Limit Laws in Transaction-Level Asset Price Models

We consider pure-jump transaction-level models for asset prices incontinuous time, driven by point processes. In a bivariate model thatadmits cointegration, we allow for time deformations to account for suche®ects as intraday seasonal patterns in volatility, and non-tradingperiods that may be di®erent for the two assets. Most assumptionsare stated directly on the point process, though we providesu±cient conditions on the corresponding inter-trade durationsfor these assumptions to hold. We obtain the asymptotic distribution ofthe log-price process. We also obtain the asymptotic distribution of theordinary least-squares estimator of the cointegrat- ing parameter basedon data sampled from an equally-spaced discretization of calendar time,in the case of weak fractional cointegration. Finally, we obtain thelimiting distribution of the ordinary least-squares estimator of theautoregressive parameter in a simpli¯ed transaction-levelunivariate model with a unit root.