1. New innovative 3-way ANOVA a-priori test for direct vs
New innovative 3-way ANOVA a-priori test for direct vs. indirect approach in Seasonal Adjustment
Workshop on Seasonal Adjustment – Luxembourg, 6 March 2012
Enrico Infante*
EUROSTAT, Unit G3: Short-Term Statistics; Tourism
Dario Buono*
EUROSTAT, Unit B1: Quality, Research and Methodology
*The views and the opinions expressed in this paper are solely of the authors and do not necessarily reflect those of the institutions for which they work
06/03/2012

2. Introduction
A generic time series Yt can be the result of an aggregation of p series:
We focus on the case of the additive function:
06/03/2012 Workshop on SA
Enrico Infante, Dario Buono

3. Introduction Direct Approach Indirect Approach
To Seasonally Adjust the aggregate, different approaches can be applied
Direct Approach
The Seasonally Adjusted data are computed directly by Seasonally Adjusting the aggregate
Indirect Approach
The Seasonally Adjusted data are computed indirectly by Seasonally Adjusting data per each series
06/03/2012 Workshop on SA
Enrico Infante, Dario Buono

4. Introduction Mixed Approach
If it is possible to divide the series into groups, then it is possible to compute the Seasonally Adjusted figures by summing the Seasonally Adjusted data of these groups
Group A
Group B
Example (two groups):
06/03/2012 Workshop on SA
Enrico Infante, Dario Buono

5. The basic idea
To use the Mixed Approach, sub-aggregates must be defined
We would like to find a criterion to divide the series into groups
The series of each group must have common regular seasonal patterns
How is it possible to decide that two or more series have common seasonal patterns?
NEW TEST!!!
06/03/2012 Workshop on SA
Enrico Infante, Dario Buono

6. Why a new test? Direct Indirect + -
Direct and indirect: there is no consensus on which is the best approach
Direct
Indirect
It could be interesting to identify which series can be aggregated in groups and decide at which level the SA procedure should be run
Transparency
Accuracy
Accounting Consistency +
No accounting consistency
Cancel-out effect
Residual Seasonality
Calculations burden
This test gives information about the approach to follow before SA of the series -
The presence of residual seasonality should always be checked in all of the Indirect and Mixed Seasonally Adjusted aggregates
06/03/2012 Workshop on SA
Enrico Infante, Dario Buono

7. The test
The classic test for moving seasonality is based on a 2-way ANOVA test, where the two factors are the time frequency (usually months or quarters) and the years. This test is based on a 3-way ANOVA model, where the three factors are the time frequency, the years and the series
The variable tested is the final estimation of the unmodified Seasonal-Irregular ratios (or differences) absolute value
Additive model
Multiplicative model
It is considered that the decomposition model is the same on all the series. The series is then considered already Calendar Adjusted
06/03/2012 Workshop on SA
Enrico Infante, Dario Buono

8. The test The model is: Where:
ai, i=1,…,M, represents the numerical contribution due to the effect of the i-th time frequency (usually M=12 or M=4)
bj, j=1,…,N, represents the numerical contribution due to the effect of the j-th year
ck, k=1,…,S, represents the numerical contribution due to the effect of the k-th series of the aggregate
The residual component term eijk (assumed to be normally distributed with zero mean, constant variance and zero covariance) represents the effect on the values of the SI of the whole set of factors not explicitly taken into account in the model
06/03/2012 Workshop on SA
Enrico Infante, Dario Buono

9. The test
The test is based on the decomposition of the variance of the observations:
Between time frequencies variance
Between years variance
Between series variance
Residual variance
06/03/2012 Workshop on SA
Enrico Infante, Dario Buono

10. The test VAR Mean Sum of Squares df The table for the ANOVA test
06/03/2012 Workshop on SA
Enrico Infante, Dario Buono

11. The test
The null hypothesis is made taking into consideration that there is no change in seasonality over the series
The test statistic is the ratio of the between series variance and the residual variance, and follows a Fisher-Snedecor distribution with (S-1) and (M-1)(N-1)(S-1) degrees of freedom
Rejecting the null hypothesis is to say that the pure Direct Approach should be avoided, and an Indirect or a Mixed one should be considered
06/03/2012 Workshop on SA
Enrico Infante, Dario Buono

12. Showing the procedure - Example
The most simple case: the aggregate is formed of two series, using the same decomposition model
Do X1t and X2t have the same seasonal patterns?
Rejecting H0: the two series have different seasonal patterns
Indirect Approach
TEST
Not rejecting H0: the two series have common regular seasonal patterns
Direct Approach
06/03/2012 Workshop on SA
Enrico Infante, Dario Buono

13. Numerical example
Let’s consider the Construction Production Index of the three French-speaking European countries: France, Belgium and Luxembourg (data are available on the EUROSTAT database). The time span is from January 2001 to December 2010
To take an example, a very simple aggregate could be the following:
VAR
Mean Square df
Months
1.5003 11
Years
0.0226 9
Series
0.1356 2
Residual
0.0117
198
There is no evidence of common seasonal patterns between the series at 5 per cent level
The Direct Approach should be avoided
06/03/2012 Workshop on SA
Enrico Infante, Dario Buono

14. Numerical example
If two of them have the same seasonal pattern, a Mixed Approach could be used. So the test is now used for each couple of series
LU - FR
BE - FR
VAR
Mean Square df
Months
2.0403 11
Years
0.0140 9
Series
0.1199 1
Residual
0.0016 99
VAR
Mean Square df
Months
1.0464 11
Years
0.0172 9
Series
0.0793 1
Residual
0.0164 99
There is no evidence of common seasonal patterns between the series at 5 per cent level
There is no evidence of common seasonal patterns between the series at 5 per cent level
06/03/2012 Workshop on SA
Enrico Infante, Dario Buono

15. Numerical example LU - BE VAR Mean Square df Months 0.9579 11 Years
0.0202 9
Series
0.0042 1
Residual
0.0181 99
Common seasonal patterns between the series present at 5 per cent level
LU and BE have the same seasonal pattern, so it is possible to Seasonally Adjust them together, using a Mixed Approach
An excel file with all the calculations is available on request
06/03/2012 Workshop on SA
Enrico Infante, Dario Buono

16. Future research line
This idea is just the start… more work needs to be done!!!
Create the theoretical base
Testing with real data
Presentation at CFE'11 & ERCIM'11, December 2011, University of London, UK
Implementation in R
06/03/2012 Workshop on SA
Enrico Infante, Dario Buono

17. Future research line + -
Theoretical review (F-ratio, trend, co-movements test)
F-ratio: re-building the test upon the ratio between the ratio of the between months variance and the residual variance (comments by Kirchner) + -
Additive and multiplicative decompositions
Moving Seasonality
A-priori estimation of the trend
Use of the co-movements test as benchmarking
06/03/2012 Workshop on SA
Enrico Infante, Dario Buono

18. Future research line Case study (IPC using Demetra+) - ongoing
Simulations (R) - ongoing
Application with a Tukey’s range test
06/03/2012 Workshop on SA
Enrico Infante, Dario Buono

19. References
[1] J. Higginson – An F Test for the Presence of Moving Seasonality When Using Census Method II-X-11 Variant – Statistics Canada, 1975
[2] R. Astolfi, D. Ladiray, G. L. Mazzi – Seasonal Adjustment of European Aggregates: Direct versus Indirect Approach – European Communities, 2001
[3] F. Busetti, A. Harvey – Seasonality Tests – Journal of Business and Economic Statistics, Vol. 21, No. 3, pp , Jul. 2003
[4] B. C. Surtradhar, E. B. Dagum – Bartlett-type modified test for moving seasonality with applications – The Statistician, Vol. 47, Part 1, 1998
[5] Centoni M., Cubbadda G. – Modelling Comovements of Economic Time Series: A Selective Survey – CEIS, 2011
[7] Maravall A. – An application of the TRAMO-SEATS automatic procedure; direct versus indirect approach – Computation Statistics & Data Analysis, 2005
[8] R. Cristadoro, R. Sabbatini - The Seasonal Adjustment of the Harmonised Index of Consumer Prices for the Euro Area: a Comparison of Direct and Indirect Method – Banca d’Italia, 2000
[9] B. Cohen – Explaning Psychological Statistics (3rd ed.), Chapter 22: Three-way ANOVA - New York: John Wiley & Sons, 2007
[10]I. Hindrayanto - Seasonal adjustment: direct, indirect or multivariate method? – Aenorm, No. 43, 2004
06/03/2012 Workshop on SA
Enrico Infante, Dario Buono

20. Questions?
Many Thanks!!!
We are really grateful for all the comments we already received
(in particular from Gatto, Kirchner, Maravall, Mazzi)
06/03/2012 Workshop on SA
Enrico Infante, Dario Buono