and Rockinger (2001), Velasco and Robinson (2001), Jurczenko et al (2002), Verhoeven and McAleer (2004), Tanaka et al (2005), León et al (2005), Bao et al (2006), Rompolis and Tzavalis (2006), León et al (2009), Polanski and Stoja (2010), Ñíguez and Perote (2012) and Ñíguez et al (2012). All these articles proposed the use of polynomial expansions of the Gaussian distribution to define density functions capable of capturing the stylized features of financial asset returns, besides of providing applications to the resulting densities to different contexts, e.g.…”