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Enrico Infante*
University of Naples Federico II
Dario Buono*
EUROSTAT, Unit B1: Quality, Research and Methodology
Euroind...
2
A generic time series Yt can be the result of an aggregation of p series:
( )pthttt XXXfY ,,,,1 =
We focus on the case...
3
To Seasonally Adjust the aggregate, different approaches can be applied
Direct Approach
Indirect Approach
The Seasonally...
4
If it is possible to divide the series into groups, then it is possible to
compute the Seasonally Adjusted figures by su...
5
To use the Mixed Approach, sub-aggregates must be defined
We would like to find a criterion to divide the series into gr...
6
Direct and indirect: there is no consensus on which is the best approach
Direct Indirect
+
-
• Transparency
• Accuracy
•...
7
The variable tested is the final estimation of the unmodified Seasonal-
Irregular ratios (or differences) absolute value...
8
The model is:
ijkkjiijk ecbaSI +++=
Where:
• ai, i=1,…,M, represents the numerical contribution due to the effect of
the...
9
The test is based on the decomposition of the variance of the
observations:
22222
RSNM SSSSS +++=
Sk ,,1 =
Nj ,,1 =
Be...
10
VAR Mean df
2
MS
2
NS
2
SS
2
RS
∑∑= =
•• =
N
j
S
k
ijki SI
NS
x
1 1
1
∑∑= =
•• =
M
i
S
k
ijkj SI
MS
x
1 1
1
( )∑∑∑= = =...
11
The null hypothesis is made taking into consideration that there is no
change in seasonality over the series
( ) ( )( )...
12
ttt XXY 21 +=
The most simple case: the aggregate is formed of two series, using the
same decomposition model
Do X1t an...
13
Let’s consider the Construction Production Index of the three French-
speaking European countries: France, Belgium and ...
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 ...
15
An excel file with all the calculations is available on request
VAR Mean Square df
Months 0.9579 11
Years 0.0202 9
Seri...
16
Theoretical review (F-ratio, trend, co-movements test)
Future research line
Enrico Infante, Dario Buono
• F-ratio: re-b...
17
Case study (IPC using Demetra+) - ongoing
Simulations (R) - ongoing
Application with a Tukey’s range test
Future resear...
18
[1] J. Higginson – An F Test for the Presence of Moving Seasonality When Using
Census Method II-X-11 Variant – Statisti...
19
Many Thanks!!!
Questions?
Enrico Infante, Dario Buono
We are really grateful for all the comments we already received
(...
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New innovative 3 way anova a-priori test for direct vs. indirect approach in seasonal adjustment

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New innovative 3 way anova a-priori test for direct vs. indirect approach in seasonal adjustment

  1. 1. Enrico Infante* University of Naples Federico II Dario Buono* EUROSTAT, Unit B1: Quality, Research and Methodology Euroindicators PEEI WG – Luxembourg, 11-12 June 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
  2. 2. 2 A generic time series Yt can be the result of an aggregation of p series: ( )pthttt XXXfY ,,,,1 = We focus on the case of the additive function: ∑= =++++= p h hthptphthtt XXXXY 1 11 ϖϖϖϖ  Introduction Enrico Infante, Dario Buono
  3. 3. 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 ( )       = ∑= p h htht XSAYSA 1 ϖ The Seasonally Adjusted data are computed indirectly by Seasonally Adjusting data per each series ( ) ( )∑= = p h htht XSAYSA 1 ϖ Introduction Enrico Infante, Dario Buono
  4. 4. 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): ∑∑ == += r u utu q l ltlt XXY 11 ϖϖ Group A Group B prq =+ ( )       +      = ∑∑ == r u utu q l ltlt XSAXSAYSA 11 ϖϖ Introduction Enrico Infante, Dario Buono
  5. 5. 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 Buono
  6. 6. 6 Direct and indirect: there is no consensus on which is the best approach Direct Indirect + - • 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 Buono The presence of residual seasonality should always be checked in all of the Indirect and Mixed Seasonally Adjusted aggregates
  7. 7. 7 The variable tested is the final estimation of the unmodified Seasonal- Irregular ratios (or differences) absolute value ijkSI 1−ijkSI 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 Buono
  8. 8. 8 The model is: ijkkjiijk ecbaSI +++= 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 The test Enrico Infante, Dario Buono
  9. 9. 9 The test is based on the decomposition of the variance of the observations: 22222 RSNM SSSSS +++= Sk ,,1 = Nj ,,1 = Between time frequencies variance Between years variance Between series variance Residual variance The test Enrico Infante, Dario Buono Mi ,,1 =
  10. 10. 10 VAR Mean df 2 MS 2 NS 2 SS 2 RS ∑∑= = •• = N j S k ijki SI NS x 1 1 1 ∑∑= = •• = M i S k ijkj SI MS x 1 1 1 ( )∑∑∑= = = •••••• +−−− M i N j S k kjiijk xxxxSI 1 1 1 2 2 ( )∑= •• − M i i xxNS 1 2 ( )∑= •• − N j j xxMS 1 2 ( )∑= •• − S k k xxMN 1 2 ∑∑= = •• = M i N j ijkk SI MN x 1 1 1 1−M 1−N 1−S ( )( )( )111 −−− SNM The table for the ANOVA test Sum of Squares The test Enrico Infante, Dario Buono
  11. 11. 11 The null hypothesis is made taking into consideration that there is no change in seasonality over the series ( ) ( )( )( )111;12 2 ~ −−−−= SNMS R S T F S S F 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 ScccH === 210 : 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 Buono
  12. 12. 12 ttt XXY 21 += 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? TEST Rejecting H0: the two series have different seasonal patterns Not rejecting H0: the two series have common regular seasonal patterns Direct Approach Indirect Approach Showing the procedure - Example Enrico Infante, Dario Buono
  13. 13. 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: tttt LUBEFRY ++= VAR Mean Square df Months 1.5003 11 Years 0.0226 9 Series 0.1356 2 Residual 0.0117 198 8122.5 0117.0 1356.0 ==− ratioF 0035.0=− valueP 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 Buono
  14. 14. 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 VAR Mean Square df Months 2.0403 11 Years 0.0140 9 Series 0.1199 1 Residual 0.0016 99 7591.75=F 0000.0=− valueP 8313.4=F 0303.0=− valueP VAR Mean Square df Months 1.0464 11 Years 0.0172 9 Series 0.0793 1 Residual 0.0164 99 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 Numerical example Enrico Infante, Dario Buono
  15. 15. 15 An excel file with all the calculations is available on request VAR Mean Square df Months 0.9579 11 Years 0.0202 9 Series 0.0042 1 Residual 0.0181 99 2314.0=F 6315.0=− valueP 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 ( ) ( ) ( )tttt LUBESAFRSAYSA ++= Numerical example Enrico Infante, Dario Buono
  16. 16. 16 Theoretical review (F-ratio, trend, co-movements test) Future research line Enrico Infante, Dario Buono • 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
  17. 17. 17 Case study (IPC using Demetra+) - ongoing Simulations (R) - ongoing Application with a Tukey’s range test Future research line Enrico Infante, Dario Buono
  18. 18. 18 [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 References Enrico Infante, Dario Buono
  19. 19. 19 Many Thanks!!! Questions? Enrico Infante, Dario Buono 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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