Semi-moments based tests of normality and the evolution of stock returns towards normality

Testing for normality is of paramount importance in many areas of science since the Gaussian distribution is a key hypothesis in many models. As the use of semi–moments is increasing in physics, economics or finance, often to judge the distributional properties of a given sample, we propose a test of normality relying on such statistics. This test is proposed in three different versions and an extensive study of their power against various alternatives is conducted in comparison with a number of powerful classical tests of normality. We find that semi–moments based tests have high power against leptokurtic and asymmetric alternatives. This new test is then applied to stock returns, to study the evolution of their normality over different horizons. They are found to converge at a “log-log” speed, as are moments and most semi–moments. Moreover, the distribution does not appear to converge to a real Gaussian.