A Pure-Jump Transaction-Level Price Model Yielding Cointegration, Leverage, and Nonsynchronous Trading Effects

Abstract

We propose a new transaction-level bivariate log-price model, which 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 thewaiting times between trades, and a pair of stationary noise processes 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 fractionalcointegration for any fixed sampling interval. We prove that the cointegrating parameter can beconsistently estimated by the ordinary least-squares estimator, and 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. We then use simulations to argue that suitably-modified versions of our model are able to capture a variety of additional properties and stylized facts, including leverage, and portfolio return autocorrelation due to nonsynchronous trading. The ability of the model to capture these effects stems in most cases from the fact that themodel treats the (stochastic) intertrade durations in a fully endogenous way.

Keywords: Tick Time, Long Memory stochastic duration, Information share