This paper deals with the regularity problem in the Almost Ideal Demand System (AIDS) in terms of the boundedness of the budget shares within the [0,1] interval. The violation of "cointegration accounting" can be seen just as another consequence of irregularity. The discussion of modifications in the PIGLOG cost functions leads to the proposition of a modified flexible PIGLOG formulation, from which a system of budget shares can be derived, which ensures regularity and validity of "cointegration accounting". A special feature of the resulting share equations is the divergence from the time path of the AIDS shares over time and therefore better budget share forecasts. This property is shown empirically with long run forecasts from demand systems estimated for consumers demand of some European economies.