Estimation of data measured with error and subject to linear restrictions

Variables are often measured subject to error, whether they are collected as part of an experiment or by sample surveys. A consequence of this is that there will be different estimates of the same variable, or, more generally, linear restrictions which the observations should satisfy but fail to. With knowledge of the variances of the various observations, it has been shown elsewhere that maximum-likelihood estimates of the observations can be produced. This paper shows how, given a sequence of such observations, estimates can be produced without knowledge of data reliabilities. The method is applied to estimates of constant price US GNP. It suggests that 64 per cent of the discrepancy should be attributed to the expenditure estimate, with only 36 per cent going to the income/output estimate. The current method of presentation, on the other hand, places the whole of the error in the income/output estimate.