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Enrico Infante* EUROSTAT, Unit G3: Short-Term Statistics; Tourism Dario Buono* EUROSTAT, Unit B1: Quality, Research and Methodology Workshop on Seasonal Adjustment – Luxembourg, 6 March 2012 *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 New innovative 3-way ANOVA a-priori test for direct vs. indirect approach in Seasonal Adjustment 06/03/2012

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2 A generic time series Y t can be the result of an aggregation of p series: We focus on the case of the additive function: Introduction Enrico Infante, Dario Buono06/03/2012Workshop on SA

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3 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 Introduction Enrico Infante, Dario Buono06/03/2012Workshop on SA

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4 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 Introduction Enrico Infante, Dario Buono06/03/2012Workshop on SA

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5 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!!! The basic idea Enrico Infante, Dario Buono06/03/2012Workshop on SA

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6 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 Why a new test? Enrico Infante, Dario Buono06/03/2012Workshop on SA The presence of residual seasonality should always be checked in all of the Indirect and Mixed Seasonally Adjusted aggregates

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7 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 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 test Enrico Infante, Dario Buono06/03/2012Workshop on SA

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8 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 The test Enrico Infante, Dario Buono06/03/2012Workshop on SA

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9 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 The test Enrico Infante, Dario Buono06/03/2012Workshop on SA

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10 VARMeandf The table for the ANOVA test Sum of Squares The test Enrico Infante, Dario Buono06/03/2012Workshop on SA

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11 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 The test Enrico Infante, Dario Buono06/03/2012Workshop on SA

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12 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 Showing the procedure - Example Enrico Infante, Dario Buono06/03/2012Workshop on SA

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13 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: VARMean Squaredf Months1.500311 Years0.02269 Series0.13562 Residual0.0117198 There is no evidence of common seasonal patterns between the series at 5 per cent level The Direct Approach should be avoided Numerical example Enrico Infante, Dario Buono06/03/2012Workshop on SA

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14 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 Months2.040311 Years0.01409 Series0.11991 Residual0.001699 VARMean Squaredf Months1.046411 Years0.01729 Series0.07931 Residual0.016499 LU - FRBE - FR There is no evidence of common seasonal patterns between the series at 5 per cent level Numerical example Enrico Infante, Dario Buono06/03/2012Workshop on SA

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15 An excel file with all the calculations is available on request VARMean Squaredf Months0.957911 Years0.02029 Series0.00421 Residual0.018199 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 Numerical example Enrico Infante, Dario Buono06/03/2012Workshop on SA

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16 This idea is just the start… more work needs to be done!!! Implementation in R Presentation at CFE'11 & ERCIM'11, 17-19 December 2011, University of London, UK Testing with real data Create the theoretical base Future research line Enrico Infante, Dario Buono06/03/2012Workshop on SA

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17 Theoretical review (F-ratio, trend, co-movements test) Future research line Enrico Infante, Dario Buono06/03/2012Workshop on SA F-ratio: re-building the test upon 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

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18 Case study (IPC using Demetra+) - ongoing Simulations (R) - ongoing Application with a Tukey’s range test Future research line Enrico Infante, Dario Buono06/03/2012Workshop on SA

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19 [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 (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 References Enrico Infante, Dario Buono06/03/2012Workshop on SA

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20 Many Thanks!!! Questions? Enrico Infante, Dario Buono06/03/2012Workshop on SA 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)

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