In a 1966 article in this Review, Maurice Allais presented a sophisticated and very successful method for estimating the demand for money. It departed from the usual investigations in (i) using the expected rate of change of outlays, which were assumed to be in fixed proportion to nominal output, rather than the expected rate of change of prices; and in (ii) using a time-variable distributed lag in the estimation of the expected rate of change of outlays.' This note concentrates on analyzing that distributed lag and its use in specifying the demand for money. However, the results of the analysis suggest that the question of what is the correct argument in the demand function for money, expected rates of change of prices or outlays, is not independent of the specification of how they are estimated. As Phillip Cagan has noted (1969, p. 428), the crucial feature of Allais' distributed lag is that the weighting pattern rises and falls with velocity. The danger in this is that the expected rate of change of outlays computed from that distributed lag is used to estimate velocity.2 The Allais procedure, therefore, may come down to regressing velocitv on its past values. But if this is the case, whether rates of change of prices or of outlays is used is relatively unimportant; either washes out in the estimation process. Hence it would not be true as Allais claims that . it is possible to choose between two different approaches only by confronting them with reality (1969b, p. 444). The fact that extrapolations of timeseries often give good predictions may be enough to explain the good results that Allais obtains. The remainder of this note is taken up with showing how Allais' formulation of the distributed lag, and the definitions of the variables in it, lead to a method of predicting velocity which is essentially an extrapolation of velocitv, and its derivatives, appropriately smoothed. The notation used is that in the original Allais piece (1966); numbered references in parentheses are to equation numbers in that work. We denote the demand for nominal money balances per dollar of transactions as Od. Transactions are assumed to be in fixed proportion to nominal output so that we identify 4d with the inverse of income velocity, V. Velocity is assumed to vary directly with z, the expected rate of change of outlays or nominal output. All of this is summarized by: