Enrico Infante* EUROSTAT, Unit G3: Short-Term Statistics; Tourism

# Enrico Infante* EUROSTAT, Unit G3: Short-Term Statistics; Tourism

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## Enrico Infante* EUROSTAT, Unit G3: Short-Term Statistics; Tourism

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1. 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 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 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 + This test gives information about the approach to follow before SA of the series • No accounting consistency • Cancel-out effect • Residual Seasonality • Calculations burden - 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 The table for the ANOVA test VAR Mean Sum of Squares df 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: 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 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 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, 17-19 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 of the between months variance and the residual variance (comments by Kirchner) + - Moving Seasonality Additive and multiplicative decompositions • 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. 420-436, 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] M. Centoni, G. Cubbadda – Modelling Comovements of Economic Time Series: A Selective Survey – CEIS, 2011 [7] A. Maravall – 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 R. Gatto, R. Kirchner, A. Maravall, G.L. Mazzi, J. Palate) 06/03/2012 Workshop on SA Enrico Infante, Dario Buono