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Conference on Seasonality, Seasonal Adjustment and their implications for Short-Term Analysis and Forecasting 10-12 May 2006 Comparison of Methods Accounting for Outliers during Seasonal Adjustment J. Aston Outliers SA Outliers Simulations

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10-12 May 2006

Comparison of Methods Accounting for Outliers during Seasonal Adjustment

  • J. Aston

Conference on Seasonality, Seasonal Adjustment and their implications for Short-Term Analysis and Forecasting

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Outliers SA Outliers Simulations Real Data Summary References

Comparison of Methods Accounting for Outliers during Seasonal Adjustment

  • J. Aston

Institute of Statistical Science Academia Sinica

Eurostat Seasonal Adjustment Conference 2006

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Outline

Outliers Seasonal Adjustment Structural Component Models ARIMA models Outliers Gaussian Methods Non-Gaussian Methods Simulations Real Data UK Housing Transactions US Automobile Retail Series US Material Handling Equipment Manufacturing Series Summary

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1 2 3 4 5 6 7 8 5 10 15 20 25 30 Data

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1 2 3 4 5 6 7 8 5 10 15 20 25 30 Data Gauss Seasonal Adjustment

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Model Based Seasonal Adjustment

Use a component model to define a seasonal component and then remove this estimated component from the data. ysa

t

= yt − St St can be defined in many ways and is also dependent on all the other components in the model. The Unobserved Component models often define St is terms of trigonometric functions, while ARIMA decomposition models give constrained ARIMA model representations.

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Structural Component Models

The Trig-1 seasonal model, with seasonal period s, is defined as St =

[s/2]

  • j=1

Sj,t Sj,t = Sj,t−1 cos λj + S∗

j,t−1 sin λj + ωj,t

S∗

j,t

= −Sj,t−1 sin λj + S∗

j,t−1 cos λj + ω∗ j,t

and ωj,t, ω∗

j,t ∼ N(0, σ2 ω) and λj = 2πj s . In addition, the local level

model, Tt = Tt−1 + ηt where ηt ∼ N(0, σ2

η), for the Trend specification, and a white

noise Gaussian irregular component (with variance σ2

ξ).

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ARIMA models

Airline Model Basics

(1 − B)(1 − Bs)yt = (1 − θB)(1 − ΘBs)ξt Using the Hillmer-Tiao decomposition we can find (for suitable parameters) (1 − B)2Tt = θT(B)ωt U(B)St = θS(B)ηt It = εt

where U(B) = (1 + B + . . . + Bs−1) and ωt, ηt, ǫt are white noise processes

then yt = St + Tt + It

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Gaussian Outlier Adjustment - Simple Procedure

  • 1. Set Outlier Threshold
  • 2. Check Residuals - perform t-test
  • 3. Include most significant term in model
  • 4. Repeat 2. and 3. until no further significant terms
  • 5. Check current outliers are still applicable

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Gaussian Outlier Identification Simulation

Variance Estimation for the differenced process

1 2 3 4 5 0.85 0.90 0.95 1.00 1.05 1.10 1.15 1.20

100 Simulated series (143 observations of monthly data) with 3 outliers. Histogram of estimated variance parameters (true value is 1.0) based on number of outliers detected by automatic Gaussian procedure (threshold value is 3.0) 9

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Non Gaussian Components

I∗

t ∼ t(0, σ2 ξ, ν),

t = 1, . . . , n, where ν > 2 is the number of degrees of freedom and σ2

ξ is the

variance, which is constant for any ν. The decomposition model with a t-distributed irregular term can be expressed in its canonical form by yt = St + Tt + I∗

t ,

I∗

t ∼ t(0, σ2 ξ, ν),

t = 1, . . . , n. where t(0, σ2

ξ, ν) refers to the t density.

Structural Component Model: Estimate σξ, ν AMB Component Model: Estimate ν. Overall model gives σξ

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Simulation Setup

  • 144/156 Series for both SC and AMB models
  • Three outliers at 5 times sd of Irr / No outliers added
  • Gaussian / Non-Gaussian Model Estimation
  • Mean Squared Seasonal Difference between 144/156 data

lengths measured for 144 coincident points.

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Comparison Histograms

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  • 5
5 10 15 20 25 30 2.5 5.0 7.5 10.0 12.5 15.0 Seasonal Stability
  • 5.0
  • 2.5
0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 5 10 15 20 25 30 35 40 45 Seasonal Stability

Trig-1 Three Outliers Trig-1 No Outliers

  • 2.5
  • 2.0
  • 1.5
  • 1.0
  • 0.5
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 2.5 5.0 7.5 10.0 12.5 15.0 Seasonal Stability
  • 0.25
0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 5 10 15 20 25 Seasonal Stability

AMB Three Outliers AMB No Outliers

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England and Wales Housing Transactions Series

1995 2000 2005 80 90 100 110 120 130 140 150 160 170 180 UK Housing Transactions Series

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Trig-1 Model

50 100 150

  • 0.2

0.0 0.2

Seasonal - Orig + 12months Seasonal - Orig

50 100 150

  • 0.10
  • 0.05

0.00 0.05

Difference between Seasonals

50 100 150

  • 0.2

0.0 0.2

Seasonal - Orig + 12months Seasonal - Orig

50 100 150

  • 0.10
  • 0.05

0.00 0.05

Difference between Seasonals

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AMB Model

50 100 150

  • 0.1

0.0 0.1

Seasonal - Orig + 12months Seasonal - Orig

50 100 150

  • 0.05

0.00 0.05

Difference between Seasonals

50 100 150

  • 0.1

0.0 0.1

Seasonal - Orig + 12months Seasonal - Orig

50 100 150

  • 0.05

0.00 0.05

Difference between Seasonals

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US Automobile Retail Series

1995 2000 2005 40000 50000 60000 70000 80000 US Automobile Retail Series

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Trig-1 Model

50 100 150

  • 0.10
  • 0.05

0.00 0.05 0.10

Seasonal - Orig + 12months Seasonal - Orig

50 100 150

  • 0.01

0.00 0.01

Difference between Seasonals

50 100 150

  • 0.10
  • 0.05

0.00 0.05 0.10

Seasonal - Orig + 12months Seasonal - Orig

50 100 150

  • 0.01

0.00 0.01

Difference between Seasonals

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AMB Model

50 100 150

  • 0.10
  • 0.05

0.00 0.05 0.10

Seasonal - Orig + 12months Seasonal - Orig

50 100 150

  • 0.002
  • 0.001

0.000 0.001 0.002 0.003

Difference between Seasonals

50 100 150

  • 0.10
  • 0.05

0.00 0.05 0.10

Seasonal - Orig + 12months Seasonal - Orig

50 100 150

  • 0.002
  • 0.001

0.000 0.001 0.002 0.003

Difference between Seasonals

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US Material Handling Equipment Manufacturing Series

1995 2000 2005 1000 1250 1500 1750 2000 2250 2500 US Material Handling Equipment Manufacturing Series

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Trig-1 Model

50 100 150

  • 0.1

0.0 0.1 0.2

Seasonal - Orig + 12months Seasonal - Orig

50 100 150

  • 0.025

0.000 0.025 0.050

Difference between Seasonals

50 100 150

  • 0.1

0.0 0.1 0.2

Seasonal - Orig + 12months Seasonal - Orig

50 100 150

  • 0.025

0.000 0.025 0.050

Difference between Seasonals

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AMB Model

50 100 150

  • 0.1

0.0 0.1 0.2

Seasonal - Orig + 12months Seasonal - Orig

50 100 150

  • 0.002
  • 0.001

0.000 0.001 0.002

Difference between Seasonals

50 100 150

  • 0.1

0.0 0.1 0.2

Seasonal - Orig + 12months Seasonal - Orig

50 100 150

  • 0.002
  • 0.001

0.000 0.001 0.002

Difference between Seasonals

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Summary

  • Critical values can lead to unstable seasonals
  • Non-Gaussian models give reasonable seasonals when no
  • utliers present
  • Non-Gaussian models reduce instability when outliers

present

  • Caveats:
  • More Parameters needed
  • Computationally more intensive

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References I

Aston, J. A. D. and S. J. Koopman (2006). A non-gaussian generalisation of the airline model for robust seasonal adjustment. Journal of Forecasting, in press. Bruce, A. G. and S. R. Jurke (1996). Non-Gaussian seasonal adjustment: X-12-ARIMA versus robust structural models. Journal of Forecasting 15, 305–328. Gómez, V. and A. Maravall (2001a). Automatic modeling methods for univariate series. In D. Peña, G. C. Tiao, and R. S. Tsay (Eds.), A Course in Time

  • Series. New York, NY: J. Wiley and Sons.

Chapter 7.

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References II

Gómez, V. and A. Maravall (2001b). Seasonal adjustment and signal extraction in economic time series. In D. Peña, G. C. Tiao, and R. S. Tsay (Eds.), A Course in Time

  • Series. New York, NY: J. Wiley and Sons.

Chapter 8. Harvey, A. C. (1989). Forecasting, Structural Time Series Models and the Kalman Filter. Cambridge: Cambridge University Press. Hillmer, S. C. and G. C. Tiao (1982). An ARIMA-model-based approach to seasonal adjustment. Journal of the American Statistical Association 77, 63–70.

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