The Properties of Co-Quantiles and Their Applications to Momentum Spillovers

Abstract

This paper introduces a generalization of quantiles, order statistics, and concomitants that we term co-quantiles, and investigates their statistical properties. The probability density functions for the co-quantiles are obtained along with their moments under the assumption that the distribution of the underlying data are multivariate normal. In contrast to the conventional order statistics that rank and record the same attribute of a population, or concomitants that consider different attributes observed over the same time period, co-quantiles allow the ranking and recording of different attributes across different time periods. The co-quantile results naturally reduce to those for order statistics and concomitants, and generalize those on the distributions of linear combinations and the maxima of vector valued random variables obtained in Arellano-Valle and Genton (2007, 2008) and those on cross sectional momentum returns obtained in Kwon and Satchell (2018). By applying the results to momentum spillover returns, we establish theoretically that these returns are susceptible to sudden changes in the skewness and the kurtosis during periods of market uncertainty. Since momentum spillover and cross sectional momentum are structurally very similar, this provides a theoretical explanation for the momentum crashes reported in the empirical literature over such periods.