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.

As part of continuing effort to research statistical methods for producing timely and accurate early estimates of national accounts statistics, this paper evaluates the ability of individual nowcasting and forecast combination techniques to reduce revisions in the first or advance estimates of U.S. quarterly personal consumption of services at the most detailed component level. At such a level, designated indicators for advance estimates are those directly relevant to the detailed components. Using the same indicators that were used in routine compilations, we show in a real time setting that nowcasting methods are able to reduce revisions in the advance estimates in over 90 percent of the detailed components, and the upper bound of the reductions reached over 60 percent. We evaluate the performances of all methods by comparing their root mean squared revisions for each component. Our study suggests that nowcasting techniques are potentially a powerful tool to reduce revisions in the early estimates in the national account statistics at the most detailed level.

This article provides a new set of empirical regularities describing the U.S. macroeconomy, focusing on business cycle fluctuations in the real GDP and eleven major aggregates from the U.S. National Income and Product Accounts (NIPAs). Patterns of the cyclical fluctuations are assessed using filtering methodologies that adapt to the properties of the series being studied. We employ a recent dataset that includes the Great Recession caused by the 2008 financial crisis and the ensuing recovery. We aim to (1) examine the lead-lag relations via cross-correlations between the aggregate cycle in the U.S. real GDP and the cyclical movements in the eleven major NIPA aggregates; (2) investigate econometrically the inter-linkage between the cyclical component of real GDP and that of each NIPA aggregate by way of the Granger-causality test; and (3) evaluate the capability or the predictive power of each NIPA aggregate to forecast real GDP growth via three forecasting models. Comparisons are made with popular nonparametric HP and Baxter-King (BK) filters. The basic conclusion from the empirical analysis is that the adaptive model-based filters have demonstrated advantages over the commonly used HP and BK filters for business cycle analysis across very diverse economic series from the U.S. national accounts, and the structural time series model using smoothed trend and cycle estimates produced more accurate forecasts than the AR(p) model that forecasts directly using the unfiltered time series.

There is an ongoing debate on whether residual seasonality is present in the estimates of real Gross Domestic Product (GDP) in U.S. national accounts and whether it explains the slower quarter-one GDP growth rate in the recent years. This article aims to bring clarity to this topic by (1) summarizing the techniques and methodologies used in these studies; (2) arguing for a sound methodological framework for evaluating claims of residual seasonality; and (3) proposing three diagnostic tests for detecting residual seasonality, applying them to different vintages and different sample spans of data on real GDP and its major components from the U.S. national accounts and making comparisons with results from the previous studies.

This paper evaluates two individual nowcasting frameworks, the bridge equation and bridging with factors model, in concert with a set of forecast combination techniques for nowcasting the advance estimates of quarterly personal consumption expenditures (PCE) of services at the detailed component level in the U.S. national accounts, using real time data from 2009:Q3 to 2019:Q4. We show that these individual nowcasting frameworks improve the accuracy of advance estimates of PCE services by reducing revisions in 74 percent of the components when quarterly source data become available. We also show that adding model-averaging techniques to nowcasting further improves the accuracy by reducing revisions in 91 percent of the detailed components. The model-averaging techniques considered in this study include simple averaging (mean, median, trimmed means), information-criterion-based averaging (AIC, BIC, log-likelihood averaging), Bates-Granger averaging with leave-one-out cross-validation errors, and covariance-minimization-based Jackknife and Mallows averaging. We evaluate the performances of all methods by comparing their root mean squared revisions (RMSR) in the advance estimates of each detailed component of PCE services. Our study demonstrates that nowcasting models and model-averaging techniques have the potential to be a powerful tool in reducing revisions in the early estimates in national economic account statistics at the detailed level.

Suppose a vector autoregressive moving‐average model is estimated formobserved variables of primary interest for an application andn–mobserved secondary variables to aid in the application. An application indicates the variables of primary interest but usually only broadly suggests secondary variables that may or may not be useful. Often, one has many potential secondary variables to choose from but is unsure which ones to include in or exclude from the application. The article proposes a method called weighted‐covariance factor decomposition (WCFD), comparable to Stock and Watson's method here called principle‐components factor decomposition (PCFD), for reducing the secondary variables to fewer factors to obtain a parsimonious estimated model that is more effective in an application. The WCFD method is illustrated in the article by forecasting quarterly observed U.S. real GDP at monthly intervals using monthly observed four coincident and eight leading indicators from the Conference Board (http://www.conference‐board.org). The results show that root mean‐squared errors of GDP forecasts of PCFD‐factor models are 0.9–11.3% higher than those of WCFD‐factor models especially as estimation‐forecasting periods pass from the pre‐2007 Great Moderation through the 2007–2009 Great Recession to the 2009–2016 Slow Recovery.

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 least-squares reconciliation procedure. The most crucial point is whether (and how) the “quality” of the preliminary series is measured or defined (and imposed) properly. One reason that researchers seem to have moved away from RAS is that it was difficult to incorporate information (when available) on the relative reliability of the data. It must be added that, in the RAS framework, considering the time is not straightforward as well. Both issues are very challenging for agencies providing official data, which are interested in providing reliable temporal profiles of the aggregates, whose estimation typically draws upon a variety of data sources. Starting from this largely shared paradigm, since the 1960s, there has been a huge contribution to the literature on these topics, which have found a very clear and useful summary in the monograph by Dagum and Cholette (2006). This special issue goes on the same track as their contribution, giving space to the most recent contributions on these topics. Through the call for a special issue on Benchmarking, Temporal Disaggregation, and Reconciliation of Systems of Time Series, we invited researchers to submit scientific papers of their research. To ensure the highest scientific quality, all submitted papers underwent a strict peer review process, and only the finally accepted papers have been published. This special issue consists of 10 articles: the papers can be grouped into three topics: (a) temporal benchmarking of time series, (b) indirect estimation through temporal disaggregation, and (c) reconciliation of (national accounts) systems of time series. The first two papers deal with temporal benchmarking of a single time series: “On the sequential benchmarking of subannual series to annual totals” by Jacco Daalmans proposes solutions to better preserve the short-term movements between sequentially benchmarked series; Umed Temursho in his paper titled “Entropy-based benchmarking methods” puts the standard “movement preservation principle” by sign preservation of the preliminary data as a possible target of the user and presents a new entropy-based benchmarking procedure able to this end. Indirect estimation and temporal disaggregation are the main topics of the next three papers. The practical nature of the problem, which is currently of interest for almost all national statistical agencies, is the main concern of the paper “Temporal disaggregation of economic time series: The view from the trenches” by Enrique Quilis, where different procedures are viewed (also) in terms of practical feasibility, ease of use, and availability of dedicated software. The experience of a national agency publishing official quarterly national accounts series using dynamic regression models is described by Laura Bisio and Filippo Moauro in their paper “Temporal disaggregation by dynamic regressions: Recent development in Italian quarterly national accounts”. Similarly, the paper “Retropolating some relevant series of Mexico's System of National Accounts at constant prices” by Victor M. Guerrero and Francisco Corona describes an exercise of estimation of missing data for the past, relevant for political decisions, through a data-driven approach that combines three different databases. The paper “Maximum likelihood estimation framework for table-balancing adjustments” by Geoffrey Brent describes a maximum-likelihood procedure able to tackle the problem of balancing a system of accounts. The last four papers consider contemporaneously constrained data observed in time: “The statistical reconciliation of time series of accounts between two benchmark revisions” by Baoline Chen, Tommaso Di Fonzo, Thomas Howells, and Marco Marini shows that the reconciliation of a system of accounts in successive years can be done at a very disaggregated level, provided that the large and sparse nature of the problem and the distinctive features of the preliminary data are taken into account. The paper “Solving large data consistency problems at Statistics Netherlands using macro-integration techniques” by Nino Mushkudiani, Jacco Daalmans, and Reinier Bikker describes in depth the practical experience of the Netherlands statistical agency in light of the most recent methodological enhancements implemented in the last years. The last two papers are devoted to the reconciliation of systems of seasonally adjusted time series, as useful complement to direct seasonally adjustment strategy: “Seasonal adjustment subject to accounting constraints” by Tucker McElroy proposes to consider also the adequacy of the component seasonal adjustments as an additional constraint. The paper “Reconciliation of seasonal adjusted data with applications to the Swedish quarterly national accounts” by Suad Elezović and Yingfu Xie discusses different movement preservation criteria when reconciling a system of directly seasonal adjusted component and total series, with a specific focus on the Swedish experience. In all the proposed contributions, what clearly emerges is the need for robust procedures able to optimally combine all the available information, at the same time exploiting the dynamic features (when available) of the data, along with the reliability of the source data used in the adjustment phase. We would like to thank the editorial board of Statistica Neerlandica for supporting the idea to launch a special issue on Benchmarking, Temporal Disaggregation, and Reconciliation of Systems of Time Series and for making us the responsible guest editors of this special issue. We would also like to thank all the authors who contributed to this special issue. Special thanks go to the referees, listed at the end of this introduction, for their valuable contribution in selecting and improving the submitted papers. The guest editors gratefully acknowledge the assistance of the following individuals who have referred the papers for this issue: Roberto Astolfi, Dario Buono, Anna Ciammola, Roberto Barcellan, Harm Jan Boonstra, David F. Findley, Michel Ferland, Esteban Fernandez-Vazquez, Suzie Fortier, Roberto Golinelli, Stefano Grassi, Paul Knottnerus, Christian Mueller-Kademann, Alain Maurin, James Mitchell, Vittorio Nicolardi, Jeroen Pannekoek, José Manuel Pavia, Tommaso Proietti, Frederic Picard, Aurelien Poissonnier, Joao F.D. Rodrigues, Sander Scholtus, Ronald van der Stegen, Alex Stuckey, Thomas Trimbur, and Ton de Waal.

The 2003–2007 U.S. annual input–output accounts, GDP‐by‐industry accounts, and expenditure‐based GDP are reconciled with the 2002 and 2007 quinquennial benchmarks and all contemporaneous constraints of the input–output accounts for the in‐between years. The reconciliation is performed at a very detailed level (six‐digit NAICS) according to feasible statistical procedures able to deal with very large systems of accounts. Our objective is to adjust the preliminary levels of the annual 2003–2007 series such that they are consistent with the quinquennial benchmarks available, fulfill all the accounting relationships for any given year, and show movements that are as close as possible to the preliminary information. We use a simultaneous least‐squares procedure based on the proportional first difference criterion, a well known movement preservation principle proposed by Denton. We evaluate the possible adoption of (i) a pure proportional (PROP) adjustment for small series and series with both negative and positive values that deteriorate the meaningfulness of growth rates, and (ii) a priori assumptions for groups of variables according to their different reliabilities, where this can reasonably be imposed.

In this study the 2003-2007 U.S. annual input-output accounts, GDP-byindustry accounts and expenditure-based GDP are reconciled with the 2002 and 2007 quinquennial benchmarks and all contemporaneous constraints of the input-output accounts for the in-between years. The series are adjusted according to statistical procedures able to deal with large systems of accounts subject to both temporal and contemporaneous constraints. Our objective is to adjust the preliminary levels of the series such that they (i) are consistent with the quinquennial benchmarks available, (ii) fulfill all the accounting relationships for any given year, and (iii) show movements that are as close as possible to the preliminary information. To this end we use a simultaneous least-squares procedure based on the proportional first difference (PFD) criterion, a movement preservation principle proposed by Denton (1971). According to our past experiences, we evaluate the possible adoption of (i) a pure proportional adjustment (PROP) for series with breaks and high volatility that deteriorate the meaningfulness of growth rates and (ii) a priori constraints for groups of variables according to their different reliability, where this can reasonably be assumed. 1 Bureau of Economic Analysis, Washington D.C., U.S.A. Email: baoline.chen@bea.gov 2 Department of Statistical Sciences, University of Padova, Italy. Email: difonzo@stat.unipd.it 3 Bureau of Economic Analysis, Washington, D.C., U.S.A. Email: thomas.howells@bea.gov 4 International Monetary Fund, Statistics Department, Washington D.C., U.S.A. Email: mmarini@imf.org The views expressed herein are those of the authors and do not represent the official policies of the Bureau of Economic Analysis and should not be attributed to the IMF, its Executive Board, or its management.

Production capital and technology (i.e., total factor productivity) in US manufacturing are fundamental for understanding output and productivity growth of the US economy but are unobserved at this level of aggregation and must be estimated before being used in empirical analysis. Previously, we developed a method for estimating production capital and technology based on an estimated dynamic structural economic model and applied the method using annual SIC data for 1947–1997 to estimate production capital and technology in US total manufacturing. In this paper, we update this work by reestimating the model and production capital and technology using annual SIC data for 1949–2001 and partly overlapping NAICS data for 1987–2005.

This paper describes and illustrates a generalized least squares (GLS) reconciliation method that can efficiently incorporate all available information on initial data in reconciling a large system of disaggregated accounts and can accurately estimate industry distribution of statistical discrepancy. The GLS reconciliation method is applied to reconciling the 1997 GDP-by-industry accounts and the Input-output accounts. The GDP-by-industry accounts measure GDP by industry using industry gross income, and the input-output accounts measure GDP by industry as the residual between gross output and intermediate inputs. The GLS method produced balanced estimates and estimated the industry distribution of the statisical discrepancy. The results show that using reliability to reconcile different accounts produces statistically meaningful balanced estimates. The study demonstrates that reconciling a large system of disaggregated accounts is empirically feasible and computationally efficient.

Fourth-order multi-step perturbation (MSP) is described and applied as a general method for numerically solving nonlinear, differentiable, algebraic equations which are first-order conditions of economic optimization problems. MSP is first described at a general level and is, then, applied to estimating production-function models, using annual US total manufacturing KLEMS data from 1949 to 2001. The application continues by comparing total factor productivity based on the best estimated model with standard Solow-residual productivity. The optimization problem is the classic firm problem of maximizing output for a given production function, given input prices, and a given cost of inputs. If started sufficiently closely to the correct solution, usual iterative methods, such as quasi-Newton methods, can quickly compute accurate solutions of such problems. However, finding good starting points can be difficult, especially in high-dimensional problems. By contrast, MSP automatically provides a good starting point and iterates a finite number of times over preset steps so that, unlike in usual iterative methods, convergence or divergence is not an issue. Although, as in any numerical method, MSP accuracy is limited by the problem's condition and floating-point accuracy, in practice, at least as implemented here, MSP can quickly compute solutions of nearly single-precision or higher accuracy.

Production capital and technology, fundamental to understanding output and productivity growth, are unobserved except at disaggregated levels and must be estimated prior to being used in empirical analysis. We develop and apply a new estimation method, based on advances in economics, statistics, and applied mathematics, which involves estimating a structural dynamic economic model of a representative production firm and using the estimated model to compute Kalman-filtered estimates of capital and technology for the sample period. We apply the method to annual data from 1947-97 for U.S. total manufacturing and compare the estimates with those reported by the Bureau of Labor Statistics.

THE Bureau of Economic Analysis has adopted a new method for the interpolation of quarterly and monthly estimates in the national accounts. The new method uses a variant of the Denton procedure. National statistical agencies routinely face the task of compiling a large number of quarterly and monthly estimates using relatively complete annual data and less complete quarterly and monthly information from various indicators. Annual data are usually detailed and of high precision, providing the most reliable in­ formation on the overall level and long-term move­ ment in the series. Quarterly or monthly data, while they are less detailed and of lower precision, provide timely and explicit information about the short-term movement in a series. The objective of interpolation in the national eco­ nomic accounts is to use annual data to derive quar­ terly or monthly estimates that preserve as much as possible the short-term movement in the indicator se­ ries while still summing to a benchmark set by the an­ nual data.1 Typically, the annual sums of the quarterly or monthly indicator values are not consistent with the annual values. This means the two series may display inconsistent movements over time. Over the years, the Bureau of Economic Analysis (BEA) has used a variety of techniques for interpola­ tion of the national accounts.2 These techniques yielded varying degrees of success, and analysts at BEA identified some technical challenges in the estimation process: ● Final quarterly or monthly series do not always fol­ low the short-term movement in the indicator series.

This study evaluates five mathematical and five statistical methods for temporal disaggregation in an attempt to select the most suitable method(s) for routine compilation of sub-annual estimates through temporal distribution and interpolation in the national accounts at BEA. The evaluation is conducted using 60 series of annual data from the National Economic Accounts, and the final sub-annual estimates are evaluated according to specific criteria to ensure high quality final estimates that are in compliance with operational policy at the national accounts. The study covers the cases of temporal disaggregation when 1) both annual and sub-annual information is available; 2) only annual data are available; 3) sub-annual estimates have both temporal and contemporaneous constraints; and 4) annual data contain negative values. The estimation results show that the modified Denton proportional first difference method outperforms the other methods, though the Casey-Trager growth preservation model is a close competitor in certain cases. Lagrange polynomial interpolation procedure is inferior to all other methods. I would like to express our great appreciation to Pierre Cholette from Statistics Canada, David Findley and Brian Monsell from the Census Bureau, Roberto Barcellan from Eurostats, and Nils Maehle from the MIF for providing us the software programs and related documentations used in the estimation experiment and for their helpful discussions on the subject. I am also very grateful to Stephen Morris and Aaron Elrod for the tremendous amount of estimation work and tedious statistical analysis they have helped with on the project. Thanks go to Brent Moulton, Marshall Reinsdorf and Ana Aizcorbe for their helpful comments.