Assessment of Diagnostics for the Presence of Seasonality

1
Assessment of Diagnosticsfor the Presence of
Seasonality

• Catherine C. H. Hood
• Catherine Hood Consulting
• Auburntown, TN

2
Acknowledgements

• Roxanne Feldpausch
• X-12-Data, programming assistance
• Kathy McDonald-Johnson
• X-12-Rvw, programming and proof-reading
  assistance
• Brian Monsell
• Modifications to X-12-ARIMA
• David Findley

3
Motivation

• Wanted to check for seasonality in quarterly
  series, some with only 8 years of data
• For series that are not too short, the spectral
  graph is the most sensitive test for residual
  seasonality
• However, 60 points recommended (15 years for a
  quarterly series)
• Other diagnostics for residual seasonality can
  also be difficult for short series, and also can
  be model-dependent

4
New Options in X-12-ARIMA

• Ability to change the spectrum used to test for
  the presence of seasonality
• spectrumseries a1 (or a19, b1)
• Ability to change the estimator for the spectrum
• maxspecar10 (default is 30)

5
Spectral Analysis

• Allows us to see the relationships between the
  frequencies
• In a quarterly time series with a significant
  seasonal component, the amplitudes that dominate
  are at the two seasonal frequencies
• An annual effect at ¼ cycles per quarter and
• A biannual effect at ½ cycles per quarter

6

• G.0 10LOG(SPECTRUM) of the differenced,
  transformed Original Series
• (Table A1). Spectrum estimated from 1991.1
  to 2005.2.
• I
  I
• -7.37I
  I -7.37
• I
  SI
• I
  SI
• I
  SI
• -10.39I
  SI -10.39
• I
  SI
• I
  SI
• I
  SI
• -13.41I
  SI -13.41
• I
  SI
• I
  SI
• I
  SI
• -16.44I
  SI -16.44
• I
  SI
• I S
  SI
• I S
  SI

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Evaluating the Estimator

• For the estimate of the spectrum, X-12-ARIMA uses
  an AR(30) model fit to each output series and
  evaluated at 61 frequencies
• In Version 0.3, the order of the AR model for the
  spectral estimate can be changed from the default

8
Reading the Peaks

• For X-12-ARIMA to flag a seasonal peak
• The frequency must be six stars or asterisks
  higher than either neighboring frequency
• The higher the peak is above its neighbors, the
  more important the peak
• A seasonal peak also must be higher than the
  median of the frequencies

9

• G.0 10LOG(SPECTRUM) of the differenced,
  transformed Original Series
• (Table A1). Spectrum estimated from 1991.1
  to 2005.2.
• I
  I
• -7.37I
  I -7.37
• I
  SI
• I
  SI
• I
  SI
• -10.39I
  SI -10.39
• I
  SI
• I
  SI
• I
  SI
• -13.41I
  SI -13.41
• I
  SI
• I
  SI
• I
  SI
• -16.44I
  SI -16.44
• I
  SI
• I S
  SI
• I S
  SI

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Example First Peak

• I
• I
  
• -16.44I
  
• I
  
• I S
  
• I S
  
• -19.46I S
  
• I S
  
• I S
  
• I S
• -22.48I S
  
• I S
  
• I S
  
• I S
  

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Peak Strength

• X-12-ARIMAs diagnostics file tells us exactly
  how many stars are in a given peak
• Original Series
• S1 4.4 S2 6.1

12
Other Diagnostics

• M7 and D8 F-test
• Chi-square test for seasonal dummies
• regression variables( seasonal )

13
D8 F-test of X-12-ARIMA

• After Table D8 is an F-test for stable
  seasonality
• Sometimes referenced as FS
• Has come to be known as the D8 F
• For time series, a cut-off value of 7.0 is
  recommended by Lothian and Morry (1978)

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M7

• A function of F-values derived from two different
  ANOVA tables to assess the amount of moving
  seasonality present in a series relative to the
  amount of stable seasonality
• Interpretation a value greater than 1.0
  indicates no identifiable seasonality

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Tests for Seasonal Regressors

• In SEATS, Agustín Maravall has used a
  significance test on the individual months or
  quarters to determine if there is significant
  seasonality in the original series
• X-12-ARIMA has a chi-square test to test for the
  significance of the set of seasonal regressors

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Potential Problems

• Changes in the model can affect the stability of
  the diagnostics, especially M7, D8 F, and the
  Chi-square test for the seasonal regressors
• Some users run the seasonally adjusted series
  back through X-12-ARIMA to look at the values for
  M7 and the D8 F, however, M7 was not designed for
  this purpose

17
Series for Study

• 314 National Accounts and Retail quarterly
  seasonal series with 10 years of data
• 432 simulated quarterly series with 15 years of
  data
• 72 simulated monthly series
• Simulated series computed from seasonal factors,
  trends, and irregular from different series (see
  Hood, Ashley, Findley 2000)

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Series Used for the Spectrum

• Assuming that the AR order is held constant, the
  results were almost identical for
• The outlier-adjusted original series
• The prior-adjusted original series
• This held true for the simulated series also
• So well focus on the outlier-adjusted original
  series

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Spectral Estimators Seasonal Quarterly Series

• With the AR(30) estimator (because there are less
  than 60 points) the peaks were often very short
  and flat and difficult to distinguish
• With the AR(10) estimator, the seasonal peaks are
  much easier to distinguish

20
Real Series Results for the S1 Peak

• Model AR of Series with Peak
• (2 1 0) 10 71 (25)
• (2 1 0) 30 9 ( 3)

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Real Series Results for the S1 Peak (2)

• Model AR Average of Stars
• (2 1 0) 10 5.3
• (2 1 0) 30 2.6

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Spectral Diagnostic Results from the Models Used

• Spectral diagnostic results suffer when using the
  model (0 0 0) with seasonal dummies (generally
  inappropriate for these series) due to the
  outlier sets chosen automatically by the program

23
Real Series Results for the S1 or S2 Peak

• Model AR of Series with Peak
• (2 1 0) 10 98 (31)
• (2 1 0) 30 29 ( 9)
• (0 0 0) 10 69 (22)
• (0 0 0) 30 28 ( 9)

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Results from the Models Used

• Most models gave very similar results
• The diagnostics were inadequate using the model
  (0 0 0) with seasonal dummies, which we expect
  would be inappropriate for these seasonal series
  

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Real Series Results for D8 F-test

• Model seasonal (D8F gt 7)
• (2 1 0) 273 (87)
• (2 1 2) 269 (86)
• (0 1 2) 271 (86)
• (0 1 0) 271 (86)
• (0 0 0) 114 (36)

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Real Series Results for the M7

• Model seasonal (M7 lt 1)
• (2 1 0) 286 (91)
• (2 1 2) 277 (88)
• (0 1 2) 281 (89)
• (0 1 0) 278 (89)
• (0 0 0) 104 (33)

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Real Series Results for the Chi-sq test

• Model seasonal (p lt 0.05)
• (2 1 0) 301 (96)
• (2 1 2) 293 (93)
• (0 1 2) 301 (96)
• (0 1 0) 283 (90)
• (0 0 0) 78 (25)

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Simulated Series

• Simulated quarterly series
• With strong seasonality
• With no seasonality
• With very little seasonality, the kind wed like
  to detect when looking for residual seasonality

29
Series with Strong Seasonality Series Found to
be Seasonal

• Diag (2 1 0) (0 0 0)
• D8 F 144 (100) 49 (34)
• Chi-sq 144 (100) 0 ( 0)
• AR10 Sp 144 (100) 144 (65)

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Series with Strong Seasonality Results for S1
or S2 Peak

• Model AR of Series with Peak
• (2 1 0) 10 144 (100)
• (2 1 0) 30 134 ( 93)
• (0 0 0) 10 144 (100)
• (0 0 0) 30 122 ( 85)

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Nonseasonal Series Series Found to be Seasonal

• Diag (2 1 0) (0 0 0)
• D8 F 56 (39) 0 ( 0)
• Chi-sq 90 (62) 0 ( 0)
• AR10 Sp 57 (40) 51 (35)

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Nonseasonal Series Results for S1 or S2 Peak

• Model AR of Series with Peak
• (2 1 0) 10 57 (40)
• (2 1 0) 30 65 (45)
• (0 0 0) 10 51 (35)
• (0 0 0) 30 72 (50)

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Series with Weak Seasonality Series Found to
be Seasonal

• Diag (2 1 0) (0 0 0)
• D8 F 111 (77) 0 ( 0)
• Chi-sq 124 (86) 0 ( 0)
• AR10 Sp 86 (60) 86 (60)

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Monthly Series

• Very limited study
• With an AR(10) estimator for the spectrum
• Trading day peaks not found
• A few more false positives (peaks for nonseasonal
  series) found

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Conclusions

• An AR(10) estimator for the spectrum gives
• Fewer false positives (peaks for nonseasonal
  series) than the default AR(30) estimator
• More true positives (peaks for seasonal series)
  than the default AR(30) estimator

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Conclusions (2)

• The diagnostics are fairly stable between various
  seasonal models
• However, there can be a lot of disagreement in
  the diagnostics when comparing runs with seasonal
  versus nonseasonal models
• The practice of using M7/D8F to look for residual
  seasonality in seasonally adjusted series is very
  much dependent on whether or not the ARIMA model
  is seasonal

37
Future Research

• There has been quite a bit of work looking at the
  six star rule for monthly series
• Similar research might be useful for AR(10)
  estimators for quarterly series

38
Contact Information

• Catherine Hood
• Catherine Hood Consulting
• 1090 Kennedy Creek Road
• Auburntown, TN 37016-9614
• Telephone [phone]
• Email cath_at_catherinechhood.net
• Web www.catherinechhood.net

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