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. Most 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 f¿kg between trades, and a pair of stationary noise processes (fekg and f´kg) which determine the jump sizes in the pure-jump log-price process. The fekg, assumed to be i:i:d: Gaussian, produce a Martingale component in log prices. We assume that the microstructure noise f´kg obeys a certain model with memory parameter d´ 2 (¡1=2; 0) (fractional cointegration case) or d´ = ¡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 + d´ 2 (0:5; 1) [ f0g. 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. We propose transaction-level method-of-moments estimators of several of the other parameters in our model. We present a data analysis, which provides evidence of fractional cointegration. We then consider special cases and generalizations of our model, mostly in simulation studies, to argue that the suitably-modified model is able to capture a variety of additional properties and stylized facts, including leverage, portfolio return autocorrelation due to nonsynchronous trading, Granger causality, and volatility feedback. The ability of the model to capture these effects stems in most cases from the fact that the model treats the (stochastic) intertrade durations in a fully endogenous way.