Cointegration from a Pure-Jump Transaction-Level Price Model

We propose a new transaction-level bivariate log-price model, which yields fractional or standard cointegration. To the best of our knowledge, all existing models for cointegration require the choice of a fixed sampling interval ∆t. By contrast, our proposed model is constructed at the transaction level, thus determining the properties of returns at all sampling frequencies. The two ingredients of our model are a Long Memory Stochastic Duration process for the waiting times {τk} between trades, and a pair of stationary noise processes ({ek} and {ηk}) which determine the jump sizes in the pure-jump log-price process. The {ek}, assumed to be i.i.d. Gaussian, produce a Martingale component in log prices. We assume that the microstructure noise {ηk} obeys a certain model with memory parameter dη ∈ (−1/2, 0) (fractional cointegration case) or dη = −1 (standard cointegration case). Our log-price model includes feedback between the disturbances of the two log-price series. This feedback yields cointegration, in that there exists a linear combination of the two series that reduces the memory parameter from 1 to 1 + dη ∈ (0.5, 1) ∪ {0}. Returns at sampling interval ∆t are asymptotically uncorrelated at any fixed lag as ∆t increases. We prove that the cointegrating parameter can be consistently estimated by the ordinary least-squares estimator, and obtain a lower bound on the rate of convergence.