IN A RECENT paper,' Frederick V. Waugh argues strongly for the use of ordinary least squares in simultaneous systems. His argument is not that least squares gives unbiased estimates of structural parameters2 but that they do (as consistent structural estimators do not) give unbiased estimates of the dependent variable in the regression equation given the values of the other variables and that this is what is needed for forecasting purposes. This note argues that Waugh's argument falls on its own grounds, that, even accepting all assumptions about no structural change and the like, least squares estimates of simultaneous equations when usedforforecasting purposes do not give unbiased forecasts of the dependent variables forecasted. The requirements of forecasting imply a different procedure. Suppose that we have made a least squares estimate of a structural equation and wish to use the result for forecasting purposes. We may assume that the system is not recursive and is in fact simultaneous, as otherwise nobody would quarrel with Waugh's position. Then, aside from the dependent variable to be forecast, there is at least one other endogenous variable in the equation. In order to forecast the dependent variable, however, one must know (or be told or forecast) the values of all the variables on the right-hand side of the equation. So long as all such variables are exogenous, this presents no problem. There is no reason why forecasts of exogenous variables (such as the weather, say) cannot be given the econometrician from outside his system. This is not the case with endogenous variables, however, for the values of endogenous variables in the system will be influenced by the value of the dependent variable being forecast and thus cannot be known before the forecast is made. They must also be forecast at the same time. Waugh states in this regard :3 .. . . I see no reason why I cannot estimate the expected future value of ce [the dependent variable to be forecast] associated with any stated value, or values, of yt [the other endogenous variable]. And unless the structure has changed, I think that in the future, as well as in the past, the least squares

The Leontief matrices of inter-industry transactions are large, a row for each industry in a nation. It would be desirable to invert such matrices of an order of from 100 to 200. The present paper suggests using the sum of a power series to approximate the inverse of a Leontief matrix with any desired degree of accuracy. This requires many more multiplications than do such direct methods as the Gauss-Doolittle process. But the method proposed in this paper is especially well adapted to automatic computation on the new electronic machines, in which case the large number of multiplications is not serious. The main advantage of the proposed method is that it provides an upper bound to the error of any element in the estimated inverse. A short cut method is also indicated for computing the approximation when the number of terms of the power series needed to obtain the desired degree of accuracy is large.

PROFESSOR HOTELLING'S PAPER, Relations between Two Sets of Variates,'l should be widely known and his method used by practical statisticians. Yet, few practical statisticians seem to know of the paper, and perhaps those few are inclined to regard it as a mathematical curiosity rather than an important and useful method of analyzing concrete problems. This may be due to two reasons: first, that Hotelling's paper makes use of rather complicated mathematics and does not spell out in detail the methods of numerical computation; and, second, that although the paper applied the methods to two sets of statistical data, the major emphasis is on mathematical theory, and only rather incidental consideration is given to the meaning of the results obtained in actual statistical work. This paper will try to do two things: first, and most important, it will apply these methods to two different kinds of problems in the hope that this will suggest other practical applications; second, it will develop the methods of analysis in somewhat simpler terms, and discuss numerical computation in greater detail than Hotelling's paper.