Design of Seasonal Adjustment Filter Robust to Variations in the Seasonal Behaviour of Time Series

Open access


Considering that many macroeconomic time series present changing seasonal behaviour, there is a need for filters that are robust to such changes. This article proposes a method to design seasonal filters that address this problem. The design was made in the frequency domain to estimate seasonal fluctuations that are spread around specific bands of frequencies. We assessed the generated filters by applying them to artificial data with known seasonal behaviour based on the ones of the real macroeconomic series, and we compared their performance with the one of X-13A-S. The results have shown that the designed filters have superior performance for series with pronounced moving seasonality, being a good alternative in these cases.

Bell, W.R. and S.C. Hillmer. 1984. “Issues Involved with the Seasonal Adjustment of Economic Time Series.” Journal of Business & Economic Statistics 2: 291-320. Doi:

Bell, W.R. and B.C. Monsell. 1992. X-11 Symmetric Linear Filters and Their Transfer Functions. Bureau of the Census, Research Report n. RR 92: 15. Available at: (accessed September 2015).

Burman, J.P. 1980. “Seasonal Adjustment by Signal Extraction.” Journal of the Royal Statistical Society: Series A (General) 143: 321-337. Doi:

Canova, F. and E. Ghysels. 1994. “Changes in Seasonal Patterns: Are They Cyclical?” Journal of Economic Dynamics and Control 18: 1143-1171. Available at: (accessed September 2015).

Canova, F. and B.E. Hansen. 1995. “Are Seasonal Patterns Constant Over Time? A Test for Seasonal Stability.” Journal of Business & Economic Statistics 13: 237-252. Doi:

Cleveland, R.B., W.S. Cleveland, J.E. McRae, and I. Terpenning. 1990. “STL: A Seasonal-Trend Decomposition Procedure Based on Loess.” Journal of Official Statistics 6: 3-73.

Dagum Bee, E. 1980. The X-11-ARIMA Seasonal Adjustment Method. Statistics Canada - Seasonal Adjustment and Time Series Staff. Available at: (accessed September 2015).

Dagum Bee, E. 1988. X-11-ARIMA/88 Seasonal Adjustment Method - Foundations and Users’ Manual. Technical Report, Statistics Canada.

Dagum Bee, E., N. Chhab, and K. Chiu. 1996. “Derivation and Properties of the X11ARIMA and Census X11 linear filters.” Journal of Official Statistics 12: 329-348.

Diniz, P., E.A. da Silva, and S.L. Netto. 2010. Digital Signal Processing: System Analysis and Design. Cambridge University Press.

Eurostat. 2015. ESS Guidelines on Seasonal Adjustment. Luxembourg: Publications Office of the European Union. Available at: (accessed April 2016). Doi:

Findley, D.F., B.C. Monsell, W.R. Bell, M.C. Otto, and B.-C. Chen. 1998. “New Capabilities and Methods of the X-12-ARIMA Seasonal-Adjustment Program.” Journal of Business & Economic Statistics 16: 127-152. Doi:

Findley, D.F. 2005. “Some Recent Developments and Directions in Seasonal Adjustment.” Journal of Official Statistics 21: 343-365. Available at: (accessed September 2015).

Findley, D.F. and D.E.K. Martin. 2006. “Frequency Domain Analyses of SEATS and X-11/12-ARIMA Seasonal Adjustment Filters for Short and Moderate Length Time Series.” Journal of Official Statistics 22: 1-34. Available at: (accessed September 2015).

Franses, P.H. and A.B. Koehler. 1998. “A Model Selection Strategy for Time Series with Increasing Seasonal Variation.” International Journal of Forecasting 14: 405-414. Doi:

Geweke, J. 1978. “The Temporal and Sectoral Aggregation of Seasonally Adjusted Time Series.” Seasonal Analysis of Economic Time Series, 411-432. NBER.

Gilbart, J. 1852. “On the Laws of the Currency in Ireland, as Exemplified in the Changes that Have Taken Place in the Amount of Bank Notes in Circulation in Ireland, Since the Passing of the Act of 1845.” Journal of the Statistical Society of London 15: 307-326.

Godfrey, M.D. and H.F. Karreman. 1964. A Spectrum Analysis of Seasonal Adjustment. Technical Report 64, Econometric Research Program - Princeton University. Available at:,erp/ERParchives/archivepdfs/M64.pdf (accessed September 2015).

Gómez, V. and A. Maravall. 1996. “Programs TRAMO (Time Series Regression with Arima Noise, Missing Observations, and Outliers) and SEATS (Signal Extraction in Arima Time Series). Instructions for the User.” Documento de Trabajo, vol. 9628.

Gómez, V. and A. Maravall. 2001. “Seasonal Adjustment and Signal Extraction in Economic Time Series.” Pen˜ a, D., et al. 202-246. Available at:¼10.1.1.470.8918rep¼rep1type¼pdf (accessed September 2015).

Hannan, E.J. 1964. “The Estimation of a Changing Seasonal Pattern.” Journal of the American Statistical Association 59: 1063-1077. Doi:

Hassani, H. 2007. “Singular Spectrum Analysis: Methodology and Comparison.” Journal of Data Science 5: 239-257.

Higginson, J. 1975. An F Test for the Presence of Moving Seasonality when Using Census Method II-X-11 Variant. Statistics Canada.

Hood, C.C., J.D. Ashley, and D.F. Findley. 2000. “An Empirical Evaluation of the Performance of TRAMO/SEATS on Simulated Series.” In of the American Statistical Association, Business and Economic Statistics Section, American Statistical Association, Alexandria, VA., 2000. Available at: (accessed September 2015).

IBGE - Instituto Brasileiro de Geografia & Estatı´stica. 2014. Available at: http://www2. (accessed June 2014).

Infante, E., D. Buono, and A. Buono. 2015. “A 3-Way ANOVA a Priori Test for Common Seasonal Patterns and Its Application to Direct Versus Indirect Methods.” Eurostat Review on National Accounts and Macroeconomic Indicators (1/2015): 67-77.

IPEA - Instituto de Pesquisa Econoˆmica Aplicada. 2014. Available at: (accessed June 2014).

Kaiser, R. and A. Maravall. 2000. An Application of Tramo-Seats: Changes in Seasonality and Current Trend-Cycle Assessment: the German Retail Trade Turnover Series. UC3M Working papers. Statistics and Econometrics 00-63, no. 29. Available at: (accessed September 2015).

Koopman, S.J., A.C. Harvey, J.A. Doornik, and N. Shephard. 2000. STAMP 6.0: Structural Time Series Analyser, Modeller and Predictor. London: Timberlake Consultants.

Kuznets, S. 1932. “Seasonal Pattern and Seasonal Amplitude: Measurement of Their Short-Time Variations.” Journal of the American Statistical Association 27: 9-20. Doi:

Maravall, A. and D. Pérez. 2011. Applying and Interpreting Model-Based Seasonal Adjustment; The Euro-Area Industrial Production Series. Technical Report 1116, Banco de Espan˜ a. Available at: (accessed September 2015).

Melnick, E.L. and J. Moussourakis. 1974. “Filter Design for the Seasonal Adjustment of a Time Series.” Communications in Statistics-Theory and Methods 3: 1171-1186. Doi:

Nerlove, M. 1964. “Spectral Analysis of Seasonal Adjustment Procedures.” Econometrica: Journal of the Econometric Society 32: 241-286. Doi:

Nettheim, N.F. 1964. A Spectral Study of Overadjustment for Seasonality. Technical Report, DTIC Document. Available at: (accessed September 2015).

Nettheim, N.F. 1965. “Fourier Methods for Evolving Seasonal Patterns.” Journal of the American Statistical Association 60: 492-502. Doi:

OECD - Organisation for Economic Co-operation and Development. 2014. Available at: (accessed July 2014).

Planas, C. 1998. “The Analysis of Seasonality in Economic Statistics: A Survey of Recent Developments.” Questiio´: Quaderns d’Estadı´stica, Sistemes, Informatica i Investigacio´ Operativa 22: 157-171. Available at: (accessed September 2015).

Shiskin, J., A.H. Young, and J.C. Musgrave. 1967. The X-11 Variant of the Census Method II Seasonal Adjustment Program. Technical Report, Economic Research and Analysis Division, US Department of Commerce, Bureau of the Census. Available at: (accessed September 2015).

SMT-UFRJ. 2014. Available at: (accessed December 2014).

Sutradhar, B.C. and E. Bee Dagum. 1998. “Bartlett-Type Modified Test for Moving Seasonality with Applications.” Journal of the Royal Statistical Society: Series D (The Statistician) 47: 191-206. Doi:

Tiller, R.T., D. Chow, and S. Scott. 2007. Empirical Evaluation of X-11 and Model-Based Seasonal Adjustment Method. Technical Report, Working Paper. Washington, DC: Bureau of Labor Statistics.

U.S. Bureau of Labor Statistics. 2014. Available at: (accessed January 2014).

U.S. Census Bureau. 2013. X-13ARIMA-SEATS Reference Manual version 1.1. Time Series Research Staff, US Census Bureau. Available at: (accessed September 2015).

U.S. Census Bureau. 2014. Available at: (accessed January 2014).

Van Dijk, D., B. Strikholm, and T. Tera¨svirta. 2003. “The Effects of Institutional and Technological Change and Business Cycle Fluctuations on Seasonal Patterns in Quarterly Industrial Production Series.” The Econometrics Journal 6: 79-98. Doi:

Wallis, K.F. 1982. “Seasonal Adjustment and Revision of Current Data: Linear Filters for the X-11 Method.” Journal of the Royal Statistical Society Series A (General) 145: 74-85. Doi:

Wells, J.M. 1997. “Modelling Seasonal Patterns and Long-Run Trends in US Time Series.” International Journal of Forecasting 13: 407-420. Doi:

Journal of Official Statistics

The Journal of Statistics Sweden

Journal Information

IMPACT FACTOR 2018: 0,837
5-year IMPACT FACTOR: 0,934

CiteScore 2018: 1.04

SCImago Journal Rank (SJR) 2018: 0.963
Source Normalized Impact per Paper (SNIP) 2018: 1.020


All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 337 256 14
PDF Downloads 160 134 7