Statistical properties of co-quantiles and their applications to momentum spillovers

This article introduces co-quantiles, a new concept in rank statistics that generalizes quantiles, order statistics, and concomitants. In contrast to 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 for the ranking and recording of different attributes, and more generally their functions, across different time periods. In the special case of affine co-quantiles (defined formally in the article), we derive explicit expressions for their probability density functions and moments under the assumption that the distribution of the underlying data is multivariate normal. In particular, we establish that the former is a sum of unified skew-normal distributions introduced in Arellano-Valle and Azzalini (2006). 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 on cross-sectional momentum returns obtained in Kwon and Satchell (2018). As an example that has not been analyzed in the financial literature, we consider momentum spill-over returns and establish theoretically that these returns are susceptible to sudden changes in skewness and kurtosis during periods of market uncertainty.