THE USE OF the geometric distributed lag in economics is widespread. The Liviatan [6] method for estimating its parameters is simple and provides consistent estimates. Moreover, it can be used to provide initial estimates for more sophisticated techniques [2]. Not much is known about the statistical properties of these estimators, especially their small sample properties. We do know that their asymptotic efficiencies are inferior to most alternatives [1], and recently Nagar and Gupta [7] have provided approximations to the small sample biases. The purpose of this note is to derive a simple way of displaying the Liviatan estimators which makes their nature clear and which allows the small sample distribution of one of them to be easily deduced. The main result is that the estimator of the parameter which defines the geometric distributed lag is a ratio of two ordinary least squares estimators. With this and the assumption that the error terms form a sequence of independent, identically distributed normal variables, it is possible, using the work of Geary [4], to derive the small sample distribution of this estimator. This result forms the main part of this note, which concludes with a brief discussion of the problem of setting confidence limits along the lines suggested by Fieller [3].
I find myself in a situation like that of Moliere's M. Jourdain who was surprised to learn that he spoke prose. I am surprised to learn that-according to Maurice Allais, at least-I commit paralogisms. If all I wanted to do was dispute that, I would not take up space with this rejoinder; the law of diminishing returns applies with special force to these running controversies. The problem is, however, that Allais does not address himself directly to my criticism, and I think the criticism is important because it is fundamental. My point was a methodological one: that the tests of Allais' theory are probably quite weak. Allais' skirting of this point is unfortunate for two reasons. The first is that one might get the idea, from reading his reply, that his hereditary and relativistic formulation of the demand for money was under attack. The second is that one might think that he provides confirmation of his theory when I think that a careful scrutiny of his test procedures suggests otherwise. On the first point, nothing was further from my mind when I wrote my comment than an attack on the ideas in the hereditary and relativistic formulation of the demand for money. Quite the opposite was the case. I had then, and I continue to have, nothing but admiration for Allais' theory, which I think is original and may ultimately prove fruitful. I addressed myself only to the second point. The nub of the problem is that, in going from theoretical to empirical specification, Allais introduced an approximation which very likely robbed his tests of much power. That approximation consists of using velocity as a measure of the rate of forgetfulness (1966, p. 1135, equations (2.38) and (2.41)). That means that the estimated coefficient of psychological expansion' is a function, among other things, of past velocity. Since that coefficient is used to predict velocity, Allais' testing comes down to estimating velocity as function of its past. It could come perilously close, in other words, to estimating velocity as an autoregressive process. And if that is true, the test of the structural content of the theory is minimal. Allais' characterization of the above argument is that I accuse him of circularity. That is simply not true. What I suggest is that his tests have little power against the naive alternative of an autoregressive specification for velocity. It may be that velocity is a good measure of the rate of forgetfulness. But I have serious doubts as to whether we are going to be able to test that-or the other parts of the theory by appealing to the behavior of velocity. Allais in fact concedes that his test must be weak, although he does not say so explicitly. He grants that my argument is correct if we can identify measured with desired velocity (what he calls observed and estimated velocity): ... Scadding unfortunatelv fails to make the distinction between the observed and estimated values V and V*. In fact, his mathematical reasonings are valid only if V and v are replaced therein by V* and v* (p. 456). Yet in his original paper, Allais makes the same assumption in deriving the empirical form of his hvpothesis: . . but it can reasonably be suggested that the discrepancy between the actual and the desired value of money holdings is always relatively small. . . . It further follows that it is possible to write as a first approximation . . . (3.2) . .. (3.3) OD . . . * (1966, p. 1138).
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: