Dr. Lisa Grace Bersales - Bangko Sentral Ng Pilipinas · Dr. Lisa Grace Bersales ... NSCB REport...

Annual BSP‐UP Professorial Chair Lectures

LECTURE NO. 7LECTURE NO. 7LECTURE NO. 7

Enhancing Seasonal Adjustment of Philippine Time Series: Procedures under Seasonal Volatilities

Dr. Lisa Grace BersalesBSP Sterling Professor of

Government and Official Statistics

Enhancing Seasonal Adjustment of Philippine Time Series:

Procedures under Seasonal Volatilities1

Lisa Grace S. Bersales,Ph.D.Lisa Grace S. Bersales,Ph.D.22

1BSP Professorial Sterling Chair on Government and Official Statistics Lecture 2009

2 Professor of Statistics, School of Statistics of the University of the Philippines in Diliman

Outline of PresentationOutline of Presentation

Motivation of the Study

Objective of the Study

Proposed Procedure for Seasonal Adjustment

Illustration using Philippine Time Series

Next Steps

Motivation of the Study

Official Seasonal Adjustment of Philippine Time Series

X11-ARIMA Seasonal Adjustment developed by Statistics Canada

Components of Time SeriesComponents of Time Series

Time Series Components – trend (T), cyclical (C), seasonal (S) and irregular (I).

These are unobserved and need to be estimated – process referred to as decomposition of time series.

Seasonal AdjustmentSeasonal Adjustment

•identification

•estimation

•removal of seasonal variations and effect of trading days and moving holidays (if present) from a time series.

Why is seasonal adjustment done?Why is seasonal adjustment done?

to read the trend of an economic time series without being hampered by seasonal movements

as part of transformations on data used for statistical modelling

preliminary to estimating business cycles

Final product of seasonal adjustment = seasonally adjusted series or deseasonalized series

NSCB REport for Q4 2009.doc

Seasonally adjusted SNA_Fourth Quarter 2009.doc

seasonally adjusted CPI.doc

Example 1 of Actual and Its Seasonally Adjusted SeriesExample 1 of Actual and Its Seasonally Adjusted SeriesChart 30. Gross Domestic Product: Original and Seasonally Adjusted Series

1988-2006

160

180

200

220

240

260

280

300

320

340

360

1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006

Original

SA

7

8

9

10

11

12

13

14

15

1988 1990 1992 1994 1996 1998 2000 2002

Unemployment_SA Unemployment

Actual Unemployment Rate (Old Definition) andSeasonally Adjusted Unemployment Rate

Example 2 of Actual and Its Seasonally Adjusted SeriesExample 2 of Actual and Its Seasonally Adjusted Series

Decomposition ModelsDecomposition Models

a. Additive Model: Y=TC+S+Ib. Multiplicative Model: Y=TC*S*I

Two other available decompositions are the log additive and the pseudo-additive decompositions

Seasonally Adjusted SeriesSeasonally Adjusted Series

The seasonally adjusted series for the additive model is,

SYSA −=

For the multiplicative model, the seasonally adjusted series is,

SYSA /=

Example of Seasonally Adjusted Series Example of Seasonally Adjusted Series

500

1000

1500

2000

2500

3000

3500

4000

1990 1992 1994 1996 1998 2000 2002 2004

EXPORT EXPORT_SA

Monthly Exports and Seasonally Adjusted Exports (X12 Default Options)From January 1990 to December 2004

Example of the TrendExample of the Trend--Cycle ComponentCycle Component

500

1000

1500

2000

2500

3000

3500

4000

1990 1992 1994 1996 1998 2000 2002 2004

EXPORT Trend-Cycle

Monthly Exports and Trend-Cycle Component (X12 Default Options)From January 1990 to December 2004

Example of the Seasonal Factors/ComponentExample of the Seasonal Factors/Component

0.84

0.88

0.92

0.96

1.00

1.04

1.08

1.12

1990 1992 1994 1996 1998 2000 2002 2004

Seasonal Factors of Export SeriesJanuary 1990 to December 2004

Example of the Irregular ComponentExample of the Irregular Component

0.85

0.90

0.95

1.00

1.05

1.10

1.15

1.20

1990 1992 1994 1996 1998 2000 2002 2004

Irregular Component of the Series ExportJanuary 1990 to December 2004

Seasonal Adjustment Procedures

Most statistical agencies use methods based on moving averages for seasonal adjustment.

The two most commonly used are the U.S. Bureau of Census’ X11-Method II Variant and Statistics Canada’s X11 ARIMA.

These two methods follow an iterative estimation procedure involving the major steps in the decomposition of a time series.

Main Steps in X11 ARIMA (Version 2000)

X11 ARIMA uses the Census X11 procedure on augmented data -the time series plus one year of monthly or quarterly forecasts and one year of backcasts from an ARIMA model. The X11 ARIMA basically consists of:

a) modeling the original series using an ARIMA or Box-Jenkins Model;

b) forecasting one year of unadjusted data at each end of the series from ARIMA models that fit and project the original series well; and

c) seasonally adjusting the augmented series using X11-Method II variant.

The Easter and trading-day adjustments are applied even before a) is done if one asks for it.

Main Steps of Multiplicative Decomposition Using X11- Method II Variant

Step 1. Estimate a preliminary trend-cycle by using a moving average, e.g., CMA that eliminates most of the other movements present including the seasonal variations.

Step 2. Subtract (a) from the original series (divide the original series by (a)) to obtain the remainder variations, that is, seasonality and the irregular component, usually known as the S-I differences (SI ratios).

Step 3. Smoothen the S-I differences (SI ratios) for each month or quarter over the whole space of the series by applying an average again, e.g., MA. This gives an estimate of the seasonal factor S (seasonal index S).

Step 4. Subtract S from S-I (divide SI by S) to get I.

Step 5. Subtract S from the original series (divide the original series by S). This will give the seasonally adjusted series.

Detailed Steps of Multiplicative Decomposition Using X11- Method II Variant

STEP 1. Compute the ratios between the original series and centered 12-term moving averages.

STEP 2. Estimate the seasonal factors by applying a weighted 5-term moving average to the SI ratios.

STEP 3. Adjust the seasonal factors to sum up to 12.STEP 4. Estimate the irregular component by dividing the SI

ratios by the seasonal factors.STEP 5. Identify and remove the “extreme” irregulars.STEP 6. Obtain preliminary seasonal factors by applying a

weighted 5-term moving average to the SI ratios with extremes replaced.


STEP 7. Adjust to sum up to 12.

STEP 8. Obtain preliminary seasonally adjusted series by dividing these values into the original observations.

STEP 9. Obtain estimates of the trend-cycle by applying a 13-term Henderson moving average to the preliminary adjusted series.

STEP 10. Estimate new SI ratios, dividing the trend-cycle into the original observations.

STEP 11. Estimate seasonal factors by applying a weighted 7-term moving average to the SI ratios.


STEP 12. Adjust to sum up to 12.

STEP 13. Divide seasonal factors into the original series to obtain a seasonal adjusted series.

X11 ARIMA was developed to produce more accurate estimates of current seasonally adjusted series when seasonality changes rapidly in a stochastic manner.

It has been noted, however, that it cannot cope with abrupt changes in the structure of the time series.

The Evaluation Measures in X11ARIMA: M Statistics

M1. The relative contribution of the irregular over months or quarters

M2. The relative contribution of the irregular to the stationary portion of the series

M3. The amount of period to period change in the irregular component as compared to the amount of period to period change in the trend-cycle

M4. The amount of autocorrelation in the irregular as described by the average duration of run

M5. The number of quarters it takes the change in the trend- cycle to surpass the amount of change in the irregular

M6. The amount of year to year change in the irregular as compared to the amount of year to year change in the seasonal

M7. The amount of moving seasonality present relative to the amount of stable seasonality

M8. The size of the fluctuations in the seasonal component throughout the whole series.

M9. The average linear movement in the seasonal component throughout the whole series

M10. Same as M8, calculated for recent years only

M11. Same as M9, calculated for recent years only

Q statistic= weighted average of M1 to M11

Q = 0.13 M1 + 0.13 M2 + 0.10 M3 + 0.05 M4 + 0.11M5 + 0.10 M6 + 0.16 M7 + 0.07 M8 + 0.07 M9 + 0.04M10 + 0.04M11

( standard: all M stats and Q < 1)

Measure of smoothness= mean of absolute value of Ya(t) – Ya(t-1), Ya(j) is the seasonally adjusted series at time j

( standard: value of measure of smoothness should be as low as possible)

Objective of the Study

A number of Philippine time series exhibit volatilities of their seasonal behavior. Such a situation results in estimated seasonal factors contaminated by irregularities which should be in the irregular component. This paper proposes a modification of seasonal adjustment to address such a situation.

Proposed ProcedureThis procedure is a three-step procedure starting with the usual X11ARIMA. This is followed by modelling the volatility of the resulting seasonal component once the evaluation statistics indicate moving seasonality..an indication of stochastic seasonality. The third and last stage is repeating X11ARIMA on the series without the irregular component derived from the model in the second stage.

The Proposed Procedure

Stage 1 1. Given a series Y, do the usual decomposition using the steps of X11ARIMA to produce the components: TC(1), S(1), I(1)

Stage 2. Build a statistical model to estimate volatility of S(1) .Stage 3. Do X11ARIMA of , Y*, the original series less the residuals from Stage 2 to produce: TC(2), S(2), I(2) .

The seasonally adjusted series, Ya, is Y*/S(2).

Another Procedure Considered

Phase 1. Do the usual decomposition using the steps of X11ARIMA to produce the components: TC(1), S(1), I(1)

Phase 2. Decompose the seasonal factor into: TC(2), S(2), I(2)

Final Decomposition ( Multiplicative);TC= TC(1) *TC(2)

S =S(2)

I = I(1 ) * I(2)

Results

When seasonal volatility is more pronounced, the proposed procedure produces smoother seasonally adjusted series compared with the usual X11ARIMA procedure and with the other procedure considered.When seasonal volatility is not pronounced, the proposed procedure and the other procedure have comparable results.Adding another iteration to the proposed procedure does not have added improvement.

IllustrationsSeries with more pronounced seasonal volatility:

Imports( monthly, Jan 1980 – Dec 2004)

Series with less pronounced seasonal volatility:

Unemployment rate ( quarterly, 1988 Q1-2006 Q4)>>>

Quarterly Unemployment Rate

7

8

9

10

11

12

13

14

15

88 90 92 94 96 98 00 02 04 06

UNE MP LOY

Unemployment Rate, P hilippines, Jan 1988 - Oct 2006

8

9

10

11

12

13

88 90 92 94 96 98 00 02 04 06

UNEMPLOY_TC

0.8

0.9

1.0

1.1

1.2

1.3

88 90 92 94 96 98 00 02 04 06

UNEMPLOY_SF

0.88

0.92

0.96

1.00

1.04

1.08

1.12

88 90 92 94 96 98 00 02 04 06

UNEMPLOY_IR

0.9985

0.9990

0.9995

1.0000

1.0005

1.0010

1.0015

88 90 92 94 96 98 00 02 04 06

UNEMPLOY_D18

Stage 1

Test for the presence of seasonality assuming stability.Sum of Dgrs.of Mean

Squares Freedom Square F-ValueBetween quarters 12436.0552 3 4145.35172 153.619**

Residual 1942.8936 72 26.98463Total 14378.9488 75

**Seasonality present at the 0.1 per cent level.Nonparametric Test for the Presence of Seasonality Assuming Stability

Kruskal-Wallis Degrees of ProbabilityStatistic Freedom Level

50.8774 3 0.000%Seasonality present at the one percent level.

Moving Seasonality TestSum of Dgrs.of Mean

Squares Freedom Square F-valueBetween Years 664.2838 18 36.904653 1.925

Error 1035.0612 54 19.167801Moving seasonality present at the five percent level.

1. 1. M1 = 0.2552. M2 = 0.2093. M3 = 0.7434. M4 = 0.7125. M5 = 0.9396. M6 = 0.2357. M7 = 0.1978. M8 = 0.3899. M9 = 0.24510. M10 = 0.82611. M11 = 0.798

Q (without M2) = 0.48 ACCEPTED.*** ACCEPTED *** at the level 0.45

Stage 2

UNEMPLOY_SF, the seasonal component from Stage 1 resulted in significant presence of ARCH-type heteroskedasticityVariance of Diff Diff4UNEMPLOY_SF modelled as GARCH(1,1) under the assumption of Generalized Error DistributionThe residuals from the model are considered as the “Seasonal Irregular Component” and removed from UNEMPLOY to produce UNEMPLOY_STAR

School of Statistics, University of the Philippines-Diliman

0.8

0.9

1.0

1.1

1.2

1.3

1.4

88 90 92 94 96 98 00 02 04 06

UNEMPLOY_STAR_SF

8

9

10

11

12

13

88 90 92 94 96 98 00 02 04 06

UNEMPLOY_STAR_TC

0.88

0.92

0.96

1.00

1.04

1.08

1.12

88 90 92 94 96 98 00 02 04 06

UNEMPLOY_STAR_IR

Stage 3

Second Iteration of the Procedure is Done to Produce UNEMPLOY_STAR2


0.8

0.9

1.0

1.1

1.2

1.3

88 90 92 94 96 98 00 02 04 06

UNEMPLOY_STAR2_SF

8

9

10

11

12

13

88 90 92 94 96 98 00 02 04 06

UNEMPLOY_STAR2_TC

0.88

0.92

0.96

1.00

1.04

1.08

1.12

88 90 92 94 96 98 00 02 04 06

UNEMPLOY_STAR2_IR

Result of the Other Procedure Considered


0.8

0.9

1.0

1.1

1.2

1.3

88 90 92 94 96 98 00 02 04 06

NEW_SF

8

9

10

11

12

13

88 90 92 94 96 98 00 02 04 06

NEW_TC

0.88

0.92

0.96

1.00

1.04

1.08

1.12

88 90 92 94 96 98 00 02 04 06

NEW_IR

Period Covered Measure of Smoothness of the

Seasonally adjusted series

through X11ARIMA

(SMOOTH_ORIG)

Measure of Smoothness of theSeasonally adjusted series through the

Proposed Procedure

(SMOOTH_STAR)

Measure of Smoothness of theSeasonally adjusted

series through second iteration of

the Proposed Procedure

(SMOOTH_STAR2)

2000‐2006 0.56 0.57 0.57

1988‐2006 0.55 0.59 0.57

7

8

9

10

11

12

13

14

15

88 90 92 94 96 98 00 02 04 06

UNEMPLOYUNEMPLOY_SANEW_SA

UNEMPLOY_STAR_SAUNEMPLOY_STAR2_SA


-15

-10

-5

0

5

10

15

20

04Q1 04Q3 05Q1 05Q3 06Q1 06Q3

CHANGE_SFCHANGE_STAR

CHANGE_STAR2CHANGE_NEW

obsCHANGE_SF

(Current Procedure)

CHANGE_STAR(Iteration 1 of

Proposed Procedure)

CHANGE_STAR2(Iteration 2 of

Proposed Procedure)

CHANGE_NEW(Additional Iteration of

Current Procedure)

2004Q1 3.66 3.775 3.75 3.892004Q2 0.98 1.08 1.10 -0.022004Q3 0.81 0.78 0.786 2.082004Q4 0.99 0.80 0.81 0.912005Q1 -1.23 -0.79 -0.80 -0.812005Q2 -6.28 -6.83 -6.83 -8.662005Q3 -0.96 -0.63 -0.62 1.552005Q4 2.86 2.82 2.84 2.762006Q1 -1.07 -0.830 -0.84 -0.882006Q2 -7.23 -7.89 -7.90 -9.922006Q3 13.37 14.15 14.20 16.842006Q4 -7.81 -7.97 -7.99 -7.84

Monthly IMPORTS

2000

2400

2800

3200

3600

4000

4400

96 97 98 99 00 01 02 03 04

IMPORT

2000

2400

2800

3200

3600

4000

96 97 98 99 00 01 02 03 04

IMPORT_TC

0.85

0.90

0.95

1.00

1.05

1.10

1.15

96 97 98 99 00 01 02 03 04

IMPORT_SF

0.84

0.88

0.92

0.96

1.00

1.04

1.08

1.12

1.16

1.20

96 97 98 99 00 01 02 03 04

IMPORT_IR

Stage 1

Imports

1. M1 = 2.2672. M2 = 0.2183. M3 = 0.7494. M4 = 0.2845. M5 = 0.6746. M6 = 0.5737. M7 = 0.6358. M8 = 1.3859. M9 = 0.360

10. M10 = 1.24911. M11 = 1.218

Q (without M2) = 0.89 CONDITIONALLY ACCEPTED AT LEVEL .81 But Check the 4 above measures which failed

Stage 2

IMPORTS_SF, the seasonal component from Stage 1 resulted in significant presence of ARCH-type heteroskedasticityVariance of IMPORTS_SF modelled as GARCH(2,1) under the assumption of Generalized Error DistributionThe residuals from the model are considered as the “Seasonal Irregular Component” and removed from IMPORTS to produce IMPORTS_STAR

Stage 3

Imports_star

1. M1 = 1.1712. M2 = 0.3773. M3 = 0.2794. M4 = 0.0605. M5 = 0.6096. M6 = 0.2397. M7 = 0.8098. M8 = 0.9369. M9 = 0.862

10. M10 = 1.37511. M11 = 1.343

*** Q (without M2) = 0.70 ACCEPTED.

*** ACCEPTED *** at the level 0.66But Check the 3 above measures which failed


0.85

0.90

0.95

1.00

1.05

1.10

1.15

96 97 98 99 00 01 02 03 04

IMPORT_STAR_SF

2000

2400

2800

3200

3600

4000

96 97 98 99 00 01 02 03 04

IMPORT_STAR_TC

0.84

0.88

0.92

0.96

1.00

1.04

1.08

1.12

1.16

1.20

96 97 98 99 00 01 02 03 04

IMPORT_STAR_IR

Second Iteration


0.85

0.90

0.95

1.00

1.05

1.10

1.15

96 97 98 99 00 01 02 03 04

IMPORT_STAR2_SF

2000

2400

2800

3200

3600

4000

96 97 98 99 00 01 02 03 04

IMPORT_STAR2_TC

0.84

0.88

0.92

0.96

1.00

1.04

1.08

1.12

1.16

1.20

96 97 98 99 00 01 02 03 04

IMPORT_STAR2_IR

Other Procedure Considered


0.84

0.88

0.92

0.96

1.00

1.04

1.08

96 97 98 99 00 01 02 03 04

NEW_SF

2000

2400

2800

3200

3600

4000

96 97 98 99 00 01 02 03 04

NEW_TC

0.84

0.88

0.92

0.96

1.00

1.04

1.08

1.12

1.16

1.20

96 97 98 99 00 01 02 03 04

NEW_IR

Period Covered Measure of Smoothness of

theSeasonally

adjusted series through

X11ARIMA (SMOOTH_ORIG)

Measure of Smoothness of the

Seasonally adjusted series through the Proposed Procedure

(SMOOTH_STAR)

Measure of Smoothness of theSeasonally adjusted

series through second iteration of

the Proposed Procedure

(SMOOTH_STAR2)

2000‐2004 159.7945 159.4536 159.4533

1996‐2004 143.9151 142.7023 142.7031

1500

2000

2500

3000

3500

4000

4500

95 96 97 98 99 00 01 02 03 04

IMPORTIMPORT_SAIMPORT_STAR_SA

IMPORT_STAR2_SANEW_SA

obsCHANGE_SF

(Current Procedure)

CHANGE_STAR(Iteration 1 of

Proposed Procedure)

CHANGE_STAR2(Iteration 2 of

Proposed Procedure)

CHANGE_NEW(Additional Iteration of

Current Procedure)

2004M01 -5.74 -4.58 -4.58 -3.892004M02 -1.98 0.54 0.54 1.222004M03 -0.09 -3.68 -3.68 1.312004M04 -0.12 0.54 0.54 -4.182004M05 -2.61 -1.77 -1.77 -2.172004M06 7.22 5.91 5.91 6.092004M07 -4.66 -3.82 -3.82 -4.412004M08 1.55 1.14 1.14 1.322004M09 4.82 4.52 4.52 4.192004M10 5.27 6.26 6.26 6.402004M11 -6.59 -7.69 -7.69 -7.352004M12 1.60 1.32 1.32 0.69

-8

-4

0

4

8

2004M01 2004M04 2004M07 2004M10

CHANGE_SACHANGE_STAR

CHANGE_STAR2CHANGE_NEW

Next Steps

Do the procedure with simulated data.

ReferencesAlbert, Jose Ramon(2002).” A Comparative Study of Seasonal Adjustment Methods for Philippine Time Series Data”, Statistical Research and Training Center publication.

Apostol, Agnes.(1993). “Seasonal Adjustment of Monetary Aggregates”, The Philippine Statistician, pp. 1-7, vol.42.

Bersales, Lisa Grace and TWG on Seasonal Adjustment of Philippine Time Series(2007), Current Methodologies and Results of Seasonally Adjusting the Selected Philippine Time Series, proceedings of the 10TH National Convention of Statistics, Mandaluyong, October 1-2, 2007.

Bersales, Lisa Grace (1993). “The Seasonal Adjustment of Philippine Time Series Using X11ARIMA”, The Philippine Statistician, pp. 1-7, vol.42.

Braganza, David, Alegria Mota and Mario Padrinao(1993).”Seasonal Adjustment of Palay Time Series Using X-11 ARIMA”, The Philippine Statistician, pp. 1-7, vol.42.

Buenaobra, Elsie, Elena Daguidi and Teresa Edora(1993). “Seasonal Adjustment of Philippine Time Series on Labor and Employment”, The Philippine Statistician, pp. 1-7, vol.42.

Cruz, Loida(1993). “Seasonal Adjustment of Quarterly National Accounts Series”, The Philippine Statistician, pp. 1-7, vol.42.

Dimaandal, Benilda and Faith Gina Asence(1993).”Seasonally Adjusted Consumer Price Index”, The Philippine Statistician, pp. 1-7, vol.42.

Foronda, Amador(2005). Seasonal Adjustment of National Income Accounts of the Philippines, proceedings of the 55th Session of the International Statistical Institute on April 5 – 12, 2005, Sydney, Australia.

Gador, Ludivina(1993).”Seasonally Adjustment of Time Series:Garments Exports”, The Philippine Statistician, pp. 1-7, vol.42.

Ghysels, Eric, Clive Granger and Pierre Siklos(1997). Seasonal Adjustment and Volatility Dynamics, CIRANO Working Papers.

Martens, Martin, Chang Yuan-Chen, and Stephen Taylor(2007?). A Comparison of Seasonal Adjustment Methods When Forecasting Intraday Volatility, CIRANO Working Papers.

Thank you!Thank you!

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