# 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

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**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