A Structural Time Series Model Facilitating Flexible Seasonality Yoshinori Kawasaki The Institute of Statistical Mathematics, Japan at Conference on Seasonality, Seasonal Adjustment and Their Implications for Short-Term Analysis and Forecasting 10 May 2006, Eurostat, Luxembourg A Structural Time Series Model Facilitating Flexible Seasonality – p.1/22
Aim of This Paper MA driven seasonal component model is considered within the framework of strctural time series model. MA driven seasonal model generally performs well, but sometimes not. Unreasonable results are found by plotting estimated components. Aim (1): to propose a measure for appropriateness of MA driven seasonal comonent model. (To process many series without visual inspection.) Aim (2): to propose a ‘model map’ to compare various seasonal component models. A Structural Time Series Model Facilitating Flexible Seasonality – p.2/22
Basic Structural Model Additive decomposition. 2nd order RW is assumed for trend. Gaussian assumption on ǫ t , η t , ω t with their variance σ 2 ǫ , σ 2 η , and σ 2 ω respectively. y t = µ t + γ t + ǫ t µ t = 2 µ t − 1 − µ t − 2 + η t s − 1 � γ t = − γ t − j + ω t j =1 State space representation is well-known (omitted). Consider alternative models for seasonal component γ t . A Structural Time Series Model Facilitating Flexible Seasonality – p.3/22
BSM-AR Introduce a stationary real root in the characteristic equation of summation type seasonal model. (1 − Φ L ) S ( L ) γ t = ω t S ( L ) = 1 + L + · · · + L s − 1 is the seasonal summation operator. | Φ | is assumed to be 0 < Φ < 1 . ‘Pink-noise driven’ Ozaki and Thomson (1992) A Structural Time Series Model Facilitating Flexible Seasonality – p.4/22
BSM-MA Seasonal summation driven by MA process. S ( L ) γ t = (1 + Θ L + Θ 2 L 2 + · · · + Θ s − 1 L s − 1 ) ω t | Θ | is assumed to be 0 < Θ < 1 . ARIMA approximation of X-11 filter → MA driven seasonal component model. Cleveland and Tiao (1976), Burridge and Wallis (1984) and so on. ‘Airline model’ accounts for wide variety of economic time series. → includes seasonal MA. State space rep. is straightforward (omitted, see 3.4). A Structural Time Series Model Facilitating Flexible Seasonality – p.5/22
Real Data Analysis Japanese data Consumption in private sector (PCSMP , 1980Q1 to 2002Q4), Machinery order (MORDER, 1987:04–2002:12), Money supply (M2CD, 1980:01–2002:12), New car registration (NEWCAR, 1980:01–2002:12), Industrial production (IIP , 1980:01–2002:12), Tokyo district sales of department stores (TDS, 1980:01–2002:12) Well-known data Coal, gas and electricity demand of other final users (UKCOAL, UKGAS, UKELEC, 1960Q1–1986Q4), Car drivers killed or seriously injured (CDKSI, 1969:01–1982:12), International airline passengers (AIRLINE, 1949:01–1960:12) A Structural Time Series Model Facilitating Flexible Seasonality – p.6/22
Results of min AIC Procedure The BSM-MA attains the minimum AIC for all series but UKCOAL, and the improvements in AIC are sometimes substantial. (See Table 1 and Table 2.) In every case, the estimated innovation variance of the seasonal component, σ 2 ω , is larger than those estimated in the BSM and the BSM-MA. MA-driven seasonal summation has successfully brought more flexibility than the BSM and the BSM-AR. A Structural Time Series Model Facilitating Flexible Seasonality – p.7/22
PCSMP: typical case of success σ 2 σ 2 PCSMP ˆ ˆ Φ or Θ AIC ω ǫ 0 . 86 × 10 − 5 0 . 35 × 10 − 4 BSM — − 508 . 60 0 . 53 × 10 − 5 0 . 39 × 10 − 4 0 . 83 × 10 − 2 BSM-AR − 509 . 70 0 . 68 × 10 − 4 0 . 31 × 10 − 6 BSM-MA 0 . 84 − 523 . 88 Large ˆ Θ . σ 2 Increase in ˆ ω . ˆ Φ is always small and BSM-AR rarely helps. (Never at least in this empirical study.) A Structural Time Series Model Facilitating Flexible Seasonality – p.8/22
PCSMP: Quarterly Plots MA part captured seasonal negative autocorrelation. Q2 −0.01 −0.025 Q1 −0.02 −0.030 −0.03 −0.035 −0.04 −0.040 −0.05 −0.045 1980 1985 1990 1995 2000 1980 1985 1990 1995 2000 Q3 BSM BSM−MA Q4 BSM−AR 0.02 0.08 0.01 0.06 0.00 0.04 1980 1985 1990 1995 2000 1980 1985 1990 1995 2000 A Structural Time Series Model Facilitating Flexible Seasonality – p.9/22
PCSMP: BSM vs. BSM-MA Original plus trend (upper-left), seasonal factors (bottom) and the difference of two seasonal factors (upper-right) for PCSMP . Original Trend (BSM−MA) Diference of Seasonal Components 11.2 0.05 11.0 0.00 10.8 −0.05 1980 1985 1990 1995 2000 1980 1985 1990 1995 2000 Seasonal (BSM) Seasonal (BSM−MA) 0.05 0.05 0.00 0.00 −0.05 −0.05 1980 1985 1990 1995 2000 1980 1985 1990 1995 2000 A Structural Time Series Model Facilitating Flexible Seasonality – p.10/22
Spectrum Offset � � �� � �� � �� � �� � �� � � � � � BSM-MA � BSM-MA � � � � BSM-AR � �� � �� � �� � �� � �� � � � � � � � � BSM BSM � � Left: Log of pseudo-spectra of the BSM and BSM-MA (with Θ = 0 . 9 ), Right: Estimated pseudo-spectra of seasonal component for private sector consumption in Japan (PCSMP). A Structural Time Series Model Facilitating Flexible Seasonality – p.11/22
Failure Case 1: UKCOAL Original Trend (BSM−MA) Difference of Seasonal Components 0.025 5 0.000 4 −0.025 1960 1965 1970 1975 1980 1985 1960 1965 1970 1975 1980 1985 Seasonal (BSM) Seasonal (BSM−MA) 0.25 0.25 0.00 0.00 −0.25 −0.25 1960 1965 1970 1975 1980 1985 1960 1965 1970 1975 1980 1985 A Structural Time Series Model Facilitating Flexible Seasonality – p.12/22
Comment on UKCOAL results σ 2 σ 2 UKCOAL ˆ ˆ Φ or Θ AIC ω ǫ 0 . 67 × 10 − 9 0 . 17 × 10 − 1 BSM — − 35 . 81 0 . 33 × 10 − 7 0 . 17 × 10 − 1 0 . 57 × 10 − 7 BSM-AR − 31 . 81 0 . 73 × 10 − 4 0 . 17 × 10 − 1 BSM-MA 0.98 − 31 . 83 Typical ‘near cancellation’ case. ( ˆ Θ ≈ 1 ) BSM-MA produces almost same results as BSM, while min AIC procedure selects BSM. → practically no problem However, fluctuation around trend does not look deterministic very much. A Structural Time Series Model Facilitating Flexible Seasonality – p.13/22
Failure Case 2: CDKSI 0.4 Original Trend (BSM−MA) Difference of Seasonal Components 7.8 0.2 7.6 0.0 7.4 −0.2 7.2 1970 1975 1980 1970 1975 1980 0.4 0.4 Seasonal (BSM) Seasonal (BSM−MA) 0.2 0.2 0.0 0.0 −0.2 −0.2 1970 1975 1980 1970 1975 1980 A Structural Time Series Model Facilitating Flexible Seasonality – p.14/22
Comment on CDKSI results σ 2 σ 2 CDKSI ˆ ˆ Φ or Θ AIC ω ǫ 0 . 24 × 10 − 8 0 . 44 × 10 − 2 BSM — − 213 . 44 0 . 11 × 10 − 3 0 . 41 × 10 − 2 0 . 21 × 10 − 8 BSM-AR − 204 . 67 0 . 48 × 10 − 2 0 . 12 × 10 − 5 BSM-MA 0.99 − 302 . 70 ˆ Θ ≈ 1 but not the near cancellation case because estimated seasonality is far from deterministic. Plot of estimated seasonal component obviously shows this is not a meaningful decomposition. σ 2 The problem is too much increase in ˆ ω . A Structural Time Series Model Facilitating Flexible Seasonality – p.15/22
BSM+AR Model Both failure cases suggest the use of additive stationary AR term in BSM. m � ψ t = ρ j ψ t − j + κ t i = j y t = µ t + γ t + ψ t + ǫ t Coefficients { ρ j } are estimated so that they should guarantee the stationarity of ψ t . A Structural Time Series Model Facilitating Flexible Seasonality – p.16/22
Improvement by BSM + AR (Fig. 7) 8.00 UKCOAL Trend(UKCOAL) CDKSI Trend(CDKSI) 5 7.75 7.50 4 7.25 1960 1965 1970 1975 1980 1985 1970 1975 1980 Seasonal(UKCOAL) Seasonal(CDKSI) 0.25 0.2 0.00 0.0 −0.25 1960 1965 1970 1975 1980 1985 1970 1975 1980 AR(1)(UKCOAL) AR(2)(CDKSI) 0.25 0.2 0.00 0.0 −0.25 1960 1965 1970 1975 1980 1985 1970 1975 1980 A Structural Time Series Model Facilitating Flexible Seasonality – p.17/22
Comments on BSM + AR results UKCOAL: AIC(BSM + AR(1)) = − 66 . 68 « − 35 . 81 Simple seasonal summation plus AR(1) component is the best. CDKSI AIC(BSM + AR(2)) = − 269 . 36 AIC(BSM-MA) = − 302 . 71 , ˆ Θ = 0 . 99 AIC(BSM-MA + AR(2)) = − 304 . 46 , ˆ Θ = 0 . 28 Still MA type specification is supported, AIC improved, and the estimated seasonal component became reasonable! (Figure 7) A Structural Time Series Model Facilitating Flexible Seasonality – p.18/22
Toward A Criterion of Failure Let s t and ¯ s t be the seasonal component derived from the BSM and the BSM-MA respectively. Then σ ∗ = � Var ( s t − ¯ s t ) is the standard deviation of the perturbation introduced by MA term. The key feature of the CDKSI case is that σ ∗ is very large in comparison with R = max s t − min s t ( = range of the seasonal pattern of BSM) On the other hand in UKCOAL case, σ ∗ is too small relative to R = max s t − min s t . Quantity to measure the impact of the fluctuation brought by MA term on the seasonal pattern: M = log 10 { R/ 6 σ ∗ } A Structural Time Series Model Facilitating Flexible Seasonality – p.19/22
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