Does the Compass Rose pattern matter for testing normality?

Abstract

Some years ago, Crack and Ledoit (1996) discovered a strikingly geometric structure when plotting US stock returns against themselves. Since this pattern, in which lines radiating from the origin pop up, resembles the navigating tool it was named “Compass Rose”. Although authors differ in opinion when explaining the causes of the phenomenon, discreteness of price jumps is unanimously indicated as driver of the structure. This paper first documents the presence of a Compass Rose Structure within the illiquid Belgian stock market, looking at both individual stocks and stock indices. We then examine whether the presence of a Compass Rose, i.e. the discreteness of prices, affects normality tests. Based on simulated Brownian Motions with rounded price increments, we notice that two commonly used normality tests react differently to discreteness in the underlying data. As the tick size increases, the popular Jarque-Bera test is not able to detect the deviations from normality. The Lilliefors test, however, clearly rejects the normality assumption when the data exhibit tick/volatility ratios in excess of 2.5.