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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
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
hhthptphthtt XXXXY
111
Introduction
Enrico Infante, Dario Buono06/03/2012 Workshop on SA
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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
hhtht XSAYSA
1
The Seasonally Adjusted data are computed indirectly by Seasonally Adjusting data per each series
p
hhtht XSAYSA
1
Introduction
Enrico Infante, Dario Buono06/03/2012 Workshop on SA
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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
uutu
q
lltlt XXY
11
Group A Group B
prq
r
uutu
q
lltlt XSAXSAYSA
11
Introduction
Enrico Infante, Dario Buono06/03/2012 Workshop on SA
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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 Buono06/03/2012 Workshop on SA
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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 Buono06/03/2012 Workshop on SA
The presence of residual seasonality should always be checked in all of the Indirect and Mixed Seasonally Adjusted aggregates
7
The variable tested is the final estimation of the unmodified Seasonal-Irregular ratios (or differences) absolute value
ijkSI
1ijkSI
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 Buono06/03/2012 Workshop on SA
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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 Buono06/03/2012 Workshop on SA
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The test is based on the decomposition of the variance of the observations:
22222RSNM SSSSS
Sk ,,1
Nj ,,1
Between time frequencies variance
Between years variance
Between series variance
Residual variance
The test
Enrico Infante, Dario Buono06/03/2012 Workshop on SA
Mi ,,1
10
VAR Mean df
2MS
2NS
2SS
2RS
N
j
S
kijki SI
NSx
1 1
1
M
i
S
kijkj SI
MSx
1 1
1
M
i
N
j
S
kkjiijk xxxxSI
1 1 1
22
M
ii xxNS
1
2
N
jj xxMS
1
2
S
kk xxMN
1
2
M
i
N
jijkk SI
MNx
1 1
1
1M
1N
1S
111 SNM
The table for the ANOVA test
Sum of Squares
The test
Enrico Infante, Dario Buono06/03/2012 Workshop on SA
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The null hypothesis is made taking into consideration that there is no change in seasonality over the series
111;12
2
~ SNMSR
ST F
S
SF
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 Buono06/03/2012 Workshop on SA
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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 Buono06/03/2012 Workshop on SA
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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
8122.50117.0
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 Buono06/03/2012 Workshop on SA
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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
7591.75F 0000.0 valueP 8313.4F 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 Buono06/03/2012 Workshop on SA
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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
2314.0F 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 Buono06/03/2012 Workshop on SA
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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 Buono06/03/2012 Workshop on SA
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Theoretical review (F-ratio, trend, co-movements test)
Future research line
Enrico Infante, Dario Buono06/03/2012 Workshop 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
18
Case study (IPC using Demetra+) - ongoing
Simulations (R) - ongoing
Application with a Tukey’s range test
Future research line
Enrico Infante, Dario Buono06/03/2012 Workshop on SA
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[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 Buono06/03/2012 Workshop on SA
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Many Thanks!!!
Questions?
Enrico Infante, Dario Buono06/03/2012 Workshop 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)