Gram–Charlier densities: a multivariate approach

Dynamic density forecasts for multivariate asset returns

“…Intuitively, the ‘negative tail’ of the multivariate distribution represents joint extreme losses on long positions in the underlying assets. Finally, consistent with Perote (2004), Leon et al (2009) and Del Brio et al (2009), our results suggest that the time‐varying conditional co‐moments yield significant improvements in the forecasting the joint return distribution.…”

“…We approximate as the product of f and a polynomial that adjusts the latter for the possibly time‐varying empirical co‐moments. In this sense, our technique is similar to the GCE expansion of the multinormal in Perote (2004), Leon et al (2009) and Del Brio et al (2009) and, as we show below, it can be seen as the dual of the latter. In particular, both techniques provide a flexible and relatively simple tool for modelling the empirical joint distribution of financial data, which, in addition to volatility, exhibit time‐varying higher co‐moments.…”

“…Owing to the mutual orthogonality of the latter, GCE can approximate any density function—under certain regularity conditions—with arbitrary accuracy. Del Brio et al (2009) exploit the orthogonality to derive marginals and moments of the CCE approximation. AJD, on the other hand, does not rely on Hermite polynomials in approximating a density and, as such, it lacks some of the theoretical properties of the GCE expansions.…”

“…In this case, the correlation coefficient adequately captures the dependence between assets. However, joint normality is not supported by empirical evidence (see, for example, Guidolin and Timmermann, 2006; Del Brio et al , 2009; Leon et al , 2009). Moreover, correlation is only a measure of linear dependence and suffers from a number of limitations (see Embrechts et al , 2002).…”

“…1 Other parametric multivariate density models include Student‐ t (Prucha and Kelejian, 1984), mixtures of normals (Vlaar and Palm, 1993), skewed normal (Azzalini and Dalla Valle, 1996), skewed Student‐ t (Sahu et al , 2003), Weibull (Malevergne and Sornette, 2004), Kotz‐type (Olcay, 2005) and normal inverse Gaussian (Aas et al , 2006). On the other hand, Perote (2004), Leon et al (2009) and Del Brio et al (2009) employ a semi‐nonparametric approach and approximate the multivariate density of returns via the GCE expansion of the multinormal. While they present evidence of satisfactory performance for two weakly correlated variables, it is not clear how these models perform in higher dimensions or for strongly correlated time series.…”