Financial density selection

Abstract

We propose and study simple but flexible methods for density selection of skewed versions of the two most popular density classes in finance, the exponential power distribution and the t distribution. For the first type of method, which simply consists of selecting a density by means of an information criterion, the Schwarz criterion stands out since it performs well across density categories, and in particular when the DGP is normal. For the second type of method, general-to-specific density selection, the simulations suggest that it can improve the recovery rate in predictable ways by changing the significance level. This is useful because it enables us to increase (reduce) the recovery rate of non-normal densities by increasing (reducing) the significance level, if one wishes to do so. The third type of method is a generalisation of the second type, such that it can be applied across an arbitrary number of density classes, nested or non-nested. Finally, the methods are illustrated in an empirical application.

Keywords:

• financial returns
• density selection
• skewed exponential power distribution
• skewed t distribution
• general-to-specific density selection

JEL Classification:

• C16
• C52
• C58

Notes

1. In financial econometrics, primarily because of Harvey (1981) and Nelson (1991), the EP distribution is also commonly known as the generalised error distribution.

2. An interesting research question is to compare the Fernández and Steel (1998) method with the Azzalini (1986) method. Unfortunately, however, a rigorous comparison is beyond the scope of this article.

3. Some of our code is adapted code from the R package fGarch, see Würtz and Chalabi (2009).

4. The SC is also known as the Bayesian information criterion or BIC.

5. http://yahoo.finance.com/ (the FTSE 100 series), the European Central bank's webpage http://www.ecb.int/ (USD/EUR) and the US Energy Information Administration's statistics webpage http://www.eia.doe.gov/emeu/international/(oilprice).

6. The estimation of the models is undertaken in EViews 6. The standard errors are those of Bollerslev and Wooldridge (1992), which are consistent under both normality and non-normality of the conditional errors.

7. Another popular ARCH model that contains leverage in the volatility specification is Nelson's (1991) EGARCH model. Unfortunately, however, Nelson's EGARCH model is practically useless when the errors are t distributed, since a necessary condition for EGARCH stability in this case is that the ARCH parameter is negative, see Nelson (1991. 365). This is a very restrictive assumption empirically and the primary reason why Nelson used an EP density instead of a T. Moreover, although Straumann and Mikosch (2006, 2452) prove consistency of QML for the first-order EGARCH in principle, it is not clear that the estimator always works in practice (even when the errors are not t distributed) due to the possibility of non-invertibility, see Sorokin (2010). For these reasons, we opt for a threshold ARCH model instead.