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 Problems:

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, multivariate techniques expand the class of univariate models, allowing the modeling of seasonal heteroscedasticity. 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.
Many published time series arise from sample surveys, and are subject to sampling error. Methodology and algorithms are needed to incorporate sampling error components into the existing seasonal adjustment framework.

Current Subprojects:

Seasonal Adjustment (McElroy/ADRM, Livsey, Pang, Roy)
Time Series Analysis (McElroy/ADRM, Livsey, Pang, Roy, Trimbur)

Potential Applications

Applications encompass the Decennial, Demographic, and Economic areas.

Accomplishments (October 2018-September 2020):

Developed and implemented new algorithms for ragged edge missing value imputation, and ad hoc filtering of multivariate time series.
Implemented and tested autoregressive diagnostics for seasonality.
Refined a benchmarking method to remove seasonality from indirect seasonal adjustments.
Added new models with stable parameterizations to multivariate time series software.
Studied an EM approach to modeling multivariate time series.
Studied outlier processes, allowing for a new approach to extreme-value adjustment of seasonal time series.
Developed methods and formulas for quadratic filtering and forecasting of time series.

Short-Term Activities (FY 2021 – FY 2023):

Continue developing diagnostics for seasonality by refining the AR diagnostic and examining forecast error and partial autocorrelation.
Continue the study of weekly and daily 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, with applications to seasonal adjustment in the presence of extremes.
Develop improved automatic model identification methods.
Develop extensions to maximum entropy extreme-value framework, allowing for more general types of outliers.
Generate an R package for Ecce Signum, and disseminate 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.
Study the impact of sampling error on seasonal adjustment.

Longer-Term Activities (beyond FY 2023):

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:

McElroy, T., Roy, A., and Hore, G. (2023). “FLIP: a Utility Preserving Privacy Mechanism for Time Series,” Journal of Machine Learning Research, 24, 1-29.

McElroy, T. and Politis, D. (2023). “Estimating the Spectral Density at Frequencies Near Zero,” Journal of the American Statistical Association, published online.

McElroy, T., Ghosh, D., and Lahiri, S. (2023). “Quadratic Prediction of Time Series via Autocumulants,” Sankhya A, published online.

McElroy, T. and Jach, A. (2023). “Identification of the Differencing Operator of a Non-stationary Time Series via Testing for Zeroes in the Spectral Density,” Computational Statistics and Data Analysis, 177, 107580.

McElroy, T. and Trimbur, T. (2022). “Variable Targeting and Reduction in Large Vector Autoregressions with Applications to Workforce Indicators,” Journal of Applied Statistics, 50, 1515-1537.

McElroy, T. and Politis, D. (2022). “Optimal Linear Interpolation of Multiple Missing Values,” Statistical Inference for Stochastic Processes, 25, 471-483.

McElroy, T. (2022). “Casting Vector Time Series: Algorithms for Forecasting, Imputation, and Signal Extraction,” Electronic Journal of Statistics, 16, 5534-5569.

McElroy, T. (2022). “Stationary Parameterization of GARCH Processes,” Economics Bulletin, 42 (4).

Davis, R.A., Fokianos, K., Holan, S., Joe, H., Livsey, J., Lund, R.B., Pipiras, V., and Ravishanker, N. (In Press). “Count Time Series: A Methodological Review,” Journal of the American Statistical Association.

Binder, C., McElroy, T., and Sheng, X. (2022). “Term Structure of Uncertainty: New Evidence from Survey Expectations,” Journal of Money, Credit, and Banking, 54(1), 39-71.

Chen, B., McElroy, T., and Pang, O. (2022). “Assessing Residual Seasonality in the U.S. National Income and Product Accounts Aggregates,” Journal of Official Statistics, Volume 38, Issue 2, 399-428.

McElroy, T. (2022). “Frequency Domain Calculation of Seasonal VARMA Autocovariances,” Journal of Computational and Graphical Statistics, 31(1), 301-303.

McElroy, T. and Politis, D. (2022). “Optimal Linear Interpolation of Multiple Missing Values,” Statistical Inference for Stochastic Processes, 1-13.

McElroy, T. and Roy, A. (2022). “A Review of Seasonal Adjustment Diagnostics,” International Statistical Review, 90(2), 259-284.

McElroy, T. and Roy, A. (2022). “Model Identification via Total Frobenius Norm of Multivariate Spectra,” Journal of the Royal Statistical Society, Series B, Volume 84, 473-495.

McElroy, T. and Trimbur, T. (2022). “Variable Targeting and Reduction in Large Vector Autoregressions with Applications to Workforce Indicators,” Journal of Applied Statistics, 1-23.

Trimbur, T. and McElroy, T. (2022). “Modelled Approximations to the Ideal Filter with Application to GDP and Its Components,” The Annals of Applied Statistics, 16(2), 627-651.

Jia, Y., Kechagias, S., Livsey, J., Lund, R., Pipiras, V. (2021). “Latent Gaussian Count Time Series Modelling,” Journal of the American Statistical Association.

McElroy, T. (2021). “A Diagnostic for Seasonality Based upon Polynomial Roots of ARMA Models,” Journal of Official Statistics, 37(2), 1-28.

McElroy, T. and Das, S. (2021). “Nonlinear Prediction via Hermite Transformation,” Statistical Theory and Related Fields 5(1), 49-54.

McElroy, T. and Roy, A. (2021). “Testing for Adequacy of Seasonal Adjustment in the Frequency Domain,” Journal of Statistical Planning and Inference, 221, 241-255.

McElroy, T., Roy, A., Livsey, J., Firestine, T., and Notis, K. (2021). “Anticipating Revisions in the Transportation Services Index,” Journal of the International Association of Official Statistics, 37, 641-653.

Baker, S., McElroy, T.S., and Sheng, X. (2020). “Expectation Formation Following Large and Unpredictable Shocks,” Review of Economics and Statistics, 14, 112-130.

McElroy, T.S. and Politis, D.N. (2020). Time Series: a First Course with Bootstrap Starter. New York: Chapman Hall.

McElroy, T.S. and Wildi, M. (2020). “Multivariate Direct Filter Analysis for Real-Time Signal Extraction Problems,” Econometrics and Statistics, 14, 112-130.

Hyatt, H. and McElroy, T.S. (2019). “Labor Reallocation, Employment, and Earnings: Vector Autoregression Evidence,” LABOUR, 33(4), 463-487.

McElroy, T.S. and Jach, A. (2019). “Testing Collinearity of Vector Time Series,” The Econometrics Journal, 22(2), 97-116.

McElroy, T. S. and Jach, A. (2019). “Subsampling Inference for the Autocorrelations of GARCH Processes,” Published online, Journal of Financial Econometrics, 17(3), 495-515.

McElroy, T. S., Pang, O., and Sheldon, G. (2019). “Custom Epoch Estimation for Surveys,” Published online, Journal of Applied Statistics, 46, 638-663.

McElroy, T.S. and Penny, R. (2019). “Maximum Entropy Extreme-Value Seasonal Adjustment,” Australian New Zealand Journal of Statistics, 61(2), 152-174.

Roy, A., McElroy, T. S., and Linton, P. (2019). “Estimation of Causal Invertible VARMA Models,” Statistica Sinica, 29(1), 455-478.

Wildi, M. and McElroy, T. S. (2019). “The Trilemma between Accuracy, Timeliness, and Smoothness in Real-Time Signal Extraction,” International Journal of Forecasting, 35, 1072-1084

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, D.C.

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. (2018). “Multivariate Integer-valued Time Series with Flexible Autocovariances and Their Application to Major Hurricane Counts,” Annals of Applied Statistics, 12(1): 408-431.

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

McElroy, T. S. (2018). “Seasonal Adjustment Subject to Accounting Constraints,” Statistica Neerlandica, 72, 574-589.

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, D.C.

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, D.C.

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.

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

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., 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, D.C.

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.

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

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, D.C.

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, D.C.

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

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

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

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

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

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

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

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

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

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

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.

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

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

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, D.C.

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

McElroy, T. and Politis, D. (2013). “Distribution Theory for the Studentized Mean for Long, Short, and Negative Memory Time Series,” Journal of Econometrics, 177, 60-74.

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

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.

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.

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.

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.

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

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.