Challenges with Seasonal Adjustment Tucker McElroy (U.S. Census - - PowerPoint PPT Presentation

Challenges with Seasonal Adjustment

Tucker McElroy (U.S. Census Bureau) FESAC Meeting December 9, 2016

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Disclaimer

This presentation is released to inform interested parties of research and to encourage discussion. The views expressed on statistical issues are those of the authors and not necessarily those

• f the U.S. Census Bureau.

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Outline

• Frequency Aggregation
• Cross-sectional Aggregation
• Seasonal Heteroscedasticity
• Improved Methodology?

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Challenges

◮ Nonseasonal monthly series that become seasonal when

aggregated to quarterly frequency

◮ Nonseasonal series that whose aggregate across cross-sections

(region, industry, etc.) is seasonal

◮ Seasonal series deemed nonseasonal by conventional

diagnostics

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Freqency Aggregation

Y ears 5 10 15 20 −2 −1 1 2 3

Figure: Simulated monthly series with salient seasonality (and trend).

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Freqency Aggregation

Y ears 5 10 15 20 −2 2 4 6

Figure: Simulated monthly series, which has been quarterly aggregated.

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Freqency Aggregation

Y ears 5 10 15 20 −5 5

Figure: Simulated monthly series, consisting of original series plus noisy “quota” series.

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Freqency Aggregation

5 10 15 20 25 30 35 0.0 0.2 0.4 0.6 0.8 1.0 Lag ACF

Figure: Autocorrelation plot of noisy monthly series. (Big spikes at lags 12, 24, and 36 would indicate seasonality.)

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Freqency Aggregation

1 2 3 4 5 6 0.05 0.10 0.20 0.50 1.00 2.00 5.00 frequency spectrum

Figure: Spectral density plot of noisy monthly series. (Big spikes at seasonal frequencies 1, 2, 3, 4, 5, 6 in red would indicate seasonality.)

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Freqency Aggregation

Y ears 5 10 15 20 −2 2 4 6

Figure: Quarterly aggregation of nonseasonal noisy monthly series.

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Frequency Aggregation

Summary:

◮ “Quota” series generated by random values of first two

months of each quarter, but with third month equal to negative of sum of other two months

◮ Exhibit noisy monthly series with NO apparent seasonality (by

conventional diagnostics)

◮ This noisy monthly series aggregates to a quarterly series with

strong seasonality

◮ How can we detect such features in the monthly data, before

quarterly aggregation?

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Cross-sectional Aggregation

−50 50 100 Series 1 −50 50 100 150 5 10 15 20 Series 2 Y ear

Figure: Simulated bivariate monthly series without salient seasonality (but mild trend).

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Cross-sectional Aggregation

5 10 15 20 25 30 35 −1.0 −0.5 0.0 0.5 1.0 Lag ACF 5 10 15 20 25 30 35 −1.0 −0.5 0.0 0.5 1.0 Lag −35 −25 −15 −5 −1.0 −0.5 0.0 0.5 1.0 Lag ACF 5 10 15 20 25 30 35 −1.0 −0.5 0.0 0.5 1.0 Lag

Figure: Acf plot of differenced bivariate monthly series. (Big spikes at lags 12, 24, and 36 would indicate seasonality.)

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Cross-sectional Aggregation

Y ear 5 10 15 20 20 40 60 80

Figure: Plot of sum of nonseasonal bivariate monthly series.

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Cross-sectional Aggregation

5 10 15 20 25 30 35 −0.4 −0.2 0.0 0.2 0.4 0.6 0.8 1.0 Lag ACF

Figure: Acf plot of differenced aggregate monthly series. (Big spikes at lags 12, 24, and 36 indicate seasonality.)

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Cross-sectional Aggregation

1 2 3 4 5 6 5e−01 5e+00 5e+01 5e+02 5e+03 frequency spectrum

Figure: Spectral density plot of aggregate monthly series. (Big spikes at seasonal frequencies 1, 2, 3, 4, 5, 6 in red indicate seasonality.)

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Cross-sectional Aggregation

Summary:

◮ Construction proceeds by adding and subtracting the same

white noise to seasonal persistency of each series

◮ Exhibit two monthly series with NO apparent seasonality (by

conventional diagnostics)

◮ These monthly series aggregate to a monthly series with

strong seasonality

◮ How can we detect such features in the monthly data, via a

bivariate analysis?

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Seasonal Heteroscedasticity

Y ear 5 10 15 20 −0.4 −0.2 0.0 0.2 0.4 0.6

Figure: Simulated quarterly series. Is it even seasonal?

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Seasonal Heteroscedasticity

2 4 6 8 10 12 −0.2 0.0 0.2 0.4 0.6 0.8 1.0 Lag ACF

Figure: Acf plot of simulated quarterly series. (Big spikes at lags 4, 8, and 12 would indicate seasonality.)

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Seasonal Heteroscedasticity

0.0 0.5 1.0 1.5 2.0 0.005 0.010 0.020 frequency spectrum

Figure: Acf plot of simulated quarterly series. (Big spikes at seasonal frequencies 1 and 2 in red would indicate seasonality.)

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Seasonal Heteroscedasticity

Series by Season Q1 Q2 Q3 Q4

Figure: Annual series for each quarter. Q1 is persistent, indicating seasonality, whereas other quarters are non-persistent, and hence non-seasonal.

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Seasonal Heteroscedasticity

Summary:

◮ Construction proceeds by making persistent Q1 effect, and

composing with noisy series for other quarters

◮ Composition appears to have some dynamic seasonality, but

diagnostics are ambiguous

◮ Isolating individual quarters’ series makes Q1 persistence clear

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Criteria for Improved Methodology

• 1. Simplicity (non model-based)
• 2. Continuum (non-seasonal to seasonal)
• 3. Idempotent (sa same as non-seasonal)
• 4. Generality (includes classical methods as special cases, and

explains their failure)

• 5. Measure (defines seasonality)
• 6. Diagnostic (tools for assessing seasonality)
• 7. Uncertainty (quantifies error in adjusting)
• 8. Flexibility (explains stable, dynamic, seasonal break, changing

amplitude, seasonal hetero, frequency agg, cross agg, et al)

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Framework for Improved Methodology

Seasonal Vector Form: series {Xt} with p seasons per cycle is embedded as p-variate vector time series Xn = U yn where U is p × r (seasonal patterns), yn is r × 1 (seasonal persistence). This can be adapted when trend structures are

• present. U and yn can be computed by SVD of data matrix (Lin,

Huang, and McElroy). Useful?: helped me construct the counter-examples of this talk – therefore useful, at a minimum, conceptually

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Framework for Improved Methodology

◮ Fixed effects in yn (e.g., linear trend) correspond to

deterministic (predictable) seasonality

◮ Stochastic portion of yn with high autocorrelation (spectral

mass at frequency zero) corresponds to persistent dynamic seasonal phenomena

◮ White noise portion of yn correponds to nonseasonal

component Seasonal Adjustment: whitening the seasonal persistence

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Questions and Discussion Points

• 1. Which agencies have data exhibiting some of these problems?

(Can you share)

• 2. How are the issues resolved now, using software of the

classical approaches (e.g., X-12-ARIMA, SEATS, STAMP)?

• 3. How do we communicate these issues to “man on the street”?

Contact: tucker.s.mcelroy@census.gov

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