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Abstract:
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Economic data are often constrained by additivity conditions, where a set of item values (detail items) are required to sum to an associated total value. The set of detail items and their respective total is referred to as a balance complex. When these additivity constraints are not met, changes must be made to either the total or the set of details. Raking proportionally adjusts each detail item by the same amount. If each item's reporting error is random and has variance proportional to its value, then raking minimizes a chi-squared statistic. However, raking was developed for strictly positive data and can produce erroneous values when negative data are included. Modifications have been developed to address this situation, but implementation is not straightforward and does not always yield a feasible solution. In this paper, we develop separate linear and nonlinear programs that minimize loss functions under specified additivity constraints that work with negative data and include item reliability weights. We apply the proposed methods to examples from the Quarterly Financial Report conducted by the U.S. Census Bureau, examining statistical properties of the resultant solutions.
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