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 1 / 26

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 of the U.S. Census Bureau. 2 / 26

Outline • Frequency Aggregation • Cross-sectional Aggregation • Seasonal Heteroscedasticity • Improved Methodology? 3 / 26

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 4 / 26

Freqency Aggregation 3 2 1 0 −1 −2 0 5 10 15 20 Y ears Figure: Simulated monthly series with salient seasonality (and trend). 5 / 26

Freqency Aggregation 6 4 2 0 −2 0 5 10 15 20 Y ears Figure: Simulated monthly series, which has been quarterly aggregated. 6 / 26

Freqency Aggregation 5 0 −5 0 5 10 15 20 Y ears Figure: Simulated monthly series, consisting of original series plus noisy “quota” series. 7 / 26

Freqency Aggregation 1.0 0.8 0.6 ACF 0.4 0.2 0.0 0 5 10 15 20 25 30 35 Lag Figure: Autocorrelation plot of noisy monthly series. (Big spikes at lags 12, 24, and 36 would indicate seasonality.) 8 / 26

Freqency Aggregation 5.00 2.00 1.00 spectrum 0.50 0.20 0.10 0.05 0 1 2 3 4 5 6 frequency Figure: Spectral density plot of noisy monthly series. (Big spikes at seasonal frequencies 1, 2, 3, 4, 5, 6 in red would indicate seasonality.) 9 / 26

Freqency Aggregation 6 4 2 0 −2 0 5 10 15 20 Y ears Figure: Quarterly aggregation of nonseasonal noisy monthly series. 10 / 26

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? 11 / 26

Cross-sectional Aggregation 100 50 Series 1 0 −50 150 100 Series 2 50 0 −50 0 5 10 15 20 Y ear Figure: Simulated bivariate monthly series without salient seasonality (but mild trend). 12 / 26

Cross-sectional Aggregation 1.0 1.0 0.5 0.5 ACF 0.0 0.0 −0.5 −0.5 −1.0 −1.0 0 5 10 15 20 25 30 35 0 5 10 15 20 25 30 35 Lag Lag 1.0 1.0 0.5 0.5 ACF 0.0 0.0 −0.5 −0.5 −1.0 −1.0 −35 −25 −15 −5 0 0 5 10 15 20 25 30 35 Lag Lag Figure: Acf plot of differenced bivariate monthly series. (Big spikes at lags 12, 24, and 36 would indicate seasonality.) 13 / 26

Cross-sectional Aggregation 80 60 40 20 0 0 5 10 15 20 Y ear Figure: Plot of sum of nonseasonal bivariate monthly series. 14 / 26

Cross-sectional Aggregation 1.0 0.8 0.6 0.4 ACF 0.2 0.0 −0.2 −0.4 0 5 10 15 20 25 30 35 Lag Figure: Acf plot of differenced aggregate monthly series. (Big spikes at lags 12, 24, and 36 indicate seasonality.) 15 / 26

Cross-sectional Aggregation 5e+03 5e+02 5e+01 spectrum 5e+00 5e−01 0 1 2 3 4 5 6 frequency Figure: Spectral density plot of aggregate monthly series. (Big spikes at seasonal frequencies 1, 2, 3, 4, 5, 6 in red indicate seasonality.) 16 / 26

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? 17 / 26

Seasonal Heteroscedasticity 0.6 0.4 0.2 0.0 −0.2 −0.4 0 5 10 15 20 Y ear Figure: Simulated quarterly series. Is it even seasonal? 18 / 26

Seasonal Heteroscedasticity 1.0 0.8 0.6 0.4 ACF 0.2 0.0 −0.2 0 2 4 6 8 10 12 Lag Figure: Acf plot of simulated quarterly series. (Big spikes at lags 4, 8, and 12 would indicate seasonality.) 19 / 26

Seasonal Heteroscedasticity 0.020 spectrum 0.010 0.005 0.0 0.5 1.0 1.5 2.0 frequency Figure: Acf plot of simulated quarterly series. (Big spikes at seasonal frequencies 1 and 2 in red would indicate seasonality.) 20 / 26

Seasonal Heteroscedasticity Q1 Q2 Q3 Q4 Series by Season Figure: Annual series for each quarter. Q1 is persistent, indicating seasonality, whereas other quarters are non-persistent, and hence non-seasonal. 21 / 26

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 22 / 26

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) 23 / 26

Framework for Improved Methodology Seasonal Vector Form : series { X t } with p seasons per cycle is embedded as p -variate vector time series X n = U y n where U is p × r (seasonal patterns), y n is r × 1 (seasonal persistence). This can be adapted when trend structures are present. U and y n 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 24 / 26

Framework for Improved Methodology ◮ Fixed effects in y n (e.g., linear trend) correspond to deterministic (predictable) seasonality ◮ Stochastic portion of y n with high autocorrelation (spectral mass at frequency zero) corresponds to persistent dynamic seasonal phenomena ◮ White noise portion of y n correponds to nonseasonal component Seasonal Adjustment : whitening the seasonal persistence 25 / 26

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 26 / 26