  We develop and justify methodology to consistently test for long-horizon
return predictability based on realized variance. To accomplish this, we
propose a parametric transaction-level model for the continuous-time log price
process based on a pure jump point process. The model determines the returns
and realized variance at any level of aggregation with properties shown to be
consistent with the stylized facts in the empirical finance literature. Under
our model, the long-memory parameter propagates unchanged from the
transaction-level drift to the calendar-time returns and the realized variance,
leading endogenously to a balanced predictive regression equation. We propose
an asymptotic framework using power-law aggregation in the predictive
regression. Within this framework, we propose a hypothesis test for long
horizon return predictability which is asymptotically correctly sized and
consistent.
