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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

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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.

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Basic Structural Model

Additive decomposition. 2nd order RW is assumed for trend. Gaussian assumption on ǫt, ηt, ωt with their variance σ2

ǫ,

σ2

η, and σ2 ω respectively.

yt = µt + γt + ǫt µt = 2µt−1 − µt−2 + ηt γt = −

s−1

• j=1

γt−j + ωt

State space representation is well-known (omitted). Consider alternative models for seasonal component γt.

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BSM-AR

Introduce a stationary real root in the characteristic equation of summation type seasonal model.

(1 − ΦL)S(L)γt = ωt S(L) = 1 + L + · · · + Ls−1 is the seasonal summation

• perator.

|Φ| is assumed to be 0 < Φ < 1.

‘Pink-noise driven’ Ozaki and Thomson (1992)

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BSM-MA

Seasonal summation driven by MA process.

S(L)γt = (1 + ΘL + Θ2L2 + · · · + Θs−1Ls−1)ωt |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).

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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)

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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, σ2

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.

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PCSMP: typical case of success

PCSMP

ˆ σ2

ω

ˆ σ2

ǫ

Φ or Θ

AIC BSM

0.86 × 10−5 0.35 × 10−4

—

−508.60

BSM-AR

0.53 × 10−5 0.39 × 10−4 0.83 × 10−2 −509.70

BSM-MA

0.68 × 10−4 0.31 × 10−6 0.84 −523.88

Large ˆ

Θ.

Increase in ˆ

σ2

ω.

ˆ Φ is always small and BSM-AR rarely helps. (Never at

least in this empirical study.)

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PCSMP: Quarterly Plots

MA part captured seasonal negative autocorrelation.

1980 1985 1990 1995 2000 −0.05 −0.04 −0.03 −0.02 −0.01

Q1

1980 1985 1990 1995 2000 −0.045 −0.040 −0.035 −0.030 −0.025

Q2

1980 1985 1990 1995 2000 0.00 0.01 0.02

Q3

1980 1985 1990 1995 2000 0.04 0.06 0.08

Q4

BSM BSM−AR BSM−MA

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PCSMP: BSM vs. BSM-MA

Original plus trend (upper-left), seasonal factors (bottom) and the difference of two seasonal factors (upper-right) for PCSMP .

1980 1985 1990 1995 2000 10.8 11.0 11.2

Original Trend (BSM−MA)

1980 1985 1990 1995 2000 −0.05 0.00 0.05

Diference of Seasonal Components

1980 1985 1990 1995 2000 −0.05 0.00 0.05

Seasonal (BSM)

1980 1985 1990 1995 2000 −0.05 0.00 0.05

Seasonal (BSM−MA)

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Spectrum Offset

• BSM-MA

BSM

• BSM

BSM-AR BSM-MA

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).

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Failure Case 1: UKCOAL

1960 1965 1970 1975 1980 1985 4 5

Original Trend (BSM−MA)

1960 1965 1970 1975 1980 1985 −0.025 0.000 0.025

Difference of Seasonal Components

1960 1965 1970 1975 1980 1985 −0.25 0.00 0.25

Seasonal (BSM)

1960 1965 1970 1975 1980 1985 −0.25 0.00 0.25

Seasonal (BSM−MA)

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Comment on UKCOAL results

UKCOAL

ˆ σ2

ω

ˆ σ2

ǫ

Φ or Θ

AIC BSM

0.67 × 10−9 0.17 × 10−1

—

−35.81

BSM-AR

0.33 × 10−7 0.17 × 10−1 0.57 × 10−7 −31.81

BSM-MA

0.73 × 10−4 0.17 × 10−1

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.

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Failure Case 2: CDKSI

1970 1975 1980 7.2 7.4 7.6 7.8

Original Trend (BSM−MA)

1970 1975 1980 −0.2 0.0 0.2 0.4

Difference of Seasonal Components

1970 1975 1980 −0.2 0.0 0.2 0.4

Seasonal (BSM)

1970 1975 1980 −0.2 0.0 0.2 0.4

Seasonal (BSM−MA)

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Comment on CDKSI results

CDKSI

ˆ σ2

ω

ˆ σ2

ǫ

Φ or Θ

AIC BSM

0.24 × 10−8 0.44 × 10−2

—

−213.44

BSM-AR

0.11 × 10−3 0.41 × 10−2 0.21 × 10−8 −204.67

BSM-MA

0.48 × 10−2 0.12 × 10−5

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. The problem is too much increase in ˆ

σ2

ω.

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BSM+AR Model

Both failure cases suggest the use of additive stationary AR term in BSM.

ψt =

m

• i=j

ρjψt−j + κt yt = µt + γt + ψt + ǫt

Coefficients {ρj} are estimated so that they should guarantee the stationarity of ψt.

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Improvement by BSM+AR (Fig. 7)

1960 1965 1970 1975 1980 1985 4 5

UKCOAL Trend(UKCOAL)

1970 1975 1980 7.25 7.50 7.75 8.00

CDKSI Trend(CDKSI)

1960 1965 1970 1975 1980 1985 −0.25 0.00 0.25

Seasonal(UKCOAL)

1970 1975 1980 0.0 0.2

Seasonal(CDKSI)

1960 1965 1970 1975 1980 1985 −0.25 0.00 0.25

AR(1)(UKCOAL)

1970 1975 1980 0.0 0.2

AR(2)(CDKSI)

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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)

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Toward A Criterion of Failure

Let st and ¯

st be the seasonal component derived from

the BSM and the BSM-MA respectively. Then σ∗ =

• Var(st − ¯

st) 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 st − min st (= range of the seasonal pattern of BSM) On the other hand in UKCOAL case, σ∗ is too small relative to R = max st − min st. Quantity to measure the impact of the fluctuation brought by MA term on the seasonal pattern:

M = log10{R/6σ∗}

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Model Map: Plot of (M, Θ)

0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 0.0 0.2 0.4 0.6 0.8 1.0

AIRLINE PCSMP UKELEC M2CD UKGAS MORDER TDS NEWCAR UKCOAL IIP CDKSI Seasonal MA parameter Seasonal Range / Change in seasonality (in logarithm)

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Comments on Model Map

Before drawing a map, BSM-MA and BSM-MA+AR should be estimated for each series. Two points connected with an arrow mean that the AIC statistic is improved by including a cyclical component. It is striking that ˆ

Θ’s are diminished and M’s are

centered around 1.00 ± 0.25 after the cyclical component is added to the model. BSM-MA models located in marginal areas like upper-left or upper-right have possibilities to be improved by introducing additive AR term. If all the models are estimated and compared in a row, perhaps drawing a map is unnecessary. But still M is worth to be calculated.

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Summary

Considered MA driven seasonal summation model. In empirical analysis, it is shown to improve trend-seasonal decomposition model, but sometimes it produces unreasonable results. Introduced M = log10{R/6σ∗}, the ratio of seasonal range to change in seasonality. Too large M (like UKCOAL) or too small M (like CDKSI) with ˆ

Θ ≈ 1 might

be cured by adding AR term in decomposition.

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