A Pure-Jump Transaction-Level Price Model Yielding Cointegration

Abstract

We propose a new transaction-level bivariate log-price model that yields fractional or standard cointegration. The model provides a link between market microstructure and lower-frequency observations. 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. Our model includes feedback between the disturbances of the two log-price series at the transaction level, which induces standard or fractional cointegration for any fixed sampling interval Δt. We prove that the cointegrating parameter can be consistently estimated by the ordinary least squares estimator, and we obtain a lower bound on the rate of convergence. We propose transaction-level method-of-moments estimators of the other parameters in our model and discuss the consistency of these estimators.