# Enhanced Methods of Seasonal Adjustment

## Abstract

**:**

## 1. Introduction

## 2. The Structure of the Paper

## 3. Comb Filters

## 4. Wiener–Kolmogorov Filters

## 5. The Finite-Sample Wiener–Kolmogorov Filter

## 6. Widening the Seasonal Stopbands

## 7. Time Domain Filters for Extracting the Trend-Cycle Function

## 8. The Frequency-Domain Methods

## 9. Stop Bands and Transition Bands

## 10. Case Study 1: The Basic Time-Domain Filter and the Trend-Cycle Function

## 11. Case Study 2: Widening the Stop Bands via the Frequency Domain Filter

## Supplementary Materials

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Attal-Toubert, Ketty, Dominique Ladiray, Marc Marini, and Jean Palate. 2018. Moving Trading-Day Effects with X-13 ARIMA-SEATS and TRAMO-SEATS. In Handbook on Seasonal Adjustment, 2018 ed. Edited by Gian Luigi Mazzi, Dominique Ladiray and Dan A. Rieser. Luxembourg: Eurostat, Publications Office of the European Union, chp. 6. [Google Scholar]
- Box, George E. P., and Gwilym M. Jenkins. 1976. Time Series Analysis: Forecasting and Control, Revised ed. San Francisco: Holden Day. [Google Scholar]
- Caporello, Gianluca, and Agustín Maravall. 2004. Program TSW, Revised Reference Manual. Madrid: Servicio de Estudios, Banco de Espa na. [Google Scholar]
- Findley, David F., Tucker S. McElroy, and Kellie C. Wills. 2005. Modifications of SEATS’ Diagnostic for Detecting over- or Underestimation of Seasonal Adjustment Decomposition Components; Suitland: U.S. Census Bureau.
- Gómez, Victor, and Agustin Maravall. 1997. TRAMO (Time Series Regression with ARIMA Noise, Missing Observations, and Outliers) and SEATS (Signal Extraction in ARIMA Time Series) Instructions for the User. Madrid: Banco de Espana. [Google Scholar]
- Gómez, Victor, and Agustin Maravall. 2001. Seasonal Adjustment and Signal Extraction in Economic Time Series. In A Course in Time Series Analysis. Edited by Daniel Peña, George C. Tiao and Ruey S. Tsay. New York: John Wiley and Sons, chp. 8. [Google Scholar]
- Grudkowska, Sylwia. 2017. JDemetra+ Reference Manual Version 2.2. Available online: https://jdemetradocumentation.github.io/JDemetra-documentation/ (accessed on 1 January 2021).
- Hillmer, Steven Craig, and George C. Tiao. 1982. An ARIMA-Model-Based Approach to Seasonal Adjustment. Journal of the American Statistical Association 77: 63–70. [Google Scholar] [CrossRef]
- Hodrick, Robert J., and Edward C. Prescott. 1997. Postwar U.S. Business Cycles: An Empirical. Investigation. Journal of Money Credit and Banking 29: 1–16. [Google Scholar] [CrossRef]
- Kaiser, Regina, and Agustín Maravall. 2001. Measuring Business Cycles in Economic Time Series. Lecture Notes in Statistics 154. New York: Springer. [Google Scholar]
- Koopman, Siem Jan, Andrew C. Harvey, Jurgen A. Doornik, and Neil Shephard. 1995. STAMP 5.0: Structural Time Series Analyser, Modeller and Predictor. London: Chapman & Hall. [Google Scholar]
- Ladiray, Dominique. 2018. Calendar Effects. In Handbook on Seasonal Adjustment, 2018 ed.; Edited by Gian Luigi Mazzi, Dominique Ladiray and Dan A. Rieser. Luxembourg: Eurostat, Publications Office of the European Union, chp. 5. [Google Scholar]
- Ladiray, Dominique, and Benoit Quenneville. 2001. Seasonal Adjustment with the X-11 Method. Springer Lecture Notes in Statistics 158. Berlin: Springer. [Google Scholar]
- Leser, Conrad Emanuel Victor. 1961. A Simple Method of Trend Construction. Journal of the Royal Statistical Society, Series B 23: 91–107. [Google Scholar] [CrossRef]
- Mazzi, Gian Luig, Dominique Ladiray, and Dan A. Rieser. 2018. Handbook on Seasonal Adjustment, 2018 ed.; Luxembourg: Eurostat, Publications Office of the European Union.
- McElroy, Tucker, and Anindya Roy. 2017. Detection of Seasonality in the Frequency Domain. Journal of Statistical Planning and Inference 221: 241–55, Forthcoming in (2021). [Google Scholar]
- Pollock, D. Stephen G. 2000. Trend Estimation and de-trending via Rational Square Wave Filters. Journal of Econometrics 99: 317–34. [Google Scholar] [CrossRef]
- Pollock, D. Stephen G. 2018. Filters, Waves and Spectra. Econometrics 6: 35. [Google Scholar] [CrossRef][Green Version]
- Proakis, John G., and Dimitris G. Manolakis. 1996. Digital Signal Processing: Principles, Algorithms, and Applications, 3rd ed. Upper Saddle River: Prentice-Hall. [Google Scholar]
- Shiskin, Julius, Allan H. Young, and John C. Musgrave. 1967. The X-11 Variant of the Census Method II Seasonal Adjustment; Technical Paper No. 15; Suitland: Bureau of the Census, U.S. Department of Commerce.
- Wildi, Marc. 2005. Signal Extraction: Efficient Estimation, Unit Root Tests and Early Detection of Turning Points. Springer Lecture Notes in Economic and Mathematical Systems 547. Berlin: Springer. [Google Scholar]

**Figure 1.**The pole-zero diagram of the unidirectional comb filter for monthly data. The poles are marked by crosses and the zeros are marked by circles.

**Figure 2.**The frequency response functions of the bidirectional comb filter for monthly data with $\rho =0.8$, giving the lesser peaks, and $\rho =0.9$, giving the higher peaks.

**Figure 3.**The frequency response functions of the basic seasonal adjustment filter for monthly data with $\lambda =0.5$ and $\rho =0.8$ (the solid line) and with $\lambda =0.5$ and $\rho =0.99$ (the dashed line).

**Figure 4.**The frequency response function of the basic time-domain seasonal-adjustment filter for quarterly data with $\lambda =0.5$. and $\rho =0.9$.

**Figure 5.**The residuals from a linear de-trending of the logarithms of an index of quarterly U.K. consumption for 1955–1994, with a superimposed seasonally-adjusted sequence, derived by the basic time-domain filter.

**Figure 6.**The seasonal component extracted by the basic time-domain filter from the logarithms of an index of quarterly U.K. consumption for 1955–1994.

**Figure 7.**The frequency response function of the double seasonal adjustment filter for monthly data with offsets of two degrees.

**Figure 8.**The frequency response function of the triple seasonal adjustment filter for monthly data with offsets of three degrees.

**Figure 9.**The frequency response of the seasonal-adjustment filter associated with the monthly airline passenger model.

**Figure 10.**The frequency response of the trend extraction filter associated with the monthly airline passenger model.

**Figure 11.**The frequency response of the composite trend extraction filter that mimics that of the monthly airline passenger model.

**Figure 12.**The effect of applying the trend extraction filter to the sequence depicted in Figure 5.

**Figure 13.**The periodogram of the residual sequence from the linear de-trending of the logarithmic consumption data.

**Figure 14.**The residual sequence from fitting a linear trend to the logarithmic consumption data with an interpolated line representing the business cycle, obtained by the frequency-domain method.

**Figure 15.**The cosine segments that give rise to: the upper-half cosine transitions (

**left**); and the lower-half cosine transitions (

**right**).

**Figure 16.**The frequency response function of a frequency-domain seasonal adjustment filter for monthly data with stop bands of six degrees in width.

**Figure 17.**The frequency response function of a low pass frequency-domain filter with a transition in the interval $[\pi /8,\pi /2]$ governed by a composite sigmoid function with $n=3$, shown by the continuous line. The frequency response of the complementary high pass filter is shown by the dashed line.

**Figure 18.**The 336 observations of the value in dollars of the monthly sales of women’s clothing is US retail stores, with a superimposed polynomial trend function of degree 4.

**Figure 19.**The periodogram of the data on the sales of clothing with the frequency response of the basic time-domain seasonal adjustment filter superimposed.

**Figure 22.**The 187 values of the monthly sales of sparkling wine in Australia in the period from January 1980 to July 1995, with a superimposed trend-cycle function.

**Figure 23.**The periodogram of the residuals from a linear de-trending of the wine data, with a superimposed frequency response function of a seasonal-adjustment filter.

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**MDPI and ACS Style**

Pollock, D.S.G.
Enhanced Methods of Seasonal Adjustment. *Econometrics* **2021**, *9*, 3.
https://doi.org/10.3390/econometrics9010003

**AMA Style**

Pollock DSG.
Enhanced Methods of Seasonal Adjustment. *Econometrics*. 2021; 9(1):3.
https://doi.org/10.3390/econometrics9010003

**Chicago/Turabian Style**

Pollock, D. Stephen G.
2021. "Enhanced Methods of Seasonal Adjustment" *Econometrics* 9, no. 1: 3.
https://doi.org/10.3390/econometrics9010003