JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org. This content downloaded from 149.150.51.The properties of various forms of Engel functions satisfyinlg tlhe additivity criterion are intvestigated. It is suiggested that a form of relatCionship used by H. Working, but recently neglected, offer.s on balance great adv,-tatges. A logical generalisation of it yields a flexible functiorn from which a good fit may be expected for most commodity groups. THE PROBLEM of finding the most appropriate form of an Engel function is an old one in econometrics, but as yet no solution appears to have found general acceptance. Generally speaking, it is perhaps true to say that the specification of the form of relationships has attracted less attention than have methods of estimating parameters for specified equations.To some extent, of course, the answer depends on the degree of emphasis placed on various properties that one desires the function to possess. Here it is assumed that a set of Engel functions for a number of commodity groups, which include all expenditure, is to be derived from family budget data, with total expeniditure as the independent variable. Bias arising from abnormal expenditure is assumed to have been removed, or at any rate reduced, by using income or another instrumental variable for grouping, as has been done by Liviatan [6].It is postulated, furthermore, that the Engel functions for the various commodity groups studied have the same mathematical form and shoukl satisfy the additivity criterion. This is somewhat restrictive but still permits a fair choice among different functions.The considerations governing the choice may be broadly divided into three categories. First, a close connection with a direct or indirect utility function may appear desirable. Secondly, the function should, ideally, be valid for all positive values of total outlay, or at any rate within a wide range. The variations in the income elasticities of demand entailed by the formula should also be plausible. Finally, there are statistical and com1putational considerations: Estimation of parameters should be simple andi convenient, allowing an assessment of reliability and goodness of fit; the error specification should also be reasonable.The weight given to the first group of considerations will necessarily be different depending upon whether the problem is approached from the viewpoint of the theoretical economist or from that of the economic statistician. In the latter case, the emphasis will be on the second and third sets of coIn-1

Summary The principle adopted here in the construction of a trend for a time series consists in minimizing a linear combination of two sums of squares, of which one refers to the second differences of the trend values, the other to the deviations of the observations from the trend values. Properties of the general solution are deduced, and the solution is explicitly obtained for up to 7 observations. In the special case in which the sum of the two sums of squares is minimized, the exact solution is derived for up to 15 observations. An approximation formula, suitable for practical use when there are 8 or more observations, is also given. The method is illustrated by examples, in which it is applied to an artificially constructed and to an actual time series.

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