Data on a number of individual units (firms, households, geographical areas, etc.) over several periods of time are becoming increasingly available. Very often we would like to use such data to estimate a behavior relationship containing an autoregressive component, due, possibly, to a distributed lag or other dynamic factor affecting economic behavior. In an earlier study [2], Balestra and the present author studied the demand for natural gas using data on 36 states of the United States, over a six year period. We encountered a number of rather serious methodological problems in attempting to estimate a distributed lag model, which appear to be of more general interest in view of the growing availability of data for individual units over several time periods. This paper reports the first of a series of experimental studies designed to explore the general methodological issues involved in studies of this type. The analysis of [2] suggested that consumer demand for natural gas was basically a derived demand from the demand for space-heating. Of the two factors of production used to produce space-heating, fuel and heating plant, only the former was systematically observable. However, the durability of the second, unobserved factor, led to a model in which there was a distributed lag in the substitution of gas for other fuels. The distribution of lag in the demand for gas was closely related to the depreciation rate for the durable factor in space-heating. This interpretation proved crucial in the assessment of various methods of estimation employed. Although the actual model used in the gas study was considerably more complicated than what follows, the latter serves well to illustrate all the essential features. Let observations be available on N individuals (e.g., consuming units, firms, geographical areas, industries, etc.) over a period of T time periods. For the i-th individual and the t-th year we suppose the following relation to hold between the endogenous variable y, the exogenous variable x, and the latent variable u:
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