Discontinuous trading in continuous-time econometrics *

Abstract

Equi-spaced sampling is the norm in applied work using high-frequency data. Due to trade intermittency, however, this traditional sampling scheme amounts to implicit random sampling. Under implicit random sampling and on continuous trajectories, we quantify the asymptotic biases of even realized moments, and the inflated asymptotic variances of even and odd realized moments, as a function of a single easily-estimable quantity capturing trade inaction. Under explicit random sampling and on continuous trajectories, we further show how all moments are, instead, asymptotically-unbiased with asymptotic variances which continue to be closed-form functions of the degree of trade inaction. We also document how a combination of thresholding and explicit random sampling leads, on discontinuous trajectories, to asymptotically-unbiased separation of continuous and discontinuous variation of all orders. Using the valuation of a vast cross section of individual stocks as an economic metric, we show that accounting for the theoretical interplay between trade intermittency and the statistical properties of realized moments is revealing about the pricing of illiquidity, skewness and kurtosis, the subjects of extensive literatures.

Keywords: Trade intermittency, random durations, liquidity, skewness, kurtosis

JEL Classification: G10, C12