SINCE KOYCK'S pioneering work, distributed lag models have come to play an important role in econometrics and much work has been done to develop methods for analyzing them-see e.g. Koyck [15], Klein [12], Solow [20], Fuller and Martin [5], Malinvaud [18], Hannan [9], Liviatan [17], Amemiya and Fuller [1], Zellner and Park [25], Thornber [21], Waud [23], Dhrymes [3], and Griliches [7]. In the present paper we present maximum likelihood and Bayesian estimation procedures for estimating the parameters of a typical distributed lag model under four alternative sets of assumptions regarding disturbance terms' properties. Then these procedures and assumptions are employed in analyses of a sample of United States quarterly consumption data to illustrate their application and to show the sensitivity of inferences to the assumptions made about disturbance terms' properties. We also compute posterior probabilities associated with four alternative models. The plan of the paper is as follows. In Section 2, the model to be analyzed is described and alternative assumptions about disturbance terms are introduced. Section 3 contains a discussion of maximum likelihood techniques and application of them in the analysis of U.S. quarterly consumption data. Then in Sections 4 and

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