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Center for Statistical Research and Methodology (CSRM)

Time Series and Seasonal Adjustment

Motivation: Seasonal adjustment is vital to the effective presentation of data collected from monthly and quarterly economic sample surveys by the Census Bureau and by other statistical agencies around the world. As the developer of the X-13ARIMA- SEATS Seasonal Adjustment Program, which has become a world standard, it is important for the Census Bureau to maintain an ongoing program of research related to seasonal adjustment methods and diagnostics, in order to keep X-13ARIMA-SEATS up-to- date and to improve how seasonal adjustment is done at the Census Bureau.

Research Problem:

  • All contemporary seasonal adjustment programs of interest depend heavily on time series models for trading day and calendar effect estimation, for modeling abrupt changes in the trend, for providing required forecasts, and, in some cases, for the seasonal adjustment calculations. Better methods are needed for automatic model selection, for detection of inadequate models, and for assessing the uncertainty in modeling results due to model selection, outlier identification and non-normality. Also, new models are needed for complex holiday and calendar effects.
  • Diagnostics of seasonality must address differing sampling frequencies (monthly versus quarterly) and multiple forms of seasonality (cycles of annual versus weekly period), and must distinguish between raw and seasonally adjusted data.
  • Multivariate modeling can not only provide increased precision of seasonal adjustments, but can also assist with series that have a low signal content. Moreover, aggregation constraints arising from accounting rules can be more readily enforced. This motivates the need to develop a viable multivariate seasonal adjustment methodology that can handle modeling, fitting, and seasonal adjustment of a large number of series.
  • Time series data are being measured at higher sampling rates or over geographical regions, requiring new seasonal adjustment methods for high frequency/space-time data.

Potential Applications:

  • Applications encompass the Decennial, Demographic, and Economic areas.

Accomplishments (October 2017 - September 2018):

  • Developed new algorithms for midcasting (imputing missing values, with uncertainty) and signal extraction of multivariate time series.
  • Developed parametrization of co-integrated VAR processes.
  • Developed methodology and code for multivariate real-time signal extraction.
  • Developed new autoregressive diagnostics for seasonality.
  • Investigated the causation of residual seasonality in indirect seasonal adjustments arising from frequency aggregation, and developed a methodology for benchmarking of direct adjustments.
  • Developed new models and filtering methods for extracting business cycles, trends, and seasonality from economic time series.

Short-Term Activities (FY 2019):

  • Further develop diagnostics for seasonality and residual seasonality in adjusted series, including the AR diagnostic and methods based on VAR embedding.
  • Continue the study of high-frequency time series, including the facets of modeling, fitting, computation, separation of low-frequency signals, identification of holiday effects, attenuating of extremes, and applications to change of support problems.
  • Develop nonlinear filtering and prediction methods based on autocumulants.
  • Finish studying the impact of weather on seasonal patterns and regARIMA models.
  • Generate an R package for sigex, and continue the dissemination of X-13 R Story.
  • Continue examining methods for estimating trading day regressors with time-varying coefficients, and determine which Census Bureau series are amenable to moving trading day adjustment.
  • Continue developing multivariate signal extraction methods, and evaluate using the EM algorithm for this purpose.

Longer-Term Activities (beyond FY 2019):

  • Further develop methods for constrained signal extraction, appropriate for multivariate data subject to accounting relations.
  • Continue investigation of Seasonal Vector Form, allowing for more exotic seasonal models, and develop the corresponding seasonal adjustment methods.
  • Expand research on multivariate seasonal adjustment in order to address the facets of co-integration, batch identification, modeling, estimation, and algorithms.
  • Improve the speed and stability of likelihood optimization in X-13ARIMA-SEATS.
  • Investigate the properties and applications of both integer time series and network time series models.
  • Develop and disseminate software to implement state space models, with the intention of treating sampling error and stochastic trading day.
  • Develop estimators for the duration of a moving holiday effect.
  • Continue investigation of cycles, band-pass filters, and signal extraction machinery for a broad array of signals.

Selected Publications:

Alexandrov, T., Bianconcini, S., Dagum, E., Maass, P., and McElroy, T. (2012). “The Review of Some Modern Approaches to the Problem of Trend Extraction,” Econometric Reviews, 31, 593-624.

Bell, W., Holan, S., and McElroy, T. (2012). Economic Time Series: Modeling and Seasonality. New York: Chapman Hall.

Blakely,C. (2012). “Extracting Intrinsic Modes in Stationary and Nonstationary Time Series Using Reproducing Kernels and Quadratic Programming,” International Journal of Computational Methods, Vol. 8, No. 3.

Blakely, C. and McElroy, T. S. (2017). “Signal Extraction Goodness-of-fit Diagnostic Tests Under Model Parameter Uncertainty: Formulations and Empirical Evaluation,” Econometric Reviews, 36 (4), 447-467.

Findley, D. F. (2013). “Model-Based Seasonal Adjustment Made Concrete with the First Order Seasonal Autoregressive Model,” Center for Statistical Research & Methodology, Research Report Series (Statistics #2013-04). U.S. Census Bureau, Washington, DC.

Findley, D.F., Lytras, D. P., and McElroy, T. S. (2017). “Detecting Seasonality in Seasonally Adjusted Monthly Time Series,” Research Report Series (Statistics #2017-03), Center for Statistical Research and Methodology, U.S. Census Bureau, Washington, DC.

Findley, D.F. and McElroy, T. S. (2018). “Background and Perspectives for ARIMA Model-Based Seasonal Adjustment,” Research Report Series (Statistics #2018-07), Center for Statistical Research and Methodology, U.S. Census Bureau, Washington, DC.

Findley, D. F., Monsell, B. C., and Hou, C.-T. (2012). “Stock Series Holiday Regressors Generated from Flow Series Holiday Regressors,” Taiwan Economic Forecast and Policy.

Holan, S. and McElroy, T. (2012). “On the Seasonal Adjustment of Long Memory Time Series,” in Economic Time Series: Modeling and Seasonality. Chapman-Hall.

Holan, S., McElroy, T. S., and Wu, G. (2017). “The Cepstral Model for Multivariate Time Series: The Vector Exponential Model,” Statistica Sinica 27, 23-42.

Jach, A., McElroy, T., and Politis, D. (2012). "Subsampling Inference for the Mean of Heavy-tailed Long Memory Time Series," Journal of Time Series Analysis, 33, 96-111.

Janicki, R. and McElroy, T. (2016). “Hermite Expansion and Estimation of Monotonic Transformations of Gaussian Data,” Journal of Nonparametric Statistics, 28(1), 207-234.

Lin, W., Huang, J., and McElroy, T. S. (2018). “Time Series Seasonal Adjustment Using Regularized Singular Value Decomposition,” Published online, Journal of Business and Economics Statistics.

Livsey, J., Lund, R. Kechagias, S., and Pipiras, V. (In Press). “Multivariate Integer-valued Time Series with Flexible Autocovariances and Their Application to Major Hurricane Counts,” Annals of Applied Statistics.

Lund, R., Holan, S., and Livsey, J. (2015). “Long Memory Discrete-Valued Time Series.” Forthcoming, Handbook of Discrete-Valued Time Series. Eds R. Davis, S. Holan, R. Lund, N. Ravishanker. CRC Press.

Lund, R. and Livsey, J. (2015). “Renewal Based Count Time Series.” Forthcoming, Handbook of Discrete-Valued Time Series. Eds R. Davis, S. Holan, R. Lund, N. Ravishanker. CRC Press.

McElroy, T. S. (2018). “Recursive Computation for Block Nested Covariance Matrices,” Journal of Time Series Analysis, 39 (3), 299-312.

McElroy, T. S. (In Press). “Seasonal Adjustment Subject to Accounting Constraints,” Statistica Neerlandica.

McElroy, T. S. (2017). “Computation of Vector ARMA Autocovariances,” Statistics and Probability Letters, 124, 92-96.

McElroy, T. S. (2017). “Multivariate Seasonal Adjustment, Economic Identities, and Seasonal Taxonomy,” Journal of Business and Economics Statistics, 35 (4), 511-525.

McElroy, T. S. (2016). “Non-nested Model Comparisons for Time Series,” Biometrika, 103, 905-914.

McElroy, T. (2016). “On the Measurement and Treatment of Extremes in Time Series,” Extremes, 19(3), 467-490.

McElroy, T. (2015). “When are Direct Multi-Step and Iterative Forecasts Identical?” Journal of Forecasting, 34, 315--336.

McElroy, T. (2013). “Forecasting CARIMA Processes with Applications to Signal Extraction,” Annals of the Institute of Statistical Mathematics, 65, 439-456.

McElroy, T. (2012). “The Perils of Inferring Serial Dependence from Sample Autocorrelation of Moving Average Series,” Statistics and Probability Letters, 82, 1632-1636.

McElroy, T. (2012). “An Alternative Model-based Seasonal Adjustment that Reduces Over-Adjustment,” Taiwan Economic Forecast and Policy 43, 33-70.

McElroy, T. and Findley, D. (2015). “Fitting Constrained Vector Autoregression Models,” in Empirical Economic and Financial Research.

McElroy, T. and Holan, S. (2014) “Asymptotic Theory of Cepstral Random Fields,” Annals of Statistics, 42, 64-86.

McElroy, T. and Holan, S. (2012). “A Conversation with David Findley,” Statistical Science, 27, 594-606.

McElroy, T. and Holan, S. (2012). “On the Computation of Autocovariances for Generalized Geganbauer Processes,” Statistica Sinica 22, 1661-1687.

McElroy, T. and Holan, S. (2012). “The Error in Business Cycle Estimates Obtained from Seasonally Adjusted Data,” in Economic Time Series: Modeling and Seasonality. Chapman-Hall.

McElroy, T. S. and Jach, A. (2018). “Subsampling Inference for the Autocorrelations of GARCH Processes,” Published online, Journal of Financial Econometrics.

McElroy, T. and Jach, A. (2012). “Subsampling inference for the autocovariances of heavy-tailed long-memory time series,” Journal of Time Series Analysis, 33, 935-953.

McElroy, T. and Jach, A. (2012). “Tail Index Estimation in the Presence of Long Memory Dynamics,” Computational Statistics and Data Analysis, 56, 266-282.

McElroy, T. and Maravall, A. (2014). “Optimal Signal Extraction with Correlated Components,” Journal of Time Series Econometrics, 6, 237--273.

McElroy, T. S. and McCracken, M. (2017). “Multi-Step Ahead Forecasting of Vector Time Series,” Econometric Reviews, 36 (5), 495-513.

McElroy, T. and McCracken, M. (2015). “Multi-Step Ahead Forecasting of Vector Time Series.” Published online, Econometrics Reviews.

McElroy, T.S. and Monsell, B. C (2017). “Issues Related to the Modeling and Adjustment of High Frequency Time Series,” Research Report Series (Statistics #2017-08), Center for Statistical Research and Methodology, U.S. Census Bureau, Washington, DC.

McElroy, T. and Monsell, B. (2014). “The Multiple Testing Problem for Box-Pierce Statistics,” Electronic Journal of Statistics, 8, 497-522.

McElroy, T. and Monsell, B. (2015). “Model Estimation, Prediction, and Signal Extraction for Nonstationary Stock and Flow Time Series Observed at Mixed Frequencies.” Journal of the American Statistical Association (Theory and Methods), 110, 1284-1303.

McElroy, T.S., Monsell B. C., and Hutchinson, R. (2018). “Modeling of Holiday Effects and Seasonality in Daily Time Series,” Research Report Series (Statistics 2018-01), Center for Statistical Research and Methodology, U.S. Census Bureau, Washington, DC.

McElroy, T. and Nagaraja, C. (2016). “Tail Index Estimation with a Fixed Tuning Parameter Fraction,” Journal of Statistical Planning and Inference, 170, 27-45.

McElroy, T. and Pang, O. (2015). “The Algebraic Structure of Transformed Time Series,” in Empirical Economic and Financial Research.

McElroy, T. S., Pang, O., and Sheldon, G. (2018). “Custom Epoch Estimation for Surveys,” Published online, Journal of Applied Statistics.

McElroy, T. and Politis, D. (2014). “Spectral Density and Spectral Distribution Inference for Long Memory Time Series via Fixed-b Asymptotics,” Journal of Econometrics, 182, 211-225.

McElroy, T. and Politis, D. (2013). “Distribution Theory for the Studentized Mean for Long, Short, and Negative Memory Time Series,” Journal of Econometrics.

McElroy, T. and Politis, D. (2012). “Fixed-b Asymptotics for the Studentized Mean for Long and Negative Memory Time Series,” Econometric Theory, 28, 471-481.

McElroy, T.S. and Roy, A. (2018). “Model Identification via Total Frobenius Norm of Multivariate Spectra,” Research Report Series (Statistics #2018-03), Center for Statistical Research and Methodology, U.S. Census Bureau, Washington, DC.

McElroy, T. S. and Roy, A. (2018). “The Inverse Kullback Leibler Method for Fitting Vector Moving Averages,” Journal of Time Series Analysis, 39, 172-191.

McElroy, T. and Trimbur, T. (2015). “Signal Extraction for Nonstationary Multivariate Time Series with Illustrations for Trend Inflation.” Journal of Time Series Analysis 36, 209--227. Also in “Finance and Economics Discussion Series," Federal Reserve Board. 2012-45. http://www.federalreserve.gov/pubs/feds/2012/201245/201245abs.html

McElroy, T. and Wildi, M. (2013). “Multi-Step Ahead Estimation of Time Series Models,” International Journal of Forecasting 29, 378-394.

Monsell, B. C. (2014) “The Effect of Forecasting on X-11 Adjustment Filters,” 2014 Proceedings American Statistical Association [CD-ROM]: Alexandria, VA.

Monsell, B. C. and Blakely, C. (2013). “X-13ARIMA-SEATS and iMetrica,” 2013 Proceedings of the World Congress of Statistics (Hong Kong), International Statistical Institute.

Nagaraja, C. and McElroy, T. S. (2018). “The Multivariate Bullwhip Effect,” European Journal of Operations Research, 267, 96-106.

Quenneville, B. and Findley, D. F. (2012). “The Timing and Magnitude Relationships Between Month-to-Month Changes and Year-to-Year Changes That Make Comparing Them Difficult,” Taiwan Economic Forecast and Policy, 43, 119-138.

Roy, A., McElroy, T. S., and Linton, P. (In Press). “Estimation of Causal Invertible VARMA Models,” Statistica Sinica.

Roy, A., McElroy, T., and Linton, P. (2014). “Estimation of Causal Invertible VARMA Models,” Cornell University Library, http://arxiv.org/pdf/1406.4584.pdf.

Sanyal, A., Mitra, P., McElroy, T.S., and Roy, A. (2017). “Holiday Effects in Indian Manufacturing Series,” Research Report Series (Statistics #2017-04), Center for Statistical Research and Methodology, U.S. Census Bureau, Washington, DC.

Trimbur, T. and McElroy, T. S. (2017). “Signal Extraction for Nonstationary Time Series with Diverse Sampling Rules,” Journal of Time Series Econometrics, 9 (1).

Trimbur, T. and McElroy, T. (2016). “Signal Extraction for Nonstationary Time Series with Diverse Sampling Rules,” Published online, Journal of Time Series Econometrics.

Wildi, M. and McElroy, T. S. (In Press). “The Trilemma Between Accuracy, Timeliness, and Smoothness in Real-Time Signal Extraction,” International Journal of Forecasting.

Wildi, M. and McElroy, T. (2016). “Optimal Real-Time Filters for Linear Prediction Problems,” Journal of Time Series Econometrics, 8(2), 155-192.

Contact: Tucker McElroy, Brian C. Monsell, James Livsey, Osbert Pang, Anindya Roy, Bill Bell (R&M), Thomas Trimbur

Funding Sources for FY 2018:

  • 0331 - Working Capital Fund / General Research Project
    Economic Projects

Annual and Quarterly Reports

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Source: U.S. Census Bureau | Research and Methodology Directorate | Center for Statistical Research & Methodology | (301) 763-9862 (or lauren.emanuel@census.gov) |   Last Revised: October 02, 2018