Mitigating Process Distortion While Preserving Accounting Relations in Hierarchical Time Series

Process distortion, that is, the corruption of time series dynamics, can be present in hierarchical time series subject to aggregation. The presence of seasonal patterns in seasonally adjusted data, known as residual seasonality (RS), is an example of process distortion. In the case of U.S. Gross Domestic Product (GDP) and its sub-aggregates, it is important that there be no RS at monthly or quarterly frequencies, while preserving aggregation relations between the various sub-components. A phenomenon of indirect seasonal adjustment is that both temporal (from monthly to quarterly frequency) and hierarchical (from specific to broad variables) aggregates of RS-free time series can themselves exhibit RS. A topological analysis of the self-intersecting lattice structure of accounting relations is developed, decomposing the lattice into ordered subsets (called terraces) that can be sequentially analyzed. Second, a reconciliation method is proposed that minimally modifies each monthly time series such that its higher aggregates—as well as its quarterly aggregates—have no process distortion. This is accomplished by minimizing a relative difference criterion subject to nonlinear constraints furnished by diagnostic measures of process distortion, proceeding from terrace to terrace of the lattice in a top-down procedure. The method is successfully applied to the Personal Consumption Expenditures sub-lattice of GDP, removing RS while maintaining all the accounting relations.