Least Quartic Regression Criterion to Evaluate Systematic Risk in the Presence of Co-Skewness and Co-Kurtosis

Abstract

This article proposes a new method for the estimation of the parameters of a simple linear regression model which is based on the minimization of a quartic loss function. The aim is to extend the traditional methodology, based on the normality assumption, to also take into account higher moments and to provide a measure for situations where the phenomenon is characterized by strong non-Gaussian distribution like outliers, multimodality, skewness and kurtosis. Although the proposed method is very general, along with the description of the methodology, we examine its application to finance. In fact, in this field, the contribution of the co-moments in explaining the return-generating process is of paramount importance when evaluating the systematic risk of an asset within the framework of the Capital Asset Pricing Model. We also illustrate a Monte Carlo test of significance on the estimated slope parameter and an application of the method based on the top 300 market capitalization components of the STOXX® Europe 600. A comparison between the slope coefficients evaluated using the ordinary Least Squares (LS) approach and the new Least Quartic (LQ) technique shows that the perception of market risk exposure is best captured by the proposed estimator during market turmoil, and it seems to anticipate the market risk increase typical of these periods. Moreover, by analyzing the out-of-sample risk-adjusted returns we show that the proposed method outperforms the ordinary LS estimator in terms of the most common performance indices. Finally, a bootstrap analysis suggests that significantly different Sharpe ratios between LS and LQ yields and Value at Risk estimates can be considered more accurate in the LQ framework. This study adds insights into market analysis and helps in identifying more precisely potentially risky assets whose extreme behavior is strongly dependent on market behavior.