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We study the effect of drift in pure-jump transaction-level models for assetprices in continuous time, driven by point processes. The drift is as-sumed toarise from a nonzero mean in the efficient shock series. It follows that thedrift is proportional to the driving point process itself, i.e. the cumulativenumber of transactions. This link reveals a mechanism by which properties ofintertrade durations (such as heavy tails and long memory) can have a strongimpact on properties of average returns, thereby poten-tially making itextremely difficult to determine long-term growth rates or to reliably detectan equity premium. We focus on a basic univariate model for log price, coupledwith general assumptions on the point process that are satisfied by severalexisting flexible models, allowing for both long mem-ory and heavy tails indurations. Under our pure-jump model, we obtain the limiting distribution forthe suitably normalized log price. This limiting distribution need not beGaussian, and may have either finite variance or infinite variance. We showthat the drift can affect not only the limiting dis-tribution for thenormalized log price, but also the rate in the corresponding normalization.Therefore, the drift (or equivalently, the properties of dura-tions) affectsthe rate of convergence of estimators of the growth rate, and can invalidatestandard hypothesis tests for that growth rate. As a rem-edy to these problems,we propose a new ratio statistic which behaves more

Drift in Transaction-Level Asset Price Models