An Iterative GLS Procedure for Estimating the Parameters of Models with Autocorrelated Errors Using Data Aggregated Over Time

Abstract

The purpose of this paper is to develop a procedure to estimate the parameters for a class of distributed lag models by making use of data aggregated over time. The general problem of aggregating economic relations over time has already received considerable attention in the literature. Theil (1954) explained the difficulties of obtaining the correct aggregate relation when lagged variables appeared in the micro relationship, but he did not consider the role of the disturbance term. Mundlak (1961) studied the effects of aggregation over time on the partial adjustment model and developed the relationship between the parameters of the micro relation and those of the mispecified (to accommodate aggregate data) time-aggregated macro model. Mundlak was also unconcerned with the full role of the disturbances in such circumstances. Morigouchi (1970). taking account of disturbances, was able to quantify both the estimation bias and the loss of efficiency resulting from temporal aggregation for certain cases of the independent variable .X (t). More recently, Rowe (1976) has demonstrated the effect of temporal aggregation on regression t-ratios and R 2s. All of these studies are similar in that they did not devise an estimation procedure to correct the bias arising from temporal aggregation. In a later secThis paperdevelops an iterative generalized least-squar-es (GILS) procedul-e for estimating the par-ameters of certain economic relations characterized by first-order autocorrelated diSturbances (!1 hxt + Et, et = /) Et-l + IIt) when the available dclata have been aggregated over time. The estimation procedlure is conditional on knowledge of the level of aggregation (the number of subilltervals) making up the aggregate data interval. An example of the estimation procedure is provided using a set of annual sales-advertising data.