Issues in Modeling and Adjusting Calendar Effects in Economic Time Series

Brian C. Monsell

U. S. Census Bureau

ICES III June 2007

U S C E N S U S B U R E A UU S C E N S U S B U R E A U

2

Acknowledgments

• David Findley

• Bill Bell

• Patrick Cantwell

• Monica Wroblewski

3

Outline

• Alternate Easter Regressors» Define Regressors» Model Diagnostics Used» Results on Retail Sales Series

• One-Coefficient Stock Trading Day» Define Regressor» Results on Industrial Inventory Series

4

Alternate Easter Regressors

• Easter is the moving holiday that has the broadest effect in U. S. economic series» Sales increase before Easter Holiday

• Determine if we can improve the Easter effect model

5

Basic Easter[w] Regressor

• Ew,t 0 only in Feb, Mar, and April

,

number of the w days before Easter in month t

the long term monthly means of

t

tw t

W

WE

w

, ,t

w t w t

WE E

w

6

Alternate 2-Part Easter Regressor

• Assumes there are two effects:» Pre-Easter effect before Good Friday» Easter duration effect from Good Friday to

Easter

• Inspired by Zhang, McLaren, Leung (2003)

7

Pre-Easter Regressor

• PEw,t 0 only in Feb, Mar, and April

,

number of the w days before

Good Friday in month t

the long term monthly means of

t

tw t

W

WPE

w

twtwt

WPE PE

w

8

Easter Duration Regressor

• EDt 0 only in March and April

number of days from Good Friday

to Easter in month t

the long term monthly means of 3

t

tt

W

WED

3t

tt

WED ED

9

Linear Easter Regressor

• Assumes change in level begins w days before Easter

• This change increases linearly until the day before Easter

• Inspired by Zhang, McLaren, Leung (2003)

10

Linear Easter[w]

Let my be the number of the w days before Easter that occur in March of year y.

2

2 ,

2

2 ,

( , , )

( , , ) 1

LEyw March

LEyw April

mLE w March y

w

mLE w April y

w

11

Weekend-Weekday Easter

• Assumes that the change in the level of the series is different for » Weekend days (Friday, Saturday, Sunday) » Weekdays (Monday through Thursday

• Construct two regressors for the proportion of weekdays and weekend days in the 16 days before Easter in each month

12

we,t

wd,t

,

,

where n is the number of weekend days for month t

in the 16 days before Easter

and n is the number of weekdays for month t

in the 16 days before Easter

( )8

( )8

we twe

wd twd

nWE t

nWD t

13

Evaluation Methods

• AICC

• Mean Sum of Squared Out of Sample Forecast Error

14

AICC

• A standard model comparison diagnostic» Based on Akaike’s Information Criteria» Smaller values are preferred

15

AICC

• AIC Corrected (for sample size)

• Note: As N gets larger, AICC approaches the AIC

Npp

LAICC NN 11

2ˆ2

16

Mean Squared Forecast Error

0

2|

0

|

ˆ( )

,1

ˆwhere = forecast for time t+h given data up to time t

actual data for time t+h

h = 1, 12

T h

t h t h tt t

h

t h t

t h

Y Y

MSSFET h t

Y

Y

17

Application to Retail Sales Series

• Sixteen (16) retail sales series » A regARIMA model with the Easter[8]

regressor is used to generate Easter adjustment factors

» January 1992 – November 2006» Model span begins in 1995

18

Current Easter Regressors

• For each series, compare AICC for 4 Easter Models» No Easter» Easter[1]» Easter[8]» Easter[15]

19

Results of Current Easter

• For all series, a model with one of the Easter regressors was preferred» Easter[8] regressor was preferred in 11 of

the 16 series» Difference in the AICC was substantial,

ranging from 6.3 to 95.5

20

Alternate Easter Regressors

• The alternate Easter regressors will be fit with the regARIMA model that differs from those used for production runs only in the Easter model

• Fit models with: » Two-Part Easter, w = 8, 15» Linear Easter, w = 8, 15» Weekend-Weekday Easter Regressor

21

For 2-Part Easter:

• Only one series (Miscellaneous Food and Beverage) preferred the best 2-Part Easter model over the best current Easter model

22

For Linear Easter Regressor:

• 11 series preferred the best linear Easter model over the best current Easter» The AICC differences were very small

(only two > 1), implying there is not much difference between the models

23

For Weekend-Weekday Easter:

• For 8 of the 16 series, the best model overall according to AICC was the Weekend-Weekday Easter model

• AICC differences indicate that this model should be preferred to the current Easter[8] model (range from 3 to 15) for these 8 series.

24

Forecast Error

• Use forecast error diagnostic to determine if the model with the best AICC » Forecasts better than a model with the current

Easter[8] regressor» Forecasts better than a model with no Easter

• Reduce out of sample forecast error using the alternate Easter models for both 1 and 12 step ahead forecasts

25

Best vs. Easter[8]

• 9 of the 16 series had smaller forecast errors for both 1 and 12 step ahead forecasts using the best alternate model over the Easter[8]

26

Best vs. No Easter

• Only 7 of the 16 series showed forecast improvement using the best alternate model over using no Easter regressor

27

• Only one of the series using the Weekend-Weekday regressor showed forecast improvement over the model with no Easter

28

Constrained Stock Trading Day

• Findley (2006) developed formulas relating flow and stock day-of-week coefficients.

• Now possible to apply flow day-of week constraints to stock trading day regressors.

• More details in Findley and Monsell (2007)

29

End of Month Stock Trading Day

• I*t (1) = 1 when the last day of the month is Monday, -1 when the last day is Sunday, and 0 otherwise

• I*t (2) = 1 when the last day of the month is Tuesday, -1 when the last day is Sunday, and 0 otherwise

• Etc.

Bell (1984,1995)

30

Constraint for 1-Coefficient Flow Trading Day

• Weekday/Weekend contrast model of TRAMO and X-12-ARIMA• Imposes separate equality constraints on weekday and weekend coefficients

1 2 3 4 5

6 7

32

Stock TD 1-coefficient Regressor

* * *

* *

3 1 1(1) (2) (3)

5 5 53

(4) (5)5

t t t t

t t

D I I I

I I

33

Manufacturing Inventory Series

• 91 inventory series of the U. S. Census Bureau’s monthly Manufacturers’ Shipments, Inventories and Orders Survey.

• Series ended in October 2006

• Starting date for models range from January 1992 to January 1995

34

Models fit for Inventory Series

• Using the regARIMA model used in production as a base, models were fit with» No trading day regressors» Unconstrained stock trading day with ω =

31» Constrained stock trading day with ω = 31

35

Model Comparisons

• Used a standard log-likelihood difference asymptotic chi-square test, with a level of significance α = 0.05 » see Taniguchi and Kakizawa (2000)

36

Log-likelihood Test Results

• Found 21 series where the model with no trading day was rejected in favor of a model with stock trading day

37

• For one of the series, the trading day model was rejected for inducing a “visually significant” trading day peak in the seasonally adjusted series.

• Rejected another series due to better forecasting performance without trading day

38

Series Breakdown

Type of Trading Day found significant

Number of Series

Unconstrained Stock TD

3

Constrained Stock TD 8

Both Constrained and Unconstrained Stock TD

8

39

Where Both TD models are significant

• Constrained Stock TD always preferred over the Unconstrained Stock TD using an appropriate chi-square test

40

However

• Preferred Unconstrained Stock TD for two of the series where the use of the Constrained Stock TD left a visually significant spectral peak in the regARIMA residuals

41

Summary:

• For 14 of the 19 series where a stock trading day regressor was found to be significant, the Constrained Stock TD was the preferred model

42

• For 12 of these 14 series, additional criteria were found to support this choice, including» Reducing visually significant spectral

peaks» Reducing the number of lags with

significant Ljung-Box statistics» Reduction in out-of sample forecast error

43

Future Work

• Further examine the forecasting results for the Easter regressors

• Apply constrained stock trading day model to retail inventories

• Incorporate one-coefficient stock trading day into X-13A-S» Currently a utility to generate these

variables from the X-12 regression matrix

44

Contact Information

• Brian Monsell– brian.c.monsell@census.gov

45

Disclaimer

This report is released to inform interested parties of ongoing research and to encourage discussion of work in progress. Any views expressed on statistical, methodological, technical, or operational issues are those of the authors and not necessarily those of the U.S. Census Bureau.

46

Much of the data analysis for this paper was generated using Base SAS® software, SAS/AF® software, and SAS/GRAPH® software, Versions 8 and 9 of the SAS System for Windows. Copyright © 1999-2003 SAS Institute Inc. SAS and all other SAS Institute Inc. product or service names are registered trademarks or trademarks of SAS Institute Inc., Cary, NC, USA.