In previous work (Zellner, 1978) structural coefficient estimates that minimize the posterior expectation of a generalized quadratic loss function were derived. On computing these minimum expected loss (MELO) estimates for coefficients of Klein's Model I and the Girschick-Haavelmo supply and demand model for food, it was found that the MELO coefficient estimates have values between corresponding direct least squares (DLS) and two stage least squares (2SLS) estimates (see Zellner and Park, 1979). This note explores the relation of MELO, DLS and 2SLS estimates. Below it is shown,that the MELO estimate of a vector of coefficients of endogenous variables in an equation of a linear, interdependent econometric model can be expressed as a matrix weighted average of the DLS and 2SLS estimates. In the scalar case, the MELO estimate can be expressed as a simple weighted average of the DLS and 2SLS estimates. Some properties of these matrix weighted averages are discussed using the results in Chamberlain and Leamer (1974).

CONSUMPTION has been cast as both villain and hero in the 1948-49 recession in the United States. It has been asserted that a weakening of consumption was important in accounting for the downturn. R. A. Gordon cites the levelling off of consumer demand as one of several factors responsible for the downturn.1 D. Hamberg has suggested an underconsumptionist explanation of the downturn.2 Even C. A. Blyth who set out to prove that the most important cause of the I948-I949 recession was a substantial fall in fixed investment . felt compelled to state in his conclusions, I accept the view that a reduced rate of growth in consumption both domestically and in the export trade in I948 caused unplanned inventory accumulation, which induced a fall in production of certain nondurables and consumer durables.3 Further, it has been argued that it was the strength of consumption which at least in part accounted for the mildness of the I949 recession. R. Fels asserts that an upward shift in the consumption function occurred in I949 which, along with continuing high levels of autonomous investment and government expenditures, accounted for the mildness of the recession.4 R. A. Gordon had adopted a similar view earlier, both with respect to the mitigating effects of the maintenance of a high level of consumption and to the strength of autonomous investment and government expenditures, although he makes no explicit reference to a shift of the consumption function.5 Hamberg points to the secular rise in the consumption function, deferred consumer replacement demand, and high levels of autonomous investment as factors explaining why this recession failed to develop into a major downturn.6 Blyth cites the continued rise in consumption during I949 as one among several factors which modified the recession.7 Since consumption figures largely in explanations of the I948-49 cycle, and in fact appears called upon to play a dual role, it may be useful to direct careful inquiry into determining what influences prompted consumption to behave as it did.

T EMPORAL aggregation problems in econometrics pose an important but relatively unexplored set of issues relevant for analyses of economic behavior and policy problems. When the behavior of individuals, firms or other economic entities is analyzed with temporally aggregated data, it is quite possible that a distorted view of parameters' values, lag structures and other aspects of economic behavior can be obtained. Since policy decisions usually depend critically on views regarding parameter values, lag structures, etc., decisions based on results marred by temporal aggregation effects can produce poor results. Further, as emphasized by Orcutt and others, aggregating data temporally or otherwise usually involves a loss of information. In the context of temporal aggregation, aggregation can lead to (a) lower precision of estimation and prediction, (b) lower power for tests, (c) inability to make short-run forecasts and (d) a reduction of the probability of discovering new hypo-theses about short-run behavior from data. It is generally appreciated that when annual data are employed in analyses, it is difficult to obtain satisfactory results pertaining to the intra-year behavior of economic units, for example seasonal effects that are often important in analyzing the variations of such variables as inventories, agricultural prices, agricultural output, etc. Previous work concerned with the theoretical analysis of the effects of temporal aggregation on estimation include Mundlak's [6] and Engle's [3] analyses of distributed lag schemes, Telser's [8] treatment of autoregressive processes and Zellner's results for stock adjustment models [11, 12]. In all these papers, it is shown that when econometric models are implemented with temporally aggregated data for flow variables or stock data pertaining to periods longer than that considered appropriate on a priori grounds, the results of analyses will usually be marred by temporal aggregation effects. Further, empirical analyses of several single equation models using temporally aggregated and disaggregated data have been reported which reveal sensitivity of inferences about lag structures to the level of data aggregation (see, e.g., Bryan [1], Laub [5] and Ranson [7]). While much previous work has concentrated attention on the adverse effects of temporal aggregation, there has not been much attention devoted to the problem of what can be done in analyses when we have to work with temporally aggregated data, perhaps because these are the only data available. The approach to be taken in this paper, also utilized in Zellner [11], is to formulate an economic relation in terms of the time unit, say a week or a month, thought to be appropriate on economic grounds and then to derive logically the implications of the model for explaining the variation of temporally aggregated data. With the implied model for the aggregated data explicitly set forth, the problem of using the aggregated data to make statistical inferences can then be approached. Below we present applications of this approach and make several theoretical and empirical comparisons of results obtained with aggregated data with those obtained from analyses based on disaggregated data. The plan of the paper is as follows: In section II we specify a simple model, derive the implied model, and examine its properties. Then inference procedures for the monthly and quarterly versions of the model are compared and some generalizations of the analysis are indicated. In section III numerical results pertaining to a moneymultiplier model are presented. Finally, in