Benchmarking, Temporal Disaggregation, and Reconciliation of Systems of Time Series

Most applications of economic models to real-world issues must deal with the problem of extracting results based on data, with noise. Data are often interconnected with economic or accounting relationships. However, due to errors in the economic data or the nature of the statistical procedures used to process the data, some (but not necessarily) linear accounting restrictions, which should be satisfied, are not met. This happens, for example, because economic data are frequently collected with different methods using different sample surveys or different pieces of measuring equipment. Many applied fields are impacted by this problem: during the complex production process of national accounts and balance of payments, data are often incomplete at some level of disaggregation. A consequence of this is that there could be different estimates of the same variable or, more generally, some expected linear restrictions on the data are not satisfied. A question arises from such situations is how to use this information efficiently to produce fit-to-use estimates of the variables. Two principles normally guide such estimation. First, estimates of the missing data are usually constrained so that they are consistent with observed and prior restrictions on their values. Second, values that more closely reflect prior estimates, in terms of a prespecified distance metric, are preferred. In this framework, the temporal dimension may play a crucial role: when time is absent (e.g., when the preliminary data to be adjusted refer to the same time period), classical methods like biproportional matrix balancing procedure (RAS) (Stone, 1961) and the least-squares adjustment procedure proposed by Stone, Champernowne, and Meade (1942) can be used. However, when the data to be adjusted form a time series, possibly with dynamic components to be accounted for by the user, there is the need to consider procedures able to get the desired result (a) while altering “as less as possible” the dynamics of the original data, (b) taking into account the information provided by other source data, and (c) considering the statistical uncertainty that characterizes all the involved values. In the time-series literature, this problem, involving either one or many time series, is generally known as temporal benchmarking or temporal disaggregation of time series (Handbook on quarterly national accounts [Eurostat, 2013] and Quarterly national accounts manual [International Monetary Fund, 2017] contain a survey, taxonomy, and description of the main temporal disaggregation methods proposed in literature). Temporal benchmarking naturally matches with the “philosophy” of the data reconciliation procedures, as it is the process of optimally combining the original high-frequency (say, subannual) series with the low-frequency (say, annual) benchmarks and with the subannual benchmarks, in order to obtain a more reliable subannual series and, depending on the assumptions made by the user, a more reliable annual series as well. The links between least-squares reconciliation procedures and benchmarking are very strict in the sense that practically any relevant benchmarking technique can be considered a particular