The role of orthogonal polynomials in adjusting hyperpolic secant and logistic distributions to analyse financial asset returns

In this paper, we will tackle the issue of accounting for skewness and potentially severe excess kurtosis of the empirical distribution of a random variable of interest by adjusting a parent leptokurtic distribution, using orthogonal polynomials. We will show that the polynomial shape adapter that allows the transformation from a given parent to a target distribution is a linear combination of the orthogonal polynomials associated to the former with coefficients depending on the difference between the moments of these two distributions. A recent work (Zoia, Commun Stat Theory Methods 39(1):52–64, 2010) has shown how to adjust the normal density by using Hermite polynomials but this application is suitable only for series with moderate kurtosis (lower than 5). This is why we provide two other parent distributions, the logistic and the hyperbolic secant which, once polynomially adjusted, can be used to reshape series with higher degrees of kurtosis. We will apply these results for modelling heavy-tailed and skewed distributions of financial asset returns by using both the conditional and unconditional approaches. We empirically demonstrate the advantages of using the polynomially adapted distributions in place of popular alternatives.