The transformed Gram Charlier distribution: Parametric properties and financial risk applications

Highlights

• We obtain the parametric properties of the transformed Gram–Charlier density.

• We study the unimodality and obtain the true ranges for skewness and kurtosis.

• Closed-form expressions for asymmetric-risk measures are obtained.

• We study the power-law tail property for asymmetric GARCH models under our density.

• The model performs well in both in-sample fitting and out-of-sample backtesting.

Abstract

In this paper we study an extension of the Gram–Charlier (GC) density in Jondeau and Rockinger (2001) which consists of a Gallant and Nychka (1987) transformation to ensure positivity without parameter restrictions. We derive its parametric properties such as unimodality, cumulative distribution, higher-order moments, truncated moments, and the closed-form expressions for the expected shortfall (ES) and lower partial moments. We obtain the analytic $k$th order stationarity conditions for the unconditional moments of the TGARCH model under the transformed GC (TGC) density. In an empirical application to asset return series, we estimate the tail index; backtest the density, VaR and ES; and implement a comparative analysis based on Hansen’s skewed-t distribution. Finally, we present extensions to time-varying conditional skewness and kurtosis, and a new class of mixture densities based on this TGC distribution.

Introduction

Densities based on polynomial expansions (PE) have drawn great attention to model the departures from normality of the empirical return distributions. For instance, in a recent study, Bagnato et al. (2015) present a simple theorem that links the higher-order moments (skewness and kurtosis) of a polynomially expanded parent distribution to those of the resulting distribution. Within this framework, the Gram–Charlier (GC) probability density function (pdf) in Jondeau-Rockinger (2001) (henceforth, JR) is a PE density with the standard Normal as parent pdf. The GC distribution is very tractable theoretically and empirically mainly because its two parameters correspond directly to skewness and excess kurtosis. The following references, without being exhaustive, provide a research line on the development of the GC distribution: Beber and Brandt, 2006, Polanski and Stoja, 2010, Cheng et al., 2011, Ñíguez and Perote, 2012, Liu and Luger, 2015, Lönnbark, 2016, León and Moreno, 2017, Zoia et al., 2018 and Del Brio et al. (2020).¹

A well-known problem of the GC density function is that it can render negative values. This issue has been mainly dealt with through two approaches. On the one hand, by means of parametric restrictions that ensure the GC pdf has positive probabilities, as in JR (2001). On the other hand, using density function transformations based on the methodology of Gallant and Nychka (1987) (GN henceforth). The latter approach was followed by León et al. 2005 (hereafter LRS) to define a positive GC density function with conditional autoregressive higher-order moments. The LRS model has been proven useful for numerous financial econometric applications; see, for instance, White et al., 2010, Alizadeh and Gabrielsen, 2013, Auer, 2015, Gabrielsen et al., 2015, Anatolyev and Petukhov, 2016, Kräussl et al., 2016, Narayan and Liu, 2018 and Wu et al. (2020).

Henceforth, the LRS density will be referred to as the transformed GC (TGC) since it is the result of transforming the GC density in JR (2001) in order to obtain a well-defined density without parameter restrictions. However, unlike the GC density parameters, the TGC ones do lose their direct interpretation as skewness and excess kurtosis. In this paper we study the TGC’s parametric properties providing its true higher-order moments and other features of the TGC pdf such as the conditions for unimodality, allowable ranges of skewness and kurtosis, closed-form formulae for (i) the cumulative distribution function (cdf), (ii) one-sided truncated TGC moments, and (iii) asymmetric risk measures such as expected shortfall (ES) and lower partial moments (LPMs).

We illustrate the practical use of this pdf through an application to modeling asset returns. For that purpose, we implement the threshold GARCH (TGARCH) model of Zakoïan (1994) for the conditional volatility. We test the performance of our model through an in-sample analysis, and out-of-sample (OOS) exercises for backtesting VaR and ES as well as for density forecasting. We derive the analytic expressions for the kth order stationarity conditions for the unconditional moments of the errors under the TGARCH with TGC density for the standardized errors (henceforth, TGC-TGARCH). Indeed, the second and fourth moments will receive special attention and so, the unconditional variance and kurtosis. Besides, we test the finiteness of the unconditional moments through the robust tail-index method of Gabaix and Ibragimov (2011). As a result, we provide stronger evidence respecting the one obtained from the second and fourth order stationarity conditions driven by the TGC-TGARCH parameter estimates. The data we use comprises stock indexes, exchange rates, a commodity and a cryptocurrency. For comparison purposes, we consider both the Normal distribution as the benchmark as well as the symmetric-GC pdf of Zoia et al. (2018), which we refer to as GCK hereafter. Our choice of the latter is based on that, unlike the TGC, the GCK does model directly kurtosis. Hence, in our comparison analyses we can easily isolate the effect of skewness on the performance of the density. Besides, we also compare the TGC’s performance with the popular skewed-t distribution of Hansen (1994). Furthermore, a robustness check of the conditional variance under the TGC is done using the nonlinear asymmetric GARCH (NAGARCH) of Engle and Ng (1993) as an alternative to the TGARCH.² We evaluate density forecasting performance through $p$-value discrepancy plots (Davidson and MacKinnon, 1998) together with proper scoring rules in Amisano and Giacomini (2007). VaR and ES forecasting accuracies are tested via the backtesting procedures in Du and Escanciano (2017).

Finally, we highlight two possible extensions left for future research although we show some previous results here. First, we extend the TGC to incorporate time-varying (TV) higher-order moments where the dynamics for the implied TGC parameters are driven by the specification in, among others, Lalancette and Simonato (2017), which accounts for asymmetric responses of conditional skewness and kurtosis to positive and negative shocks. Our in-sample analysis results show that modeling the TGC with TV higher-order moments clearly contributes to improving goodness-of-fit. The empirical evidence from our daily conditional skewness series is also reinforced by using an asymmetry measure beyond the third central moment and specifically, the one based on the return distribution’s tails proposed in Jiang et al. (2020). Second, we present a mixture of TGC (MTGC hereafter) densities that features higher flexibility than the TGC for capturing large ranges of kurtosis under a rather similar model framework to that in Alexander and Lazar (2006).

The remainder of the paper is structured as follows. Section 2 deals with the GC pdf as a set-up base of our analysis. In Section 3 we characterize the TGC pdf and study its parametric properties. In Section 4 we apply the TGC for modeling the conditional distribution of asset returns under the TGARCH structure, and analyze the power-law tail properties. Section 5 provides an empirical application to asset return series. Section 6 presents two possible extensions of the TGC linked to lines for further research. Section 7 provides a summary of the conclusions. Appendix A includes some properties of the Hermite polynomials used throughout the paper. Appendix B includes all proofs, and also the kth order stationarity conditions for the unconditional moments under the NAGARCH model. Appendix C presents the estimation results of the NAGARCH under the TGC pdf (henceforth, TGC-NAGARCH). The robust tail-index estimation results for the asset returns are provided in Appendix D. Finally, Appendix E shows a sensitivity analysis based on a simulation procedure that aims to study the theoretical behavior of the implied tail index from Kesten’s equation.