3-way anova test test for direct vs. indirect approach in Seasonal Adjustment Enrico Infante * – Università degli studi di Napoli Federico II Dario Buono * – EUROSTAT, Unit B1: Quality, Research and Methodology SAUG, 8 February 2012, Vilnius, INS of Lithuania *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

2 INTRODUCTION A generic time series Y t can be the result of an aggregation of p series: SAUG, 9 February 2012, Vilnius, INS of Lithuania Buono and Infante test for direct vs. indirect approach in Seasonal Adjustment We focus on the case of the additive function:

3 INTRODUCTION To Seasonally Adjust the aggregate, different approaches can be applied Direct Approach Indirect Approach The Seasonally Adjusted data are computed directly by Seasonally Adjusting the aggregate The Seasonally Adjusted data are computed indirectly by Seasonally Adjusting data per each series SAUG, 9 February 2012, Vilnius, INS of Lithuania Buono and Infante test for direct vs. indirect approach in Seasonal Adjustment

4 INTRODUCTION 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 Mixed Approach Example (two groups): Group AGroup B SAUG, 9 February 2012, Vilnius, INS of Lithuania Buono and Infante test for direct vs. indirect approach in Seasonal Adjustment

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!!! SAUG, 9 February 2012, Vilnius, INS of Lithuania Buono and Infante test for direct vs. indirect approach in Seasonal Adjustment

6 WHY A NEW TEST? Direct and indirect: there is no consensus on which is the best approach DirectIndirect + - Transparency Accuracy Accounting Consistency No accounting consistency Cancel-out effect Residual Seasonality Calculations burden It could be interesting to identify which series can be aggregated in groups and decide at which level the SA procedure should be run This test gives information about the approach to follow before SA of the series SAUG, 9 February 2012, Vilnius, INS of Lithuania Buono and Infante test for direct vs. indirect approach in Seasonal Adjustment

7 THE TEST The variable tested is the final estimation of the unmodified Seasonal- Irregular differences (or ratios) absolute value Additive model Multiplicative model If the series are not modelled with the same decomposition model, then the data must be normalized by the column (the time frequency factor). The series is then considered already Calendar Adjusted 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 SAUG, 9 February 2012, Vilnius, INS of Lithuania Buono and Infante test for direct vs. indirect approach in Seasonal Adjustment

8 THE TEST The model is: Where: a i, i=1,…,M, represents the numerical contribution due to the effect of the i-th time frequency (usually M=12 or M=4) b j, j=1,…,N, represents the numerical contribution due to the effect of the j-th year c k, k=1,…,S, represents the numerical contribution due to the effect of the k-th series of the aggregate The residual component term e ijk (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 SAUG, 9 February 2012, Vilnius, INS of Lithuania Buono and Infante test for direct vs. indirect approach in Seasonal Adjustment

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 SAUG, 9 February 2012, Vilnius, INS of Lithuania Buono and Infante test for direct vs. indirect approach in Seasonal Adjustment

10 THE TEST VARMeandf The table for the ANOVA test Sum of Squares SAUG, 9 February 2012, Vilnius, INS of Lithuania Buono and Infante test for direct vs. indirect approach in Seasonal Adjustment

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 SAUG, 9 February 2012, Vilnius, INS of Lithuania Buono and Infante test for direct vs. indirect approach in Seasonal Adjustment

12 SHOWING THE PROCEDURE - EXAMPLE The most simple case: the aggregate is formed of two series, using the same decomposition model Do X 1t and X 2t have the same seasonal patterns? TEST Rejecting H 0 : the two series have different seasonal patterns Not rejecting H 0 : the two series have common regular seasonal patterns Direct Approach Indirect Approach SAUG, 9 February 2012, Vilnius, INS of Lithuania Buono and Infante test for direct vs. indirect approach in Seasonal Adjustment

13 NUMERICAL EXAMPLE Let’s consider the Construction Production of the three French speaker European counties: France, Belgium and Luxembourg (data are available on the EUROSTAT database). The time span is from Jan-01 to Dec-10 To take an example, a very simple aggregate could be the following: VARMean Squaredf Months Years Series Residual There is no evidence of common seasonal patterns between the series at 5 per cent level The Direct Approach should be avoided SAUG, 9 February 2012, Vilnius, INS of Lithuania Buono and Infante test for direct vs. indirect approach in Seasonal Adjustment

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 VARMean Squaredf Months Years Series Residual VARMean Squaredf Months Years Series Residual LU - FRBE - FR There is no evidence of common seasonal patterns between the series at 5 per cent level SAUG, 9 February 2012, Vilnius, INS of Lithuania Buono and Infante test for direct vs. indirect approach in Seasonal Adjustment

15 NUMERICAL EXAMPLE An excel file with all the calculations is available on request VARMean Squaredf Months Years Series Residual 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 SAUG, 9 February 2012, Vilnius, INS of Lithuania Buono and Infante test for direct vs. indirect approach in Seasonal Adjustment

16 FUTURE RESEARCH LINE Starting from this idea, there is still work to do!!! Implementation in R Presentation at CFE'11 & ERCIM'11, December 2011, University of London, UK Testing with real data Create the theoretical base SAUG, 9 February 2012, Vilnius, INS of Lithuania Buono and Infante test for direct vs. indirect approach in Seasonal Adjustment

17 FUTURE RESEARCH LINE Starting from this idea, there is still work to do!!! Case study (IPC using Demetra+) Simulations (R) Application with a Tukey’s range test Theoretical review (co-movements test) SAUG, 9 February 2012, Vilnius, INS of Lithuania Buono and Infante test for direct vs. indirect approach in Seasonal Adjustment

18 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 [4]B. C. Surtradhar, E. B. Dagum – Bartlett-type modified test for moving seasonality with applications – The Statistician, Vol. 47, Part 1, 1998 [5]R. Astolfi, D. Ladiray, G. L. Mazzi – Business cycle extraction of Euro-zone GDP: direct versus indirect approach – European Communities, 2001 [7]J. Lothian, M. Morry - A set of Quality Control Statistics for the X-11-ARIMA Seasonal Adjustment Method – Statistics Canada, 1978 [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 (3 rd 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 SAUG, 9 February 2012, Vilnius, INS of Lithuania Buono and Infante test for direct vs. indirect approach in Seasonal Adjustment

19 QUESTIONS? Many Thanks!!! SAUG, 9 February 2012, Vilnius, INS of Lithuania Buono and Infante test for direct vs. indirect approach in Seasonal Adjustment