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

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  • León, Ángel
  • Ñíguez, Trino-Manuel

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 kth 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.

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  • León, Ángel & Ñíguez, Trino-Manuel, 2021. "The transformed Gram Charlier distribution: Parametric properties and financial risk applications," Journal of Empirical Finance, Elsevier, vol. 63(C), pages 323-349.
    • Handle: RePEc:eee:empfin:v:63:y:2021:i:c:p:323-349
    DOI: 10.1016/j.jempfin.2021.07.004
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    More about this item

    Keywords

    Backtesting; Expected shortfall; Kurtosis; Skewness; Tail index; Unimodality;
    All these keywords.

    JEL classification:

    • C2 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables
    • C5 - Mathematical and Quantitative Methods - - Econometric Modeling
    • G1 - Financial Economics - - General Financial Markets

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